CS2403 DIGITAL SIGNAL PROCESSING L T P C

Transcription

1 CS403 DIGITAL SIGNAL PROCESSING L T P C UNIT I SIGNALS AND SYSTEMS 9 Basic elemets of DSP cocepts of frequecy i Aalog ad Digital Sigals samplig theorem Discrete time sigals, systems Aalysis of discrete time LTI systems Z trasform Covolutio liear ad circular Correlatio. UNIT II FREQUENCY TRANSFORMATIONS 9 Itroductio to DFT Properties of DFT Filterig methods based o DFT FFT Algorithms Decimatio i time Algorithms, Decimatio i frequecy Algorithms Use of FFT i Liear Filterig DCT. UNIT III IIR FILTER DESIGN 9 Structures of IIR Aalog filter desig Discrete time IIR filter from aalog filter IIR filter desig by Impulse Ivariace, Biliear trasformatio, Approximatio of derivatives HPF, BPF, BRF filter desig usig frequecy traslatio UNIT IV FIR FILTER DESIGN 9 Structures of FIR Liear phase FIR filter Filter desig usig widowig techiques, Frequecy samplig techiques Fiite word legth effects i digital Filters UNIT V APPLICATIONS 9 Multirate sigal processig Speech compressio Adaptive filter Musical soud processig Image ehacemet. TEXT BOOKS:. Joh G. Proakis & Dimitris G.Maolakis, Digital Sigal Processig Priciples, Algorithms & Applicatios, Fourth editio, Pearso educatio / Pretice Hall, Emmauel C..Ifeachor, & Barrie.W.Jervis, Digital Sigal Processig, Secod editio, Pearso Educatio / Pretice Hall, 00. REFERENCES:. Ala V.Oppeheim, Roald W. Schafer & Hoh. R.Back, Discrete Time Sigal Processig, Pearso Educatio, d editio, Adreas Atoiou, Digital Sigal Processig, Tata McGraw Hill, 00

2 S.No Topic Page No. UNIT - SIGNALS AND SYSTEM INTRODUCTION. CLASSIFICATION OF SIGNAL PROCESSING ADVANTAGES OF DSP OVER ASP. CLASSIFICATION OF SIGNALS.3 DISCRETE TIME SIGNALS AND SYSTEM 6.4 ANALYSIS OF DISCRETE LINEAR TIME INVARIANT LTI/LSI SYSTEM.5 A/D CONVERSION.6 BASIC BLOCK DIAGRAM OF A/D CONVERTER.7 ANALYSIS OF LTI SYSTEM 6.7. Z TRANFORM 6 RELATIONSHIP BETWEEN FOURIER TRANSFORM AND Z 9 TRANSFORM INVERSE Z TRANSFORM IZT 0 POLE ZERO PLOT 7 SOLUTION OF DIFFERENTIAL EQUATION 30.8 CONVOLUTION LINEAR CONVOLUTION SUM METHOD PROPERTIES OF LINEAR CONVOLUTION 34.9 CORRELATION DIFFERENCE BETWEEN LINEAR CONVOLUTION AND 37 CORRELATION.9. TYPES OF CORRELATION PROPERTIES OF CORRELATION 38 UNIT II - FREQUENCY TRANSFORMATIONS 4. INTRODUCTION 4. DIFFERENCE BETWEEN FT & DFT 4.3 CALCULATION OF DFT & IDFT 4.4 DIFFERENCE BETWEEN DFT & IDFT 43.5 PROPERTIES OF DFT 44.6 APPLICATION OF DFT 50.7 FAST FOURIER ALGORITHM FFT 53.8 RADIX- FFT ALGORITHMS.9 COMPUTATIONAL COMPLEXITY FFT V/S DIRECT COMPUTATION.0 BIT REVERSAL 59. DECIMATION IN FREQUENCY DIFFFT 60. GOERTZEL ALGORITHM 66 UNIT III - IIR FILTER DESIGN INTRODUCTION TYPES OF DIGITAL FILTER STRUCTURES FOR FIR SYSTEMS STRUCTURES FOR IIR SYSTEMS 76 58

3 3.5 CONVERSION OF ANALOG FILTER INTO DIGITAL FILTER IIR FILTER DESIGN - BILINEAR TRANSFORMATION 84 METHOD BZT 3.7 LPF AND HPF ANALOG BUTTERWORTH FILTER TRANSFER 87 FUNCTION 3.8 METHOD FOR DESIGNING DIGITAL FILTERS USING BZT BUTTERWORTH FILTER APPROXIMATION FREQUENCY RESPONSE CHARACTERISTIC FREQUENCY TRANSFORMATION FREQUENCY TRANSFORMATION ANALOG FILTER FREQUENCY TRANSFORMATION DIGITAL FILTER 88 UNIT IV - FIR FILTER DESIGN FEATURES OF FIR FILTER SYMMETRIC AND ANTI-SYMMETRIC FIR FILTERS GIBBS PHENOMENON DESIGNING FILTER FROM POLE ZERO PLACEMENT NOTCH AND COMB FILTERS DIGITAL RESONATOR FIXED-POINT QUANTIZATION ERRORS FLOATING-POINT QUANTIZATION ERRORS ROUNDOFF NOISE 4.8 ROUNDOFF NOISE IN FIR FILTERS 4.9 LIMIT CYCLE OSCILLATIONS OVERFLOW OSCILLATIONS 8 4. REALIZATION CONSIDERATIONS 0 UNIT V - APPLICATIONS OF DSP 3 5. SPEECH RECOGNITION 3 5. LINEAR PREDICTION OF SPEECH SYNTHESIS SOUND PROCESSING ECHO CANCELLATION VIBRATION ANALYSIS MULTISTAGE IMPLEMENTATION OF DIGITAL FILTERS SPEECH SIGNAL PROCESSING SUBBAND CODING ADAPTIVE FILTER AUDIO PROCESSING MUSIC SOUND PROCESSING SPEECH GENERATION SPEECH RECOGNITION IMAGE ENHANCEMENT 39

4 UNIT I SIGNALS AND SYSTEM PRE REQUISITE DISCUSSION: The Various types of Sigals ad Types of Systems are to be aalyed to get a idea to process the icomig sigal. The Quality output is provided by proper operatio o sigal ad the maiteace of the System. The system coditios are to be aalyed such that the Iput sigal get ito a perfect processig. INTRODUCTION A SIGNAL is defied as ay physical quatity that chages with time, distace, speed, positio, pressure, temperature or some other quatity. A SIGNAL is physical quatity that cosists of may siusoidal of differet amplitudes ad frequecies. Ex xt 0t Xt 5x 0xy30y A System is a physical device that performs a operatios or processig o a sigal. Ex Filter or Amplifier.. CLASSIFICATION OF SIGNAL PROCESSING ASP Aalog sigal Processig : If the iput sigal give to the system is aalog the system does aalog sigal processig. Ex Resistor, capacitor or Iductor, OP-AMP etc. Aalog Iput ANALOG SYSTEM Aalog Output DSP Digital sigal Processig : If the iput sigal give to the system is digital the system does digital sigal processig. Ex Digital Computer, Digital Logic Circuits etc. The devices called as ADC aalog to digital Coverter coverts Aalog sigal ito digital ad DAC Digital to Aalog Coverter does vice-versa. Aalog sigal ADC DIGITAL SYSTEM DAC Aalog sigal Most of the sigals geerated are aalog i ature. Hece these sigals are coverted to digital form by the aalog to digital coverter. Thus AD Coverter geerates a array of samples ad gives it to the digital sigal processor. This array of samples or sequece of samples is the digital equivalet of iput aalog sigal. The DSP performs sigal processig operatios like filterig, multiplicatio, trasformatio or amplificatio etc operatios over these digital sigals. The digital output sigal from the DSP is give to the DAC. ADVANTAGES OF DSP OVER ASP. Physical sie of aalog systems is quite large while digital processors are more compact ad light i weight.. Aalog systems are less accurate because of compoet tolerace ex R, L, C ad active compoets. Digital compoets are less sesitive to the evirometal chages, oise ad disturbaces. 3. Digital system is most flexible as software programs & cotrol programs ca be easily modified. 4. Digital sigal ca be stores o digital hard disk, floppy disk or magetic tapes. Hece becomes trasportable. Thus easy ad lastig storage capacity. 5. Digital processig ca be doe offlie.

5 6. Mathematical sigal processig algorithm ca be routiely implemeted o digital sigal processig systems. Digital cotrollers are capable of performig complex computatio with costat accuracy at high speed. 7. Digital sigal processig systems are upgradeable sice that are software cotrolled. 8. Possibility of sharig DSP processor betwee several tasks. 9. The cost of microprocessors, cotrollers ad DSP processors are cotiuously goig dow. For some complex cotrol fuctios, it is ot practically feasible to costruct aalog cotrollers. 0. Sigle chip microprocessors, cotrollers ad DSP processors are more versatile ad powerful. Disadvatages of DSP over ASP. Additioal complexity A/D & D/A Coverters. Limit i frequecy. High speed AD coverters are difficult to achieve i practice. I high frequecy applicatios DSP are ot preferred..classification OF SIGNALS. Sigle chael ad Multi-chael sigals. Sigle dimesioal ad Multi-dimesioal sigals 3. Cotiuous time ad Discrete time sigals. 4. Cotiuous valued ad discrete valued sigals. 5. Aalog ad digital sigals. 6. Determiistic ad Radom sigals 7. Periodic sigal ad No-periodic sigal 8. Symmetricaleve ad Ati-Symmetricalodd sigal 9. Eergy ad Power sigal.. Sigle chael ad Multi-chael sigals If sigal is geerated from sigle sesor or source it is called as sigle chael sigal. If the sigals are geerated from multiple sesors or multiple sources or multiple sigals are geerated from same source called as Multi-chael sigal. Example ECG sigals. Multichael sigal will be the vector sum of sigals geerated from multiple sources...sigle Dimesioal -D ad Multi-Dimesioal sigals M-D If sigal is a fuctio of oe idepedet variable it is called as sigle dimesioal sigal like speech sigal ad if sigal is fuctio of M idepedet variables called as Multi- dimesioal sigals. Gray scale level of image or Itesity at particular pixel o black ad white TV is examples of M-D sigals...3 Cotiuous time ad Discrete time sigals. S Cotiuous Time CTS No This sigal ca be defied at ay time istace & they ca take all values i the cotiuous itervala, b where a ca be - & b ca be Discrete time DTS This sigal ca be defied oly at certai specific values of time. These time istace eed ot be equidistat but i practice they are usually takes at equally spaced itervals. These are described by differece equatio. These are described by differetial equatios. 3 This sigal is deoted by xt. These sigals are deoted by x or

6 4 The speed cotrol of a dc motor usig a tacho geerator feedback or Sie or expoetial waveforms. otatio xt caalso be used. Microprocessors ad computer based systems uses discrete time sigals...4 Cotiuous valued ad Discrete Valued sigals. S Cotiuous Valued No If a sigal takes o all possible values o a fiite or ifiite rage, it is said to be cotiuous valued sigal. Cotiuous Valued ad cotiuous time sigals are basically aalog sigals. Discrete Valued If sigal takes values from a fiite set of possible values, it is said to be discrete valued sigal. Discrete time sigal with set of discrete amplitude are called digital sigal...5 Aalog ad digital sigal Sr Aalog sigal Digital sigal No These are basically cotiuous time & cotiuous amplitude sigals. ECG sigals, Speech sigal, Televisio sigal etc. All the sigals geerated from various sources i ature are aalog. These are basically discrete time sigals & discrete amplitude sigals. These sigals are basically obtaied by samplig & quatiatio process. All sigal represetatio i computers ad digital sigal processors are digital. 3

7 Note: Digital sigals DISCRETE TIME & DISCRETE AMPLITUDE are obtaied by samplig the ANALOG sigal at discrete istats of time, obtaiig DISCRETE TIME sigals ad the by quatiig its values to a set of discrete values & thus geeratig DISCRETE AMPLITUDE sigals. Samplig process takes place o x axis at regular itervals & quatiatio process takes place alog y axis. Quatiatio process is also called as roudig or trucatig or approximatio process...6 Determiistic ad Radom sigals Sr No Determiistic sigals Radom sigals Determiistic sigals ca be represeted or described by a mathematical equatio or lookup table. Determiistic sigals are preferable because for aalysis ad processig of sigals we ca use mathematical model of the sigal. 3 The value of the determiistic sigal ca be evaluated at time past, preset or future Radom sigals that caot be represeted or described by a mathematical equatio or lookup table. Not Preferable. The radom sigals ca be described with the help of their statistical properties. The value of the radom sigal ca ot be evaluated at ay istat of time. without certaity. 4 Example Sie or expoetial waveforms. Example Noise sigal or Speech sigal..7 Periodic sigal ad No-Periodic sigal The sigal x is said to be periodic if xn x for all where N is the fudametal period of the sigal. If the sigal does ot satisfy above property called as No-Periodic sigals. Discrete time sigal is periodic if its frequecy ca be expressed as a ratio of two itegers. f k/n where k is iteger costat. a cos 0.0 Periodic N00 samples per cycle. b cos 3 Periodic N samples c si3 No-periodic d cos/8 cos /8 No-Periodic..8 SymmetricalEve ad Ati-Symmetricalodd sigal A sigal is called as symmetricaleve if x x- ad if x- -x the sigal is odd. X cosω ad x siω are good examples of eve & odd sigals respectively. Every discrete sigal ca be represeted i terms of eve & odd sigals. 4

8 X sigal ca be writte as X X X Rearragig the above terms we have X- - X- X X X- X - X- Thus X Xe Xo Eve compoet of discrete time sigal is give by Xe X X- 5

9 Odd compoet of discrete time sigal is give by Xo X - X- Ex..:Test whether the followig CT waveforms is periodic or ot. If periodic fid out the fudametal period. a si/3t 4 cos /t 5 cos/3t As: Period of xt b a cost b sit/4 As: No-Periodic a Fid out the eve ad odd parts of the discrete sigal x{,4,3,,} b Fid out the eve ad odd parts of the discrete sigal x{,,,}..9 Eergy sigal ad Power sigal Discrete time sigals are also classified as fiite eergy or fiite average power sigals. The eergy of a discrete time sigal x is give by E x - The average power for a discrete time sigal x is defied as Lim P N N x - If Eergy is fiite ad power is ero for x the x is a eergy sigal. If power is fiite ad eergy is ifiite the x is power sigal. There are some sigals which are either eergy or a power sigal. a Fid the power ad eergy of u uit step fuctio. b Fid the power ad eergy of r uit ramp fuctio. c Fid the power ad eergy of a u..3 DISCRETE TIME SIGNALS AND SYSTEM There are three ways to represet discrete time sigals. Fuctioal Represetatio 4 for,3 x - for 0 elsewhere Tabular method of represetatio x Sequece Represetatio X { 0, 4, -, 4, 0, } 0.3.STANDARD SIGNAL SEQUENCES Uit sample sigal Uit impulse sigal δ i.e δ{} Uit step sigal 6

10 u 0 0 <0 3 Uit ramp sigal ur 0 0 <0 4 Expoetial sigal x a re j Ø r e j Ø r cos Ø j si Ø 5 Siusoidal waveform x A Si w.3. PROPERTIES OF DISCRETE TIME SIGNALS Shiftig : sigal x ca be shifted i time. We ca delay the sequece or advace the sequece. This is doe by replacig iteger by -k where k is iteger. If k is positive sigal is delayed i time by k samples Arrow get shifted o left had side Ad if k is egative sigal is advaced i time k samples Arrow get shifted o right had side X {, -, 0, 4, -, 4, 0, } Delayed by samples : 0 X- {, -, 0, 4, -, 4, 0, } Advaced by samples : 0 X {, -, 0, 4, -, 4, 0, } 0 Foldig / Reflectio : It is foldig of sigal about time origi 0. I this case replace by. Origial sigal: X {, -, 0, 4, -, 4, 0} 0 Folded sigal: X- { 0, 4, -, 4, 0, -, } 0 3 Additio : Give sigals are x ad x, which produces output y where y x x. Adder geerates the output sequece which is the sum of iput sequeces. 4 Scalig: Amplitude scalig ca be doe by multiplyig sigal with some costat. Suppose origial sigal is x. The output sigal is A x 4 Multiplicatio : The product of two sigals is defied as y x * x. 7

11 .3.3 SYMBOLS USED IN DISCRETE TIME SYSTEM. Uit delay x Z - y x-. Uit advace 3. Additio x x Z y x x 4. Multiplicatio x x 5. Scalig costat multiplier x A y A x y xx y x*x.3.4 CLASSIFICATION OF DISCRETE TIME SYSTEMS STATIC v/s DYNAMIC Sr No STATIC DYNAMIC Dyamicity property Static systems are those systems whose output at ay Dyamic systems output istace of time depeds at most o iput depeds upo past or sample at same time. future samples of iput. Static systems are memory less systems. They have memories for memorie all samples. It is very easy to fid out that give system is static or dyamic. Just check that output of the system solely depeds upo preset iput oly, ot depedet upo past or future. 8

12 Sr No System [y] Static / Dyamic x Static A- Dyamic 3 X Static 4 X Dyamic 5 x x Static 6 X x- x Dyamic TIME INVARIANT v/s TIME VARIANT SYSTEMS Sr No TIME INVARIANT TIV / SHIFT INVARIANT A System is time ivariat if its iput output characteristic do ot chage with shift of time. Liear TIV systems ca be uiquely characteried by Impulse respose, frequecy respose or trasfer fuctio. 3 a. Thermal Noise i Electroic compoets b. Pritig documets by a priter TIME VARIANT SYSTEMS / SHIFT VARIANT SYSTEMS Shift Ivariace property A System is time variat if its iput output characteristic chages with time. No Mathematical aalysis ca be performed. a. Raifall per moth b. Noise Effect It is very easy to fid out that give system is Shift Ivariat or Shift Variat. Suppose if the system produces output y by takig iput x x y If we delay same iput by k uits x-k ad apply it to same systems, the system produces output y-k x-k y-k x SYSTEM DELAY y x SYSTEM DELAY y 3 LINEAR v/s NON-LINEAR SYSTEMS Sr LINEAR No A System is liear if it satisfies superpositio theorem. Let x ad x are two iput sequeces, the the system is said to be liear if ad oly if T[ax ax]at[x]at[x] NON-LINEAR Liearity Property A System is No-liear if it does ot satisfies superpositio theorem. 9

13 x a x a SYSTEM y T[ax[] ax ] x SYSTEM a yt[axax] x SYSTEM a hece T [ a x a x ] T [ a x ] T [ a x ] It is very easy to fid out that give system is Liear or No-Liear. Respose to the system to the sum of sigal sum of idividual resposes of the system. Sr No System y Liear or No-Liear e x No-Liear x No-Liear 3 m x c No-Liear 4 cos [ x ] No-Liear 5 X- Liear 6 Log 0 x No-Liear 4 CAUSAL v/s NON CAUSAL SYSTEMS Sr No CAUSAL NON-CAUSAL Causality Property A System is causal if output of system at ay time depeds oly past ad preset iputs. A System is No causal if output of system at ay time depeds o future iputs. I Causal systems the output is the fuctio of x, x-, x-.. ad so o. I No-Causal System the output is the fuctio of future iputs also. X x.. ad so o 3 Example Real time DSP Systems Offlie Systems It is very easy to fid out that give system is causal or o-causal. Just check that output of the system depeds upo preset or past iputs oly, ot depedet upo future. Sr No System [y] Causal /No-Causal x x-3 Causal X Causal 3 X x3 No-Causal 4 x Causal 5 X No-Causal 6 X x- x No-Causal 0

14 5 STABLE v/s UNSTABLE SYSTEMS Sr STABLE No A System is BIBO stable if every bouded iput produces a bouded output. The iput x is said to bouded if there exists some fiite umber Mx such that x Mx < The output y is said to bouded if there exists some fiite umber My such that y My < UNSTABLE Stability Property A System is ustable if ay bouded iput produces a ubouded output. STABILITY FOR LTI SYSTEM It is very easy to fid out that give system is stable or ustable. Just check that by providig iput sigal check that output should ot rise to. The coditio for stability is give by h k < k - Sr No System [y] Stable / Ustable Cos [ x ] Stable x- Stable 3 x Stable 4 x u Stable 5 X x Ustable.4 ANALYSIS OF DISCRETE LINEAR TIME INVARIANT LTI/LSI SYSTEM.5 A/D CONVERSION.6 BASIC BLOCK DIAGRAM OF A/D CONVERTER Aalog sigal Xat Sampler x Quatier Ecoder Discrete time Quatied Digital sigal sigal sigal SAMPLING THEOREM It is the process of covertig cotiuous time sigal ito a discrete time sigal by takig samples of the cotiuous time sigal at discrete time istats. X[] Xat where t Ts /Fs. Whe samplig at a rate of fs samples/sec, if k is ay positive or egative iteger, we caot distiguish betwee the samples values of fa H ad a sie wave of fa kfs H. Thus fa kfs wave is alias or image of a wave.

15 Thus Samplig Theorem states that if the highest frequecy i a aalog sigal is Fmax ad the sigal is sampled at the rate fs > Fmax the xt ca be exactly recovered from its sample values. This samplig rate is called Nyquist rate of samplig. The imagig or aliasig starts after Fs/ hece foldig frequecy is fs/. If the frequecy is less tha or equal to / it will be represeted properly. Example: Case : Xt cos 0 t Fs 40 H i.e t /Fs x[] cos /4 cos / Case : Xt cos 50 t Fs 40 H i.e t /Fs x[] cos 5/4 cos ¼ cos / Thus the frequecy 50 H, 90 H, 30 H are alias of the frequecy 0 H at the samplig rate of 40 samples/sec QUANTIZATION The process of covertig a discrete time cotiuous amplitude sigal ito a digital sigal by expressig each sample value as a fiite umber of digits is called quatiatio. The error itroduced i represetig the cotiuous values sigal by a fiite set of discrete value levels is called quatiatio error or quatiatio oise. Example: x[] 50.9 u where 0 < < & fs H N [] Xq [] Roudig Xq [] Trucatig eq [] Quatiatio Step/Resolutio : The differece betwee the two quatiatio levels is called quatiatio step. It is give by XMax xmi / L- where L idicates Number of quatiatio levels. CODING/ENCODING Each quatiatio level is assiged a uique biary code. I the ecodig operatio, the quatiatio sample value is coverted to the biary equivalet of that quatiatio level. If 6 quatiatio levels are preset, 4 bits are required. Thus bits required i the coder is the smallest iteger greater tha or equal to Log L. i.e b Log L Thus Samplig frequecy is calculated as fsbit rate / b. ANTI-ALIASING FILTER Whe processig the aalog sigal usig DSP system, it is sampled at some rate depedig upo the badwidth. For example if speech sigal is to be processed the frequecies upo 3kh ca be used. Hece the samplig rate of 6kh ca be used. But the speech sigal also cotais some frequecy compoets more tha 3kh. Hece a samplig rate of 6kh will itroduce aliasig. Hece sigal should be bad limited to avoid aliasig.

16 The sigal ca be bad limited by passig it through a filter LPF which blocks or atteuates all the frequecy compoets outside the specific badwidth. Hece called as Ati aliasig filter or prefilter. Block Diagram SAMPLE-AND-HOLD CIRCUIT: The samplig of a aalogue cotiuous-time sigal is ormally implemeted usig a device called a aalogue-to- digital coverter A/D. The cotiuous-time sigal is first passed through a device called a sample-ad-hold S/H whose fuctio is to measure the iput sigal value at the clock istat ad hold it fixed for a time iterval log eough for the A/D operatio to complete. Aalogue-to-digital coversio is potetially a slow operatio, ad a variatio of the iput voltage durig the coversio may disrupt the operatio of the coverter. The S/H prevets such disruptio by keepig the iput voltage costat durig the coversio. This is schematically illustrated by Figure. After a cotiuous-time sigal has bee through the A/D coverter, the quatied output may differ from the iput value. The maximum possible output value after the quatiatio process could be up to half the quatiatio level q above or q below the ideal output value. This deviatio from the ideal output value is called the quatiatio error. I order to reduce this effect, we icreases the umber of bits. Q Calculate Nyquist Rate for the aalog sigal xt xt 4 cos 50 t 8 si 300 t cos 00 t F300 H xt cos 000 t 3 si 6000 t 8 cos 000 t FKH 3 xt 4 cos 00 t F00 H Q The followig four aalog siusoidal are sampled with the fs40h. Fid out correspodig time sigals ad commet o them Xt cos 0t Xt cos 50t 3

17 X3t cos 90t X4t cos 30t Q Sigal xt0cos 000t 5 cos 5000t. Determie Nyquist rate. If the sigal is sampled at 4kh will the sigal be recovered from its samples. Q Sigal xt3 cos 600 t cos800 t. The lik is operated at 0000 bits/sec ad each iput sample is quatied ito 04 differet levels. Determie Nyquist rate, samplig frequecy, foldig frequecy & resolutio. DIFFERENCE BETWEEN FIR AND IIR Sr Fiite Impulse Respose FIR Ifiite Impulse Respose No IIR FIR has a impulse respose that is ero outside of some IIR has a impulse respose o fiite time iterval. ifiite time iterval. Covolutio formula chages to M Covolutio formula chages to y x k h k -M y x k h k For causal FIR systems limits chages to 0 to M. - For causal IIR systems limits 3 The FIR system has limited spa which views oly most recet M iput sigal samples formig output called as Widowig. chages to 0 to. The IIR system has ulimited spa. 4 FIR has limited or fiite memory requiremets. IIR System requires ifiite memory. 5 Realiatio of FIR system is geerally based o Covolutio Sum Method. Discrete time systems has oe more type of classificatio.. Recursive Systems. No-Recursive Systems Sr Recursive Systems No I Recursive systems, the output depeds upo past, preset, future value of iputs as well as past output. Realiatio of IIR system is geerally based o Differece Method. No-Recursive systems Recursive Systems has feedback from output to iput. No Feedback. I No-Recursive systems, the output depeds oly upo past, preset or future values of iputs. 3 Examples y x y- Y x x- 4

18 .7 ANALYSIS OF LTI SYSTEM.7. Z TRANFORM INTRODUCTION TO Z TRANSFORM For aalysis of cotiuous time LTI system Laplace trasform is used. Ad for aalysis of discrete time LTI system trasform is used. Z trasform is mathematical tool used for coversio of time domai ito frequecy domai domai ad is a fuctio of the complex valued variable Z. The trasform of a discrete time sigal x deoted by X ad give as X x -Trasform. - Z trasform is a ifiite power series because summatio idex varies from - to. But it is useful for values of for which sum is fiite. The values of for which f is fiite ad lie withi the regio called as regio of covergece ROC. ADVANTAGES OF Z TRANSFORM. The DFT ca be determied by evaluatig trasform.. Z trasform is widely used for aalysis ad sythesis of digital filter. 3. Z trasform is used for liear filterig. trasform is also used for fidig Liear covolutio, cross-correlatio ad auto-correlatios of sequeces. 4. I trasform user ca characterie LTI system stable/ustable, causal/aticausal ad its respose to various sigals by placemets of pole ad ero plot. ADVANTAGES OF ROCREGION OF CONVERGENCE. ROC is goig to decide whether system is stable or ustable.. ROC decides the type of sequeces causal or ati-causal. 3. ROC also decides fiite or ifiite duratio sequeces. Z TRANSFORM PLOT Z-Plae Imagiary Part of Im >a <a Re Real part of Fig show the plot of trasforms. The trasform has real ad imagiary parts. Thus a plot of imagiary part versus real part is called complex -plae. The radius of circle is called as uit circle. This complex plae is used to show ROC, poles ad eros. Complex variable is also expressed i polar form as Z re jω where r is radius of circle is give by ad ω is the frequecy of the sequece i radias ad give by. 5

19 Sr Time Domai Property Trasform ROC No Sequece δ Uit sample complete plae δ-k Time shiftig -k except0 3 δk Time shiftig k except 4 u Uit step /- - /- > 5 u- Time reversal /- < 6 -u-- Time reversal /- < 7 u Uit ramp Differetiatio - / - - > 8 a u Scalig /- a - > a 9 -a u--left side /- a - < a expoetial sequece 0 a u Differetiatio a - / - a - > a - a u-- Differetiatio a - / - a - < a a for 0 < < N- - a - N / - a - a - < except0 3 for 0<<N- or Liearity - -N / - - > u u-n Shiftig 4 cosω0 u cosω0 - > 5 siω0 u - siω0 - - cosω0 - > 6 a cosω0 u Time scalig - /a - - /a - cosω0/a - > a 7 a siω0 u Time scalig /a - siω0 - /a - cosω0/a - > a Q Determie trasform of followig sigals. Also draw ROC. i x {,,3,4,5} ii x{,,3,4,5,0,7} Q Determie trasform ad ROC for x -/3 u / u--. Q Determie trasform ad ROC for x [ ] u. Q Determie trasform ad ROC for x / u-. Q Determie trasform ad ROC for x / {u u-0}. Q Fid liear covolutio usig trasform. X{,,3} & h{,} PROPERTIES OF Z TRANSFORM ZT Liearity The liearity property states that if x x X Ad X The 6

20 The a x a x a X a X Trasform of liear combiatio of two or more sigals is equal to the same liear combiatio of trasform of idividual sigals. Time shiftig The Time shiftig property states that if x X Ad The x-k X k Thus shiftig the sequece circularly by k samples is equivalet to multiplyig its trasform by k 3 Scalig i domai This property states that if x X Ad The a x x/a Thus scalig i trasform is equivalet to multiplyig by a i time domai. 4 Time reversal Property The Time reversal property states that if x X Ad The x- x - It meas that if the sequece is folded it is equivalet to replacig by - i domai. 5 Differetiatio i domai The Differetiatio property states that if x X Ad The x - d/d X 6 Covolutio Theorem The Circular property states that if x X Ad x X The The x * x X X N Covolutio of two sequeces i time domai correspods to multiplicatio of its Z trasform sequece i frequecy domai. 7

21 7 Correlatio Property The Correlatio of two sequeces states that if x X Ad x X The the x l x-l X x Iitial value Theorem Iitial value theorem states that if x X Ad the x0 lim XZ 9 Fial value Theorem Fial value theorem states that if x X Ad the lim x lim- X RELATIONSHIP BETWEEN FOURIER TRANSFORM AND Z TRANSFORM. There is a close relatioship betwee Z trasform ad Fourier trasform. If we replace the complex variable by e jω, the trasform is reduced to Fourier trasform. Z trasform of sequece x is give by X x Defiitio of -Trasform - Fourier trasform of sequece x is give by Xω x e jω Defiitio of Fourier Trasform - Complex variable is expressed i polar form as Z re jω where r ad ω is. Thus we ca be writte as X [ x r ] e jω - X e jw x e jω - X e jw xω at uit circle. 8

22 Thus, X ca be iterpreted as Fourier Trasform of sigal sequece x r. Here r grows with if r< ad decays with if r>. X coverges for r. hece Fourier trasform may be viewed as Z trasform of the sequece evaluated o uit circle. Thus The relatioship betwee DFT ad Z trasform is give by X e j k xk The frequecy ω0 is alog the positive Re axis ad the frequecy / is alog the positive Im axis. Frequecy is alog the egative Re axis ad 3 / is alog the egative Im axis. Im ω / 0,j re jω ω ω0 -,0,0 Re ω3 / 0,-j Frequecy scale o uit circle X Xω o uit circle INVERSE Z TRANSFORM IZT The sigal ca be coverted from time domai ito domai with the help of trasform ZT. Similar way the sigal ca be coverted from domai to time domai with the help of iverse trasformizt. The iverse trasform ca be obtaied by usig two differet methods. Partial fractio expasio Method PFE / Applicatio of residue theorem Power series expasio Method PSE. PARTIAL FRACTION EXPANSION METHOD I this method X is first expaded ito sum of simple partial fractio. a0 m a m-. am X for m b0 b -. b First fid the roots of the deomiator polyomial a0 m a m-. am X - p - p - p The above equatio ca be writte i partial fractio expasio form ad fid the coefficiet AK ad take IZT. 9

23 SOLVE USING PARTIAL FRACTION EXPANSION METHOD PFE Sr No Fuctio ZT Time domai sequece Commet 3 - a / a N u for > a causal sequece -a u-- for < a ati-causal sequece - N u for > causal sequece -- u-- for < a ati-causal sequece -3 u u stable system for 0.5< <3 3 u 0.5 u causal system for >3-3 u u-- for ati-causal system <0.5 - u u stable system for 0.5< < u 0.5 u causal system for > - u u-- for ati-causal system <0.5 δ8 u u causal system for > /-j3/ /j/ u /j3/ /j/ u causal system /4 - /8 - - / - - / / 4-/ u 3 -/4 u for >/ causal system -/ u for >/ causal system δ u /3 u for > 5 - for > 4-3/ for > causal system causal system causal system. RESIDUE THEOREM METHOD I this method, first fid G - XZ ad fid the residue of G at various poles of X. 0

24 SOLVE USING RESIDUE THEOREM METHOD Sr No Fuctio ZT Time domai Sequece For causal sequece a u a - u - 3 u u 3. POWER-SERIES EXPANSION METHOD The trasform of a discrete time sigal x is give as X x - Expadig the above terms we have x..x-z x-z x0 x Z - x Z.. This is the expasio of trasform i power series form. Thus sequece x is give as x {..,x-,x-,x0,x,x,..}. Power series ca be obtaied directly or by log divisio method. SOLVE USING POWER SERIES EXPANSION METHOD Sr No Fuctio ZT Time domai Sequece -a For causal sequece a u For Ati-causal sequece -a u-- {,3/,7/4,5,8,.} For > {.4,6,,0,0} For < 0.5 {0,,4,9,..} For > X {,-0.5,-,0.5} 5 loga - - a / for ad > a 4. RECURSIVE ALGORITHM The log divisio method ca be recast i recursive form. a0 a - a - X Their IZT is give as b 0 b - b - x /b0 [ a - x-i bi] i Thus X0 a0/b0 X /b0 [ a- x0 b] X /b0 [ a- x b - x0 b] for,,.

25 SOLVE USING RECURSIVE ALGORITHM METHOD Sr No Fuctio ZT Time domai Sequece X {,3,3.6439,.} - -5/6 - /6 - X {,/6,49/36,.} 3 4 X { 3/6,63/64, } -3/4 /8 Example : Example :Fid the magitude ad phase plot of

26 Example 3: 3

27 Example 4: 4

28 Example 5:Fid the iverse Z Trasform 5

29 POLE ZERO PLOT. X is a ratioal fuctio, that is a ratio of two polyomials i - or. The roots of the deomiator or the value of for which X becomes ifiite, defies locatios of the poles. The roots of the umerator or the value of for which X becomes ero, defies locatios of the eros.. ROC dos ot cotai ay poles of X. This is because x becomes ifiite at the locatios of the poles. Oly poles affect the causality ad stability of the system. 3. CASUALTY CRITERIA FOR LSI SYSTEM LSI system is causal if ad oly if the ROC the system fuctio is exterior to the circle. i. e > r. This is the coditio for causality of the LSI system i terms of trasform. The coditio for LSI system to be causal is h 0.. <0 4. STABILITY CRITERIA FOR LSI SYSTEM Bouded iput x produces bouded output y i the LSI system oly if h < - With this coditio satisfied, the system will be stable. The above equatio states that the LSI system is stable if its uit sample respose is absolutely summable. This is ecessary ad sufficiet coditio for the stability of LSI system. H h Z-Trasform. - Takig magitude of both the sides H h... - Magitudes of overall sum is less tha the sum of magitudes of idividual sums. 6

30 H h - - H h If H is evaluated o the uit circle -. Hece LSI system is stable if ad oly if the ROC the system fuctio icludes the uit circle. i.e r <. This is the coditio for stability of the LSI system i terms of trasform. Thus For stable system < For ustable system > Margially stable system Poles iside uit circle gives stable system. Poles outside uit circle gives ustable system. Poles o uit circle give margially stable system. 6. A causal ad stable system must have a system fuctio that coverges for > r <. STANDARD INVERSE Z TRANSFORMS Sr No Fuctio ZT Causal Sequece Ati-causal sequece > a < a a u -a u-- a u u-- 3 a a -a 4 k a k /k-! a -/k-! a 5 δ δ 6 Z k δk δk 7 Z -k δ-k δ-k ONE SIDED Z TRANSFORM Sr Trasform Bilateral No trasform is a ifiite power series because summatio idex varies from to -. Thus Z trasform are give by X x - Oe sided Trasform Uilateral Oe sided trasform summatio idex varies from 0 to. Thus Oe sided trasform are give by X x 0 7

31 trasform is applicable for relaxed systems havig ero iitial coditio. 3 trasform is also applicable for ocausal systems. 4 ROC of x is exterior or iterior to circle hece eed to specify with trasform of sigals. Oe sided trasform is applicable for those systems which are described by differetial equatios with o ero iitial coditios. Oe sided trasform is applicable for causal systems oly. ROC of x is always exterior to circle hece eed ot to be specified. Properties of oe sided trasform are same as that of two sided trasform except shiftig property. Time delay x X Ad k The x-k k [ X x- ] k>0 Time advace x X Ad 8

32 k- The xk k [ X - x - ] k>0 0 Examples: Q Determie oe sided trasform for followig sigals x{,,3,4,5} x{,,3,4,5} SOLUTION OF DIFFERENTIAL EQUATION Oe sided Z trasform is very efficiet tool for the solutio of differece equatios with oero iitial coditio. System fuctio of LSI system ca be obtaied from its differece equatio. Similarly Z{x-} x- - Oe sided Z trasform 0 x- x0 - x - x -3 x- - [x0 - x - x -3 ] Z{ x- } - X x- Z{ x- } - X - x- x- Z{ x } X - x0 Z{ x } X - x0 x. Differece equatios are used to fid out the relatio betwee iput ad output sequeces. It is also used to relate system fuctio H ad Z trasform.. The trasfer fuctio Hω ca be obtaied from system fuctio H by puttig e jω. Magitude ad phase respose plot ca be obtaied by puttig various values of ω. First order Differece Equatio y x a y- where y Output Respose of the recursive system x Iput sigal a Scalig factor y- Uit delay to output. Now we will start at 0 0 y0 x0 a y-. y x a y0. x a [ x0 a y- ] a y- a x0 x.3 hece y a y- a k x -k 0 k 0 9

33 The first part A is respose depedig upo iitial coditio. The secod Part B is the respose of the system to a iput sigal. Zero state respose Forced respose : Cosider iitial coditio are ero. System is relaxed at time 0 i.e y- 0 Zero Iput respose Natural respose : No iput is forced as system is i o- relaxed iitial coditio. i.e y-! 0 Total respose is the sum of ero state respose ad ero iput respose. Q Determie ero iput respose for y 3y- 4y-0; Iitial Coditios are y-5 & y- 0 Aswer: y Q A differece equatio of the system is give below Y 0.5 y- x Determie a System fuctio b Pole ero plot c Uit sample respose Q A differece equatio of the system is give below Y 0.7 y- 0. y- x- x- a System Fuctio b Pole ero plot c Respose of system to the iput x u d Is the system stable? Commet o the result. Q A differece equatio of the system is give below Y 0.5 x 0.5 x- Determie a System fuctio b Pole ero plot c Uit sample respose d Trasfer fuctio e Magitude ad phase plot Q A differece equatio of the system is give below a. Y 0.5 y- x x- b. Y x 3x- 3x- x-3 a System Fuctio b Pole ero plot c Uit sample respose d Fid values of y for 0,,,3,4,5 for x δ for o iitial coditio. Q Solve secod order differece equatio x- 3x- x 3 - with x--4/9 ad x--/3. Q Solve secod order differece equatio x 3x x with x00 ad x. 30

34 3

35 .8 CONVOLUTION.8. LINEAR CONVOLUTION SUM METHOD. This method is powerful aalysis tool for studyig LSI Systems.. I this method we decompose iput sigal ito sum of elemetary sigal. Now the elemetary iput sigals are take ito accout ad idividually give to the system. Now usig liearity property whatever output respose we get for decomposed iput sigal, we simply add it & this will provide us total respose of the system to ay give iput sigal. 3. Covolutio ivolves foldig, shiftig, multiplicatio ad summatio operatios. 4. If there are M umber of samples i x ad N umber of samples i h the the maximum umber of samples i y is equals to M-. Liear Covolutio states that y x * h y x k h k x k h[ -k- ] k - k - Example : h {,,, - } & x {,, 3, } Fid y METHOD : GRAPHICAL REPRESENTATION Step Fid the value of x h - Startig Idex of x startig idex of h Step y { y-, y0, y, y,.} It goes up to legthx legthy -. i.e - y- xk * h--k 0 y0 xk * h0-k y xk * h-k. ANSWER : y {, 4, 8, 8, 3, -, - } METHOD : MATHEMATICAL FORMULA Use Covolutio formula y x k h k k - k 0 to 3 start idex to ed idex of x y x0 h x h- x h- x3 h-3 METHOD 3: VECTOR FORM TABULATION METHOD X {x,x,x3} & h { h,h,h3} h h X x x3 hx hx hx3 hx hx hx3 h3 h3x h3x h3x3 y- h x y0 h x h x y h x3 hx h3 x 3

36 METHOD 4: SIMPLE MULTIPLICATION FORM X {x,x,x3} & h { h,h,h3} x x x3 y y y y3.8.properties OF LINEAR CONVOLUTION x Excitatio Iput sigal y Output Respose h Uit sample respose. Commutative Law: Commutative Property of Covolutio x * h h * x X Uit Sample Respose h Respose y x *h h Uit Sample Respose x Respose y h * x. Associate Law: Associative Property of Covolutio [ x * h ] * h x * [ h * h ] X Uit Sample Resposeh Uit Sample Resposeh h Respose X Uit Sample Respose h h * h Respose 3 Distribute Law: Distributive property of covolutio x * [ h h ] x * h x * h CAUSALITY OF LSI SYSTEM The output of causal system depeds upo the preset ad past iputs. The output of the causal system at 0 depeds oly upo iputs x for 0. The liear covolutio is give as y hk x k k- At 0,the output y0 will be y0 hk x0 k k- Rearragig the above terms... - y0 hk x0 k hk x0 k k0 k- The output of causal system at 0 depeds upo the iputs for < 0 Hece h-h- h-30 Thus LSI system is causal if ad oly if h 0 for <0 33

37 This is the ecessary ad sufficiet coditio for causality of the system. Liear covolutio of the causal LSI system is give by y x k h k k0 STABILITY FOR LSI SYSTEM A System is said to be stable if every bouded iput produces a bouded output. The iput x is said to bouded if there exists some fiite umber Mx such that x Mx <. The output y is said to bouded if there exists some fiite umber My such that y My <. Liear covolutio is give by y x k h k k- Takig the absolute value of both sides y hk x-k k- The absolute values of total sum is always less tha or equal to sum of the absolute values of idividually terms. Hece y hk x k k- y hk x k k- The iput x is said to bouded if there exists some fiite umber Mx such that x Mx <. Hece bouded iput x produces bouded output y i the LSI system oly if hk < k- With this coditio satisfied, the system will be stable. The above equatio states that the LSI system is stable if its uit sample respose is absolutely summable. This is ecessary ad sufficiet coditio for the stability of LSI system. Example : 34

38 Solutio- 35

39 Example : 36

40 SELF-STUDY: Exercise No. Q Show that the discrete time sigal is periodic oly if its frequecy is expressed as the ratio of two itegers. Q Show that the frequecy rage for discrete time siusoidal sigal is - to radias/sample or -½ cycles/sample to ½ cycles/sample. Q3 Prove δ u u-. Q4 Prove u δk k- Q5 Prove u δ-k k0 Q6 Prove that every discrete siusoidal sigal ca be expressed i terms of weighted uit impulse. Q7 Prove the Liear Covolutio theorem..9 CORRELATION: It is frequetly ecessary to establish similarity betwee oe set of data ad aother. It meas we would like to correlate two processes or data. Correlatio is closely related to covolutio, because the correlatio is essetially covolutio of two data sequeces i which oe of the sequeces has bee reversed. Applicatios are i Images processig for robotic visio or remote sesig by satellite i which data from differet image is compared I radar ad soar systems for rage ad positio fidig i which trasmitted ad reflected waveforms are compared. 3 Correlatio is also used i detectio ad idetifyig of sigals i oise. 4 Computatio of average power i waveforms. 5 Idetificatio of biary codeword i pulse code modulatio system..9. DIFFERENCE BETWEEN LINEAR CONVOLUTION AND CORRELATION Sr Liear Covolutio No I case of covolutio two sigal sequeces iput sigal ad impulse respose give by the same system is calculated Our mai aim is to calculate the respose give by the system. Correlatio I case of Correlatio, two sigal sequeces are just compared. Our mai aim is to measure the degree to which two sigals are similar ad thus to extract some iformatio that depeds to alarge extet o the applicatio 37

41 3 Liear Covolutio is give by the equatio y x * h & calculated as y x k h k k - 4 Liear covolutio is commutative Not commutative. Received sigal sequece is give as Y α x-d ω Where α Atteuatio Factor D Delay ω Noise sigal.9. TYPES OF CORRELATION Uder Correlatio there are two classes. CROSS CORRELATION: Whe the correlatio of two differet sequeces x ad y is performed it is called as Cross correlatio. Cross-correlatio of x ad y is rxyl which ca be mathematically expressed as rxyl x y l - OR rxyl x l y - AUTO CORRELATION: I Auto-correlatio we correlate sigal x with itself, which ca be mathematically expressed as rxxl x x l - OR rxxl x l x PROPERTIES OF CORRELATION The cross-correlatio is ot commutative. rxyl ryx-l The cross-correlatio is equivalet to covolutio of oe sequece with folded versio of aother sequece. rxyl xl * y-l. 3 The autocorrelatio sequece is a eve fuctio. rxxl rxx-l Examples: Q Determie cross-correlatio sequece x{, -, 3, 7,,, -3} & y{, -,, -, 4,, -,5} Aswer: rxyl {0, -9, 9, 36, -4, 33, 0,7, 3, -8, 6, -7, 5, -3} Q Determie autocorrelatio sequece x{,,, } Aswer: rxxl {, 3, 5, 7, 5, 3, } 38

42 GLOSSARY:. Sigal. A sigal is a fuctio of oe or more idepedet variables which cotai some iformatio. Eg: Radio sigal, TV sigal, Telephoe sigal etc.. System. A system is a set of elemets or fuctioal block that are coected together ad produces a output i respose to a iput sigal. Eg: A audio amplifier, atteuator, TV set etc. 3. Cotiuous Time sigals. Cotiuous time sigals are defied for all values of time. It is also called as a aalog sigal ad is represeted by xt. Eg: AC waveform, ECG etc. 4. Discrete Time sigals. Discrete time sigals are defied at discrete istaces of time. It is represeted by x. Eg: Amout deposited i a bak per moth. 5. Samplig. Samplig is a process of covertig a cotiuous time sigal ito discrete time Sigal. After samplig the sigal is defied at discrete istats of time ad the time Iterval betwee two subsequet samplig istats is called samplig iterval. 6.Ati-aliasig filter. A filter that is used to reject high frequecy sigals before it is sampled to reduce the aliasig is called ati-aliasig filter. 7. Nyquist rate: Whe the samplig rate becomes exactly equal to W samples/sec, for a give badwidth of fm or W Hert, the it is called as Nyquist rate. Nyquist rate f m samples/secod. 8. Z-trasform ad Iverse -trasform. The -trasform of a geeral discrete-time sigal x is defied as, X x It is used to represet complex siusoidal discrete time sigal. The iverse -trasform is give by, x 9. Regio of covergece. πj X d By defiitio, x X The -trasform exists whe the ifiite sum i equatio coverges. A ecessary coditio for covergece is absolute summability of x -. The 39

43 value of for which the -trasform coverges is called regio of covergece. 0. Left sided sequece ad Right sided sequece Left sided sequece is give by X x For this type of sequece the ROC is etire -plae except at. Right sided sequece is give by X 0 x For this type of sequece the ROC is etire -plae except 0.. Parseval s theorem of -trasform. If x ad x are complex valued sequeces, the the Parseval s relatio states that, x x X v X v dv π j v. LTI System. A System is said to be liear ad Time Ivariat if it Satisfies the Superpositio Priciple ad if the iput is delayed tha the output also delayed by the same amout. 40

44 UNIT II FREQUENCY TRANSFORMATIONS PRE REQUISITE DISCUSSION: The Discrete Fourier trasform is performed for Fiite legth sequece whereas DTFT is used to perform trasformatio o both fiite ad also Ifiite legth sequece. The DFT ad the DTFT ca be viewed as the logical result of applyig the stadard cotiuous Fourier trasform to discrete data. From that perspective, we have the satisfyig result that it's ot the trasform that varies.. INTRODUCTION Ay sigal ca be decomposed i terms of siusoidal or complex expoetial compoets. Thus the aalysis of sigals ca be doe by trasformig time domai sigals ito frequecy domai ad vice-versa. This trasformatio betwee time ad frequecy domai is performed with the help of Fourier TrasformFT But still it is ot coveiet for computatio by DSP processors hece Discrete Fourier TrasformDFT is used. Time domai aalysis provides some iformatio like amplitude at samplig istat but does ot covey frequecy cotet & power, eergy spectrum hece frequecy domai aalysis is used. For Discrete time sigals x, Fourier Trasform is deoted as xω & give by Xω x e jω - DFT is deoted by xk ad give by ω k/n N- Xk x e j k / N DFT. 0 IDFT is give as N- FT. 4

45 x /N X k e j k / N IDFT 3 k0. DIFFERENCE BETWEEN FT & DFT Sr Fourier Trasform FT No FT xω is the cotiuous fuctio of x. Discrete Fourier Trasform DFT DFT xk is calculated oly at discrete values of ω. Thus DFT is discrete i ature. The rage of ω is from - to or 0 to. Samplig is doe at N equally spaced poits over period 0 to. Thus DFT is sampled versio of FT. 3 FT is give by equatio DFT is give by equatio 4 FT equatios are applicable to most of ifiite sequeces. DFT equatios are applicable to causal, fiite duratio sequeces 5 I DSP processors & computers I DSP processors ad computers DFTs are applicatios of FT are limited mostly used. because xω is cotiuous APPLICATION fuctio of ω. a Spectrum Aalysis b Filter Desig Q Prove that FT xω is periodic with period. Q Determie FT of x a u for -< a <. Q Determie FT of x A for 0 L-. Q Determie FT of x u Q Determie FT of x δ Q Determie FT of x e at ut.3 CALCULATION OF DFT & IDFT For calculatio of DFT & IDFT two differet methods ca be used. First method is usig mathematical equatio & secod method is 4 or 8 poit DFT. If x is the sequece of N samples the cosider WN e j / N twiddle factor Four POINT DFT 4-DFT Sr No WNW4e j / Agle Real Imagiary Total W W4 - / 0 -j -j 3 W W4 3-3 / 0 J J 0 3 k0 W4 0 W4 0 W4 0 W4 0 [WN] k W4 0 W4 W4 W4 3 k 4

46 W4 0 W4 W4 4 W4 6 k3 W4 0 W4 3 W4 6 W4 9 Thus 4 poit DFT is give as XN [WN ] XN [WN] j - j - - j - -j EIGHT POINT DFT 8-DFT Sr No WN W8 e j /4 Agle Magitude Imagiary Total W W8 - /4 / -j / / -j / 3 W8 - / 0 -j -j 4 W8 3-3 /4 -/ -j / - / -j / 5 W W8 5-5 / 4 -/ j / - / j / 7 W8 6-7 / 4 0 J J 8 W8 7 - / j / / j / Remember that W8 0 W8 8 W8 6 W8 4 W8 3 W8 40 Periodic Property Magitude ad phase of xk ca be obtaied as, xk sqrt Xrk XIk Agle xk ta - XIk / XRk Examples: Q Compute DFT of x {0,,,3} As: x4[6, -j, -, --j ] Q Compute DFT of x {,0,0,} As: x4[, j, 0, -j ] Q Compute DFT of x {,0,,0} As: x4[, 0,, 0 ] Q Compute IDFT of xk {, j, 0, -j } As: x4[,0,0,].4 DIFFERENCE BETWEEN DFT & IDFT Sr DFT Aalysis trasform No DFT is fiite duratio discrete frequecy sequece that is obtaied by samplig oe period of FT. DFT equatios are applicable to causal fiite duratio sequeces. IDFT Sythesis trasform IDFT is iverse DFT which is used to calculate time domai represetatio Discrete time sequece form of xk. IDFT is used basically to determie sample respose of a filter for which we kow oly trasfer fuctio. 43

47 3 Mathematical Equatio to calculate DFT is give by N- Xk x e j k / N 0 4 Thus DFT is give by Xk [WN][x] Mathematical Equatio to calculate IDFT is give by N- x /N X ke j k / N 0 I DFT ad IDFT differece is of factor /N & sig of expoet of twiddle factor. Thus x /N [ WN] - [XK].5 PROPERTIES OF DFT DFT x N. Periodicity Let x ad xk be the DFT pair the if xn x XkN Xk Thus periodic sequece xp ca be give as xp x-ln l-. Liearity The liearity property states that if DFT x xk Xk Ad for all the for all k 44

48 N DFT x Xk The N The DFT a x a x a Xk a Xk N DFT of liear combiatio of two or more sigals is equal to the same liear combiatio of DFT of idividual sigals. 3. Circular Symmetries of a sequece A A sequece is said to be circularly eve if it is symmetric about the poit ero o the circle. Thus XN- x B A sequece is said to be circularly odd if it is ati symmetric about the poit ero o the circle. Thus XN- - x C A circularly folded sequece is represeted as x-n ad give by x-n xn-. D Aticlockwise directio gives delayed sequece ad clockwise directio gives advace sequece. Thus delayed or advaces sequece x` is related to x by the circular shift. 4. Symmetry Property of a sequece A Symmetry property for real valued x i.e xi0 This property states that if x is real the XN-k X * kx-k B Real ad eve sequece x i.e xi0 & XIK0 This property states that if the sequece is real ad eve x xn- the DFT becomes N- Xk x cos k/n 0 C Real ad odd sequece x i.e xi0 & XRK0 This property states that if the sequece is real ad odd x-xn- the DFT becomes N- Xk -j x si k/n 0 D Pure Imagiary x i.e xr0 This property states that if the sequece is purely imagiary xj XI the DFT becomes N- XRk xi si k/n 0 45

49 N- XIk xi cos k/n 0 5. Circular Covolutio The Circular Covolutio property states that if DFT x N Xk Ad DFT x Xk The N DFT The x N x xk xk N It meas that circular covolutio of x & x is equal to multiplicatio of their DFTs. Thus circular covolutio of two periodic discrete sigal with period N is give by N- ym x x m-n.4 0 Multiplicatio of two sequeces i time domai is called as Liear covolutio while Multiplicatio of two sequeces i frequecy domai is called as circular covolutio. Results of both are totally differet but are related with each other. There are two differet methods are used to calculate circular covolutio Graphical represetatio form Matrix approach DIFFERENCE BETWEEN LINEAR CONVOLUTION & CIRCULAR CONVOLUTION Sr No Liear Covolutio Circular Covolutio I case of covolutio two sigal sequeces iput sigal x ad impulse respose h give by the same system, output y is calculated Multiplicatio of two sequeces i time domai is called as Liear covolutio 3 Liear Covolutio is give by the equatio y x * h & calculated as y x k h k Multiplicatio of two DFTs is called as circular covolutio. Multiplicatio of two sequeces i frequecy domai is called as circular covolutio. Circular Covolutio is calculated as N- ym x x m-n 0 46

50 k - 4 Liear Covolutio of two sigals returs N- elemets where N is sum of elemets i both sequeces. Circular covolutio returs same umber of elemets that of two sigals. Q The two sequeces x{,,,} & x{,,3,4}. Fid out the sequece x3m which is equal to circular covolutio of two sequeces. As: X3m{4,6,4,6} Q x{,,,,-,-,-,-} & x{0,,,3,4,3,,}. Fid out the sequece x3m which is equal to circular covolutio of two sequeces. As: X3m{-4,-8,-8,-4,4,8,8,4} Q Perform Liear Covolutio of x{,} & h{,} usig DFT & IDFT. Q Perform Liear Covolutio of x{,,,} & h{,,3} usig 8 Pt DFT & IDFT. DIFFERENCE BETWEEN LINEAR CONVOLUTION & CIRCULAR CONVOLUTION 47

51 6. Multiplicatio The Multiplicatio property states that if DFT X N xk Ad X DFT N xk The DFT The x x /N xk N N xk 48

52 It meas that multiplicatio of two sequeces i time domai results i circular covolutio of their DFTs i frequecy domai. 7. Time reversal of a sequece The Time reversal property states that if DFT X xk Ad N DFT The x-n xn- x-kn xn-k N It meas that the sequece is circularly folded its DFT is also circularly folded. 8. Circular Time shift The Circular Time shift states that if DFT X xk Ad The x-ln N DFT xk e j k l / N N Thus shiftig the sequece circularly by l samples is equivalet to multiplyig its DFT by e j k l / N 9. Circular frequecy shift The Circular frequecy shift states that if DFT X xk Ad The x e j l / N N DFT N x-ln Thus shiftig the frequecy compoets of DFT circularly is equivalet to multiplyig its time domai sequece by e j k l / N 0. Complex cojugate property The Complex cojugate property states that if DFT X N DFT x * -N x * N-k x *. Circular Correlatio N DFT N xk the x * -kn x * N-k Ad x * k The Complex correlatio property states DFT rxyl Rxyk xk Y * k N Here rxyl is circular cross correlatio which is give as

53 rxyl N- x y * l N 0 This meas multiplicatio of DFT of oe sequece ad cojugate DFT of aother sequece is equivalet to circular cross-correlatio of these sequeces i time domai.. Parseval s Theorem The Parsevals theorem states N- N- X y * /N x k y * k 0 0 This equatio give eergy of fiite duratio sequece i terms of its frequecy compoets..6 APPLICATION OF DFT. DFT FOR LINEAR FILTERING Cosider that iput sequece x of Legth L & impulse respose of same system is h havig M samples. Thus y output of the system cotais N samples where NLM-. If DFT of y also cotais N samples the oly it uiquely represets y i time domai. Multiplicatio of two DFTs is equivalet to circular covolutio of correspodig time domai sequeces. But the legth of x & h is less tha N. Hece these sequeces are appeded with eros to make their legth N called as Zero paddig. The N poit circular covolutio ad liear covolutio provide the same sequece. Thus liear covolutio ca be obtaied by circular covolutio. Thus liear filterig is provided by DFT. Whe the iput data sequece is log the it requires large time to get the output sequece. Hece other techiques are used to filter log data sequeces. Istead of fidig the output of complete iput sequece it is broke ito small legth sequeces. The output due to these small legth sequeces are computed fast. The outputs due to these small legth sequeces are fitted oe after aother to get the fial output respose. METHOD : OVERLAP SAVE METHOD OF LINEAR FILTERING Step > I this method L samples of the curret segmet ad M- samples of the previous segmet forms the iput data block. Thus data block will be X {0,0,0,0,0,,x0,x,.xL-} X {xl-m,.xl-,xl,xl,,,,,,,,,,,,,xl-} X3 {xl-m,.xl-,xl,xl,,,,,,,,,,,,,x3l-} Step> Uit sample respose h cotais M samples hece its legth is made N by paddig eros. Thus h also cotais N samples. h{ h0, h,.hm-, 0,0,0, L- eros} Step3> The N poit DFT of h is Hk & DFT of m th data block be xmk the correspodig DFT of output be Y`mk

54 Y`mk Hk xmk Step 4> The sequece ym ca be obtaied by takig N poit IDFT of Y`mk. Iitial M- samples i the correspodig data block must be discarded. The last L samples are the correct output samples. Such blocks are fitted oe after aother to get the fial output. Sie L X of Sie N X M- Sie L Zeros X Sie L X3 Y Discard M- Poits Y Y3

55 Discard M- Poits Discard M- Poits Y of Sie N METHOD : OVERLAP ADD METHOD OF LINEAR FILTERING Step > I this method L samples of the curret segmet ad M- samples of the previous segmet forms the iput data block. Thus data block will be X {x0,x,.xl-,0,0,0,.} X {xl,xl,xl-,0,0,0,0} X3 {xl,xl,,,,,,,,,,,,,x3l-,0,0,0,0} Step> Uit sample respose h cotais M samples hece its legth is made N by paddig eros. Thus h also cotais N samples. h{ h0, h,.hm-, 0,0,0, L- eros} Step3> The N poit DFT of h is Hk & DFT of m th data block be xmk the correspodig DFT of output be Y`mk Y`mk Hk xmk Step 4> The sequece ym ca be obtaied by takig N poit IDFT of Y`mk. Iitial M- samples are ot discarded as there will be o aliasig. The last M- samples of curret output block must be added to the first M- samples of ext output block. Such blocks are fitted oe after aother to get the fial output. X of Sie N Sie L X M- Zeros Sie L X M- Zeros Sie L X3 M- Zeros Y Y 47

56 M- Poits add together Y of Sie N DIFFERENCE BETWEEN OVERLAP SAVE AND OVERLAP ADD METHOD Sr OVERLAP SAVE METHOD OVERLAP ADD METHOD No I this method, L samples of the curret segmet ad M- samples of the previous segmet forms the iput data block. Iitial M- samples of output sequece are discarded which occurs due to aliasig effect. 3 To avoid loss of data due to aliasig last M- samples of each data record are saved.. SPECTRUM ANALYSIS USING DFT I this method L samples from iput sequece ad paddig M- eros forms data block of sie N. There will be o aliasig i output data blocks. Last M- samples of curret output block must be added to the first M- samples of ext output block. Hece called as overlap add method. DFT of the sigal is used for spectrum aalysis. DFT ca be computed o digital computer or digital sigal processor. The sigal to be aalyed is passed through ati-aliasig filter ad samples at the rate of Fs Fmax. Hece highest frequecy compoet is Fs/. Frequecy spectrum ca be plotted by takig N umber of samples & L samples of waveforms. The total frequecy rage is divided ito N poits. Spectrum is better if we take large value of N & L But this icreases processig time. DFT ca be computed quickly usig FFT algorithm hece fast processig ca be doe. Thus most accurate resolutio ca be obtaied by icreasig umber of samples..7 FAST FOURIER ALGORITHM FFT. Large umber of the applicatios such as filterig, correlatio aalysis, spectrum aalysis require calculatio of DFT. But direct computatio of DFT require large umber of computatios ad hece processor remai busy. Hece special algorithms are developed to compute DFT quickly called as Fast Fourier algorithms FFT. 48

57 . The radix- FFT algorithms are based o divide ad coquer approach. I this method, the N-poit DFT is successively decomposed ito smaller DFTs. Because of this decompositio, the umber of computatios are reduced..8 RADIX- FFT ALGORITHMS. DECIMATION IN TIME DITFFT There are three properties of twiddle factor WN W N W K N Periodicity Property kn W N - kn/ WN K Symmetry Property 3 WN WN/. N poit sequece x be splitted ito two N/ poit data sequeces f ad f. f cotais eve umbered samples of x ad f cotais odd umbered samples of x. This splitted operatio is called decimatio. Sice it is doe o time domai sequece it is called Decimatio i Time. Thus fmxm fmxm N poit DFT is give as where 0,,.N/- where 0,,.N/- N- k Xk x WN 0 Sice the sequece x is splitted ito eve umbered ad odd umbered samples, thus N/- N/- mk km Xk x m WN x m WN m0 m0 Xk Fk k WN Fk 3 XkN/ Fk - k WN Fk Symmetry property 4 Fig shows that 8-poit DFT ca be computed directly ad hece o reductio i computatio. x0 x x x3 x7 8 Poit DFT X0 X X X3 X7 49

58 Fig. DIRECT COMPUTATION FOR N8 50

59 x0 f0 X0 x N/ Poit f X x4 f X x6 DFT f3 X3 x f0 w8 0 X4 x3 f w8 X5 x5 N/ Poit f w8 X6 x7 DFT f3 w8 3 X7 Fig. FIRST STAGE FOR FFT COMPUTATION FOR N8 Fig 3 shows N/ poit DFT base separated i N/4 boxes. I such cases equatios become gk Pk k WN Pk 5 gkn/ pk - k WN Pk 6 x0 x4 N/4 Poit DFT F0 F x w8 0 F x6 N/4 Poit DFT w8 F3 Fig 3. SECOND STAGE FOR FFT COMPUTATION FOR N8 a A a W r N b b WN r B a - W r b Fig 4. BUTTERFLY COMPUTATION THIRD STAGE x0 A X0 x w8 0 B X x C w8 0 X x3 w8 0 D w8 X3 N 5

60 Fig 5. SIGNAL FLOW GRAPH FOR RADIX- DIT FFT N4 x0 A A X0 x4 w8 0 B B X x C w8 0 C X x6 w8 0 D w8 D X3 x E E w8 0 X4 x5 w8 0 F F w8 X5 x3 G w8 0 G w8 X6 x7 w8 0 H w8 H w8 3 X7 Fig 6. SIGNAL FLOW GRAPH FOR RADIX- DIT FFT N8

61 .9 COMPUTATIONAL COMPLEXITY FFT V/S DIRECT COMPUTATION For Radix- algorithm value of N is give as N V Hece value of v is calculated as V log0 N / log0 log N Thus if value of N is 8 the the value of v3. Thus three stages of decimatio. Total umber of butterflies will be Nv/. If value of N is 6 the the value of v4. Thus four stages of decimatio. Total umber of butterflies will be Nv/ 3. Each butterfly operatio takes two additio ad oe multiplicatio operatios. Direct computatio requires N multiplicatio operatio & N N additio operatios. N Direct computatio DIT FFT algorithm Improvemet i Complex Multiplicatio Complex Additio Complex Multiplicatio Complex Additio processig speed for multiplicatio N N - N N/ log N N log N times times times MEMORY REQUIREMENTS AND IN PLACE COMPUTATION a A a W r b N N b W r N B a - W r b Fig. BUTTERFLY COMPUTATION From values a ad b ew values A ad B are computed. Oce A ad B are computed, there is o eed to store a ad b. Thus same memory locatios ca be used to store A

62 ad B where a ad b were stored hece called as I place computatio. The advatage of i place computatio is that it reduces memory requiremet. Thus for computatio of oe butterfly, four memory locatios are required for storig two complex umbers A ad B. I every stage there are N/ butterflies hece total N memory locatios are required. N locatios are required for each stage. Sice stages are computed successively these memory locatios ca be shared. I every stage N/ twiddle factors are required hece maximum storage requiremets of N poit DFT will be N N/..0 BIT REVERSAL For 8 poit DIT DFT iput data sequece is writte as x0, x4, x, x6, x, x5, x3, x7 ad the DFT sequece Xk is i proper order as X0, X, X, X3, X4, x5, X6, x7. I DIF FFT it is exactly opposite. This ca be obtaied by bit reversal method. Decimal Memory Address x i biary Natural Order Memory Address i bit reversed order New Address i decimal Table shows first colum of memory address i decimal ad secod colum as biary. Third colum idicates bit reverse values. As FFT is to be implemeted o digital computer simple iteger divisio by method is used for implemetig bit reversal algorithms. Flow chart for Bit reversal algorithm is as follows DECIMAL NUMBER B TO BE REVERSED I BB BR0 BItB/ BR*BR B- * B] BB; I 53

63 Is I > logn Store BR as Bit reversal of B. DECIMATION IN FREQUENCY DIFFFT I DIF N Poit DFT is splitted ito N/ poits DFTs. Xk is splitted with k eve ad k odd this is called Decimatio i frequecydif FFT. N poit DFT is give as N- Xk x k WN 0 Sice the sequece x is splitted N/ poit samples, thus N/- N/- Xk x WN x N/ WN m0 m0 N/- N/- N Xk x W k N W kn/ N x N/ W k m0 m0 N/- N/- N Xk x W k N - k x N/ W k m0 m0 N/- Xk m0 x - k x N/ WN k 3 Let us split Xk ito eve ad odd umbered samples N/- Xk m0 x - k x N/ WN k 4 54

64 N/- Xk m0 x - k x N/WN k 5 Equatio 4 ad 5 are thus simplified as g g x x N/ x - x N/ WN Fig shows Butterfly computatio i DIF FFT. a A a b b W N r B a bw r N Fig. BUTTERFLY COMPUTATION Fig shows sigal flow graph ad stages for computatio of radix- DIF FFT algorithm of N4 x0 A X0 x B w4 0 X x w4 0 C X x3 w4 D w4 0 X3 Fig. SIGNAL FLOW GRAPH FOR RADIX- DIF FFT N4 55

65 Fig 3 shows sigal flow graph ad stages for computatio of radix- DIF FFT algorithm of N8 x0 A A X0 x B B w8 0 X4 x C w8 0 C X 56

66 x3 D w8 D w8 0 X6 x4 w8 0 E E X x5 w8 F F w8 0 X5 x6 w8 G w8 0 G X3 x7 w8 3 H w8 H w8 0 X7 Fig 3. SIGNAL FLOW GRAPH FOR RADIX- DIF FFT N8 DIFFERENCE BETWEEN DITFFT AND DIFFFT Sr No DIT FFT DIF FFT DITFFT algorithms are based upo decompositio of the iput sequece ito smaller ad smaller sub sequeces. I this iput sequece x is splitted ito eve ad odd umbered samples 3 Splittig operatio is doe o time domai sequece. 4 I DIT FFT iput sequece is i bit reversed order while the output sequece is i atural order. DIFFFT algorithms are based upo decompositio of the output sequece ito smaller ad smaller sub sequeces. I this output sequece Xk is cosidered to be splitted ito eve ad odd umbered samples Splittig operatio is doe o frequecy domai sequece. I DIFFFT, iput sequece is i atural order. Ad DFT should be read i bit reversed order. DIFFERENCE BETWEEN DIRECT COMPUTATION & FFT Sr No Direct Computatio Radix - FFT Algorithms Direct computatio requires large umber of computatios as compared with FFT algorithms. Processig time is more ad more for large umber of N hece processor remais busy. Radix- FFT algorithms requires less umber of computatios. Processig time is less hece these algorithms compute DFT very quickly as compared with direct computatio. 57

67 3 Direct computatio does ot requires splittig operatio. 4 As the value of N i DFT icreases, the efficiecy of direct computatio decreases. 5 I those applicatios where DFT is to be computed oly at selected values of kfrequecies ad whe these values are less tha logn the direct computatio becomes more efficiet tha FFT. Splittig operatio is doe o time domai basis DIT or frequecy domai basis DIF As the value of N i DFT icreases, the efficiecy of FFT algorithms icreases. Applicatios Liear filterig Digital filter desig Q x{,,,} Fid Xk usig DITFFT. Q x{,,,} Fid Xk usig DIFFFT. Q x{0.3535,0.3535,0.6464,.0607,0.3535,-.0607,-.3535, } Fid Xk usig DITFFT. Q Usig radix FFT algorithm, plot flow graph for N8. 58

68 59

69 . GOERTZEL ALGORITHM FFT algorithms are used to compute N poit DFT for N samples of the sequece x. This requires N/ logn umber of complex multiplicatios ad N logn complex additios. I some applicatios DFT is to be computed oly at selected values of frequecies ad selected values are less tha logn, the direct computatios of DFT becomes more efficiet tha FFT. This direct computatios of DFT ca be realied through liear filterig of x. Such liear filterig for computatio of DFT ca be implemeted usig Goertel algorithm. 60

70 By defiitio N poit DFT is give as N- Xk x m WN m0 km Multiplyig both sides by WN -kn which is always equal to. N- Xk x m W kn-m N m0 Thus for LSI system which has iput x ad havig uit sample respose k - hk WN u X hk W -k u yk N Liear covolutio is give by y x k h k k- yk x m WN m- u m 3 As xm is give for N values yk N- x m - WN m0 4 The output of LSI system at N is give by yk N x m WN -kn-m 5 m- Thus comparig equatio ad 5, Xk yk N Thus DFT ca be obtaied as the output of LSI system at N. Such systems ca give Xk at selected values of k. Thus DFT is computed as liear filterig operatios by Goertel Algorithm. GLOSSARY: Fourier Trasform: The Trasform that used to aalye the sigals or systems Characteristics i frequecy domai, which is difficult i case of Time Domai. 6

71 Laplace Trasform: Laplace Trasform is the basic cotiuous Trasform. The it is developed to represet the cotiuous sigals i frequecy domai. Discrete Time Fourier Trasform: For aalyig the discrete sigals, the DTFT Discrete Time Fourier Trasform is used. The output, that the frequecy is cotiuous i DTFT. But the Trasformed Value should be discrete. Sice the Digital Sigal Processors caot work with the cotiuous frequecy sigals. So the DFT is developed to represet the discrete sigals i discrete frequecy domai. Discrete Fourier Trasform: Discrete Fourier Trasform is used for trasformig a discrete time sequece of fiite legth N ito a discrete frequecy sequece of the same fiite legth N. Periodicity: If a discrete time sigal is periodic the its DFT is also periodic. i.e. if a sigal or sequece is repeated after N Number of samples, the it is called periodic sigal. Symmetry: If a sigal or sequece is repeated its waveform i a egative directio after N/ umber of Samples, the it is called symmetric sequece or sigal. Liearity: A System which satisfies the superpositio priciple is said to be a liear system. The DFT have the Liearity property. Sice the DFT of the output is equal to the sum of the DFT s of the Iputs. Fast Fourier Trasform: Fast Fourier Trasform is a algorithm that efficietly computes the discrete fourier trasform of a sequece x. The direct computatio of the DFT requires N evaluatios of trigometric fuctios. 4N real multiplicatios ad 4NN- real additios. 6

72 UNIT III IIR FILTER DESIGN PREREQUISITE DISCUSSION: Basically a digital filter is a liear time ivariat discrete time system. The terms Fiite Impulse respose FIR ad Ifiite Impulse Respose IIR are used to distiguish filter types. The FIR filters are of No-Recursive type whereas the IIR Filters are of recursive type. 3. INTRODUCTION To remove or to reduce stregth of uwated sigal like oise ad to improve the quality of required sigal filterig process is used. To use the chael full badwidth we mix up two or more sigals o trasmissio side ad o receiver side we would like to separate it out i efficiet way. Hece filters are used. Thus the digital filters are mostly used i. Removal of udesirable oise from the desired sigals. Equaliatio of commuicatio chaels 3. Sigal detectio i radar, soar ad commuicatio 4. Performig spectral aalysis of sigals. Aalog ad digital filters I sigal processig, the fuctio of a filter is to remove uwated parts of the sigal, such as radom oise, or to extract useful parts of the sigal, such as the compoets lyig withi a certai frequecy rage. The followig block diagram illustrates the basic idea. There are two mai kids of filter, aalog ad digital. They are quite differet i their physical makeup ad i how they work. A aalog filter uses aalog electroic circuits made up from compoets such as resistors, capacitors ad op amps to produce the required filterig effect. Such filter circuits are widely used i such applicatios as oise reductio, video sigal ehacemet, graphic equaliers i hi-fi systems, ad may other areas. I aalog filters the sigal beig filtered is a electrical voltage or curret which is the direct aalogue of the physical quatity e.g. a soud or video sigal or trasducer output ivolved. 63

73 A digital filter uses a digital processor to perform umerical calculatios o sampled values of the sigal. The processor may be a geeral-purpose computer such as a PC, or a specialied DSP Digital Sigal Processor chip. The aalog iput sigal must first be sampled ad digitied usig a ADC aalog to digital coverter. The resultig biary umbers, represetig successive sampled values of the iput sigal, are trasferred to the processor, which carries out umerical calculatios o them. These calculatios typically ivolve multiplyig the iput values by costats ad addig the products together. If ecessary, the results of these calculatios, which ow represet sampled values of the filtered sigal, are output through a DAC digital to aalog coverter to covert the sigal back to aalog form. I a digital filter, the sigal is represeted by a sequece of umbers, rather tha a voltage or curret. The followig diagram shows the basic setup of such a system. BASIC BLOCK DIAGRAM OF DIGITAL FILTERS Aalog sigal Xa t Sampler Quatier & Ecoder Digital Filter Discrete time sigal Digital sigal. Samplers are used for covertig cotiuous time sigal ito a discrete time sigal by takig samples of the cotiuous time sigal at discrete time istats.. The Quatier are used for covertig a discrete time cotiuous amplitude sigal ito a digital sigal by expressig each sample value as a fiite umber of digits. 64

74 3. I the ecodig operatio, the quatiatio sample value is coverted to the biary equivalet of that quatiatio level. 4. The digital filters are the discrete time systems used for filterig of sequeces. These digital filters performs the frequecy related operatios such as low pass, high pass, bad pass ad bad reject etc. These digital Filters are desiged with digital hardware ad software ad are represeted by differece equatio. DIFFERENCE BETWEEN ANALOG FILTER AND DIGITAL FILTER Sr Aalog Filter Digital Filter No Aalog filters are used for filterig aalog sigals. Digital filters are used for filterig digital sequeces. Aalog filters are desiged with various compoets like resistor, iductor ad capacitor Digital Filters are desiged with digital hardware like FF, couters shift registers, ALU ad softwares like C or assembly laguage. 3 Aalog filters less accurate & because of compoet tolerace of active compoets & more sesitive to evirometal chages. Digital filters are less sesitive to the evirometal chages, oise ad disturbaces. Thus periodic calibratio ca be avoided. Also they are extremely stable. 4 Less flexible These are most flexible as software programs & cotrol programs ca be easily modified. Several iput sigals ca be filtered by oe digital filter. 5 Filter represetatio is i terms of Digital filters are represeted by the differece system compoets. 6 A aalog filter ca oly be chaged by redesigig the filter circuit. FILTER TYPES AND IDEAL FILTER CHARACTERISTIC equatio. A digital filter is programmable, i.e. its operatio is determied by a program stored i the processor's memory. This meas the digital filter ca easily be chaged without affectig the circuitry hardware. Filters are usually classified accordig to their frequecy-domai characteristic as lowpass, highpass, badpass ad badstop filters.. Lowpass Filter A lowpass filter is made up of a passbad ad a stopbad, where the lower frequecies Of the iput sigal are passed through while the higher frequecies are atteuated. H ω -ωc ωc ω 65

75 . Highpass Filter A highpass filter is made up of a stopbad ad a passbad where the lower frequecies of the iput sigal are atteuated while the higher frequecies are passed. Hω ω -ωc ωc 3. Badpass Filter A badpass filter is made up of two stopbads ad oe passbad so that the lower ad higher frequecies of the iput sigal are atteuated while the iterveig frequecies are passed. Hω -ω -ω ω ω ω 4. Badstop Filter A badstop filter is made up of two passbads ad oe stopbad so that the lower higher frequecies of the iput sigal are passed while the iterveig frequecies are atteuated. A idealied badstop filter frequecy respose has the followig shape. Hω ad ω 5. Multipass Filter A multipass filter begis with a stopbad followed by more tha oe passbad. By default, a multipass filter i Digital Filter Desiger cosists of three passbads ad four stopbads. The frequecies of the iput sigal at the stopbads are atteuated while those at the passbads are passed. 6. Multistop Filter 66

76 A multistop filter begis with a passbad followed by more tha oe stopbad. By default, a multistop filter i Digital Filter Desiger cosists of three passbads ad two stopbads. 7. All Pass Filter A all pass filter is defied as a system that has a costat magitude respose for all frequecies. Hω for 0 ω < The simplest example of a all pass filter is a pure delay system with system fuctio H Z -k. This is a low pass filter that has a liear phase characteristic. All Pass filters fid applicatio as phase equaliers. Whe placed i cascade with a system that has a udesired phase respose, a phase equaliers is desiged to compesate for the poor phase characteristic of the system ad therefore to produce a overall liear phase respose. IDEAL FILTER CHARACTERISTIC. Ideal filters have a costat gai usually take as uity gai passbad characteristic ad ero gai i their stop bad.. Ideal filters have a liear phase characteristic withi their passbad. 3. Ideal filters also have costat magitude characteristic. 4. Ideal filters are physically urealiable. 3. TYPES OF DIGITAL FILTER Digital filters are of two types. Fiite Impulse Respose Digital Filter & Ifiite Impulse Respose Digital Filter DIFFERENCE BETWEEN FIR FILTER AND IIR FILTER Sr FIR Digital Filter No FIR system has fiite duratio uit sample respose. i.e h 0 for <0 ad M Thus the uit sample respose exists for the duratio from 0 to M-. FIR systems are o recursive. Thus output of FIR filter depeds upo preset ad past iputs. 3 Differece equatio of the LSI system for FIR filters becomes M y bk x k k0 4 FIR systems has limited or fiite memory requiremets. IIR Digital Filter IIR system has ifiite duratio uit sample respose. i. e h 0 for <0 Thus the uit sample respose exists for the duratio from 0 to. IIR systems are recursive. Thus they use feedback. Thus output of IIR filter depeds upo preset ad past iputs as well as past outputs Differece equatio of the LSI system for IIR filters becomes N M y- ak y k bk x k k k0 IIR system requires ifiite memory. 67

77 5 FIR filters are always stable Stability caot be always guarateed. 6 FIR filters ca have a exactly liear phase respose so that o phase distortio is itroduced i the sigal by the filter. IIR filter is usually more efficiet desig i terms of computatio time ad memory requiremets. IIR systems usually requires less processig time ad storage as compared with FIR. 7 The effect of usig fiite word legth to implemet filter, oise ad quatiatio errors are less severe i FIR tha i IIR. Aalogue filters ca be easily ad readily trasformed ito equivalet IIR digital filter. But same is ot possible i FIR because that have o aalogue couterpart. 8 All ero filters Poles as well as eros are preset. 9 FIR filters are geerally used if o phase distortio is desired. Example: System described by Y 0.5 x 0.5 x- is FIR filter. h{0.5,0.5} IIR filters are geerally used if sharp cutoff ad high throughput is required. Example: System described by Y y- x is IIR filter. ha u for STRUCTURES FOR FIR SYSTEMS FIR Systems are represeted i four differet ways. Direct Form Structures. Cascade Form Structure 3. Frequecy-Samplig Structures 4. Lattice structures.. DIRECT FORM STRUCTURE OF FIR SYSTEM The covolutio of h ad x for FIR systems ca be writte as M- y hk x k k0 The above equatio ca be expaded as, Y h0 x h x- h x- hm- x-m Implemetatio of direct form structure of FIR filter is based upo the above equatio. x Z - x- Z - x-m h0 h hm- h0x h0x hx FIG - DIRECT FORM REALIZATION OF FIR SYSTEM y 68

78 There are M- uit delay blocks. Oe uit delay block requires oe memory locatio. Hece direct form structure requires M- memory locatios. The multiplicatio of hk ad x-k is performed for 0 to M- terms. Hece M multiplicatios ad M- additios are required. 3 Direct form structure is ofte called as trasversal or tapped delay lie filter.. CASCADE FORM STRUCTURE OF FIR SYSTEM I cascade form, stages are cascaded coected i series. The output of oe system is iput to aother. Thus total K umber of stages are cascaded. The total system fuctio 'H' is give by H H. H. Hk H Y/X. Y/X. Yk/Xk k Hπ Hk 3 k xx yx yx3 yky H H Hk FIG- CASCADE FORM REALIZATION OF FIR SYSTEM Each H, H etc is a secod order sectio ad it is realied by the direct form as show i below figure. System fuctio for FIR systems M- H bk -k k0 Expadig the above terms we have H H. H. Hk where HK bk0 bk - bk - Thus Direct form of secod order system is show as x Z - x- Z - bk0 bk bk y FIG - DIRECT FORM REALIZATION OF FIR SECOND ORDER SYSTEM 69

79 3. 4 STRUCTURES FOR IIR SYSTEMS IIR Systems are represeted i four differet ways. Direct Form Structures Form I ad Form II. Cascade Form Structure 3. Parallel Form Structure 4. Lattice ad Lattice-Ladder structure. DIRECT FORM STRUCTURE FOR IIR SYSTEMS IIR systems ca be described by a geeralied equatios as N M y- ak y k bk x k k k0 Z trasform is give as M N H bk k / ak k K0 k M N Here H bk k Ad H ak k K0 k0 Overall IIR system ca be realied as cascade of two fuctio H ad H. Here H represets eros of H ad H represets all poles of H. x b0 DIRECT FORM - I y Z - b -a Z - Z - Z - b -a bm- Z - bm -an- -an Z - 70

80 FIG - DIRECT FORM I REALIZATION OF IIR SYSTEM. Direct form I realiatio of H ca be obtaied by cascadig the realiatio of H which is all ero system first ad the H which is all pole system.. There are MN- uit delay blocks. Oe uit delay block requires oe memory locatio. Hece direct form structure requires MN- memory locatios. 3. Direct Form I realiatio requires MN umber of multiplicatios ad MN umber of additios ad MN umber of memory locatios. DIRECT FORM - II. Direct form realiatio of H ca be obtaied by cascadig the realiatio of H which is all pole system ad H which is all ero system.. Two delay elemets of all pole ad all ero system ca be merged ito sigle delay elemet. 3. Direct Form II structure has reduced memory requiremet compared to Direct form I structure. Hece it is called caoic form. 4. The direct form II requires same umber of multiplicatiosmn ad additios MN as that of direct form I. X b0 y Z - -a b Z - -a b -an- bn- Z - -an bn FIG - DIRECT FORM II REALIZATION OF IIR SYSTEM 7

81 CASCADE FORM STRUCTURE FOR IIR SYSTEMS I cascade form, stages are cascaded coected i series. The output of oe system is iput to aother. Thus total K umber of stages are cascaded. The total system fuctio 'H' is give by H H. H. Hk H Y/X. Y/X. Yk/Xk k Hπ Hk 3 k xx yx yx3 yky H H Hk FIG - CASCADE FORM REALIZATION OF IIR SYSTEM Each H, H etc is a secod order sectio ad it is realied by the direct form as show i below figure. System fuctio for IIR systems M N H bk k / ak k K0 k Expadig the above terms we have H H. H. Hk where HK bk0 bk - bk - / ak - ak - Thus Direct form of secod order IIR system is show as X bk0 y Z - -ak bk Z - -ak bk 7

82 FIG - DIRECT FORM REALIZATION OF IIR SECOND ORDER SYSTEM CASCADE PARALLEL FORM STRUCTURE FOR IIR SYSTEMS System fuctio for IIR systems is give as M N H bk k / ak k K0 k b0 b - b -.. bm -M / a - a - an -N The above system fuctio ca be expaded i partial fractio as follows H C H H. Hk 3 Where C is costat ad Hk is give as Hk bk0 bk - / ak - ak - 4 C H H X k y FIG - PARALLEL FORM REALIZATION OF IIR SYSTEM

83 IIR FILTER DESIGN. IMPULSE INVARIANCE. BILINEAR TRANSFORMATION 3. BUTTERWORTH APPROXIMATION 4. IIR FILTER DESIGN - IMPULSE INVARIANCE METHOD Impulse Ivariace Method is simplest method used for desigig IIR Filters. Importat Features of this Method are. I impulse variace method, Aalog filters are coverted ito digital filter just by replacig uit sample respose of the digital filter by the sampled versio of impulse respose of aalog filter. Sampled sigal is obtaied by puttig tt hece h hat 0,,.. where h is the uit sample respose of digital filter ad T is samplig iterval.. But the mai disadvatage of this method is that it does ot correspod to simple algebraic mappig of S plae to the Z plae. Thus the mappig from aalog frequecy to digital frequecy is may to oe. The segmets k- /T Ω k /T of jω axis are all mapped o the uit circle ω. This takes place because of samplig. 3. Frequecy aliasig is secod disadvatage i this method. Because of frequecy aliasig, the frequecy respose of the resultig digital filter will ot be idetical to the origial aalog frequecy respose. 4. Because of these factors, its applicatio is limited to desig low frequecy filters like LPF or a limited class of bad pass filters. RELATIONSHIP BETWEEN Z PLANE AND S PLANE Z is represeted as re jω i polar form ad relatioship betwee Z plae ad S plae is give as Ze ST where s σ j Ω. Z e ST Relatioship Betwee Z plae ad S plae Z e σ j Ω T e σ T. e j Ω T Comparig Z value with the polar form we have. r e σ T ad ω Ω T Here we have three coditio If σ 0 the r If σ < 0 the 0 < r < 3 If σ > 0 the r> Thus 73

84 Left side of s-plae is mapped iside the uit circle. Right side of s-plae is mapped outside the uit circle. 3 jω axis is i s-plae is mapped o the uit circle. Im jω Re σ 3 74

85 Im jω 75

86 Re σ CONVERSION OF ANALOG FILTER INTO DIGITAL FILTER Let the system fuctio of aalog filter is Has Σ Ck / s-pk k where pk are the poles of the aalog filter ad ck are the coefficiets of partial fractio expasio. The impulse respose of the aalog filter hat is obtaied by iverse Laplace trasform ad give as hat Σ Ck e pkt k The uit sample respose of the digital filter is obtaied by uiform samplig of hat. h hat 0,,.. h Σ Ck e pkt 3 k System fuctio of digital filter H is obtaied by Z trasform of h. N H Σ Ck Σ e pkt - 4 k 0 Usig the stadard relatio ad comparig equatio ad 4 system fuctio of digital filter is give as 76

87 s - pk - e pkt - STANDARD RELATIONS IN IIR DESIGN Sr No Aalog System Fuctio Digital System fuctio s - a - e at - 3 s a sa b - e -at cos bt - -e -at cos bt - e -at - b sa b e -at si bt - -e -at cos bt - e -at - EXAMPLES - IMPULSE INVARIANCE METHOD Sr No Aalog System Fuctio Digital System fuctio s 0. s0. 9 s s for samplig frequecy of 5 samples/sec 3 0 s for samplig time is 0.0 sec - e -0.T cos3t - -e -0.T cos 3T - e -0.T IIR FILTER DESIGN - BILINEAR TRANSFORMATION METHOD BZT The method of filter desig by impulse ivariace suffers from aliasig. Hece i order to overcome this drawback Biliear trasformatio method is desiged. I aalogue domai frequecy axis is a ifiitely log straight lie while sampled data plae it is uit circle radius. The biliear trasformatio is the method of squashig the ifiite straight aalog frequecy axis so that it becomes fiite. Importat Features of Biliear Trasform Method are. Biliear trasformatio method BZT is a mappig from aalog S plae to digital Z plae. This coversio maps aalog poles to digital poles ad aalog eros to digital eros. Thus all poles ad eros are mapped.. This trasformatio is basically based o a umerical itegratio techiques used to simulate a itegrator of aalog filter. 77

88 3. There is oe to oe correspodece betwee cotiuous time ad discrete time frequecy poits. Etire rage i Ω is mapped oly oce ito the rage - ω. 4. Frequecy relatioship is o-liear. Frequecy warpig or frequecy compressio is due to o-liearity. Frequecy warpig meas amplitude respose of digital filter is expaded at the lower frequecies ad compressed at the higher frequecies i compariso of the aalog filter. 5. But the mai disadvatage of frequecy warpig is that it does chage the shape of the desired filter frequecy respose. I particular, it chages the shape of the trasitio bads. CONVERSION OF ANALOG FILTER INTO DIGITAL FILTER Z is represeted as re jω i polar form ad relatioship betwee Z plae ad S plae i BZT method is give as S S S S - T re jω - T re jω r cos ω j si ω - T r cos ω j si ω r - r j r si ω p T r r cos ω r r cos ω Comparig the above equatio with S σ j Ω. We have σ Ω r - T r r cos ω r si ω T r r cos ω Here we have three coditio If σ < 0 the 0 < r < If σ > 0 the r > 3 If σ 0 the r Whe r Ω si ω 78

89 T cos ω Ω ω /T ta ω/ ta - ΩT/ The above equatios shows that i BZT frequecy relatioship is o-liear. The frequecy relatioship is plotted as ω ta - ΩT/ ΩT FIG - MAPPING BETWEEN FREQUENCY VARIABLE ω AND Ω IN BZT METHOD. DIFFERENCE - IMPULSE INVARIANCE Vs BILINEAR TRANSFORMATION Sr Impulse Ivariace No I this method IIR filters are desiged havig a uit sample respose h that is sampled versio of the impulse respose of the aalog filter. I this method small value of T is selected to miimie the effect of aliasig. Biliear Trasformatio This method of IIR filters desig is based o the trapeoidal formula for umerical itegratio. The biliear trasformatio is a coformal mappig that trasforms the j Ω axis ito the uit circle i the plae oly oce, thus avoidig aliasig of frequecy compoets. 79

90 3 They are geerally used for low frequecies like desig of IIR LPF ad a limited class of badpass filter For desigig of LPF, HPF ad almost all types of Bad pass ad bad stop filters this method is used. 4 Frequecy relatioship is liear. Frequecy relatioship is o-liear. Frequecy warpig or frequecy compressio is due to o-liearity. 5 All poles are mapped from the s plae to the plae by the relatioship Z k e pkt. But the eros i two domai does ot satisfy the same relatioship. All poles ad eros are mapped. 3.7 LPF AND HPF ANALOG BUTTERWORTH FILTER TRANSFER FUNCTION Sr No Order of the Filter Low Pass Filter High Pass Filter / s s / s / s s s / s s 3 3 / s 3 s s s 3 / s 3 s s 3.8 METHOD FOR DESIGNING DIGITAL FILTERS USING BZT step. Fid out the value of ω *. c ω * c cc /T ta ωc Ts/ step. Fid out the value of frequecy scaled aalog trasfer fuctio Normalied aalog trasfer fuctio is frequecy scaled by replacig s by s/ωp *. step 3. Covert ito digital filter Apply BZT. i.e Replace s by the -/. Ad fid out the desired trasfer fuctio of digital fuctio. Example: Q Desig first order high pass butterworth filter whose cutoff frequecy is kh at samplig frequecy of 0 4 sps. Use BZT Method Step. To fid out the cutoff frequecy ωc f 000 rad/sec Step. To fid the prewarp frequecy ωc * ta ωc Ts/ ta /0 Step 3. Scalig of the trasfer fuctio For First order HPF trasfer fuctio Hs s/s Scaled trasfer fuctio H * s Hs ss/ωc* 80

91 H * s s/s 0.35 Step 4. Fid out the digital filter trasfer fuctio. Replace s by -/ H Q Desig secod order low pass butterworth filter whose cutoff frequecy is kh at samplig frequecy of 0 4 sps. Q First order low pass butterworth filter whose badwidth is kow to be rad/sec. Use BZT method to desig digital filter of 0 H badwidth at samplig frequecy 60 sps. Q Secod order low pass butterworth filter whose badwidth is kow to be rad/sec. Use BZT method to obtai trasfer fuctio H of digital filter of 3 DB cutoff frequecy of 50 H ad samplig frequecy.8 kh. Q The trasfer fuctio is give as s / s s The fuctio is for Notch filter with frequecy rad/sec. Desig digital Notch filter with the followig specificatio Notch Frequecy 60 H Samplig frequecy 960 sps. 3.9 BUTTERWORTH FILTER APPROXIMATION The filter passes all frequecies below Ωc. This is called passbad of the filter. Also the filter blocks all the frequecies above Ωc. This is called stopbad of the filter. Ωc is called cutoff frequecy or critical frequecy. No Practical filters ca provide the ideal characteristic. Hece approximatio of the ideal characteristic are used. Such approximatios are stadard ad used for filter desig. Such three approximatios are regularly used. a Butterworth Filter Approximatio b Chebyshev Filter Approximatio c Elliptic Filter Approximatio Butterworth filters are defied by the property that the magitude respose is maximally flat i the passbad. HΩ ΩC Ω 8

92 HaΩ Ω/Ωc N The squared magitude fuctio for a aalog butterworth filter is of the form. HaΩ Ω/Ωc N N idicates order of the filter ad Ωc is the cutoff frequecy -3DB frequecy. At s j Ω magitude of Hs ad H-s is same hece Has Ha-s -s /Ω N c To fid poles of Hs. H-s, fid the roots of deomiator i above equatio. -s - /N Ωc As e jk - where k 0,,,.. N-. -s e jk Ωc /N s - Ω c e jk / N Takig the square root we get poles of s. p k As e j / j - Ωc [ e jk / N ] / Pk j Ωc e jk / N Pk Ωc e j / e jk / N Pk Ωc e jnk / N This equatio gives the pole positio of Hs ad H-s. 3.0 FREQUENCY RESPONSE CHARACTERISTIC The frequecy respose characteristic of HaΩ is as show. As the order of the filter N icreases, the butterworth filter characteristic is more close to the ideal characteristic. Thus at higher orders like N6 the butterworth filter characteristic closely approximate ideal filter characteristic. Thus a ifiite order filter N is required to get ideal characteristic. 8

93 HaΩ N8 83

94 N6 N Ω HaΩ Ap 0.5 As Ωp Ωc Ωs Ω Ap atteuatio i passbad. As atteuatio i stopbad. Ωp passbad edge frequecy Ωs stopbad edge frequecy Specificatio for the filter is HaΩ Ap for Ω Ωp ad HaΩ As for Ω Ωs. Hece we have Ωp/Ωc N Ap Ωs/Ωc N As 8

95 To determie the poles ad order of aalog filter cosider equalities. Ωp/Ωc N /Ap Ωs/Ωc N /As - Ωs N Ωp /As - /Ap - Hece order of the filter N is calculated as N 0.5 log /As - /Ap - log Ωs/ Ωp N 0.5 log/as - log Ωs/ Ωc A Ad cutoff frequecy Ωc is calculated as Ωp 3 Ωc [/Ap -] /N If As ad Ap values are give i DB the As DB - 0 log As log As -As /0 As 0 -As/0 As - 0 As/0 As As DB Hece equatio is modified as N 0.5 log 0 0. As Ap - 4 log Ωs/ Ωp 83

96 Q Desig a digital filter usig a butterworth approximatio by usig impulse ivariace. Example HaΩ Filter Type - Low Pass Filter Ap As Ωp - 0. Ωs Ω Step To covert specificatio to equivalet aalog filter. I impulse ivariace method frequecy relatioship is give as ω Ω T while i Biliear trasformatio method frequecy relatioship is give as Ω /T ta ω/ If Ts is ot specified cosider as HaΩ for Ω 0. /T ad HaΩ for Ω 0.3 /T. Step To determie the order of the filter. N 5.88 log /As - /Ap - N 0.5 log Ωs/ Ωp A Order of the filter should be iteger. B Always go to earest highest iteger vale of N. Hece N6 Step 3 To fid out the cutoff frequecy -3DB frequecy Ωc Ωp [/Ap -] /N

97 cutoff frequecy Ωc Step 4 To fid out the poles of aalog filter system fuctio. Pk Ωc e jnk / N As N6 the value of k 0,,,3,4,5. K Poles 0 P e j7 / -0.8 j j P e j9 / j j P e j / j j P e j3 / j j P e j5 / j j P e j7 / j j For stable filter all poles lyig o the left side of s plae is selected. Hece S -0.8 j S * j S j S * j S j 0.8 S3 * j 0.8 Step 5 To determie the system fuctio Aalog Filter Hece Has Ωc 6 s-ss-s * s-ss-s * s-s3s-s3 * Has Has s0.8-j0.679s0.8j0.679 s0.497-j0.497 s0.497j0.497 s0.679-j0.8s0.679-j [s ] [s ] [s ] Has [s ] [s ] [s ]

98 Step 6 To determie the system fuctio Digital Filter I Biliear trasformatio replace s by the term -/ ad fid out the trasfer fuctio of digital fuctio Step 7 Represet system fuctio i cascade form or parallel form if asked. Q Give for low pass butterworth filter Ap - db at 0. As -5 db at 0.3 Calculate N ad Pole locatio Desig digital filter usig BZT method. Q Obtai trasfer fuctio of a lowpass digital filter meetig specificatios Cutoff 0-60H Stopbad > 85H Stopbad atteuatio > 5 db Samplig frequecy 56 H. use butterworth characteristic. Q Desig secod order low pass butterworth filter whose cutoff frequecy is kh at samplig frequecy of 0 4 sps. Use BZT ad Butterworth approximatio. 3. FREQUENCY TRANSFORMATION Whe the cutoff frequecy Ωc of the low pass filter is equal to the it is called ormalied filter. Frequecy trasformatio techiques are used to geerate High pass filter, Badpass ad badstop filter from the lowpass filter system fuctio. 3.. FREQUENCY TRANSFORMATION ANALOG FILTER Sr No Type of trasformatio Trasformatio Replace s by Low Pass High Pass 3 Bad Pass s ωlp ωlp -Password edge frequecy of aother LPF ωhp s ωhp Password edge frequecy of HPF s ωl ωh s ωh - ωl ωh - higher bad edge frequecy ωl - Lower bad edge frequecy 4 Bad Stop s ωh - ωl s ωh ωl ωh - higher bad edge frequecy ωl - Lower bad edge frequecy 86

99 FREQUENCY TRANSFORMATION DIGITAL FILTER Sr No Type of trasformatio Trasformatio Replace - by Low Pass - - a - a - High Pass - - a a - 3 Bad Pass a - a a - - a - 4 Bad Stop - - a - a a - - a - Example: Q Desig high pass butterworth filter whose cutoff frequecy is 30 H at samplig frequecy of 50 H. Use BZT ad Frequecy trasformatio. Step. To fid the prewarp cutoff frequecy ωc * ta ωcts/ Step. LPF to HPF trasformatio For First order LPF trasfer fuctio Hs /s Scaled trasfer fuctio H * s Hs sωc*/s H * s s/s Step 4. Fid out the digital filter trasfer fuctio. Replace s by -/ H Q Desig secod order bad pass butterworth filter whose passbad of 00 H ad 300 H ad samplig frequecy is 000 H. Use BZT ad Frequecy trasformatio. Q Desig secod order bad pass butterworth filter which meet followig specificatio Lower cutoff frequecy 0 H Upper cutoff frequecy 330 H Samplig Frequecy 960 sps Use BZT ad Frequecy trasformatio. 87

100 GLOSSARY: System Desig: Usually, i the IIR Filter desig, Aalog filter is desiged, the it is trasformed to a digital filter the coversio of Aalog to Digital filter ivolves mappig of desired digital filter specificatios ito equivalet aalog filter. Warpig Effect: The aalog Frequecy is same as the digital frequecy respose. At high frequecies, the relatio betwee ω ad Ω becomes No-Liear. The Noise is itroduced i the Digital Filter as i the Aalog Filter. Amplitude ad Phase resposes are affected by this warpig effect. Prewarpig: The Warpig Effect is elimiated by prewarpig of the aalog filter. The aalog frequecies are prewarped ad the applied to the trasformatio. Ifiite Impulse Respose: Ifiite Impulse Respose filters are a Type of Digital Filters which has ifiite impulse respose. This type of Filters are desiged from aalog filters. The Aalog filters are the trasformed to Digital Domai. Biliear Trasformatio Method: I Biliear trasformatio method the trasform of filters from Aalog to Digital is carried out i a way such that the Frequecy trasformatio produces a Liear relatioship betwee Aalog ad Digital Filters. Filter: A filter is oe which passes the required bad of sigals ad stops the other uwated bad of frequecies. Pass bad: The Bad of frequecies which is passed through the filter is termed as passbad. Stopbad: The bad of frequecies which are stopped are termed as stop bad. 88

101 89

102 UNIT IV FIR FILTER DESIGN PREREQUISITE DISCUSSION: The FIR Filters ca be easily desiged to have perfectly liear Phase. These filters ca be realied recursively ad No-recursively. There are greater flexibility to cotrol the Shape of their Magitude respose. Errors due to roud off oise are less severe i FIR Filters, maily because Feed back is ot used. 4. Features of FIR Filter. FIR filter always provides liear phase respose. This specifies that the sigals i the pass bad will suffer o dispersio Hece whe the user wats o phase distortio, the FIR filters are preferable over IIR. Phase distortio always degrade the system performace. I various applicatios like speech processig, data trasmissio over log distace FIR filters are more preferable due to this characteristic.. FIR filters are most stable as compared with IIR filters due to its o feedback ature. 3. Quatiatio Noise ca be made egligible i FIR filters. Due to this sharp cutoff FIR filters ca be easily desiged. 4. Disadvatage of FIR filters is that they eed higher ordered for similar magitude respose of IIR filters. FIR SYSTEM ARE ALWAYS STABLE. Why? Proof: Differece equatio of FIR filter of legth M is give as M- y bk x k k0 Ad the coefficiet bk are related to uit sample respose as H b for 0 M- 0 otherwise. We ca expad this equatio as Y b0 x b x-.. bm- x-m System is stable oly if system produces bouded output for every bouded iput. This is stability defiitio for ay system. Here h{b0, b, b, } of the FIR filter are stable. Thus y is bouded if iput x is bouded. This meas FIR system produces bouded output for every bouded iput. Hece FIR systems are always stable. 4. Symmetric ad Ati-symmetric FIR filters. Uit sample respose of FIR filters is symmetric if it satisfies followig coditio. h hm-- 0,,.M-. Uit sample respose of FIR filters is Ati-symmetric if it satisfies followig coditio h -hm-- 0,,.M- 90

103 FIR Filter Desig Methods The various method used for FIR Filer desig are as follows. Fourier Series method. Widowig Method 3. DFT method 4. Frequecy samplig Method. IFT Method 4.3 GIBBS PHENOMENON Cosider the ideal LPF frequecy respose as show i Fig with a ormaliig agular cut off frequecy Ωc. Impulse respose of a ideal LPF is as show i Fig.. I Fourier series method, limits of summatio idex is - to. But filter must have fiite terms. Hece limit of summatio idex chage to -Q to Q where Q is some fiite iteger. But this type of trucatio may result i poor covergece of the series. Abrupt trucatio of ifiite series is equivalet to multiplyig ifiite series with rectagular sequece. i.e at the poit of discotiuity some oscillatio may be observed i resultat series.. Cosider the example of LPF havig desired frequecy respose Hd ω as show i figure. The oscillatios or rigig takes place ear bad-edge of the filter. 3. This oscillatio or rigig is geerated because of side lobes i the frequecy respose Wω of the widow fuctio. This oscillatory behavior is called "Gibbs Pheomeo". 9

104 Trucated respose ad rigig effect is as show i fig 3. WINDO WING TECHNIQUE W[] Widowig is the quickest method for desigig a FIR filter. A widowig fuctio simply trucates the ideal impulse respose to obtai a causal FIR approximatio that is o causal ad ifiitely log. Smoother widow fuctios provide higher out-of bad rejectio i the filter respose. However this smoothess comes at the cost of wider stopbad trasitios. Various widowig method attempts to miimie the width of the mai lobe peak of the frequecy respose. I additio, it attempts to miimie the side lobes ripple of the frequecy respose. Rectagular Widow: Rectagular This is the most basic of widowig methods. It does ot require ay operatios because its values are either or 0. It creates a abrupt discotiuity that results i sharp roll-offs but large ripples. Rectagular widow is defied by the followig equatio. for 0 N 0 otherwise Triagular Widow: The computatioal simplicity of this widow, a simple covolutio of two rectagle widows, ad the lower sidelobes make it a viable alterative to the rectagular widow. 9

105 Kaiser Widow: This widowig method is desiged to geerate a sharp cetral peak. It has reduced side lobes ad trasitio bad is also arrow. Thus commoly used i FIR filter desig. Hammig Widow: This widowig method geerates a moderately sharp cetral peak. Its ability to geerate a maximally flat respose makes it coveiet for speech processig filterig. Haig Widow: This widowig method geerates a maximum flat filter desig. 93

106 4.4 DESIGNING FILTER FROM POLE ZERO PLACEMENT Filters ca be desiged from its pole ero plot. Followig two costraits should be imposed while desigig the filters.. All poles should be placed iside the uit circle o order for the filter to be stable. However eros ca be placed aywhere i the plae. FIR filters are all ero filters hece they are always stable. IIR filters are stable oly whe all poles of the filter are iside uit circle.. All complex poles ad eros occur i complex cojugate pairs i order for the filter coefficiets to be real. I the desig of low pass filters, the poles should be placed ear the uit circle at poits correspodig to low frequecies ear ω0ad eros should be placed ear or o uit circle at poits correspodig to high frequecies ear ω. The opposite is true for high pass filters. 4.5 NOTCH AND COMB FILTERS A otch filter is a filter that cotais oe or more deep otches or ideally perfect ulls i its frequecy respose characteristic. Notch filters are useful i may applicatios where specific frequecy compoets must be elimiated. Example Istrumetatio ad recordig systems required that the power-lie frequecy 60H ad its harmoics be elimiated. To create ulls i the frequecy respose of a filter at a frequecy ω0, simply itroduce a pair of complex-cojugate eros o the uit circle at a agle ω0. comb filters are similar to otch filters i which the ulls occur periodically across the frequecy bad similar with periodically spaced teeth. Frequecy respose characteristic of otch filter Hω is as show 94

107 4.6 DIGITAL RESONATOR ωo ω ω A digital resoator is a special two pole badpass filter with a pair of complex cojugate poles located ear the uit circle. The ame resoator refers to the fact that the filter has a larger magitude respose i the viciity of the pole locatios. Digital resoators are useful i may applicatios, icludig simple badpass filterig ad speech geeratios. IDEAL FILTERS ARE NOT PHYSICALLY REALIZABLE. Why? Ideal filters are ot physically realiable because Ideal filters are ati-causal ad as oly causal systems are physically realiable. Proof: Let take example of ideal lowpass filter. Hω for - ωc ω ωc 0 elsewhere The uit sample respose of this ideal LPF ca be obtaied by takig IFT of Hω. _ h Hω e jω dω - h ωc _ e jω dω -ωc _ h e jω j ωc - ωc j [e jωc - e -jωc ] Thus h si ωc / for 0 Puttig 0 i equatio we have ωc h _ dω 3 -ωc 95

108 _ [ω] ωc -ωc ad h ωc / for 0 i.e si ωc for 0 h ωc Hece impulse respose of a ideal LPF is as show i Fig for 0 LSI system is causal if its uit sample respose satisfies followig coditio. h 0 for <0 I above figure h exteds - to. Hece h 0 for <0. This meas causality coditio is ot satisfied by the ideal low pass filter. Hece ideal low pass filter is o causal ad it is ot physically realiable. EXAMPLES OF SIMPLE DIGITAL FILTERS: The followig examples illustrate the essetial features of digital filters.. UNITY GAIN FILTER: y x Each output value y is exactly the same as the correspodig iput value x:. SIMPLE GAIN FILTER: y Kx K costat Amplifier or atteuator This simply applies a gai factor K to each iput value: 3. PURE DELAY FILTER: y x- The output value at time t h is simply the iput at time t -h, i.e. the sigal is delayed by time h: 4. TWO-TERM DIFFERENCE FILTER: y x - x- The output value at t h is equal to the differece betwee the curret iput x ad the previous iput x-: 5. TWO-TERM AVERAGE FILTER: y x x- / The output is the average arithmetic mea of the curret ad previous iput: 96

109 6. THREE-TERM AVERAGE FILTER: y x x- x- / 3 This is similar to the previous example, with the average beig take of the curret ad two previous iputs. 7. CENTRAL DIFFERENCE FILTER: y x - x- / This is similar i its effect to example 4. The output is equal to half the chage i the iput sigal over the previous two samplig itervals: ORDER OF A DIGITAL FILTER The order of a digital filter ca be defied as the umber of previous iputs stored i the processor's memory used to calculate the curret output. This is illustrated by the filters give as examples i the previous sectio. Example : y x This is a ero order filter, sice the curret output y depeds oly o the curret iput x ad ot o ay previous iputs. Example : y Kx The order of this filter is agai ero, sice o previous outputs are required to give the curret output value. Example 3: y x- This is a first order filter, as oe previous iput x- is required to calculate y. Note that this filter is classed as first-order because it uses oe previous iput, eve though the curret iput is ot used. Example 4: y x - x- This is agai a first order filter, sice oe previous iput value is required to give the curret output. Example 5: y x x- / The order of this filter is agai equal to sice it uses just oe previous iput value. Example 6: y x x- x- / 3 To compute the curret output y, two previous iputs x- ad x- are eeded; this is therefore a secod-order filter. Example 7: y x - x- / The filter order is agai, sice the processor must store two previous iputs i order to compute the curret output. This is uaffected by the absece of a explicit x- term i the filter expressio. Q For each of the followig filters, state the order of the filter ad idetify the values of its coefficiets: a y x - x- A Order : a0, a - b y x- B Order : a0 0, a 0, a c y x - x- x- x-3 C Order 3: a0, a -, a, a3 97

110 98

111 99

112 00

113 0

114 0

115 03

116 04

117 05

118 06

119 07

120 Number Represetatio I digital sigal processig, B -bit fixed-poit umbers are usually represeted as two s- complemet siged fractios i the format bo b-ib- b-b The umber represeted is the X -bo b-i - b- - b-b -B 3. where bo is the sig bit ad the umber rage is <X <. The advatage of this represetatio is that the product of two umbers i the rage from to is aother umber i the same rage. Floatig-poit umbers are represeted as X - s m c 3. where s is the sig bit, m is the matissa, ad c is the characteristic or expoet. To make the represetatio of a umber uique, the matissa is ormalied so that 0.5 <m <. Although floatig-poit umbers are always represeted i the form of 3., the way i which this represetatio is actually stored i a machie may differ. Sice m > 0.5, it is ot ecessary to store the - -weight bit of m, which is always set. 08

121 Therefore, i practice umbers are usually stored as X - s 0.5 f c 3.3 where f is a usiged fractio, 0 <f < 0.5. Most floatig-poit processors ow use the IEEE Stadard bit floatigpoit format for storig umbers. Accordig to this stadard the expoet is stored as a usiged iteger p where p c Therefore, a umber is stored as X - s 0.5 f p where s is the sig bit, f is a 3-b usiged fractio i the rage 0 <f < 0.5, ad p is a 8-b usiged iteger i the rage 0 <p < 55. The total umber of bits is For example, i IEEE format 3/4 is writte so s 0, p 6, ad f 0.5. The value X 0 is a uique case ad is represeted by all bits ero i.e., s 0, f 0, ad p 0. Although the - -weight matissa bit is ot actually stored, it does exist so the matissa has 4 b plus a sig bit Fixed-Poit Quatiatio Errors I fixed-poit arithmetic, a multiply doubles the umber of sigificat bits. For example, the product of the two 5-b umbers 0.00 ad 0.00 is the 0-b umber The extra bit to the left of the decimal poit ca be discarded without itroducig ay error. However, the least sigificat four of the remaiig bits must ultimately be discarded by some form of quatiatio so that the result ca be stored to 5 b for use i other calculatios. I the example above this results i quatiatio by roudig or quatiatio by trucatig. Whe a sum of products calculatio is performed, the quatiatio ca be performed either after each multiply or after all products have bee summed with doublelegth precisio. We will examie three types of fixed-poit quatiatio roudig, trucatio, ad magitude trucatio. If X is a exact value, the the rouded value will be deoted Q r X, the trucated value Q t X, ad the magitude trucated value Q mt X. If the quatied value has B bits to the right of the decimal poit, the quatiatio step sie is A -B 3.6 Sice roudig selects the quatied value earest the uquatied value, it gives a value which is ever more tha ± A / away from the exact value. If we deote the roudig error by fr QrX - X 3.7 the AA <f r < Trucatio simply discards the low-order bits, givig a quatied value that is always less tha or equal to the exact value so 09

122 - A < f t < Magitude trucatio chooses the earest quatied value that has a magitude less tha or equal to the exact value so A < f mt < A 3. 0 The error resultig from quatiatio ca be modeled as a radom variable uiformly distributed over the appropriate error rage. Therefore, calculatios with roudoff error ca be cosidered error-free calculatios that have bee corrupted by additive white oise. The mea of this oise for roudig is f A/ m r E{fr } x/ frdfr 0 3. A J-A/ where E{} represets the operatio of takig the expected value of a radom variable. Similarly, the variace of the oise for roudig is A/ A a E{fr - m r } fr - m r dfr 3. A -A/ Likewise, for trucatio, A m f t E{f t } - y A a E{ft - mft} 3.3 m f mt E{f mt } 0 ad, for magitude trucatio A a f-mt E{f mt - m m } Floatig-Poit Quatiatio Errors With floatig-poit arithmetic it is ecessary to quatie after both multiplicatios ad additios. The additio quatiatio arises because, prior to additio, the matissa of the smaller umber i the sum is shifted right util the expoet of both umbers is the same. I geeral, this gives a sum matissa that is too log ad so must be quatied. We will assume that quatiatio i floatig-poit arithmetic is performed by roudig. Because of the expoet i floatig-poit arithmetic, it is the relative error that is importat. The relative error is defied as Sice X - s m c, Q r X - s Q r m c ad Q r m - m QrX - X e r e r r XX e 0

123 er mm If the quatied matissa has B bits to the right of the decimal poit, e < A/ where, as before, A -B. Therefore, sice 0.5 <m <, er I < A 3.7 If we assume that e is uiformly distributed over the rage from - A/ to A/ ad m is uiformly distributed over 0.5 to, m Sr E\ } 0 m a er E { f' A 4 dedm I'm/ J A J/J-a/ m A B I practice, the distributio of m is ot exactly uiform. Actual measuremets of roudoff oise i [] suggested that al r «0.3A 3.9 while a detailed theoretical ad experimetal aalysis i [] determied a «0.8A 3.0 From 3.5 we ca represet a quatied floatig-poit value i terms of the uquatied value ad the radom variable er usig QrX X er 3. Therefore, the fiite-precisio product XX ad the sum X X ca be writte f IXX XXU e r 3. ad flx X X X er 3.3 where e r is ero-mea with the variace of Roudoff Noise: To determie the roudoff oise at the output of a digital filter we will assume that the oise due to a quatiatio is statioary, white, ad ucorrelated with the filter iput, output, ad iteral variables. This assumptio is good if the filter iput chages from sample to sample i a sufficietly complex maer. It is ot valid for ero or costat iputs for which the effects of roudig are aalyed from a limit cycle perspective. To satisfy the assumptio of a sufficietly complex iput, roudoff oise i digital filters is ofte calculated for the case of a ero-mea white oise filter iput sigal x of variace a. This simplifies calculatio of the output roudoff oise because expected values of the form E{xx k} are ero for k 0 ad give a whe k 0. This approach to aalysis has bee foud to give estimates of B al 3.8

124 the output roudoff oise that are close to the oise actually observed for other iput sigals. Aother assumptio that will be made i calculatig roudoff oise is that the product of two quatiatio errors is ero. To justify this assumptio, cosider the case of a 6-b fixed-poit processor. I this case a quatiatio error is of the order 5, while the product of two quatiatio errors is of the order 3 0, which is egligible by compariso. If a liear system with impulse respose g is excited by white oise with mea m x ad variace a, the output is oise of mea [3, pp ] TO my mx ^g 3.4 TO ad variace TO ay al ^ g 3.5 TO Therefore, if g is the impulse respose from the poit where a roudoff takes place to the filter output, the cotributio of that roudoff to the variace measquare value of the output roudoff oise is give by 3.5 with a replaced with the variace of the roudoff. If there is more tha oe source of roudoff error i the filter, it is assumed that the errors are ucorrelated so the output oise variace is simply the sum of the cotributios from each source. 4.8 Roudoff Noise i FIR Filters: The simplest case to aalye is a fiite impulse respose FIR filter realied via the covolutio summatio N y E hkx k 3.6 k0 Whe fixed-poit arithmetic is used ad quatiatio is performed after each multiply, the result of the N multiplies is N-times the quatiatio oise of a sigle multiply. For example, roudig after each multiply gives, from 3.6 ad 3., a output oise variace of B a N 3.7 Virtually all digital sigal processor itegrated circuits cotai oe or more doublelegth accumulator registers which permit the sum-of-products i 3.6 to be accumulated without quatiatio. I this case oly a sigle quatiatio is ecessary followig the summatio ad For the floatig-poit roudoff oise case we will cosider 3.6 for N 4 ad the geeralie the result to other values of N. The fiite-precisio output ca be writte as the exact output plus a error term e. Thus,

125 y e {[h0x[ E] hx -[ ]][ S3] hx -[ 4]}{ s 5 } h3x - 3[ 6][ j] 3.9 I 3.9, represets the errori the first product, the error i the secod product, 3 the error i the firstadditio, etc.notice that it has bee assumed that the products are summed i the order implied by the summatio of 3.6. Expadig 3.9, igorig products of error terms, ad recogiig y gives e h0x[ 3 $ i] hx-[ 3 5 j] hx-[ 4 5 i] h3x- 3[ 6 j] 3.30 Assumig that the iput is white oise of variace a^ so that E{xx - k} is ero for k 0, ad assumig that the errors are ucorrelated, E{e } [4h 0 4h 3h h 3]a a 3.3 I geeral, for ay N, N- ao E{e } Nh 0 J N - a a r 3. 3 kh k k Notice that if the order of summatio of the product terms i the covolutio summatio is chaged, the the order i which the hk s appear i 3.3 chages. If the order is chaged so that the hkwith smallest magitude is first, followed by the ext smallest, etc., the the roudoff oise variace is miimied. However, performig the covolutio summatio i osequetial order greatly complicates data idexig ad so may ot be worth the reductio obtaied i roudoff oise. Roudoff Noise i Fixed-Poit IIR Filters To determie the roudoff oise of a fixed-poit ifiite impulse respose IIR filter realiatio, cosider a causal first-order filter with impulse respose h a u 3.33 realied by the differece equatio y ay - x 3.34 Due to roudoff error, the output actually obtaied is y Q{ay - x} ay - x e

126 where e is a radom roudoff oise sequece. Sice e is ijected at the same poit as the iput, it propagates througha system with impulse respose h. Therefore, forfixed-poit arithmetic with roudig,the outputroudoff oise variace from 3.6, 3., 3.5, ad 3.33 is A A -B a > h > a o ^ ^ - a <x 0 With fixed-poit arithmetic there is the possibility of overflow followig additio. To avoid overflow it is ecessary to restrict the iput sigal amplitude. This ca be accomplished by either placig a scalig multiplier at the filter iput or by simply limitig the maximum iput sigal amplitude. Cosider the case of the first-order filter of The trasfer fuctio of this filter is,v, Ye jm He jm XeJ m ej m a so \He jm \ a a cos, ad,7, He^ max \a\ The peak gai of the filter is / \a\ so limitig iput sigal amplitudes to \x\ < \ a will make overflows ulikely. A expressio for the output roudoff oise-to-sigal ratio ca easily be obtaied for the case where the filter iput is white oise, uiformly distributed over the iterval from \ a \ to \ a \ [4,5]. I this case? f \ a \?? a x a x dx 3 \ a \ \ a \ J \a \ 3 so, from 3.5, \a \ ay 3, 3 ' 4 y 3 a Combiig 3.36 ad 3.4 the gives 0^ _ l^\ 3^0L^ B 3 4 a \ a \ \a \ \ a \. Notice that the oise-to-sigal ratio icreases without boud as \ a \ ^. Similar results ca be obtaied for the case of the causal secod-order filter 4

127 realied by the differece equatio y r cos0y r y x 3.43 This filter has complex-cojugate poles at re± j0 ad impulse respose h r si[ 0]u si 0 Due to roudoff error, the output actually obtaied is y r cos0y r y x e

128 There are two oise sources cotributig to e if quatiatio is performed after each multiply, ad there is oe oise source if quatiatio is performed after summatio. Sice the output roudoff oise is oo r r 4r cos B a V r 3.47 r r 4r where V for quatiatio cos after 9 summatio, ad V for quatiatio after each multiply. To obtai a output oise-to-sigal ratio we ote that He jw 3.48 r cos9e j m r e jm ad, usig the approach of [6], ihemmax r sat cos9^ cos9 si 9 wher e I > sat i <I< 3.50 Followig the same approach as for the first-order case the gives B r 3 V y r r 4r cos 9 X ^cos 3.5 4r sat 9 cos r si9 9 Figure3. is a cotour plot showig the oise-to-sigal ratio rof 3.5 for v i uits of the oise variace of a sigle quatiatio, B /. The plot is symmetrical about 9 90, so oly the rage from 0 to 90 is show. Notice that as r ^, the roudoff oise icreases without boud. Also otice that the oise icreases as 9 ^ 0. It is possible to desig state-space filter realiatios that miimie fixed-poit roudoff oise [7] - [0]. Depedig o the trasfer fuctio beig realied, these structures may provide a roudoff oise level that is orders-of-magitude lower tha for a ooptimal realiatio. The price paid for this reductio i roudoff oise is a icrease i the umber of computatios required to implemet the filter. For a Nth-order filter the icrease is from roughly N multiplies for a direct form realiatio to roughly N for a optimal realiatio. However, if the filter is realied by the parallel or cascade coectio of first- ad secodorder optimal subfilters, the icrease is oly to about 4N multiplies. Furthermore, ear-optimal realiatios exist that icrease the umber of multiplies to oly about 3N [0]. 6

129 Normalied fixed-poit roudoff oise variace. 4.9 Limit Cycle Oscillatios: A limit cycle, sometimes referred to as a multiplier roudoff limit cycle, is a lowlevel oscillatio that ca exist i a otherwise stable filter as a result of the oliearity associated with roudig or trucatig iteral filter calculatios []. Limit cycles require recursio to exist ad do ot occur i orecursive FIR filters. As a example of a limit cycle, cosider the secod-order filter realied by 7 5 y Qr{ ^y 8y x where Q r {} represets quatiatio by roudig. This is stable filter with poles at ± j Cosider the implemetatio of this filter with 4-b 3-b ad a sig bit two s complemet fixed-poit arithmetic, ero iitial coditios y y 0, ad a iput sequece x S, where S is the uit impulse or uit sample. The followig sequece is obtaied;

130 Notice that while the iput is ero except for the first sample, the output oscillates with amplitude /8 ad period 6. Limit cycles are primarily of cocer i fixed-poit recursive filters. As log as floatig-poit filters are realied as the parallel or cascade coectio of first- ad secod-order subfilters, limit cycles will geerally ot be a problem sice limit cycles are practically ot observable i first- ad secod-order systems implemeted with 3-b floatig-poit arithmetic []. It has bee show that such systems must have a extremely small margi of stability for limit cycles to exist at aythig other tha uderflow levels, which are at a amplitude of less tha 0 38 []. There are at least three ways of dealig with limit cycles whe fixedpoit arithmetic is used. Oe is to determie a boud o the maximum limit cycle amplitude, expressed as a itegral umber of quatiatio steps [3]. It is the possible to choose a word legth that makes the limit cycle amplitude acceptably low. Alterately, limit cycles ca be preveted by radomly roudig calculatios up or dow [4]. However, this approach is complicated to implemet. The third approach is to properly choose the filter realiatio structure ad the quatie the filter calculatios usig magitude trucatio [5,6]. This approach has the disadvatage of producig more roudoff oise tha trucatio or roudig [see ]. 4.0 Overflow Oscillatios: With fixed-poit arithmetic it is possible for filter calculatios to overflow. This happes whe two umbers of the same sig add to give a value havig magitude greater tha oe. Sice umbers with magitude greater tha oe are ot represetable, the result overflows. For example, the two s complemet umbers 0.0 5/8 ad /8 add to give.00 which is the two s complemet represetatio of -7/8. The overflow characteristic of two s complemet arithmetic ca be represeted as R{} where X - X> For the example just cosidered, R{X} R{9/8} X 7/8. - < X < 3.7 A overflow oscillatio, sometimes X also X referred <- to as a adder overflow limit cycle, is a high- level oscillatio that ca exist i a otherwise stable fixed-poit filter due to the gross oliearity associated with the overflow of iteral filter calculatios [7]. Like limit cycles, overflow oscillatios require recursio to exist ad do ot occur i orecursive FIR filters. Overflow oscillatios also do ot occur with floatig-poit arithmetic due to the virtual impossibility of overflow. As a example of a overflow oscillatio, oce agai cosider the filter of 3.69 with 4-b fixed-poit two s complemet arithmetic ad with the two s complemet overflow characteristic of 3.7: 75 y Qr\R 8y - - 8y - x 3.7 [Type text]

131 I this case we apply the iput 35 x -4 & - ^& - 3 5,, , 0, 4 8 s to scale the filter calculatios so as to reder overflow impossible. However, this may uacceptably restrict the filter dyamic rage. Aother method is to force completed sums-of- products to saturate at ±, rather tha overflowig [8,9]. It is importat to saturate oly the completed sum, sice itermediate overflows i two s complemet arithmetic do ot affect the accuracy of the fial result. Most fixed-poit digital sigal processors provide for automatic saturatio of completed sums if their saturatio arithmetic feature is eabled. Yet aother way to avoid overflow oscillatios is to use a filter structure for which ay iteral filter trasiet is guarateed to decay to ero [0]. Such structures are desirable ayway, sice they ted to have low roudoff oise ad be isesitive to coefficiet quatiatio []. Coefficiet Quatiatio Error:

132 Re Z FIGURE: Realiable pole locatios for the differece equatio of The sparseess of realiable pole locatios ear ± will result i a large coefficiet quatiatio error for poles i this regio. Figure3.4 gives a alterative structure to 3.77 for realiig the trasfer fuctio of Notice that quatiig the coefficiets of this structure correspods to quatiig X r ad Xi. As show i Fig.3.5 from [5], this results i a uiform grid of realiable pole locatios. Therefore, large coefficiet quatiatio errors are avoided for all pole locatios. It is well established that filter structures with low roudoff oise ted to be robust to coefficiet quatiatio, ad visa versa []- [4]. For this reaso, the uiform grid structure of Fig.3.4 is also popular because of its low roudoff oise. Likewise, the lowoise realiatios of [7]- [0] ca be expected to be relatively isesitive to coefficiet quatiatio, ad digital wave filters ad lattice filters that are derived from low-sesitivity aalog structures ted to have ot oly low coefficiet sesitivity, but also low roudoff oise [5,6]. It is well kow that i a high-order polyomial with clustered roots, the root locatio is a very sesitive fuctio of the polyomial coefficiets. Therefore, filter poles ad eros ca be much more accurately cotrolled if higher order filters are realied by breakig them up ito the parallel or cascade coectio of first- ad secod-order subfilters. Oe exceptio to this rule is the case of liear-phase FIR filters i which the symmetry of the polyomial coefficiets ad the spacig of the filter eros aroud the uit circle usually permits a acceptable direct realiatio usig the covolutio summatio. Give a filter structure it is ecessary to assig the ideal pole ad ero locatios to the realiable locatios. This is geerally doe by simplyroudig or trucatigthe filter coefficiets to the available umber of bits, or by assigig the ideal pole ad ero locatios to the earest realiable locatios. A more complicated alterative is to cosider the origial filter desig problem as a problem i discrete

133 FIGURE 3.4: Alterate realiatio structure. FIGURE 3.5: Realiable pole locatios for the alterate realiatio structure. optimiatio, ad choose the realiable pole ad ero locatios that give the best approximatio to the desired filter respose [7]- [30]. 4. Realiatio Cosideratios: Liear-phase FIR digital filters ca geerally be implemeted with acceptable coefficiet quatiatio sesitivity usig the direct covolutio sum method. Whe implemeted i this way o a digital sigal processor, fixed-poit arithmetic is ot oly acceptable but may actually be preferable to floatig-poit arithmetic. Virtually all fixed-poit digital sigal processors accumulate a sum of products i a double-legth accumulator. This meas that oly a sigle quatiatio is ecessary to compute a output. Floatig-poit arithmetic, o

134 the other had, requires a quatiatio after every multiply ad after every add i the covolutio summatio. With 3-b floatig-poit arithmetic these quatiatios itroduce a small eough error to be isigificat for may applicatios. Whe realiig IIR filters, either a parallel or cascade coectio of first- ad secod-order subfilters is almost always preferable to a high-order direct-form realiatio. With the availability of very low-cost floatig-poit digital sigal processors, like the Texas Istrumets TMS30C3, it is highly recommeded that floatig-poit arithmetic be used for IIR filters. Floatig-poit arithmetic simultaeously elimiates most cocers regardig scalig, limit cycles, ad overflow oscillatios. Regardless of the arithmetic employed, a low roudoff oise structure should be used for the secod- order sectios. Good choices are give i [] ad [0]. Recall that realiatios with low fixed-poit roudoff oise also have low floatig-poit roudoff oise. The use of a low roudoff oise structure for the secod-order sectios also teds to give a realiatio with low coefficiet quatiatio sesitivity. First-order sectios are ot as critical i determiig the roudoff oise ad coefficiet sesitivity of a realiatio, ad so ca geerally be implemeted with a simple direct form structure. GLOSSARY: FIR Filters: I the Fiite Impulse Respose Filters the No.of. Impulses to be cosidered for filterig are fiite. There are o feed back Coectios from the Output to the Iput. There are o Equivalet Structures of FIR filters i the Aalog Regime. Symmetric FIR Filters: Symmetric FIR Filters have their Impulses that occur as the mirror image i the first quadrat ad secod quadrat or Third quadrat ad fourth quadrat or both. Ati Symmetric FIR Filters: The Atisymmetric FIR Filters have their impulses that occur as the mirror image i the first quadrat ad third quadrat or secod quadrat ad Fourth quadrat or both. Liear Phase: The FIR Filters are said to have liear i phase if the filter have the impulses that icreases accordig to the Time i digital domai. Frequecy Respose: The Frequecy respose of the Filter is the relatioship betwee the agular frequecy ad the Gai of the Filter. Gibbs Pheomeo: The abrupt trucatio of Fourier series results i oscillatio i both passbad ad stop bad. These oscillatios are due to the slow covergece of the fourier series. This is termed as Gibbs Pheomeo. Widowig Techique: To avoid the oscillatios istead of trucatig the fourier co-efficiets we are multiplyig the fourier series with a fiite weighig sequece called a widow which has o-ero values at the required iterval ad ero values for other Elemets.

135 Quatiatio: Total umber of bits i x is reduced by usig two methods amely Trucatio ad Roudig. These are kow as quatiatio Processes. Iput Quatiatio Error: The Quatied sigal are stored i a b bit register but for earest values the same digital equivalet may be represeted. This is termed as Iput Quatiatio Error. Product Quatiatio Error: The Multiplicatio of a b bit umber with aother b bit umber results i a b bit umber but it should be stored i a b bit register. This is termed as Product Quatiatio Error. Co-efficiet Quatiatio Error: The Aalog to Digital mappig of sigals due to the Aalog Co-efficiet Quatiatio results i error due to the Fact that the stable poles marked at the edge of the jω axis may be marked as a ustable pole i the digital domai. Limit Cycle Oscillatios: If the iput is made ero, the output should be made ero but there is a error occur due to the quatiatio effect that the system oscillates at a certai bad of values. Overflow limit Cycle oscillatios: Overflow error occurs i additio due to the fact that the sum of two umbers may result i overflow. To avoid overflow error saturatio arithmetic is used. Dead bad: The rage of frequecies betwee which the system oscillates is termed as Deadbad of the Filter. It may have a fixed positive value or it may oscillate betwee a positive ad egative value. Sigal scalig: The iputs of the summer is to be scaled first before executio of the additio operatio to fid for ay possibility of overflow to be occurred after additio. The scalig factor s 0 is multiplied with the iputs to avoid overflow.

136 UNIT V APPLICATIONS OF DSP PRE REQUISITE DISCUSSION: The time domai waveform is trasformed to the frequecy domai usig a filter bak. The stregth of each frequecy bad is aalyed ad quatied based o how much effect they have o the perceived decompressed sigal. 5.. SPEECH RECOGNITION: Basic block diagram of a speech recogitio system is show i Fig. I speech recogitio system usig microphoe oe ca iput speech or voice. The aalog speech sigal is coverted to digital speech sigal by speech digitier. Such digital sigal is called digitied speech.. The digitied speech is processed by DSP system. The sigificat features of speech such as its formats, eergy, liear predictio coefficiets are extracted. The template of this extracted features are compared with the stadard referece templates. The closed matched template is cosidered as the recogied word. 3. Voice operated cosumer products like TV, VCR, Radio, lights, fas ad voice operated telephoe dialig are examples of DSP based speech recogied devices. Impulse Trai Geerator Radom umber geerator Voiced Uvoiced Time varyig digital filter Sythetic speech 5.. LINEAR PREDICTION OF SPEECH SYNTHESIS Fig shows block diagram of speech sythesier usig liear predictio.. For voiced soud, pulse geerator is selected as sigal source while for uvoiced souds oise geerator is selected as sigal source.. The liear predictio coefficiets are used as coefficiets of digital filter. Depedig upo these coefficiets, the sigal is passed ad filtered by the digital filter. 3. The low pass filter removes high frequecy oise if ay from the sythesied speech. Because of liear phase characteristic FIR filters are mostly used as digital filters.

137 Pitch Period Pulse Geerator White Noise geerator Voiced Uvoiced Digital filter Time varyig digital filter Sythetic speech

138 Filter Coefficiets 5.3. SOUND PROCESSING:. I soud processig applicatio, Music compressiomp3 is achieved by covertig the time domai sigal to the frequecy domai the removig frequecies which are o audible.. The time domai waveform is trasformed to the frequecy domai usig a filter bak. The stregth of each frequecy bad is aalyed ad quatied based o how much effect they have o the perceived decompressed sigal. 3. The DSP processor is also used i digital video disk DVD which uses MPEG- compressio, Web video cotet applicatio like Itel Ideo, real audio. 4. Soud sythesis ad maipulatio, filterig, distortio, stretchig effects are also doe by DSP processor. ADC ad DAC are used i sigal geeratio ad recordig ECHO CANCELLATION I the telephoe etwork, the subscribers are coected to telephoe exchage by two wire circuit. The exchages are coected by four wire circuit. The two wire circuit is bidirectioal ad carries sigal i both the directios. The four wire circuit has separate paths for trasmissio ad receptio. The hybrid coil at the exchage provides the iterface betwee two wire ad four wire circuit which also provides impedace matchig betwee two wire ad four wire circuits. Hece there are o echo or reflectios o the lies. But this impedace matchig is ot perfect because it is legth depedet. Hece for echo cacellatio, DSP techiques are used as follows.. A DSP based acoustic echo caceller works i the followig fashio: it records the soud goig to the loudspeaker ad substract it from the sigal comig from the microphoe. The soud goig through the echo-loop is trasformed ad delayed, ad oise is added, which complicate the substractio process.. Let be the iput sigal goig to the loudspeaker; let be the sigal picked up by the microphoe, which will be called the desired sigal. The sigal after

139 substractio will be called the error sigal ad will be deoted by. The adaptive filter will try to idetify the equivalet filter see by the system from the loudspeaker to the microphoe, which is the trasfer fuctio of the room the loudpeaker ad microphoe are i. 3. This trasfer fuctio will deped heavily o the physical characteristics of the eviromet. I broad terms, a small room with absorbig walls will origiate just a few, first order reflectios so that its trasfer fuctio will have a short impulse respose. O the other had, large rooms with reflectig walls will have a trasfer fuctio whose impulse respose decays slowly i time, so that echo cacellatio will be much more difficult VIBRATION ANALYSIS:. Normally machies such as motor, ball bearig etc systems vibrate depedig upo the speed of their movemets.. I order to detect fault i the system spectrum aalysis ca be performed. It shows fixed frequecy patter depedig upo the vibratios. If there is fault i the machie, the predetermied spectrum is chages. There are ew frequecies itroduced i the spectrum represetig fault. 3. This spectrum aalysis ca be performed by DSP system. The DSP system ca also be used to moitor other parameters of the machie simultaeously. Etire Z Plae except Z0 Etire Z Plae except Z Etire Plae except Z 0 & Z

140

141 5.6 Multistage Implemetatio of Digital Filters: I some applicatios we wat to desig filters where the badwidth is just a small fractio of the overall samplig rate. For example, suppose we wat to desig a lowpass filter with badwidth of the order of a few hert ad a samplig frequecy of the order of several kilohert. This filter would require a very sharp trasitio regio i the digital frequecy a>, thus requirig a high-complexity filter. Example < As a example of applicatio, suppose you wat to desig a Filter with the fallowig specificatios: Passbad F p 450 H Stopbad F s 500 H Samplig frequecy F s ~96 kh Notice that the stopbad is several orders of magitude smaller tha the samplig frequecy. This leads to a filter with a very short trasitio regio of high complexity. I Speech sigals From prehistory to the ew media of the future, speech has bee ad will be a primary form of commuicatio betwee humas. Nevertheless, there ofte occur coditios uder which we measure ad the trasform the speech to aother form, speech sigal, i order to ehace our ability to commuicate. The speech sigal is exteded, through techological media such as telephoy, movies, radio, televisio, ad ow Iteret. This tred reflects the primacy of speech commuicatio i huma psychology. Speech will become the ext major tred i the persoal computer market i the ear future. 5.7 Speech sigal processig: The topic of speech sigal processig ca be loosely defied as the maipulatio of sampled speech sigals by a digital processor to obtai a ew sigal with some desired properties. Speech sigal processig is a diverse field that relies o kowledge of laguage at the levels of Sigal processig Acoustics P Phoetics ^ ^ ^ Laguage-idepedet Phoology ^^ Morphology i^^^ Sytax ^, Laguage-depedet Sematics \%X Pragmatics if,ffl^ 7 layers for describig speech From Speech to Speech Sigal, i terms of Digital Sigal Processig [Type text] At-* Acoustic ad perceptual features {traits - fudametal freouecy FO pitch - amplitude loudess - spectrum timber

142 It is based o the fact that - Most of eergy betwee 0 H to about 7KH, - Huma ear sesitive to eergy betwee 50 H ad 4KH I terms of acoustic or perceptual, above features are cosidered. From Speech to Speech Sigal, i terms of Phoetics Speech productio, the digital model of Speech Sigal will be discussed i Chapter. Motivatio of covertig speech to digital sigals:

143 Speech codig, A PC ad SBC fie structure Adaptive predictive codig APC is a techique used for speech codig, that is data compressio of spccch sigals Typic al Voice d speec pitch, APC assumes that the iput speech sigal is repetitive with a period sigificatly loger tha the average frequecy cotet. Two predictors arc used i APC. The high frequecy compoets up to 4 kh are estimated usig h a 'spectral or 'format prcdictor ad the low frequecy compoets H by a pitch or fie structure prcdictor see figure 7.4. The spcctral estimator may he of order - 4 ad the pitch estimator about order 0. The low-frequecy compoets of the spccch sigal are due to the movemet of the togue, chi ad spectral evelope, formats Figure 7.4 Ecoder for adaptive, predictive codig of speech sigals. The decoder is maily a mirrored versio of the ecoder The high-frequecy compoets origiate from the vocal chords ad the oise-like souds like i s produced i the frot of the mouth. The output sigal ytogether with the predictor parameters, obtaied adaptively i the ecoder, are trasmitted to the decoder, where the spcech sigal is recostructed. The decoder has the same structure as the ecoder but the predictors arc ot adaptive ad arc ivoked i the reverse order. The predictio parameters are adapted for blocks of data correspodig to for istace 0 ms time periods. A PC' is used for codig spcech at 9.6 ad 6 kbits/s. The algorithm works well i oisy eviromets, but ufortuately the quality of the processed speech is ot as good as for other methods like CELP described below. 5.8 Subbad Codig: Aother codig method is sub-bad codig SBC see Figure 7.5 which belogs to the class of waveform codig methods, i which the frequecy domai properties of the iput sigal arc utilied to achieve data compressio. The basic idea is that the iput speech sigal is split ito sub-bads usig bad-pass filters. The sub-bad sigals arc the ecoded usig ADPCM or other techiques. I this way, the available data trasmissio capacity ca be allocated betwee bads accordig to pcrccptual criteria, ehacig the speech quality as pcrceivcd by listeers. A sub-bad that is more importat from the huma listeig poit of view ca be allocated more bits i the data stream, while less importat sub-bads will use fewer bits. A typical setup for a sub-bad codcr would be a bak of Ndigital badpass filters followed

144 by dccimators, ecoders for istace ADPCM ad a multiplexer combiig the data bits comig from the sub-bad chaels. The output of the multiplexer is the trasmitted to the sub-bad dccodcr havig a demultiplexer splittig the multiplexed data stream back ito Nsubbad chaels. Every sub-bad chael has a dccodcr for istace ADPCM, followed by a iterpolator ad a bad-pass filter. Fially, the outputs of the bad-pass filters are summed ad a recostructed output sigal results. Sub-bad codig is commoly used at bit rates betwee 9.6 kbits/s ad 3 kbits/s ad performs quite well. The complexity of the system may however be cosiderable if the umber of sub-bads is large. The desig of the bad-pass filters is also a critical topic whe workig with sub-bad codig systems.

145 Figure 7.5 A example of a sub-bad codig system Vocoders ad LPC I the methods described above APC, SBC ad ADPCM, speech sigal applicatios have bee used as examples. By modifyig the structure ad parameters of the predictors ad filters, the algorithms may also be used for other sigal types. The mai objective was to achieve a reproductio that was as faithful as possible to the origial sigal. Data compressio was possible by removig redudacy i the time or frequecy domai. The class of vocoders also called source coders is a special class of data compressio devices aimed oly at spcech sigals. The iput sigal is aalysed ad described i terms of speech model parameters. These parameters are the used to sythesie a voice patter havig a acceptable level of perceptual quality. Hece, waveform accuracy is ot the mai goal, as i the previous methods discussed.

146 pitch periodic excitatio 9 V Noise sythetic speech voiced/uvoiced Figure 7.6 The LPC model The first vocoder was desiged by H. Dudley i the 930s ad demostrated at the New York Fair i 939* Vocoders have become popular as they achieve reasoably good speech quality at low data rates, from A kbits/s to 9,6 kbits/s. There arc may types of vocoders Marvc ad Ewers, 993, some of the most commo techiques will be briefly preseted below. Most vocoders rely o a few basic priciples. Firstly, the characteristics of the spccch sigal is assumed to be fairly costat over a time of approximately 0 ms, hccc most sigal processig is performed o overlappig data blocks of 0 40 ms legth. Secodly, the spccch model cosists of a time varyig filter correspodig to the acoustic properties of the mouth ad a excitatio sigal. The cxeilalio sigal is cither a periodic waveform, as crcatcd by the vocal chords, or a radom oise sigal for productio of uvoiced' souds, for example s ad T. The filter parameters ad excitatio parameters arc assumed to be idepedet of each other ad are commoly coded separately. Liear predictive codig LPC is a popular method, which has however bee replaced by ewer approaches i may applicatios. LPC works exceedigly well at low bit rates ad the LPC parameters cotai sufficiet iformatio of the spccch sigal to be used i spccch recogitio applicatios. The LPC model is show i Figure 7*6. LPC is basically a aut-regressive model sec Chapter 5 ad the vocal tract is modelled as a time-varyig all-pole filter HR filter havig the trasfer fuctio H 7*7 -k ki where p is the order of the filter. The excitatio sigal *?«, beig either oise or a periodic waveform, is fed to the filter via a variable gai factor G. The output sigal ca be expressed i the time domai as y ~ Geri - a,y- - a y- %y{-p. IK The output sample at time is a liear combiatio of p previous samples

147 ad the excitatio sigal liear predictive codig. The filter coefficiets a k arc time varyig. The model above describes how to sythesie the speech give the pitch iformatio if oise or pet iodic excitatio should be used, the gai ad the filter parameters. These parameters must be determied by the ccoder or the aalyser, takig the origial spccch sigal x as iput. The aalyser widows the spccch sigal i blocks of 0-40 ms. usually with a Hammig widow see Chapter 5. These blocks or frames arc repeated every 0 30 ms, hece there is a certai overlap i time. Every frame is the aalysed with respect to the parameters metioed above. Firstly, the pitch frequecy is determied. This also tells whether we arc dealig with a voiced or uvoiccd spccch sigal. This is a crucial part of the system ad may pitch detectio algorithms have bee proposed. If the segmet of the spccch sigal is voiced ad has a dear periodicity or if it is uvoiced ad ot pet iodic, thigs arc quite easy* Segmets havig properties i betwee these two extremes are difficult to aalyse. No algorithm has bee foud so far that is perfect* for all listeers. Now, the secod step of the aalyser is to determie the gai ad the filter parameters. This is doe by estimatig the spccch sigal usig a adaptive predictor. The predictor has the same structure ad order as the filter i the sythesier, Hecc, the output of the predictor is -i -tf]jt/7 a x... OpX p 7-9 where irt is the predicted iput spcech sigal ad jcrt is the actual iput sigal. The filter coefficiets a k are determied by miimiig the square error This ca be doe i differet ways, cither by calculatig the auto-corrc- latio coefficiets ad solvig the Yule Walker equatios see Chapter 5 or by usig some recursive, adaptive filter approach see Chapter 3, So, for every frame, all the parameters above arc determied ad iras- mittcd to the sythesiser, where a sythetic copy of the spccch is geerated. A improved versio of LPC is residual excited liear predictio RELP. Let us take a closer look at the error or residual sigal rfi resultig from the predictio i the aalyser equatio 7.9. The residual sigal wc arc try ig to miimie ca be expressed as r *«-irt jfrt a^x a x -h *.. a p xft p <7- From this it is straightforward to fid out that the correspodig expressio usig the -trasforms is Hccc, the prcdictor ca be regarded as a iverse filter to the LPC model filter. If we ow pass this residual sigal to the sythesier ad use it to excite the LPC filter, that is E - R, istead of usig the oise or periodic waveform sources we get Y EH RH XH~\H X 7.3 I the ideal case, we would hece get the origial speech sigal back. Whe miimiig the variace of the residual sigal equatio 7.0, we gathered as much iformatio about the spccch sigal as possible usig this model i the filter coefficiets a k. The residual sigal cotais the remaiig iformatio. If

148 the model is well suited for the sigal type speech sigal, the residual sigal is close to white oise, havig a flat spectrum. I such a case we ca get away with codig oly a small rage of frequecies, for istace 0- kh of the residual sigal. At the sythesier, this basebad is the repeated to geerate higher frequecies. This sigal is used to excite the LPC filter Vocoders usig RELP are used with trasmissio rates of 9.6 kbits/s. The advatage of RELP is a better speech quality compared to LPC for the same bit rate. However, the implemetatio is more computatioally demadig. Aother possible extesio of the origial LPC approach is to use multipulse excited liear predictive codig MLPC. This extesio is a attempt to make the sythesied spcech less mechaical, by usig a umber of differet pitches of the excitatio pulses rather tha oly the two periodic ad oise used by stadard LPC. The MLPC algorithm sequetially detects k pitches i a speech sigal. As soo as oe pitch is foud it is subtracted from the sigal ad detectio starts over agai, lookig for the ext pitch. Pitch iformatio detectio is a hard task ad the complexity of the required algorithms is ofte cosiderable. MLPC however offers a better spcech quality tha LPC for a give bit rate ad is used i systems workig with 4.S-9.6 kbits/s. Yet aother extesio of LPC is the code excited liear predictio CELP. The mai feature of the CELP compared to LPC is the way i which the filter coefficiets are hadled. Assume that we have a stadard LPC system, with a filter of the order p. If every coefficiet a k requires N bits, we eed to trasmit N-p bits per frame for the filter parameters oly. This approach is all right if all combiatios of filter coefficiets arc equally probable. This is however ot the case. Some combiatios of coefficiets are very probable, while others may ever occur. I CELP, the coefficiet combiatios arc represeted by p dimesioal vectors. Usig vector quatiatio techiques, the most probable vectors are determied. Each of these vectors are assiged a idex ad stored i a codebook. Both the aalyser ad sythesier of course have idetical copies of the codebook, typically cotaiig 56-5 vectors. Hccc, istead of trasmittig N-p bits per frame for the filter parameters oly 8-9 bits arc eeded. This method offers high-quality spcech at low-bit rates but requires cosiderable computig power to be able to store ad match the icomig spcech to the stadard souds stored i the codebook. This is of course especially true if the codebook is large. Speech quality degrades as the codebook sie decreases. Most CELP systems do ot perform well with respect to higher frequecy compoets of the spccch sigal at low hit rates. This is coutcractcd i There is also a variat of CELP called vector sum excited liear predictio VSELP. The mai differece betwee CELP ad VSELP is the way the codebook is orgaied. Further, sice VSELP uses fixed poit arithmetic algorithms, it is possible to implemet usig cheaper DSP chips tha Adaptive Filters The sigal degradatio i some physical systems is time varyig, ukow, or possibly both. For example,cosider a high-speed modem for trasmittig ad receivig data over telephoe chaels. It employs a filter called a chael equalier to compesate for the chael distortio. Sice the dial-up commuicatio chaels have differet ad time-varyig characteristics o each coectio, the equalier must be a adaptive filter.

149 5.9 Adaptive Filter: Adaptive filters modify their characteristics to achieve certai objectives by automatically updatig their coefficiets. May adaptive filter structures ad adaptatio algorithms have bee developed for differet applicatios. This chapter presets the most widely used adaptive filters based o the FIR filter with the least-mea-square LMS algorithm. These adaptive filters are relatively simple to desig ad implemet. They are well uderstood with regard to stability, covergece speed, steady-state performace, ad fiite-precisio effects. Itroductio to Adaptive Filterig A adaptive filter cosists of two distict parts - a digital filter to perform the desired filterig, ad a adaptive algorithm to adjust the coefficiets or weights of the filter. A geeral form of adaptive filter is illustrated i Figure 7., where d is a desired or primary iput sigal, y is the output of a digital filter drive by a referece iput sigal x, ad a error sigal e is the differece betwee d ad y. The adaptive algorithm adjusts the filter coefficiets to miimie the mea-square value of e. Therefore, the filter weights are updated so that the error is progressively miimied o a sample-bysample basis. I geeral, there are two types of digital filters that ca be used for adaptive filterig: FIR ad IIR filters. The FIR filter is always stable ad ca provide a liear-phase respose. O the other had, the IIR

150 filter ivolves both eros ad poles. Uless they are properly cotrolled, the poles i the filter may move outside the uit circle ad result i a ustable system durig the adaptatio of coefficiets. Thus, the adaptive FIR filter is widely used for practical real-time applicatios. This chapter focuses o the class of adaptive FIR filters. The most widely used adaptive FIR filter is depicted i Figure 7.. The filter output sigal is computed 7.3

151 DIGITAL SIGNAL PROCESSING IV YEAR / VII SEM CSE, where the filter coefficiets wl are time varyig ad updated by the adaptive algorithms that will be discussed ext. We defie the iput vector at time as x [xx -... x - L ]T, 7.4 ad the weight vector at time as w [w0w... wl-]t. 7.5 Equatio 7.3 ca be expressed i vector form as y wt x xt w. 7.6 The filter outputy is compared with the desired d to obtai the error sigal e d - y d - wt x. 7.7 Our objective is to determie the weight vector w to miimie the predetermied performace or cost fuctio. Performace Fuctio: The adaptive filter show i Figure 7. updates the coefficiets of the digital filter to optimie some predetermied performace criterio. The most commoly used performace fuctio is based o the mea-square error MSE. 5.0 Audio Processig: The two pricipal huma seses are visio ad hearig. Correspodigly,much of DSP is related to image ad audio processig. People liste toboth music ad speech. DSP has made revolutioary chages i both these areas Music Soud processig: The path leadig from the musicia's microphoe to the audiophile's speaker is remarkably log. Digital data represetatio is importat to prevet the degradatio commoly associated with aalog storage ad maipulatio. This is very familiar to ayoe who has compared the musical quality of cassette tapes with compact disks. I a typical sceario, a musical piece is recorded i a soud studio o multiple chaels or tracks. I some cases, this eve ivolves recordig idividual istrumets ad sigers separately. This is doe to give the soud egieer greater flexibility i creatig the fial product. The complex process of combiig the idividual tracks ito a fial product is called mix dow. DSP ca provide several importat fuctios durig mix dow, icludig: filterig, sigal additio ad subtractio, sigal editig, etc. Oe of the most iterestig DSP applicatios i music preparatio is artificial reverberatio. If the idividual chaels are simply added together, the resultig piece souds frail ad diluted, much as if the musicias were playig outdoors. This is because listeers are greatly iflueced by the echo or reverberatio cotet of the music, which is usually miimied i the soud studio. DSP allows artificial echoes ad reverberatio to be added durig mix dow to simulate various ideal listeig eviromets. Echoes with delays of a few hudred millisecods give the impressio of cathedral likelocatios. Addig echoes with delays of 0-0 millisecods provide the perceptio of more modest sie listeig rooms Speech geeratio: Speech geeratio ad recogitio are used to commuicate betwee humas ad machies. Rather tha usig your hads ad eyes, you use your mouth ad ears. This is very coveiet whe your hads ad 35 MUTHUKUMAR.A 04 05

152 DIGITAL SIGNAL PROCESSING IV YEAR / VII SEM CSE eyes should be doig somethig else, such as: drivig a car, performig surgery, or ufortuately firig your weapos at the eemy. Two approaches are used for computer geerated speech: digital recordig ad vocal tract simulatio. I digital recordig, the voice of a huma speaker is digitied ad stored, usually i a compressed form. Durig playback, the stored data are ucompressed ad coverted back ito a aalog sigal. A etire hour of recorded speech requires oly about three me gabytes of storage, well withi the capabilities of eve small computer systems. This is the most commo method of digital speech geeratio used today. Vocal tract simulators are more complicated, tryig to mimic the physical mechaisms by which humas create speech. The huma vocal tract is a acoustic cavity with resoate frequecies determied by the sie ad shape of the chambers. Soud origiates i the vocal tract i oe of two basic ways, called voiced ad fricative souds. With voiced souds, vocal cord vibratio produces ear periodic pulses of air ito the vocal cavities. I compariso, fricative souds origiate from the oisy air turbulece at arrow costrictios, such as the teeth ad lips. Vocal tract simulators operate by geeratig digital sigals that resemble these two types of excitatio. The characteristics of the resoate chamber are simulated by passig the excitatio sigal through a digital filter with similar resoaces. This approach was used i oe of the very early DSP success stories, the Speak & Spell, a widely sold electroic learig aid for childre Speech recogitio: The automated recogitio of huma speech is immesely more difficult tha speech geeratio. Speech recogitio is a classic example of thigs that the huma brai does well, but digital computers do poorly. Digital computers ca store ad recall vast amouts of data, perform mathematical calculatios at blaig speeds, ad do repetitive tasks without becomig bored or iefficiet. Ufortuately, preset day computers perform very poorly whe faced with raw sesory data. Teachig a computer to sed you a mothly electric bill is easy. Teachig the same computer to uderstad your voice is a major udertakig. Digital Sigal Processig geerally approaches the problem of voice recogitio i two steps: feature extractio followed by feature matchig. Each word i the icomig audio sigal is isolated ad the aalyed to idetify the type of excitatio ad resoate frequecies. These parameters are the compared with previous examples of spoke words to idetify the closest match. Ofte, these systems are limited to oly a few hudred words; ca oly accept speech with distict pauses betwee words; ad must be retraied for each idividual speaker. While this is adequate for may commercialapplicatios, these limitatios are humblig whe compared to the abilities of huma hearig. There is a great deal of work to be doe i this area, with tremedous fiacial rewards for those that produce successful commercial products. 5. Image Ehacemet: Spatial domai methods: Suppose we have a digital image which ca be represeted by a two dimesioal radom field A y ^. t r A image processig operator i the spatial domai may be expressed as a mathematical fuctio L applied to the image f to produce a ew image - v^ ^ ^ ' X ' - v^ - as follows. g{x,y T\{x,y_ The operator T applied o fx, y may be defied over: i A sigle pixel x,y. I this case T is a grey level trasformatio or mappig fuctio. 36 MUTHUKUMAR.A 04 05

153 DIGITAL SIGNAL PROCESSING IV YEAR / VII SEM CSE ii Some eighbourhood of x, y. iii T may operate to a set of iput images istead of a sigle image. EXample The result of the trasformatio show i the figure below is to produce a image of higher cotrast tha the origial, by darkeig the levels below m ad brighteig the levels above m i the origial image. This techique is kow as cotrast stretchig. Example The result of the trasformatio show i the figure below is to produce a biary image. s Tr Frequecy domai methods Let g x, y be a desired image formed by the covolutio of a image f x, y ad a liear, positio ivariat operator hx,y, that is: gx,y hx,yfx,y The followig frequecy relatioship holds: Gu, i H a, vfu, i We ca select H u, v so that the desired image gx,y 3 _ i$ii,vfu,v exhibits some highlighted features of f x,y. For istace, edges i f x,y ca be accetuated by usig a fuctio Hu,v that emphasises the high frequecy compoets of Fu,v. Glossary: Samplig Rate: The No. of samples per cycle give i the sigal is termed as samplig rate of the sigal.the samples occur at T equal itervals of Time. 37 MUTHUKUMAR.A 04 05

154 DIGITAL SIGNAL PROCESSING IV YEAR / VII SEM CSE Samplig Theorem: Samplig Theorem states that the o. of samples per cycle should be greater tha or equal to twice that of the frequecy of the iput message sigal. Samplig Rate Coversio: The Samplig rate of the sigal may be icreased or decreased as per the requiremet ad applicatio. This is termed as samplig rate Coversio. Decimatio: The Decrease i the Samplig Rate are termed as decimatio or Dowsamplig. The No. of Samples per Cycle is reduced to M- o. of terms. Iterpolatio: The Icrease i the Samplig rate is termed as Iterpolatio or Up samplig. The No. of Samples per Cycle is icreased to L- No. of terms. Polyphase Implemetatio: If the Legth of the FIR Filter is reduced ito a set of smaller filters of legth k. Usual upsamplig process Iserts I- eros betwee successive Values of x. If M Number of Iputs are there, The oly K Number of Outputs are o-ero. These k Values are goig to be stored i the FIR Filter. Narrow bad Filters: If we wat to desig a arrow passbad ad a arrow trasitio bad, the a lowpass liear phase FIR filters are more efficietly implemeted i a Multistage Decimator Iterpolator. 38 MUTHUKUMAR.A 04 05

155 DIGITAL SIGNAL PROCESSING IV YEAR / VII SEM CSE CS 403-DIGITAL SIGNAL PROCESSING UNIT I CLASSIFICATION OF SIGNALS AND SYSTEMS. Defie Sigal. A sigal is a fuctio of oe or more idepedet variables which cotai some iformatio. Eg: Radio sigal, TV sigal, Telephoe sigal etc.. Defie System. A system is a set of elemets or fuctioal block that are coected together ad produces a output i respose to a iput sigal. Eg: A audio amplifier, atteuator, TV set etc. 3. Defie CT sigals. Cotiuous time sigals are defied for all values of time. It is also called as a aalog sigal ad is represeted by xt. Eg: AC waveform, ECG etc. 4. Defie DT sigal. Discrete time sigals are defied at discrete istaces of time. It is represeted by x. Eg: Amout deposited i a bak per moth. 5. Give few examples for CT sigals. AC waveform, ECG,Temperature recorded over a iterval of time etc. 6. Give few examples of DT sigals. Amout deposited i a bak per moth, 39 MUTHUKUMAR.A 04 05

156 DIGITAL SIGNAL PROCESSING IV YEAR / VII SEM CSE 7. Defie uit step,ramp ad delta fuctios for CT. Uit step fuctio is defied as Ut for t > 0 0 otherwise Uit ramp fuctio is defied as rt t for t>0 0 for t<0 Uit delta fuctio is defied as δt for t0 0 otherwise 8. State the relatio betwee step, ramp ad delta fuctios CT. The relatioship betwee uit step ad uit delta fuctio is δ t ut The relatioship betwee delta ad uit ramp fuctio is δ t.dt rt 9. State the classificatio of CT sigals. The CT sigals are classified as follows i Periodic ad o periodic sigals ii Eve ad odd sigals iii Eergy ad power sigals iv Determiistic ad radom sigals. 0. Defie determiistic ad radom sigals. [Madras Uiverstiy, April -96] A determiistic sigal is oe which ca be completely represeted by mathematical equatio at ay time. I a determiistic sigal there is o ucertaity with respect to its value at ay time. Eg: xtcoswt xwft A radom sigal is oe which caot be represeted by ay mathematical equatio. Eg: Noise geerated i electroic compoets, trasmissio chaels, cables etc.. Defie Radom sigal. [ Madras Uiversity, April 96 ] There is o ucertaity about the determiistic sigal. It is completely represeted by mathematical expressio.. Defie power ad eergy sigals. The sigal xt is said to be power sigal, if ad oly if the ormalied average power p is fiite ad o-ero. ie. 0<p<4 40 MUTHUKUMAR.A 04 05

157 DIGITAL SIGNAL PROCESSING IV YEAR / VII SEM CSE A sigal xt is said to be eergy sigal if ad oly if the total ormalied eergy is fiite ad o-ero. ie. 0<E< 4 3. Compare power ad eergy sigals.[aa Uiversity May -0] Sl.No. POWER SIGNAL ENERGY SIGNALS The ormalied average power is fiite ad o-ero Practical periodic sigals are power sigals Total ormalied eergy is fiite ad o- ero. No-periodic sigals are eergy sigals 4. Defie odd ad eve sigal. A DT sigal x is said to be a eve sigal if x-x ad a odd sigal if x--x. A CT sigal is xt is said to be a eve sigal if xtx-t ad a odd sigal if x-t-xt. 5. Defie periodic ad aperiodic sigals. A sigal is said to be periodic sigal if it repeats at equal itervals. Aperiodic sigals do ot repeat at regular itervals. A CT sigal which satisfies the equatio xt xtt0 is said to be periodic ad a DT sigal which satisfies the equatio x xn is said to be periodic. 6. State the classificatio or characteristics of CT ad DT systems. The DT ad CT systems are accordig to their characteristics as follows i. Liear ad No-Liear systems ii. Time ivariat ad Time varyig systems. iii. Causal ad No causal systems. iv. Stable ad ustable systems. v. Static ad dyamic systems. vi. Iverse systems. 7. Defie liear ad o-liear systems. A system is said to be liear if superpositio theorem applies to that system. If it does ot satisfy the superpositio theorem, the it is said to be a oliear system. 4 MUTHUKUMAR.A 04 05

158 DIGITAL SIGNAL PROCESSING IV YEAR / VII SEM CSE 8. What are the properties liear system should satisfy? [Madras Uiverstiy, April-95] A liear system should follow superpositio priciple. A liear system should satisfy, f [ a x t a x t] a y t a y t where y t f [ x t ] y t f[ x t ] 9. What is the criterio for the system to possess BIBO stability? A system is said to be BIBO stable if it produces bouded output for every bouded iput. 0. Defie shift ivariace. [Madras Uiversity, April-95, Oct-95] If the system produces same shift i the output as that of iput, the it is called shift ivariace or time ivariace system. i.e., f [ x t - t ] y t - t. Defie Causal ad o-causal systems. [Madras Uiversity, April 95,99, Oct.-95] A system is said to be a causal if its output at aytime depeds upo preset ad past iputs oly. A system is said to be o-causal system if its output depeds upo future iputs also.. Defie time ivariat ad time varyig systems. A system is time ivariat if the time shift i the iput sigal results i correspodig time shift i the output. A system which does ot satisfy the above coditio is time variat system. 3. Defie stable ad ustable systems. Whe the system produces bouded output for bouded iput, the the system is called bouded iput, bouded output stable. A system which does ot satisfy the above coditio is called a ustable system. 4. Defie Static ad Dyamic system. A system is said to be static or memory less if its output depeds upo the preset iput oly. The system is said to be dyamic with memory if its output depeds upo the preset ad past iput values. 5. Check causality of the system give by, y x- o [Madras Uiversity, April-00] If o 0, the output y depeds upo preset or past iput. Hece the system is causal. If o < 0, the system become ocausal. 6. Check whether the give system is causal ad stable. [Madras Uiversity, Oct.-98] y 3 x - 3 x 4 MUTHUKUMAR.A 04 05

159 DIGITAL SIGNAL PROCESSING IV YEAR / VII SEM CSE Sice y depeds upo x, this system is ocausal. As log as x ad x are bouded, the output y will be bouded. Hece this system is stable. 7. Whe the discrete sigal is said to be eve? [Madras Uiversity, April 99] A discrete time sigal is said to be eve whe, x- x. For example cosω is a eve sigal. 8. Is diode a liear device? Give your reaso. Nov/Dec 003 Diode is oliear device sice it operates oly whe forward biased. For egative bias, diode does ot coduct. 9. Defie power sigal. A sigal is said to be power sigal if its ormalied power is oero ad fiite. i.e., 0 < P < 30. Defie sigal. What are classificatios of sigals? May/Jue-006 A fuctio of oe or more idepedet variables which cotai some iformatio is called sigal. 3. What is the periodicity of xt e j00πt 30 o? Here xt e j00πt 30 o Comparig above equatio with e jωtф, we get ω 00 Π. Therefore period T is give as, T Π/ ω Π /00 Π / sec. 3. Fid the fudametal period of the sigal x 3 e j3π/ Nov./Dec 006 X 3/5 e j 3Π. e j3π/ -j3/5 e j3π Here, ω3π, hece, f3/k/n. Thus the fudametal period is N. 33. Is the discrete time system describe by the equatio y x- causal or o causal? Why? Here y x-. If - the, y- x Thus the output depeds upo future iputs. Hece system is ocausal. 34. Is the system describe by the equatio yt xt Time ivariat or ot? Why? 43 MUTHUKUMAR.A 04 05

160 DIGITAL SIGNAL PROCESSING IV YEAR / VII SEM CSE Output for delayed iputs becomes, yt,t xt-t Delayed output will be, yt-t x[t-t ] Sice yt,t yt-t. The system is shift variat. 35. What is the period T of the sigal xt cos /4? Here, x cos /4. Compare x with Acos f. This gives, f /4 > F /8. Which is ot ratioal. Hece this is ot periodic sigal. 36. Is the system yt yt- t yt- time ivariat? Here yt-t yt--t t yt--t ad yt t yt-t- t-t yt-t-. Here yt-t yt t. This is time variat system. 37. Check Whether the give system is causal ad stable.[madras Uiversity, Oct.-98] y 3 x - 3 x Sice y depeds upo x, this system is ocausal. As log as x ad x are bouded, the output y will be bouded. Hece this system is stable. 38. What is the periodicity of xt e j00πt 30 o? Here xt e j00πt 30 o Comparig above equatio with e jωtф, we get ω 00 Π. Therefore period T is give as, T Π/ ω Π /00 Π / sec. 39. Fid the fudametal period of the sigal x 3 e j3π/ Nov./Dec 006 X 3/5 e j 3Π. e j3π/ -j3/5 e j3π Here, ω3π, hece, f3/k/n. Thus the fudametal period is N. 40. Defie Samplig. Samplig is a process of covertig a cotiuous time sigal ito discrete time Sigal. After samplig the sigal is defied at discrete istats of time ad the time Iterval betwee two subsequet samplig istats is called samplig iterval. 44 MUTHUKUMAR.A 04 05

161 DIGITAL SIGNAL PROCESSING IV YEAR / VII SEM CSE 4. What is a ati-aliasig filter? A filter that is used to reject high frequecy sigals before it is sampled to reduce the aliasig is called ati-aliasig filter. 4. Defie Samplig Theorem..[Aa Uiversity May -0] It states that, i A bad limited sigal of fiite eergy, which has o frequecy compoets higher tha W hert is completely described by specifyig the value of the sigal at istats of time separated by /W secods. ii A bad limited sigal of fiite eergy which has o frequecy compoets higher tha W hert is completely recovered from the kowledge of its samples take at the rate of W samples per secod, where W f m. These statemets ca be combiedly stated as A badlimited sigal xt with Xjω 0 for ω > ω m is uiquely determied from its samples xt, if the samplig frequecy f s f m, ie., samplig frequecy must be atleast twice the highest frequecy preset i the sigal. 43.Defie Nyquist rate ad Nyquist iterval. Nyquist rate: Whe the samplig rate becomes exactly equal to W samples/sec, for a give badwidth of fm or W Hert, the it is called as Nyquist rate. Nyquist rate f m samples/secod Nyquist iterval: It is the time iterval betwee ay two adjacet samples whe samplig rate is Nyquist rate. Nyquist iterval /W or /f m 44.Defie aliasig. Whe the high frequecy iterferes with low frequecy ad appears as low frequecy, the the pheomeo is called aliasig. 45.What are the effects of aliasig? The effects of aliasig are, iii Sice high ad low frequecies iterferes with each other distortio is geerated iv The data is lost ad it caot be recovered 46.What is meat by a bad limited sigals? A bad limited sigal is a sigal xt for which the Fourier trasform of xt is Zero above certai frequecy ω m. xt -- Xj ω 0 for ω > ω m πf m 47.What is the trasfer fuctio of a ero order hold? The trasfer fuctio of a ero order hold is give by e H s s st 45 MUTHUKUMAR.A 04 05

162 DIGITAL SIGNAL PROCESSING IV YEAR / VII SEM CSE 48.A sigal xt whose spectrum is show i figure is sampled at a rate of 300 samples/sec. What is the spectrum of the sampled discrete time sigal. April/May 003. Solutio: f m 00 H Nyquist rate f m 00 H Samplig frequecy f s 300 H f s > f m, Therefore o aliasig takes place The spectrum of the sampled sigal repeats for every 300 H. 49. A sigal havig a spectrum ragig from ear dc to 0 KH is be sampled ad Coverted to discrete form. What is the miimum umber of samples per secod that must be take esure recovery? Solutio: Give: f m 0 KH From Nyquist rate the miimum o. of samples per secod that must be take to esure recovery is, f s f m 0 K 0000samples/sec 50. A sigal xt sic 50πt is sampled at a rate of a 00 H, b 00 H, ad c 300H. For each of these cases, explai if you ca recover the sigal xt from the sampled sigal. Solutio: Give xt sic 50πt The spectrum of the sigal xt is a rectagular pulse with a badwidth maximum frequecy compoet of 50π rad/sec as show i figure. πf m 50π f m 75 H X 0 50 ω 46 MUTHUKUMAR.A 04 05

163 DIGITAL SIGNAL PROCESSING IV YEAR / VII SEM CSE Nyquist rate is f m 50 H a For the first case the samplig rate is 00H, which is less tha Nyquist rate uder samplig. Therefore xt caot be recovered from its samples. b Ad c i both cases the samplig rate is greater tha Nyquist rate. Therefore xt ca be recovered from its sample. 5. How ca you miimie the aliasig error? Aliasig error ca be reduced by choosig a samplig rate higher tha Nyquist rate to avoid ay spectral overlap. Prefilterig is doe i order to esure about the Nyquist frequecy compoet so as to decide the appropriate samplig rate. By doig this we ca reduce aliasig error. 5. Defie -trasform ad iverse -trasform. The -trasform of a geeral discrete-time sigal x is defied as, X x It is used to represet complex siusoidal discrete time sigal. The iverse -trasform is give by, x πj X d 53. What do you mea by ROC? or Defie Regio of covergece. Nov/Dec 003 By defiitio, x X The -trasform exists whe the ifiite sum i equatio coverges. A ecessary coditio for covergece is absolute summability of x -. The value of for which the -trasform coverges is called regio of covergece. 54. What are left sided sequece ad right sided sequece? Left sided sequece is give by X x For this type of sequece the ROC is etire -plae except at. Example: x {-3, -, -, 0} Right sided sequece is give by 47 MUTHUKUMAR.A 04 05

164 DIGITAL SIGNAL PROCESSING IV YEAR / VII SEM CSE X 0 x For this type of sequece the ROC is etire -plae except 0. Example: x {,0,3,-,} 55. Defie two sided sequece or sigal. A sigal that has fiite duratio o both left ad right sides is kow as Two sided sequece. X x For such a type of sequeces the ROC is etire -plae except at 0 ad. Example: x {,-,3,,,0,,3,-} 56. List the properties of regio of covergece for the -trasform.. The ROC of X cosists of a rig i the -plae cetered about the origi.. The ROC does ot cotai ay poles. 3. If x is of fiite duratio the the ROC is the etire -plae except 0 ad / or. 4. If x is a right sided sequece ad if the circle r 0 is i the ROC the all fiite values of for which > r 0 will also be i the ROC. 5. If x is a left sided sequece ad if the circle r 0 is i the ROC the all values of for which 0< < r 0 will also be i the ROC. 6. If x is two sided sequece ad if the circle r 0 is i the ROC the the ROC will cosist of a rig i the -plae that icludes the circle r Determie the -trasform of the sigal x α u ad also the ROC ad pole & ero locatios of X i the -plae. Solutio: Give x α u By defiitio X x α u 0 α 48 MUTHUKUMAR.A 04 05

165 DIGITAL SIGNAL PROCESSING IV YEAR / VII SEM CSE Usig geometric series, a 0 a X ; > α α ; > α α There is a pole at α & ero at List the properties of -trasform. v Liearity vi Time shiftig vii Time reversal viii Time scalig ix Multiplicatio x Covolutio xi Parseval s relatio 59. What are the methods used to fid iverse -trasform? xii Log divisio method or Power series expasio xiii Residue method xiv Partial fractio method xv Covolutio method 60. State the Parseval s theorem of -trasform. If x ad x are complex valued sequeces, the the Parseval s relatio states that, x x X v X v dv π j v 6.Determie the -trasform of uit step sequece. The uit step sequece is give by ; 0 u 0; otherwise Here x u X x 49 MUTHUKUMAR.A 04 05

166 DIGITAL SIGNAL PROCESSING IV YEAR / VII SEM CSE 0 X u X ; > The ROC is at > ie., the area outside the uit circle 6. What is the relatioship betwee the -trasform ad FourierTrasform?April/May 003 The Fourier trasform represetatio of a system is give by, jω jω H e h e The -trasform represetatio of a system is give by, h H O comparig & we ca say that r e jω Whe r, equatios & are equal. jω i.e., H e H jω e 63.Determie the -trasform of the sigal x -a u-- & plot the ROC. Solutio: Give: x -a u-- By defiitio of -trasform, X x X [ a u ] a 50 MUTHUKUMAR.A 04 05

167 DIGITAL SIGNAL PROCESSING IV YEAR / VII SEM CSE a 0 a a X a X a ROC: < a, Pole: a & ero: Fid the left sided -trasform of the give sequece x[]{,4,-6,3,8,-} Solutio: By defiitio of -trasform X x X 6 x a x X x-6 6 x-5 5 x-4 4 x-3 3 x- x- X The ROC is etire -plae except at 65. How the stability of a system ca be foud i -trasform? Let h be a impulse respose of a causal or o-causal LTI system ad H be the -trasform ofh. The stability of a system ca be foud from ROC usig the theorem which states, A Liear time system with the system fuctio H is BIBO stable if ad oly if the ROC for H cotais the uit circle. The coditio for the stable system is, h < 66. What is the coditio for causality i terms of -trasform? The coditio for Liear time ivariat system to be causal is give as, 5 MUTHUKUMAR.A 04 05

168 DIGITAL SIGNAL PROCESSING IV YEAR / VII SEM CSE 5 MUTHUKUMAR.A h 0, <0. Where h is the sample respose of the system. Whe the sequece is causal, its ROC is the exterior of the circle. i.e., >r. LTI system is causal if ad oly if the ROC of the system fuctio is exterior of the circle. 67. Fid the system fuctio ad the impulse respose of the system Described by the differece equatio, y xx--4x-x-3 Solutio: The give differece equatio is y xx--4x-x-3 Takig -trasform, Y X - X 4 - X -3 X 3 4 X Y H The system fuctio, 3 4 H The impulse respose is h {,,-4,} 68. Cosider a LTI system with differece equatio, y-3/4 y-/8 y- x. Fid H. Nov/Dec 003 Solutio: Give, y-3/4 y-/8 y- x Takig -trasform, X Y Y Y 8 3 X Y X Y H 69. State ad prove the time shiftig property of -trasform. Nov/Dec003 The time shiftig property of -trasform states that if X {x}the x-k -k X where k is a iteger. Proof: x x X } {

169 DIGITAL SIGNAL PROCESSING IV YEAR / VII SEM CSE { x k} x k put -k m, km ad hece the limits of m will also be - to, the { x k} m m x m x m k m m k k X Thus by shiftig the sequece by k samples i time domai is equivalet to multiplyig its -trasform by -k. 70.State iitial value theorem of -trasform. April/May 004 If x is a causal sequece the its iitial value is give by lim x 0 X Proof: By defiitio of -trasform X x x 0 for <0, sice x is a causal sequece. X 0 x x0x - x - Applyig limit o both sides, lim X x0 7. Determie the system fuctio of the discrete system described by differece equatio y-/ y-/4 y- x-x-. April/May003 Solutio: y-/ y-/4 y- x-x- Takig -trasform, Y -/ - Y /4 - Y X- - X Y [ ] 4 X 53 MUTHUKUMAR.A 04 05

170 DIGITAL SIGNAL PROCESSING IV YEAR / VII SEM CSE Y H X 4 7. What is the trasfer fuctio of a system whose poles are at -0.3±j0.4 ad a ero at -0.? Nov/Dec 004 Poles are at -0.3±j0.4 Deomiator polyomial [--0.3j0.4][--0.3-j0.4] Zero at -0. Numerator polyomial 0. The trasfer fuctio is give by, 0. H 0.3 j j H Defie oe-sided ad two-sided -trasform. Nov/Dec 004 Oe sided -trasform is defied as, X 0 x Two sided -trasform is defied as X x 74. State ay two properties of autocorrelatio fuctio.[aa uiversity Nov 00] i A fudametal property of the autocorrelatio is symmetry,, which is easy to prove from the defiitio. ii The autocorrelatio of a periodic fuctio is, itself, periodic with the same period. 75. State covolutio property. [Aa Uiversity Nov 03] Covolutio property states that if x X ad x X x X X The x Part B 54 MUTHUKUMAR.A 04 05

171 DIGITAL SIGNAL PROCESSING IV YEAR / VII SEM CSE. Fid the fudametal period T of the followig sigal. Oct / Nov 00 Here the three frequecy compoets are, f Π/ > f /4 therefore N 4 Π f Π/8 > f /6 therefore N 6 Π f 3 Π/4 >f 3 /8 therefore N 3 4 Here f, f ad f 3 are ratioal, hece idividual sigals are periodic. The overall sigal will also be periodic sice N /N 4/6 /4 ad N / N 3 6/8. The period of the sigal will be least commo multiple of N, N ad N 3 which is 6. Thus fudametal period, N 6.. Fid whether the followig sigal is periodic or ot. x 5cos6Π Nov/Dec 003, 4marks Compare the give sigal with, x A cos Πf. We get, Πf6Π>f3, which is ratioal. Hece this sigal is periodic. Its period is give as, F k/n 3/ > N. 3. What is the periodicity of the sigal xt si 00πt cos50πt? Nov/Dec 004, 3 Marks Compare the give sigal with, Xt si πf t cos πf t πf t 00 πt > f 50 T πf t 50 πt > f 75 T Sice i.e. ratioal, the sigal is periodic. The fudametal period will be, TT 3 T, i.e. least commo multiple of T ad T. Here TT 3T /5. 4. What are the basic cotiuous time sigals? Draw ay four waveforms ad write their equatios. Nov/Dec 004, 9 Marks Basic cotiuous time sigals: These are the stadard test sigals which are used to study the properties of systems. Basic CT sigals posses some specific characteristics that are used for systems aalysis. The Stadard sigals are, i. Uit Step sigal ii. Uit Impulse sigal iii. Uit ramp Sigal 55 MUTHUKUMAR.A 04 05

172 DIGITAL SIGNAL PROCESSING IV YEAR / VII SEM CSE iv. Complex Expoetial Sigal i. Uit Step sigal ii. Uit Impulse sigal iii. Uit ramp Sigal iv. Complex Expoetial Sigal 5. Determie eergy of the discrete time sigal. April/May-005, 8 Marks 56 MUTHUKUMAR.A 04 05

173 DIGITAL SIGNAL PROCESSING IV YEAR / VII SEM CSE x, 0 3, < 0 Eergy of the sigal is give as, E - Here /3. Hece above equatio will be, E Here let us use, for a < for the secod summatio. i.e., E [ 3 4. ] /3[ 3. ] The term iside the brackets is geometric series which ca be writte as, aa a 3., a <. Thus, E /3, ½.5 6. Test whether the system described by the equatio y x is i Liear ii Shift ivariat. Nov./Dec 0, 44 Marks a To check liearity y T{x x y T{x x y 3 T{a x a x } [a x a x ] a x a x ad y 3 a y a y a x a x Here y 3 y 3 Hece this system is liear. 57 MUTHUKUMAR.A 04 05

174 DIGITAL SIGNAL PROCESSING IV YEAR / VII SEM CSE b To check shift ivariace y T{x x y,k T{x-k x-k} y-k -k x-k Sice y,k y-k the system is shift variat. 7. Verify the liearity, causality ad time ivariace of the system, y a x b x3 Nov./Dec. 0, 9 Marks Puttig - for i give differece equatio, y- a x- b x-3 y a x- b x i Liearity y T {x a x - b x y T{x a x - b x y 3 T {a x a x } a[a x - a x -] b[a x a x ] a a x - a a x - ba x b a x y 3 a y a y a [a x - b x ] a [a x - b x ] a a x - b a x a a x - b a x Sice y 3 y 3, the system is liear. ii Causality y depeds upo x, which is future iput. Hece the system is ocausal. iii Time ivariace y,k T{x-k} a x--k b x-k y,k a x-k- b x-k Sice y,k y-k, the system is time ivariat 8. Determie whether the systems are liear, time ivariat, causal ad stable.. y x. yt xt xt- for t 0 0 for t < 0 May / Jue -006, 8 Marks; 5 Marks. y x y x ad y x 58 MUTHUKUMAR.A 04 05

175 DIGITAL SIGNAL PROCESSING IV YEAR / VII SEM CSE ad y 3 T [a x a x ] [a x a x ] a x a x y 3 a y a y a x a x Sice y 3 y 3, the system is liear. Sice, y is fuctio of, the system is time variat. As ->,y ->. Hece the output is ot bouded eve if x is bouded. Hece the system is ustable. Output is fuctio of preset iput oly, hece the system is causal.. yt xt xt- for t 0 0 for t < 0 y t x t x t- y t x t x t- ad y 3 t T[a x t a x t] T[a x t]t[a x t] a x t a x t- a x t- y 3 t a y t a y t a x t a x t-a x t a x t- Sice, y 3 t y 3 t, the system is liear. yt is ot the fuctio of time directly. Hece the system is time ivariat. Output depeds upo preset ad past iput. Hece this is causal system. As log as xt is bouded xt- will be bouded ad hece yt will be bouded. Hece this is stable system. Sice the term xt- is preset, the system requires memory. 9. Cosider a cotiuous time sigal xt t- t- Calculate the value of E α for the sigals. Nov. / Dec. 006, 4 Marks yt d yt d Sice d ut, above equatio becomes, yt ut ut- for - t 0 elsewhere. E t dt t dt dt 4 59 MUTHUKUMAR.A 04 05

176 DIGITAL SIGNAL PROCESSING IV YEAR / VII SEM CSE 0. Prove that u-u- We kow that u for 0 0 for <0 ad u- for 0 0 for < Hece, u -u - 0 for i.e., >0 for 0 0 for <0 The above equatio is othig but δ.i.e., u-u- δ for 0 0 for 0. What is the periodicity of the sigal xt si 00πt cos 50πt? Nov/Dec 004 Compare the give sigal with xt si πf t cos πf t siπf t 00πt f50 T/f/50 siπft 50πt f75 T/f/75 Sice T /T /50//75 3/ i.e. ratioal, the sigal is periodic. The fudametal period will be, TT 3T, i.e. least commo multiple of T ad T. Here TT 3T /5.. A Discrete time sigal is as show below: Sketch the followig: i. x-3 ii. x3-60 MUTHUKUMAR.A 04 05

177 DIGITAL SIGNAL PROCESSING IV YEAR / VII SEM CSE iii. x i. x-3 ii. x3- iii. x 3. Determie whether the followig sigals are eergy or power sigals ad evaluate their ormalied eergy or power? 6 MUTHUKUMAR.A 04 05

178 DIGITAL SIGNAL PROCESSING IV YEAR / VII SEM CSE 4. State ad prove the properties of -trasform or Explai the Properties of -Trasform. The Properties of -trasform are. Liearity. Time shiftig 3. Time reversal 4. Scalig i time domai 5. Differetiatio i -domai 6 MUTHUKUMAR.A 04 05

179 DIGITAL SIGNAL PROCESSING IV YEAR / VII SEM CSE 6. Covolutio 7. Multiplicatio 8. Correlatio of two sequece 9. Cojugatio ad cojugate symmetry property 0. Parseval s Relatio. Iitial value theorem & Fial value theorem. Liearity: The liearity property states that if X {x} ad X {x }, The {a x a x } a X a X Where a ad a are arbitrary costats Proof: Let x a x a x [ a x ax a x a x a X a X X x X ] X Hece, {a x a x } a X a X. Time shiftig: The time shiftig property of -trasform states that if X{x} k The x k X, where k is a iteger. Proof: { x k} x k puttig -km, mk ad hece the limits of m will be - to. { x k} k x m x m k m m 63 MUTHUKUMAR.A 04 05

180 DIGITAL SIGNAL PROCESSING IV YEAR / VII SEM CSE -k X Thus the shiftig the sequece by k samples i time domai is equivalet to multiplyig its -trasform by -k. 3. Time reversal: This property states that if x X ; ROC: r < <r The x X ; ROC: /r < </r Proof: By defiitio, I the RHS put m -, { x x { x x m m m x m m m x m x X ROC for X is r < <r ROC for X - is r < - <r i.e., /r < </r m 4. Scalig i -domai: This property states that if x X, the a x X ; a ROC: a r < < a r Proof: By defiitio, { a x } a x x a Xa - X/a Thus the scalig i -domai is equivalet to multiplyig by a i time domai 5. Differetiatio i -domai: 64 MUTHUKUMAR.A 04 05

181 DIGITAL SIGNAL PROCESSING IV YEAR / VII SEM CSE This property states that if x X the x X Proof: By defiitio, X x Differetiatig with respect to o both sides, d X d d x d d x X d d d d X x X { x } d d { x } X d The ROC is same as that of X. 6. Covolutio of two sequece: Covolutio property states that if x X ad x X The x x X X Proof: Let x represets the covolutio of the sequeces x & x i.e., x x x By defiitio of -trasform, X x { x x } X Usig covolutio sum, x x x k x k x k d d 65 MUTHUKUMAR.A 04 05

182 DIGITAL SIGNAL PROCESSING IV YEAR / VII SEM CSE [ x x x ] x x k x k k Iterchagig the order of summatio, x k x k x k k x k X k x X. X x usig time shiftig property { x x } X. X The ROC of the product of X X is the itersectio or overlap of ROC of two idividual sequeces. 7. Correlatio of Two sequeces: This property states that if x X ad x X the x x X X Proof: The correlatio of two sequeces x ad x is deoted as r xx l ad it is give as l r x x x x l Rearrage the term x -l as x [ l ] l i the above equatio, r x x { l } x x The RHS of equatio represets the covolutio of x l ad x -l. r x x l x l x l 66 MUTHUKUMAR.A 04 05

183 DIGITAL SIGNAL PROCESSING IV YEAR / VII SEM CSE { r } { x l x l} { x l} x { r l } X X x x 8. Multiplicatio of Two sequeces: This property states that if x X ad x X x x X v X v dv j π v c Here c is the closed cotour. It ecloses the origi ad lies i the ROC which is commo to both X v ad X. v Proof: The iverse -trasform of X is x X d π j The iverse -trasform of X is c x X d π j c Let us defie discrete sigal w which is multiplicatio of discrete sigals x ad x, w x x By defiitio of -trasform, W w x x } { W The -trasform of discrete time sigal x is x X v v dv π j usig 4 i 3 W c x πj X v c v dv 67 MUTHUKUMAR.A 04 05

184 DIGITAL SIGNAL PROCESSING IV YEAR / VII SEM CSE 68 MUTHUKUMAR.A dv v v x v X j W c π dv v v X v X j W c π 9. Cojugatio ad Cojugate Symmetry Property: This property states that if x is a complex sequece ad if X x, the -trasform of complex cojugate x is X x. Proof: { } x x { } [ ] x x { } [ ] X x { } X x 0. Parseval s Relatio: If x ad x are complex valued sequees the the Parseval s relatio states that, dv v v X v X j x x c π Proof: By defiitio of iverse -trasform, dv v v X j x c π dvx v v X j x x c π dv v x v X j c π v - v v - v - - v - /v - v -

185 DIGITAL SIGNAL PROCESSING IV YEAR / VII SEM CSE 69 MUTHUKUMAR.A dv v v x v X j x x π v x x x v X v X Hece, dv v v X v X j x x c π. Iitial Value Theorem: If x is causal sequece the its iitial value is give by lim 0 X x Proof: By defiitio, x X x0 for <0, sice x is a causql sequece. 0 x X x0 x - x -. Applyig the limits o both sides,... lim 0 lim lim x x X lim x 0 lim x X. Fial Value Theorem:

186 DIGITAL SIGNAL PROCESSING IV YEAR / VII SEM CSE lim This states that, x X This theorem is meaigful oly if the poles of -X are iside the uit Circle. 5. Explai the effect of Aliasig or Uder-samplig. Defiitio: Whe the high frequecy iterferes with low frequecy & appears as low frequecy, the the pheomeo is called Aliasig. Xf X δ f Effect of Aliasig: Sice high & low frequecies iterfere with each other distortio is geerated. The data is lost & it caot be recovered. Methods to avoid aliasig: i Samplig rate f s W ii Strictly bad-limited the sigal to W H. Samplig rate f s W: The spectrum will ot overlap ad there will be sufficiet gap betwee the Idividual spectrums. X δ f 70 MUTHUKUMAR.A 04 05

187 DIGITAL SIGNAL PROCESSING IV YEAR / VII SEM CSE Bad limitig the sigal: The samplig rate is fs W. Theoretically there should be o aliasig but there ca be few compoets higher tha W. These compoets create aliasig. Hece a Low pass filter is used before samplig the sigals as show i figure. x t Bad limitig LPF Sampler X δ t The output of low pass filter is strictly bad limited ad there are o Frequecy compoets higher tha W. The there will be o aliasig. 6. Explai the discrete time processig of cotiuous time sigals. Figure shows the system for processig of cotiuous time sigals. A/D coverter DT Processig D/A coverter Most of the sigals geerated are aalog. For example soud, video, temperature, pressure, seismic sigals, biomedical sigals, etc. Cotiuous time sigals are coverted to discrete time sugals by Aalog to digital A/D coverter. I above figure xt is aalog sigal ad x is discrete time sigal. Discrete time processig ca be amplificatio, filterig, atteuatio, etc. It is ormally performed with the help of digital sigal processor. The processed DT sigal y is coverted back to aalog sigal by digital to aalog D/A coverter. The fial output sigal is yt which is digitally processed. PROBLEMS IN Z-TRANSFORM:. Fid the -trasform of the sequece, x u u 4 Nov/Dec 004. Solutio: X x 7 MUTHUKUMAR.A 04 05

188 DIGITAL SIGNAL PROCESSING IV YEAR / VII SEM CSE 7 MUTHUKUMAR.A u u 4 u u

189 DIGITAL SIGNAL PROCESSING IV YEAR / VII SEM CSE 73 MUTHUKUMAR.A X ; <. Fid the -trasform of{ } 0 u u. April/May 003 Solutio: Give: { } 0 u u x x X { } u u X 0 u X 0 0 Usig time shiftig property, X k x k X X 3. Fid the -trasform of the followig, i cos 0 u x ω ii ;0 si si < < r u x ω ω Solutio:

190 DIGITAL SIGNAL PROCESSING IV YEAR / VII SEM CSE 74 MUTHUKUMAR.A i cos 0 u x ω By defiitio, x X u X cos 0 ω 0 0 cos ω j j e e 0 ω 0 ω 0 j j e e 0 0 ω 0 ω e e j j ω ω Both the series coverges if, & 0 0 < < e e j j ω ω ROC is > 0 0 e e j j ω ω e e e e j j j j ω ω ω ω e e e e j j j j ω ω ω ω 0 0 cos cos X ω ω

191 DIGITAL SIGNAL PROCESSING IV YEAR / VII SEM CSE 75 MUTHUKUMAR.A ii ;0 si si < < r u r x ω ω Solutio: By defiitio x X u r X si si ω ω 0 si si r X ω ω j j j e e r si 0 ω ω ω 0 0 si j j j j e e r e e r j ω ω ω ω ω 0 0 si j j j j re e e r e j ω ω ω ω ω si re e re e j j j j j ω ω ω ω ω Both the series coverges if the ROC is, re jω re jω < r - < < r si re re re e re e j j j j j j j ω ω ω ω ω ω ω si re re e e j j j j j ω ω ω ω ω si si r e e r j j ω ω ω ω

192 DIGITAL SIGNAL PROCESSING IV YEAR / VII SEM CSE X r cosω r INVERSE TRANSFORM Iverse trasform ca be obtaied by followig methods: a Power series expasio b Partial fractio method c Cotour itegratio / Residue method d Covolutio method POWER SERIES EXPANSION: It is possible to express X as a power series i - or of the form X x The value of the sigal x is the give by the coefficiet associated with -, The power series method is limited to oe sided sigals whose ROC is either < a or > a. If the ROC of X is > a, the X is expressed as a power series of - so that we obtai right sided sigal iverse -trasform. If the ROC of X is < a, the X is expressed as a power series of so that we obtai left sided sigal iverse -trasform. Here we use Log divisio for power series expasio. PROBLEMS:. Fid the iverse -trasform of the fuctio X for i ROC: > ; ii ROC: <0.5 & iii ROC: 0.5< < April/May 003 Solutio: X Multiply ad divide by, X Usig partial fractio expasio, X 0.5 X A A MUTHUKUMAR.A 04 05

193 DIGITAL SIGNAL PROCESSING IV YEAR / VII SEM CSE A A X 0.5 A X A A A X 0.5 X X 0.5 a u > a a ; i ROC is >, The sigal is causal X 0.5 Takig iverse -trasform, x u-0.5 u ii ROC is <0.5, The sigal is ati-causal X 0.5 Takig iverse -trasform, x - u--0.5 u-- iii ROC is0.5< <, X 0.5 Takig iverse -trasform, x u u 77 MUTHUKUMAR.A 04 05

194 DIGITAL SIGNAL PROCESSING IV YEAR / VII SEM CSE 78 MUTHUKUMAR.A What are the three possible sequeces whose -trasform is give by X Nov/Dec 004 Solutio: X X ± ± a ac b b ± 3, X B A X X A

195 DIGITAL SIGNAL PROCESSING IV YEAR / VII SEM CSE 79 MUTHUKUMAR.A A A X B B 3 69 B X X X i ROC: 4 3 > ; u u x ii ROC: 3 < ; u u x

196 DIGITAL SIGNAL PROCESSING IV YEAR / VII SEM CSE 80 MUTHUKUMAR.A u x iii ROC: < < ; u u x CONVOLUTION METHOD The covolutio property ca be used to fid the iverse -trasform of X. X X X x x For the give X we obtai X & X ad the take iverse - Trasform. PROBLEMS:. Fid the iverse -trasform of 6 5 X usig covolutio method. Solutio: 6 5 X By factoriig the equatio, 3 X X X where 3 & X X By takig iverse -trasform, x u ad x 3 u By defiitio of covolutio, X X X x x k k x k x x x x 0 3 k k k k k

197 DIGITAL SIGNAL PROCESSING IV YEAR / VII SEM CSE 8 MUTHUKUMAR.A u x u x [ ] 3 u x CONTOUR INTEGRATION METHOD: Cauchy itegral theorem is used to calculate iverse -trasform. Step : Defie the fuctio X 0, which is ratioal ad its deomiator is expaded ito product of poles. 0 X X m i m p i N X 0 Here m is the order of poles Step : i For simple poles, i.e. m, the residue of X 0 at pole p i is give as, [ ] lim Re 0 0 X p p X p s i i i p i i X p ii For multiple poles of order m 0, the residue of X 0 ca be calculated as, p i m i m m i X p d d m X p s Re iii If X 0 has simple pole at the origi, i.e. 0, the x0 ca be calculated idepedetly. Step 3: i Usig residue theorem, calculate x for poles iside the uit circle. i.e., Re 0 X p s x N i i ii For poles outside the cotour itegratio,

198 DIGITAL SIGNAL PROCESSING IV YEAR / VII SEM CSE 8 MUTHUKUMAR.A Re 0 X p s x N i i With < 0 PROBLEMS:. Determie the iverse -trasform of a X, ROC: a < usig cotour itegratioi.e. residue method. Solutio: Step : 0 X X a a Step : Here the pole is at a ad it has order m. Hece usig equatio, p i m i m m i X p d d m X p s Re 0 0 we ca calculate residue of X 0 at a. a i X a d d X p s Re 0 0 a a d d a Step 3: By fidig the residues, the sequece x is give as, Re 0 X a s x i u a x Sice ROC: a < UNIT II FREQUENCY TRANSFORMATIONS. Defie DTFT. APRIL/MAY008 The discrete-time Fourier trasform or DTFT of is usually writte:. Defie Periodicity of DTFT.

199 DIGITAL SIGNAL PROCESSING IV YEAR / VII SEM CSE Samplig causes its spectrum DTFT to become periodic. I terms of ordiary frequecy, cycles per secod, the period is the sample rate,. I terms of ormalied frequecy, cycles per sample, the period is. Ad i terms of radias per sample, the period is π, which also follows directly from the periodicity of. That is: Where both ad k are arbitrary itegers. Therefore: 3. Differece betwee DTFT ad other trasform. NOV/DEC 00 The DFT ad the DTFT ca be viewed as the logical result of applyig the stadard cotiuous Fourier trasform to discrete data. From that perspective, we have the satisfyig result that it's ot the trasform that varies; it's just the form of the iput: If it is discrete, the Fourier trasform becomes a DTFT. If it is periodic, the Fourier trasform becomes a Fourier series. If it is both, the Fourier trasform becomes a DFT. 4. Write about symmetry property of DTFT. APRIL/MAY009 The Fourier Trasform ca be decomposed ito a real ad imagiary or ito eve ad odd. or Time Domai Frequecy Domai 5.Defie DFT pair. The sequece of N complex umbers x 0,..., x N is trasformed ito the sequece of N complex umbers X 0,..., X N by the DFT accordig to the formula: where is a primitive N'th root of uity. The trasform is sometimes deoted by the symbol, as i or or. The iverse discrete Fourier trasform IDFT is give by 83 MUTHUKUMAR.A 04 05

200 DIGITAL SIGNAL PROCESSING IV YEAR / VII SEM CSE 6.How will you express IDFT iterms of DFT. MAY/JUNE 00 A useful property of the DFT is that the iverse DFT ca be easily expressed i terms of the forward DFT, via several well-kow "tricks". For example, i computatios, it is ofte coveiet to oly implemet a fast Fourier trasform correspodig to oe trasform directio ad the to get the other trasform directio from the first. First, we ca compute the iverse DFT by reversig the iputs: As usual, the subscripts are iterpreted modulo N; thus, for 0, we have x N 0 x 0. Secod, oe ca also cojugate the iputs ad outputs: Third, a variat of this cojugatio trick, which is sometimes preferable because it requires o modificatio of the data values, ivolves swappig real ad imagiary parts which ca be doe o a computer simply by modifyig poiters. Defie swapx as x with its real ad imagiary parts swapped that is, if x a bi the swapx is b ai. Equivaletly, swapx equals. The 7.Write about Bilateral Z trasform. APRIL/MAY009 The bilateral or two-sided Z-trasform of a discrete-time sigal x[] is the fuctio X defied as where is a iteger ad is, i geeral, a complex umber: Ae jφ OR Acosjφ sijφ where A is the magitude of, ad φ is the complex argumet also referred to as agle or phase i radias 8. Write about Uilateral Z trasforms. Alteratively, i cases where x[] is defied oly for 0, the sigle-sided or uilateral Z-trasform is defied as I sigal processig, this defiitio is used whe the sigal is causal. 9. Defie Regio Of Covergece. The regio of covergece ROC is the set of poits i the complex plae for which the Z-trasform summatio coverges. 0.Write about the output respose of Z trasform APRIL/MAY MUTHUKUMAR.A 04 05

201 DIGITAL SIGNAL PROCESSING IV YEAR / VII SEM CSE If such a system is drive by a sigal the the output is. By performig partial fractio decompositio o ad the takig the iverse Z-trasform the output ca be foud. I practice, it is ofte useful to fractioally decompose before multiplyig that quatity by to geerate a form of which has terms with easily computable iverse Z-trasforms.. Defie Twiddle Factor. MAY/JUNE 00 A twiddle factor, i fast Fourier trasform FFT algorithms, is ay of the trigoometric costat coefficiets that are multiplied by the data i the course of the algorithm.. State the coditio for existece of DTFT? MAY/JUNE 007 The coditios are, If xis absolutely summable the x < If x is ot absolutely summable the it should have fiite eergy for DTFT to exit. 3. List the properties of DTFT. Periodicity, Liearity, Time shift, Frequecy shift, Scalig, Differetiatio i frequecy domai, Time reversal, Covolutio, Multiplicatio i time domai, Parseval s theorem 4. What is the DTFT of uit sample? NOV/DEC 00 The DTFT of uit sample is for all values of w. 5. Defie Zero paddig. The method of appedig ero i the give sequece is called as Zero paddig. 6. Defie circularly eve sequece. A Sequece is said to be circularly eve if it is symmetric about the poit ero o the circle. xn-x,<<n-. 7. Defie circularly odd sequece. A Sequece is said to be circularly odd if it is ati symmetric about poit x0 o the circle 8. Defie circularly folded sequeces. A circularly folded sequece is represeted as x-n. It is obtaied by plottig x i clockwise directio alog the circle. 9. State circular covolutio. NOV/DEC 009 This property states that multiplicatio of two DFT is equal to circular covolutio of their sequece i time domai. 0. State parseval s theorem. NOV/DEC 009 Cosider the complex valued sequeces x ad y.if x Xk, y Yk the xy*/n XkY*k. Defie Z trasform. The Z trasform of a discrete time sigal x is deoted by X ad is give by X xz-.. Defie ROC. The value of Z for which the Z trasform coverged is called regio of covergece. 3. Fid Z trasform of x{,,3,4} x {,,3,4} X x /3/4/3. 85 MUTHUKUMAR.A 04 05

202 DIGITAL SIGNAL PROCESSING IV YEAR / VII SEM CSE 4. State the covolutio property of Z trasforms. APRIL/MAY00 The covolutio property states that the covolutio of two sequeces i time domai is equivalet to multiplicatio of their Z trasforms. 5. What trasform of -m? By time shiftig property Z[A -m]az-m siz[ ] 6. State iitial value theorem. If x is causal sequece the its iitial value is give by x0lim X 7. List the methods of obtaiig iverse Z trasform. Partial fractio expasio. Cotour itegratio Power series expasio Covolutio. 8. Obtai the iverse trasform of X/-a, > a APRIL/MAY00 Give X-/-a- By time shiftig property Xa.u- 6-MARKS, Determie the 8-Poit DFT of the Sequece x{,,,,,,0,0} Nov / 03 Sol: N- Xk Σ xe -jπk/n, k0,,...n X0 Σ xe -jπk/8, k0,, X0 6 7 X Σ xe -jπk/8, k0,, X j X Σ xe -jπk/8, k0,, X-j 7 X3 Σ xe -jπk/8, k0,, X30.707j X4 Σ xe -jπk/8, k0,, MUTHUKUMAR.A 04 05

203 DIGITAL SIGNAL PROCESSING IV YEAR / VII SEM CSE X40 7 X5 Σ xe -jπk/8, k0,, X j X6 Σ xe -jπk/8, k0,, X6j 7 X7 Σ xe -jπk/8, k0,, X j.707. Xk{6, j.707,-j, 0.707j0.93, 0, j0.93, j, j.707}, Determie the 8-Poit DFT of the Sequece x{,,,,,,,} April 04 Sol: N- Xk Σ xe -jπk/n, k0,,...n X0 Σ xe -jπk/8, k0,, X0 8 7 X Σ xe -jπk/8, k0,, X0. 7 X Σ xe -jπk/8, k0,, X0. 7 X3 Σ xe -jπk/8, k0,, X30 7 X4 Σ xe -jπk/8, k0,, X40 7 X5 Σ xe -jπk/8, k0,, MUTHUKUMAR.A 04 05

204 DIGITAL SIGNAL PROCESSING IV YEAR / VII SEM CSE X50 7 X6 Σ xe -jπk/8, k0,, X60 7 X7 Σ xe -jπk/8, k0,, X7 0. Xk{8, 0, 0, 0, 0, 0, 0, 0} 3, Determie the 8-Poit DFT of the Sequece x{,,3,4,4,3,,} Nov / 04 Sol: N- Xk Σ xe -jπk/n, k0,,...n X0 Σ xe -jπk/8, k0,, X0 0 7 X Σ xe -jπk/8, k0,, X-5.88-j.44 7 X Σ xe -jπk/8, k0,, X0. 7 X3 Σ xe -jπk/8, k0,, X3-0.7-j X4 Σ xe -jπk/8, k0,, X40 7 X5 Σ xe -jπk/8, k0,, X5-0.7j X6 Σ xe -jπk/8, k0,, MUTHUKUMAR.A 04 05

205 DIGITAL SIGNAL PROCESSING IV YEAR / VII SEM CSE X60 7 X7 Σ xe -jπk/8, k0,, X7-5.88j.44 Xk{0,-5.88-j.44, 0, -0.7-j0.44, 0, -0.7j0.44, 0, -5.88j.44} 4, Determie the 8-Poit IDFT of the Sequece x{5,0,-j,0,,0,j,0} Nov / 04 Sol: N- x/n Σ Xke -jπk/n, 0,,...N-. k0 7 x0/8 Σ Xke -jπk/8, 0,,...7. k0 x0 7 x/8 Σ Xke -jπk/8, 0,,...7. k0 x x/8 Σ Xke -jπk/8, 0,,...7. k0 x0.5 7 x3/8 Σ Xke -jπk/8, 0,,...7. k0 x x4/8 Σ Xke -jπk/8, 0,,...7. k0 x4 7 x5/8 Σ Xke -jπk/8, 0,,...7. k0 x x6/8 Σ Xke -jπk/8, 0,,...7. k0 x MUTHUKUMAR.A 04 05

206 DIGITAL SIGNAL PROCESSING IV YEAR / VII SEM CSE 7 x7/8 Σ Xke -jπk/8, 0,,...7. k0 x7 0.5 x{,0.75, 0.5, 0.5,,0.75,0.5,0.5} 5. Derive ad draw the radix - DIT algorithms for FFT of 8 poits. 6 DEC 009 RADIX- FFT ALGORITHMS The /V-poit DFT of a fv-poit sequece J is jv-l rjfl Because r v/i may be either real or complex, evaluatig - A requires o the order of jv complex multiplicatios ad N complex additios for each value of jt. Therefore, because there are Nvalues of Xkh computig a N-poit DFT requires N complex multiplicatios ad additios. The basic strategy thaiis used i the FFT algorithm is oe of "divide ad coquer." which ivolves decomposig a /V-poit DFT ilo successively smaller DFTs. To see how this works, suppose lhat the legth of is eve i.e,,n is divisible by.if jr is decimated ito two sequeces of legth N/, computig the jv/-poit DFT of each of these sequeces requires approximatelymultiplicatios ad thesame umber of additios, Thus, ihe two DFTs require iv/ ^ /V multiplies ad adds, Therefore, if it is possible to fid the fy-poim DFT of jth from these two /V/-poit DFTs i fewer iha N foperatios, a savigs has bee realied. Deetmatio-t-Tlme FFT The decimatio'i-time FFT algorilhm is based o splittig decimatig jt{a ito smaller sequeces ad fidig Xkfrom the DFTsof these decimated sequeces This sectio describes how this decimatio leads to a efficiet algorithm whe the sequece legth is a power of. Lei.r be a sequece of legth N ', ad suppose thal v is splil decimated ito iwo subsequeces, each of legth N/.As illustrated i Fig. the first sequece, %.is formed frum the cve-idex tcrmx, g x 0. I N - I ad ihe secod, hri, is formed from ihe odd-idex terms, I terms of these sequeces, the JV-poit DFT of is N - I h{ x 0. I - N- I 90 MUTHUKUMAR.A 04 05

207 DIGITAL SIGNAL PROCESSING IV YEAR / VII SEM CSE Xk ^xwtf x{wtf 4 x{wj? Odd-Idex Ttraii Because W$ k y X{t amii 4- w*jwk fl 0 lto Note that the firs! term is the A//-poim DFT of ad ho secod is Ihe N/-poit DFT of h; X{k Gik W l Hk k0,in - I Although the N/-poit DFTs of gad hare sequeces of legth Nt[, the periodicity of the complex expoetials allows us to write G*G*y H*>/m-y Therefore, Xk may be computed from the A '/-poit DFTs Gk ad Hk, Note that because IV* _ U/* LV' v^ IV* IT N iy iv w iv vv iv the IV.*' 4 * Hit j} - W* H [k ad it is oly ecessarv to form the products W#HJt tor k 0,... N/!. The complex expoetials multiplyigti{k are called twiddle factors.a block diagram showig the computatios thaiare ecessaryfor the first stageof a eight-poit dedmatio-i-time FFT is show i Fig. If N/ is eve, #tft adart may agai be decimated. For example, GA may be evaluated as follows: : - ^ i - Gk V V gi«w$ ir I Ji eve J.II Id 9 MUTHUKUMAR.A 04 05

208 DIGITAL SIGNAL PROCESSING IV YEAR / VII SEM CSE 9 MUTHUKUMAR.A 04 05

209 DIGITAL SIGNAL PROCESSING IV YEAR / VII SEM CSE X 0 Xl xm X 3 *4 > xs X 6 X 7 A complete eight-poit radix- decimaliot-iihue FFT. Computig a /V-poit DFT usig a radix- decimatio-i-lime FFT is much more efficiet lhacalculatig the DFTdirectly. For example, if N '\ there are log, X v stagesof compulatio. Bee a use each stage requires N/complex multiplies by the twiddle factors W r Nad Ncomplex additios, there are a total of \Xlog ; N complex multiplicatios ad Xlog-i Ncomplex additios. From the structure of the dec i mat io-i-time FFT algorithm, ote that oce a butterfly operatio has bee performed o a pair of complex umbers, there is o eed to save the iput pair. Therefore, ihe output pair may be stored i he same registers as the iput. Thus, oly oe airay of sie Xis required, ad il is said that the computatios may be performed i place.to perform the compulatios i place, however, the iput sequece ah must he stored or accessed i osequetial order as see i Fig. The shufflig of the ipul sequece that takes place is due to ihe successive decimatios of The orderig that results correspods to a bil-reversed idexig of Ihe origial sequece. I oilier words, if the idex is writte i biary fo. Ihe order i which i the ipul sequece musi be accessed is foud by readig the biary represetatio for /? I reverse order as illustrated i ihe table below for N 8; 93 MUTHUKUMAR.A 04 05

210 DIGITAL SIGNAL PROCESSING IV YEAR / VII SEM CSE Biary Bil-Reversed Biary Oil Alterate forms FFFT algorithms may he derived from the decimalio-i-time FFT hy maipulatig the flowgrapli ad rearragig the order i which ihe results of each stage of Ihe computatio are stored. For example, the txjes of the fluwgraph may be rearraged su that the iput sequeuc,r/i is i ormal order. What is lost with this reorderig, however, is the.ibililv iti perform the computatios i place. 6. Derive DIF radix FFT algorithm Decimatio-i-Frequecy FFT Aother class of FFT algorithms may be derived by decimatig the output sequece X k ito smaller ad smaller subsequeces. These algorithms are called decimatio-i-frequecyffts ad may be derived as follows. Let N be a power of, /V '. ad cosider separately evaluatig the eve-idex ad odd-idcx samples of Xk. The eve samples are fi~ I Xk fl0 Separatig this sum ito the first N / poits ad the last N / poits, ad usig the fact that Wjf k W^, this becomes S-i ftf-r Xk ' *W*/ xw* N k, 0 A// With a chage i the idexig o the secod sum we have Xk J x{w N/ x ll 7^/^ l 0 > ^ Fially, because ^" 5 ** W'Jv/ 94 MUTHUKUMAR.A 04 05

211 DIGITAL SIGNAL PROCESSING IV YEAR / VII SEM CSE X* fn\] x K. L H k N / which is the N/-poit DFT of the sequece that is formed by addig the first N/ poits of.* to the last N/. Proceedig i the same way for the odd samples of Xk leads to A completeeight-poit decimatio-i- frequecy FFT is show i Fig The complexity of the decimatio-i-frequecy FFT is the same as the decimatio-i-time, ad the computatios may be performed i place. Fially, ote that although the iput sequece x is i ormal order, the frequecy samples Xk are i bit-reversed order. 95 MUTHUKUMAR.A 04 05

212 DIGITAL SIGNAL PROCESSING IV YEAR / VII SEM CSE 7, State ad prove properties of DFT. PROPERTIES OF DFT DFT x N. Periodicity Let x ad xk be the DFT pair the if x N x XkN Xk Thus periodic sequece xp ca be give as ra xp Z x-ln l-ra xk for all the for all k 96 MUTHUKUMAR.A 04 05

213 DIGITAL SIGNAL PROCESSING IV YEAR / VII SEM CSE N DFT x * ^ Xk The N The a x a x DFT ^^ a Xk a Xk N DFT of liear combiatio of two or more sigals is equal to the same liear combiatio of DFT of idividual sigals.. Circular Symmetries of a sequece A A sequece is said to be circularly eve if it is symmetric about the poit ero o the circle. Thus XN x B A sequece is said to be circularly odd if it is ati symmetric about the poit ero o the circle. Thus XN- - x C A circularly folded sequece is represeted as x- N ad give by x- N xn-. D Aticlockwise directio gives delayed sequece ad clockwise directio gives advace sequece. Thus delayed or advaces sequece x' is related to x by the circular shift.. Symmetry Property of a sequece A Symmetry property for real valued x i.e xi0 This property states that if x is real the XN-k X kx-k B Real ad eve sequece x i.e xi0 & XiK0 This property states that if the sequece is real ad eve x xn- the DFT becomes N- Xk I x cos k/n 0 C Real ad odd sequece x i.e xi0 & XrK0 This property states that if the sequece is real ad odd x-xn- the DFT becomes N- Xk -j I x si k/n 0 D Pure Imagiary x i.e x R 0 This property states that if the sequece is purely imagiary xj Xi the DFT becomes N- XRk I xi si k/n 0 d x i. 97 MUTHUKUMAR.A 04 05

214 DIGITAL SIGNAL PROCESSING IV YEAR / VII SEM CSE N- Xik I xi cos k/n 0 3. Circular Covolutio The Circular Covolutio property states that if DFT x ^ Xk Ad N DFT x Xk The N DFT The x x -^k xk It meas that circular N covolutio of x & x is equal to multiplicatio of their DFT's. Thus circular covolutio of two periodic discrete sigal with period N is give by N- ym I x x m-n Multiplicatio of two sequeces i time domai is called as Liear covolutio while Multiplicatio of two sequeces i frequecy domai is called as circular covolutio. Results of both are totally differet but are related with each other. UNIT III IIR FILTER DESIGN.Give the expressio for locatio of poles of ormalied Butterworth filtermay07,nov0 The poles of the Butterworth filter is give by S-SS- S..S-SN Where N is the order of filter..what are the parametersspecificatios of a Chebyshev filter?may/jue-07 From the give chebyshev filter specificatios we ca obtai the parameters like the order of the filter N, ε, trasitio ratio k, ad the poles of the filter. 3.Fid the digital trasfer fuctio H by usig impulse ivariat method for the aalog trasfer fuctio Hs/s.Assume T0.5secMay/jue-07 H / -e What is Warpig effect?nov-07,may-09 The relatio betwee the aalog ad digital frequecies i ω biliear trasformatio is give byω ta. For smaller T 98 MUTHUKUMAR.A 04 05

215 values of ω there exist liear relatioship betwee ω ad Ω. But for large values of ω the relatioship is o-liear. This o-liearity itroduces distortio i the frequecy axis. This is kow as warpig effect. This effect compresses the magitude ad phase respose at high frequecies. 5.Compare Butterworth & Chebyshev filter.nov-07,may-09 S.o Butterworth Chebyshev Type- All pole desig All pole desig The poles lie o a circle i S-plae. 3 The magitude respose is maximally flat at the origi ad mootoically decreasig fuctio of Ω 4 The ormalied magitude respose has a value of / at the cutoff frequecy Ωc 5 Oly few parameters has to be calculated to determie the trasfer fuctio The poles lie o ellipse i S-plae. The magitude respose is equiriple i passbad ad mootoically decreasig i the stopbad. The ormalied magitude respose has a value of / ε at the cutoff frequecy Ωc. A large umber of parameter has to be calculated to determie the trasfer fuctio 6.What is the relatioship betwee aalog & digital freq. i impulse ivariat trasformatio?apr/may-08 If Hs Σ Ck / S-Pk the H Σ Ck / -e PkT Z - 7. What is biliear trasformatio?nov/dec-08 The biliear trasformatio is a mappig that trasforms the left half of s plae ito the uit circle i the -plae oly oce, thus avoidig aliasig of frequecy compoets. The mappig from the s- plae to the -plae i biliear trasformatio is s. T 8.What is the mai disadvatage of direct form-i realiatio?nov/dec-08 The direct form realiatio is extremely sesitive to parameter quatiatio. Whe the order of the system N is large, a small chage i a filter coefficiet due to parameter quatiatio, results i a large chage i the locatio of the pole ad eros of the system.

216 9.What is Prewarpig?May-09,Apr-0,May- The effect of the o-liear compressio at high frequecies ca be compesated. Whe the desired magitude respose is piece-wise costat over frequecy, this compressio ca be compesated by itroducig a suitable prescalig, or prewarpig the critical frequecies by usig the formula, Ω/T ta ω/. 0.List the features that make a aalog to digital mappig for IIR filter desig coefficiet.may/jue-00 The biliear trasformatio provides oe-to-oe mappig. Stable cotiuous systems ca be mapped ito realiable, stable digital systems. There is o aliasig..state the limitatios of impulse ivariace mappig techique.nov-09 I impulse ivariat method, the mappig from s-plae to - plae is may to oe i.e., all the poles i the s-plae betwee the itervals [k-π]/t to [kπ]/t for k0,, map ito the etire -plae. Thus, there are a ifiite umber of poles that map to the same locatio i the -plae, producig aliasig effect. Due to spectrum aliasig the impulse ivariace method is iappropriate for desigig high pass filters. That is why the impulse ivariace method is ot preferred i the desig of IIR filter other tha low pass filters..fid digital trasfer fuctio usig approximate derivative techique for the aalog trasfer fuctio Hs/s3.Assume T0.sec May-0 H / Ze Give the square magitude fuctio of Butterworth filter.nov-00 The magitude fuctio of the butter worth filter is give by H j Ω N,,3,... N Ω Ω c Where N is the order of the filter ad Ω c is the cutoff frequecy. The magitude respose of the butter worth filter closely approximates the ideal respose as the order N

217 icreases. The phase respose becomes more o-liear as N icreases. 4. Fid the digital trasfer fuctio H by usig impulse ivariat method for the aalog trasfer fuctio Hs/s.Assume Tsec.Apr- H / -e Give the equatio for the order of N ad cut-off frequecy Ωc of butter worth filter.nov-06 0.α s 0 log 0. α p The order of the filter N 0 Ωs log Ω Where α stop bad atteuatio at stop bad frequecy Ω s s α pass bad atteuatio at pass bad frequecy Ω p p Ω Ω c p N 0.α 0 6.What are the properties of the biliear trasformatio?apr- 06 The mappig for the biliear trasformatio is a oe-tooe mappig; that is for every poit, there is exactly oe correspodig poit s, ad vice versa. The jω-axis maps o to the uit circle, the left half of the s-plae maps to the iterior of the uit circle ad the right half of the s-plae maps o to the exterior of the uit circle. 7.Write a short ote o pre-warpig.apr-06 The effect of the o-liear compressio at high frequecies ca be compesated. Whe the desired magitude respose is piece-wise costat over frequecy, this compressio ca be compesated by itroducig a suitable pre-scalig, or pre-warpig the critical frequecies by usig the formula. Ω ta ω T 8.What are the differet types of structure for realiatio of IIR systems?nov-05 The differet types of structures for realiatio of IIR system are p

218 i. Direct-form-I structure ii. Direct-form-II structure iii. Trasposed direct-form II structure iv. Cascade form structure v. Parallel form structure vi. Lattice-Ladder structure 9.Draw the geeral realiatio structure i direct-form I of IIR system.may-04 0.Give direct form II structure.apr-05.draw the parallel form structure of IIR filter.may-06.metio ay two techiques for digitiig the trasfer fuctio of a aalog filter. Nov

219 The two techiques available for digitiig the aalog filter trasfer fuctio are Impulse ivariat trasformatio ad Biliear trasformatio. 3.Write a brief otes o the desig of IIR filter. Or how a digital IIR filter is desiged? For desigig a digital IIR filter, first a equivalet aalog filter is desiged usig ay oe of the approximatio techique for the give specificatios. The result of the aalog filter desig will be a aalog filter trasfer fuctio Has. The aalog filter trasfer fuctio is trasformed to digital filter trasfer fuctio H usig either Biliear or Impulse ivariat trasformatio. 4.Defie a IIR filter The filters desiged by cosiderig all the ifiite samples of impulse respose are called IIR filers. The impulse respose is obtaied by takig iverse Fourier trasform of ideal frequecy respose. 5. Compare IIR ad FIR filters. S.No IIR Filter FIR Filter i. All the ifiite samples of impulse Oly N samples of impulse respose are cosidered. respose are cosidered. ii. The impulse respose caot be The impulse respose ca be directly coverted to digital filter directly coverted to digital filter trasfer fuctio. trasfer fuctio. iii. iv. The desig ivolves desig of aalog filter ad the trasformig aalog filter to digital filter. The specificatios iclude the desired characteristics for magitude respose oly. v. Liear phase characteristics caot be achieved. The digital filter ca be directly desiged to achieve the desired specificatios. The specificatios iclude the desired characteristics for both magitude ad phase respose. Liear phase filters ca be easily desiged. PART- B6 MARKS. Explai Frequecy Trasformatio i Aalog domai ad frequecy trasformatio i digital domai. Nov/Dec-07 i. Frequecy trasformatio i aalog domai

220 I this trasformatio techique ormalied Low Pass filter with cutoff freq of Ω p rad /sec is desiged ad all other types of filters are obtaied from this prototype. For example, ormalied LPF is trasformed to LPF of specific cutoff freq by the followig trasformatio formula, Normalied LPF to LPF of specific cutoff: s s ' Ω p Ω p H s H p s ' Ω p Where, Ω p ormalied cutoff freq rad/sec Ω p Desired LP cutoff freq at Ω Ω p it is Hj The other trasformatios are show i the below table. ii. Frequecy Trasformatio i Digital Domai: This trasformatio ivolves replacig the variable Z - by a ratioal fuctio g -, while doig this followig properties eed to be satisfied:. Mappig Z - to g - must map poits iside the uit circle i the Z- plae oto the uit circle of -

221 plae to preserve causality of the filter.. For stable filter, the iside of the uit circle of the Z - plae must map oto the iside of the uit circle of the -plae. The geeral form of the fuctio g. that satisfy the above requiremets of " all-pass " type is The differet trasformatios are show i the below table.

222 . Represets the trasfer fuctio of a low-pass filter ot butterworth with a pass-bad of rad/sec. Use freq trasformatio to fid the trasfer fuctio of the followig filters: Let Hs s s Represets the trasfer fuctio of a low pass filter ot butterworth with a passbad of rad/sec. Use freq trasformatio to fid the trasfer fuctio of the followig filters: Apr/May-08. A LP filter with a pass bad of 0 rad/sec. A HP filter with a cutoff freq of rad/sec 3. A HP filter with a cutoff freq of 0 rad/sec 4. A BP filter with a pass bad of 0 rad/sec ad a corer freq of 00 rad/sec 5. A BS filter with a stop bad of rad/sec ad a ceter freq of 0 rad/sec

223 Solutio: Give a. LP LP Trasform replace b. LP HPormalied Trasform c. LP HPspecified cutoff Trasform replace d. LP BP Trasform replace e. LP BS Trasform replace s s s H Ω s s s s s H s H sub s s s s s a p Ω s s s s s s H s H sub s s s s s a u Ω s s s s s s H s H sub s s s s s a u Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω s s s s s s H s H sub B ad where sb s s s s s s s a l u o l u o o l u l u Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω s s s s s s H s H sub B ad where s sb s s s s s s a l u o l u o o l u l u

224 3. Covert sigle pole LP Bufferworth filter with system 0.45 fuctio H ito BPF with upper & lower cutoff frequecy ω u & ωl respectively, The LPF has 3-dB badwidth ω 0. π Nov-07 8 p. Solutio: We have the trasformatio formula give by, applyig this to the give trasfer fuctio, Note that the resultig filter has eros at ± ad a pair of poles that deped o the choice of ωl ad ωu This filter has poles at ±j0.73 ad hece resoates at ωπ/ The followig observatios are made, It is show here that how easy to covert oe form of filter desig to aother form.

225 What we require is oly prototype low pass filter desig steps to trasform to ay other form. 4. Explai the differet structures for IIR Filters. May-09 6 The IIR filters are represeted by system fuctio; M k HZ b k 0 N k a k k k Ad correspodig differece equatio give by, y N k a y k k N k 0 b x k Differet realiatios for IIR filters are, k. Direct form-i. Direct form-ii 3. Cascade form 4. Parallel form 5. Lattice form Direct form-i This is a straight forward implemetatio of differece equatio which is very simple. Typical Direct form I realiatio is show below. The upper brach is forward path ad lower brach is feedback path. The umber of delays depeds o presece of most previous iput ad output samples i the differece equatio. Direct form-ii The give trasfer fuctio H ca be expressed as,

226 . V Y X V X Y H where V is a itermediate term. We idetify, N k k a k X V all poles M k k b k V Y all eros The correspodig differece equatios are, N k k k v a x v M k k v b v y

227 This realiatio requires MN! multiplicatios, MN additio ad the maximum of {M, N} memory locatio Cascade Form The trasfer fuctio of a system ca be expressed as, H H H... H k Where H k Z could be first order or secod order sectio realied i Direct form II form i.e., bk 0 bkz bk Z H k Z akz ak Z Where K is the iteger part of N/ Similar to FIR cascade realiatio, the parameter b 0 ca be distributed equally amog the k filter sectio B 0 that b 0 b 0 b 0..b k0. The secod order sectios are required to realie sectio which has complex-cojugate poles with real coefficiets. Pairig the two complex-cojugate poles with a pair of complex-cojugate eros or real-valued eros to form a subsystem of the type show above is doe arbitrarily. There is o specific rule used i the combiatio. Although all cascade realiatios are equivalet for ifiite precisio

228 arithmetic, the various realiatios may differ sigificatly whe implemeted with fiite precisio arithmetic. Parallel form structure I the expressio of trasfer fuctio, if N M we ca express system fuctio N N Ak H Z C C H k Z k pk Z k Where {p k } are the poles, {A k } are the coefficiets i the partial fractio expasio, ad the costat C is defied as C b N a N, The system realiatio of above form is show below. bk 0 bkz Where H k Z akz ak Z Oce agai choice of usig first- order or secod-order sectios depeds o poles of the deomiator polyomial. If there are complex set of poles which are cojugative i ature the a secod order sectio is a must to have real coefficiets. Lattice Structure for IIR System: Cosider a All-pole system with system fuctio. H Z N k a N k Z k A Z N

229 The correspodig differece equatio for this IIR system is, OR y N k a x y N k y k x N k a N k y k For N x y a y Which ca realied as, We observe x f y f f k g 0 0 x k y k a g k f 0 g 0 k y y For N, the y x a y a y This output ca be obtaied from a two-stage lattice filter as show i below fig

230 f g f g f x f k g k f g f k g 0 0 k f 0 g 0 y 0 f k f g 0 g k f k 0 g g [ k f 0 g 0 ] kg k [ k y y ] k y f k 0 x x k k y k y Similarly g k y k k y y We observe a 0 ; a k k ; a k N-stage IIR filter realied i lattice structure is, 0

231 x f N g k f f m m m m mn, N-,--- g f k g m m m m mn, N-, g f y 5. Realie the followig system fuctios Dec Z j Z j Z Z Z Z Z Z H a. Direct form I b. Direct form-ii c. Cascade d. Parallel Solutio: Z j Z j Z Z Z Z Z Z H Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z H

232 H a. Direct form-i c. Cascade Form H H H Where H

233 0 H Parallel Form H H H ; H Obtai the direct form I, direct form-ii, Cascade ad parallel form realiatio for the followig system, y- 0. y-0.y-3x3.6 x-0.6 x-. May/jue-07 Solutio: The Direct form realiatio is doe directly from the give i/p o/p equatio, show i below diagram

234 Direct form II realiatio Takig ZT o both sides ad fidig H Y H X Cascade form realiatio The trasformer fuctio ca be expressed as: H which ca be re writte as where H ad H Parallel Form realiatio The trasfer fuctio ca be expressed as H C H H where H & H is give by,

235 H Covert the followig pole-ero IIR filter ito a lattice ladder structure,apr-0 3 Z Z Z H Z Z 8 Z 3 Z Solutio: 3 Give b M Z Z Z Z Ad Z Z Z Z a A N ; a3 ; a 4 3 ; a 8 33 k 3 a3 3 3 Usig the equatio am k am m am m k am k a m m For m3, k 3 5 a3 a3 3 a a a3 3 3 For m3, & k a3 a33 a3 a k a for m, & k a a a a k a 3 3 8

236 for lattice structure k, k 4, k 3 3 For ladder structure M3 C m C b 3 m. 3 M i m C. a m mm, M-,,0 b ; C b C a b ca i m i [ ca c3a3 ] 3 5 [.4583 ] C m b c b0 ca i m i [ ca ca c3 33 ] [ ] 0695 b a 4 To covert a lattice- ladder form ito a direct form, we fid a equatio to obtai a N k From k m m,, N the equatio for c m is recursively used to computeb m m0,,, M. 3

237 UNIT-IV FIR FILTER DESIGN.What is the coditio satisfied by liear phase FIR filter?may/jue-09 The coditio for costat phase delay are Phase delay, α N-/ i.e., phase delay is costat Impulse respose, h hn-- i.e., impulse respose is symmetric.what are the desirable characteristics of the frequecy respose of widow fuctio?nov-07,08ov-0 Advatages:. FIR filters have exact liear phase.. FIR filters are always stable. 3. FIR filters ca be realied i both recursive ad o recursive structure. 4. Filters with ay arbitrary magitude respose ca be tackled usig FIR sequecy. Disadvatages:. For the same filter specificatios the order of FIR filter desig ca be as high as 5 to 0 times that of IIR desig.. Large storage requiremets eeded. 3.Powerful computatioal facilities required for the implemetatio. 3.What is meat by optimum equiripple desig criterio.nov-07

238 The Optimum Equiripple desig Criterio is used for desigig FIR Filters with Equal level filteratio throughout the Desig. 4.What are the merits ad demerits of FIR filters?ap/may-08 Merits:. FIR Filter is always stable.. FIR Filter with exactly liear phase ca easily be desiged. Demerits:. High Cost..Require more Memory. 5.For what type of filters frequecy samplig method is suitable?ov/dec-08 FIR FIlters 6.What are the properties of FIR filters?nov/dec-09. FIR Filter is always stable.. A Realiable filter ca always be obtaied. 7.What is kow as Gibbs pheomeo?ap/may-0, I the filter desig by Fourier series method the ifiite duratio impulse respose is trucated to fiite duratio impulse respose at N-/. The abrupt trucatio of impulse itroduces oscillatios i the pass bad ad stop bad. This effect is kow as Gibb s pheomeo. 8.Metio various methods available for the desig of FIR filter.also list a few widow for the desig of FIR filters.may/jue-00 There are three well kow method of desig techique for liear phase FIR filter. They are. Fourier series method ad widow method. Frequecy samplig method 3. Optimal filter desig methods. Widows: i.rectagular ii.hammig iii.haig iv.blackma v.kaiser 9.List ay two advatages of FIR filters.nov/dec-0. FIR filters have exact liear phase.. FIR filters are always stable.

239 3. FIR filters ca be realied i both recursive ad o recursive structure. 0.Metio some realiatio methods available to realie FIR filternov/dec-0 i.direct form. ii.cascade form iii.liear phase realiatio..metio some desig methods available to desig FIR filter.nov/dec-0 There are three well kow method of desig techique for liear phase FIR filter. They are. Fourier series method ad widow method. Frequecy samplig method 3. Optimal filter desig methods. Widows: i.rectagular ii.hammig iii.haig iv.blackma v.kaiser.what are FIR filters?ov/dec-0 The specificatios of the desired filter will be give i terms of ideal frequecy respose Hdω. The impulse respose hd of desired filter ca be obtaied by iverse Fourier trasform of hdω, which cosists of ifiite samples. The filters desiged by selectig fiite umber of samples of impulse respose are called FIR filters. 3.What are the coditios to be satisfied for costat phase delay i liear phase FIR filter?apr-06 The coditio for costat phase delay are Phase delay, α N-/ i.e., phase delay is costat Impulse respose, h hn-- i.e., impulse respose is symmetric 4.What is the reaso that FIR filter is always stable?ov-05 FIR filter is always stable because all its poles are at the origi. 5.What are the possible types of impulse respose for liear phase FIR filter?nov- There are four types of impulse respose for liear phase FIR filters. Symmetric impulse respose whe N is odd.. Symmetric impulse respose whe N is eve. 3. Atisymmetric impulse respose whe N is odd 4. Atisymmetric impulse respose whe is eve. 6.Write the procedure for desigig FIR filter usig widow.may-06. Choose the desired frequecy respose of the filter Hd.

240 . Take iverse Fourier trasform of Hd to obtai the desired impulse respose hd. 3.Choose a widow sequece w ad multiply hd by w to covert the ifiite duratio impulse respose to fiite duratio impulse respose h. 4. The Trasfer fuctio H of the filter is obtaied by takig -trasform of h. 7.Write the procedure for FIR filter desig by frequecy samplig method.may-05. Choose the desired frequecy respose Hd.. Take N-samples of Hd to geerate the sequece H K Here H bar of k should come 3. Take iverse of DFT of H k to get the impulse respose h. 4. The trasfer fuctio H of the filter is obtaied by takig -trasform of impulse respose. 8.List the characteristic of FIR filter desiged usig widow.ov-04. The width of the trasitio bad depeds o the type of widow.. The width of the trasitio bad ca be made arrow by icreasig the value of N where N is the legth of the widow sequece. 3. The atteuatio i the stop bad is fixed for a give widow, except i case of Kaiser Widow where it is variable. 9.List the features of haig widow spectrum.ov-04. The mailobe width is equal to 8π/N.. The maximum sidelobe magitude is -3db. 3. The sidelobe magitude decreases with icreasig. 0. Compare the rectagular widow ad haig widow.dec- 07

241 Rectagular widow. The width of mailobe i widow spectrum is 4π/N..The maximum sidelobe magitude i widow spectrum is -3db 3. I widow spectrum the sidelobe magitude slightly decreases with icreasig. 4. I FIR filter desiged usig rectagular widow the miimum stopbad atteuatio is db. Hammig widow. The width of mailobe i widow spectrum is 8π/N..The maximum sidelobe magitude i widow spectrum is -4db 3. I widow spectrum the sidelobe magitude remais costat. 4. I FIR filter desiged usig hammig widow the miimum stopbad atteuatio is 5db..Compare Hammig widow with Kaiser Widow.Nov-06 Hammig widow Kaiser widow.the mai lobe width is equal to8π/n The mai lobe width,the peak side lobe ad the peak side lobe level is 4dB. level ca be varied by varyig the parameter α ad N..The low pass FIR filter desiged will have first side lobe peak of 53 db The side lobe peak ca be varied by varyig the parameter α. Part B6 MARKS. Desig a ideal high-pass filter with a frequecy respose usig a haig widow with M ad plot the frequecy respose. Nov-07,09 H d e j ω Solutio: 0 π for ω π 4 π ω < 4

242 ] [ 4 / 4 / ω ω π π π ω π π ω d e d e h j j d ] [ 0 0 ] 4 si [si 4 / 4 / ω ω π π π π π π π π d d h ad for h d d h d h d h d h d h d 3 h d h d 4 h d -4 0 h d 5 h d The hammig widow fuctio is give by cos cos 0.5 w N for otherwise M M M w h h π π w h 0 w h w h w h w h w h 3 w h w h 4 w h w h 5 w h -50 h w h h d h[ ]. Desig a filter with a frequecy respose: usig a Haig widow with M 7 π ω π π ω π ω ω < for e e H j j d

243 8Apr/0 8 Solutio: The freq resp is havig a term e jωm-/ which gives h symmetrical about M-/ 3 i.e we get a causal sequece. The Haig widow fuctio values are give by w h 0 w h 6 0 w h w h w h w h w h 3 hh d w h h[ ] 3. Desig a LP FIR filter usig Freq samplig techique havig cutoff freq of π/ rad / sample. The filter should have liear phase ad legth of 7. May-07 The desired respose ca be expressed as / 7 0 π ω ω ω ω ω c ad M with otherwise c for e e H M j j d π ω π π ω ω ω / 0 / 0 8 for for e e H j j d Selectig 6 0,,... 7 k for k M k k π π ω si 4 / 4 / 3 d d d d d d d j j d h h h h h h h gives this d e e h π π ω π ω π π ω

244 H jω k H d e πk ω 7 H k e 0 H k e 0 πk j 8 7 6πk j 7 for for πk π for 0 7 πk π / π 7 7 for 0 k k 4 The rage for k ca be adjusted to be a iteger such as 0 k 4 ad 5 k 8 The freq respose is give by πk j 8 7 H k e for 0 k 4 0 for 5 k 8 Usig these value of Hk we obtai h from the equatio M / jπk / M h H 0 Re H k e M k i.e., 4 j6πk /7 j πk /7 h Re e e 7 k 4 πk 8 h H 0 cos for 0,, k Eve though k varies from 0 to 6 sice we cosidered ω varyig betwee 0 ad π/ oly k values from 0 to 8 are cosidered While fidig h we observe symmetry i h such that varyig 0 to 7 ad 9 to 6 have same set of h 4.Desig a Ideal Differetiator usig a Rectagular widow 6

245 b Hammig widow 6 with legth of the system 7.Nov-09 Solutio: From differetiator frequecy characteristics He jω jω betwee π to π π jω cosπ hd 0 jω e dω ad π π The h d is a add fuctio with h d -h d - ad h d 00 a rectagular widow hh d w r h-h-hd- h-h-hd0.5 h3-h-3hd h h-3 for causal system thus, H ' Also from the M 3 / j M H r e h si equatio ω ω 0 For M7 ad h as foud above we obtai this as jω H e 0.66 si 3ω si ω si ω r jω jω H e jh e j0.66si 3ω si ω siω r b Hammig widow hh d w h where w h is give by π w h cos M 0 otherwise M / M / 3 For the preset problem π w h cos The widow fuctio coefficiets are give by for -3 to

246 -0.067] Wh [ ] Thus h h-5 [0.067, -0.55, 0.77, 0, -0.77, 0.55, Similar to the earlier case of rectagular widow we ca write the freq respose of differetiator as jω jω H e jh e j0.0534si 3ω 0.3si ω.54siω r W i t h rectagular widow, the effect of ripple is more ad trasitio bad width is small compared with hammig widow With hammig widow, effect of ripple is less whereas trasitio bad is more 5.Justify Symmetric ad Ati-symmetric FIR filters givig out Liear Phase characteristics.apr-08 0 Symmetry i filter impulse respose will esure liear phase A FIR filter of legth M with i/p x & o/p y is described by the differece equatio: y b 0 x b x-.b M- x-m- M k k 0 - b x k - Alteratively, it ca be expressed i covolutio form y M k 0 h k x k i.e bk hk, k0,,..m- Filter is also characteried by

247 0 M k k k h H -3 polyomial of degree M- i the variable -. The roots of this polyomial costitute eros of the filter. A FIR filter has liear phase if its uit sample respose satisfies the coditio h ± hm-- 0,,.M- -4 Icorporatig this symmetry & ati symmetry coditio i eq 3 we ca show liear phase characteristics of FIR filters... 0 M M M h M h h h h H If M is odd M M M M M M h M h M h M h M h h h H M M M M M h M h M h M h h h Applyig symmetry coditios for M odd h M h M h M h M h M h M h h M h h ± ± ± ± ±

248 ± ± 0 / / 3 0 / / } { } { M M M M M M M M h H M eve for similarly h M h H 6.What are the structure of FIR filter systems ad explai it. Dec-0 FIR system is described by, 0 M k k k x b y Or equivaletly, the system fuctio 0 M k k b k Z Z H Where we ca idetify otherwise b h 0 0 Differet FIR Structures used i practice are,. Direct form. Cascade form 3. Frequecy-samplig realiatio 4. Lattice realiatio Direct form It is No recursive i structure A s c a be see from the above implemetatio it requires M- memory locatios for storig the M- previous iputs

249 It requires computatioally M multiplicatios ad M- additios per output poit It is more popularly referred to as tapped delay lie or trasversal system Efficiet structure with liear phase characteristics are possible where h ± h M Cascade form The system fuctio H Z is factored ito product of secod order FIR system H Z K k H k Z Where H k Z bk 0 bkz bk Z k,.. K ad K iteger part of M / The filter parameter b 0 may be equally distributed amog the K filter sectio, such that b 0 b 0 b 0. b k0 or it may be assiged to a sigle filter sectio. The eros of H are grouped i pairs to produce the secod order FIR system. Pairs of complex-cojugate roots are formed so that the coefficiets {b ki } are real valued. I case of liear phase FIR filter, the symmetry i h implies that the eros of H also exhibit a form of symmetry. If k ad k* are pair of complex cojugate eros the /k ad /k* are also a pair complex cojugate eros. Thus simplified fourth order sectios are formed. This is show below,

250 4 3 0 * 0 / / * C C C C C H k k k k k k k k k k Frequecy samplig method We ca express system fuctio H i terms of DFT samples Hk which is give by 0 N k k N N W k H N H This form ca be realied with cascade of FIR ad IIR structures. The term - -N is realied as FIR ad the term 0 N k k W N k H N as IIR structure. The realiatio of the above freq samplig form shows ecessity of complex arithmetic. Icorporatig symmetry i h ad symmetry properties of DFT of real sequeces the realiatio ca be modified to have oly real coefficiets.

251 Lattice realiatio Lattice structures offer may iterestig features:. Upgradig filter orders is simple. Oly additioal stages eed to be added istead of redesigig the whole filter ad recalculatig the filter coefficiets.. These filters are computatioally very efficiet tha other filter structures i a filter bak applicatios eg. Wavelet Trasform 3. Lattice filters are less sesitive to fiite word legth effects. Cosider m Y i H am i X i m is the order of the FIR filter ad a m 0 whe m Y/ X a - y x a x- f is kow as upper chael output ad ras lower chael output. f 0 r 0 x

252 The outputs are, y f the a k if b r f k r a r k f f If m r k f y x a x a x y a a X Y Substitutig a ad b i ] si ] ] [ x k x k k k x x k x k k x k x y x r f ce r k f k k r k f r f k k r k f y We recogie k a k k k a Solvig the above equatio we get 4 a k ad a a k Equatio 3 meas that, the lattice structure for a secodorder filter is simply a cascade of two first-order filters with k ad k as defied i eq 4

253 Similar to above, a M th order FIR filter ca be implemeted by lattice structures with M Stages 7.Realie the followig system fuctio usig miimum umber of multiplicatio HZ Z Z Z Z Z 6Dec- Solutio: We recogie h,,,,, M is eve 6, ad we observe h h M-- h h 5- i.e h 0 h 5 h h 4 h h 3 Direct form structure for liear phase FIR ca be realied 3

254 8. Realie the followig usig system fuctio usig miimum umber of multiplicatio. 8May Z Z 4 Z 3 Z Z Z 3 Z 4 HZ Solutio: m9, 4, 3,,, 3, 4, h Odd symmetry h -hm--; h -h 8-; h m-/ h 4 0 h 0 -h 8; h -h 7; h -h 6; h 3 -h 5

255 9. Realie the differece equatio i cascade form May-07 6 y x 0.5x 0.5x 0.75x 3 x 4 Solutio: 3 4 Y X { H 0.5 H.9 H H H _ Give FIR filter H Z Z 3 Z obtai lattice structure for the same 4Apr-08 Solutio: Give a, a 3 Usig the recursive equatio for m M, M-, Here M therefore m,

256 If m k a 3 If m k a Also, whe m ad i a a a 3 Hece k a 3 3 FINITE WORDLENGTH EFFECTS:.Determie Dead bad of the filter.may-07, Dec-07, Dec-09 The limit cycle occur as a result of quatiatio effects i multiplicatios. The amplitudes of output durig a limit cycle are cofied to a rage of values that is called the dead bad of the filter..why roudig is preferred to trucatio i realiig digital filter?may-07 I digital system the product quatiatio is performed by roudig due to the followig desirable characteristics of roudig.. The roudig error is idepedet of the type of arithmetic.. The mea value of roudig error sigal is ero. 3. The variace of the roudig error sigal is least. 3.What is Sub bad codig?may-07 Sub bad codig is a method by which the sigal speech sigal is sub divided i to several frequecy bads ad each bad is digitally ecoded separately. 4.Idetify the various factors which degrade the performace of the digital filter implemetatio whe fiite word legth is used.may-07,may-00 5.What is meat by trucatio & roudig?nov/dec-07 The trucatio is the process of reducig the sie of biary umber by discardig all bits less sigificat tha the least sigificat bit that is retaied.

257 Roudig is the process of reducig the sie of a biary umber to fiite word sies of b-bits such that, the rouded b-bit umber is closest to the origial uquatied umber. 6.What is meat by limit cycle oscillatio i digital filters?may- 07,08,Nov-0 I recursive system whe the iput is ero or some oero costat value, the oliearities due to fiite precisio arithmetic operatio may cause periodic oscillatios i the output. These oscillatios are called limit cycles. 7.What are the three types of quatiatio error occurred i digital systems?apr-08,nov-0 i. Iput quatiatio error ii. Product quatiatio error iii. Coefficiet quatiatio error. 8.What are the advatages of floatig poit arithmeticnov-08 i. Larger dyamic rage. ii. Overflow i floatig poit represetatio is ulikely. 9.Give the IEEE754 stadard for the represetatio of floatig poit umbers.may-09 The IEEE-754 stadard for 3-bit sigle precisio floatig poit umber is give by Floatig poit umber, N f -* ^E-7*M S E M S -bit field for sig of umbers E 8-bit field for expoet M 3-bit field for matissa 0.Compare the fixed poit & floatig poit arithmetic.may/jue-09 Fixed poit arithmetic Floatig poit arithmetic. The accuracy of the result is less due to. The accuracy of the result will be smaller dyamic rage. higher due to larger dyamic rage.. Speed of processig is high.. Speed of processig is low. 3. Hardware implemetatio is cheaper. 3. Hardware implemetatio is costlier. 4. Fixed poit arithmetic ca be used for real; time computatios. 5. quatiatio error occurs oly i multiplicatio 4. Floatig poit arithmetic caot be used for real time computatios 5. Quatiatio error occurs i both multiplicatio ad additio..defie Zero iput limit cycle oscillatio.dec-09

258 I recursive system, the product quatiatio may create periodic oscillatios i the output. These oscillatios are called limit cycles. If the system output eters a limit cycle, it will cotiue to remai i limit cycle eve whe the iput is made ero. Hece these limit cycles are called ero iput limit cycles..what is the effect of quatiatio o pole locatios?apr-00 Quatiatio of coefficiets i digital filters lead to slight chages i their value. These chages i value of filter coefficiets modify the pole-ero locatios. Some times the pole locatios will be chaged i such a way that the system may drive ito istability. 3.What is the eed for sigal scalig?apr-00 To prevet overflow, the sigal level at certai poits i the digital filters must be scaled so that o overflow occurs i the adder. 4.What are the results of trucatio for positive & egative umbers?nov-06 cosider the real umbers , , To trucate these umbers to 4 decimal digits, we oly cosider the 4 digits to the right of the decimal poit. The result would be: 5.634,3.438, Note that i some cases, trucatig would yield the same result as roudig, but trucatio does ot roud up or roud dow the digits; it merely cuts off at the specified digit. The trucatio error ca be twice the maximum error i roudig. 5.What are the differet quatiatio methods?nov-0 The two types of quatiatio are: i. Trucatio ad ii. Roudig. 6. List out some of the fiite word legth effects i digital filter.apr-06. Errors due to quatiatio of iput data.. Errors due to quatiatio of filter coefficiets. 3. Errors due to roudig the product i multiplicatios. 4. Limit cycles due to product quatiatio ad overflow i additio. 7. What are the differet formats of fixed poit s represetatio?may-05 I fixed poit represetatio, there are three differet formats for represetig biary umbers.. Siged-magitude format. Oe s-complemet format 3. Two s-complemet format.

259 I all the three formats, the positive umber is same but they differ oly i represetig egative umbers. 8. Explai the floatig poit represetatio of biary umber.dec-06 The floatig umbers will have a matissa part ad expoet part. I a give word sie the bits allotted for matissa ad expoet are fixed. The matissa is used to represet a biary fractio umber ad the expoet is a positive or egative biary iteger. The value of the expoet ca be adjusted to move the positio of biary poit i matissa. Hece this represetatio is called floatig poit. The floatig poit umber ca be expressed as, Floatig poit umber, N f M*^E Where M Matissa ad EExpoet. 9. What is quatiatio step sie?apr-07, I digital system, the umber is represeted i biary. With b- bit biary we ca geerate ^b differet biary codes. Ay rage of aalog value to be represeted i biary should be divided ito ^b levels with equal icremet. The ^b level are called quatiatio levels ad the icremet i each level is called quatiatio step sie. It R is the rage of aalog sigal the. Quatiatio step sie, q R/^b 0. What is meat by product quatiatio error?nov- I digital computatio, the output of multipliers i.e., the products is quatied to fiite word legth i order to store them i register ad to be used i subsequet calculatio. The error due to the quatiatio of the output of multipliers is referred to as product quatiatio error.. What is overflow limit cycle?may/jue-0 I fixed poit additio the overflow occurs whe the sum exceeds the fiite word legth of the register used to store the sum. The overflow i additio may be lead to oscillatio i the output which is called overflow limit cycle. 3. What is iput quatiatio error?nov-04 The filter coefficiets are computed to ifiite precisio i theory. But i digital computatio the filter coefficiets are represeted i biary ad are stored i registers. If a b bit register is used the filter coefficiets must be rouded or trucated to b bits, which produces a error. 4.What are the differet types of umber represetatio?apr- There are three forms to represet umbers i digital computer or ay other digital hardware. i. Fixed poit represetatio

260 ii. Floatig poit represetatio iii. Block floatig poit represetatio. 5. Defie white oise?dec-06 A statioary radom process is said to be white oise if its power desity spectrum is costat. Hece the white oise has flat frequecy respose spectrum. 8. What are the methods used to prevet overflow?may-05 There are two methods to prevet overflow i saturatio arithmetic ii scalig 9. What is meat by A/D coversio oise?mu-04 A DSP cotais a device, A/D coverter that operates o the aalog iput xt to produce xqt which is biary sequece of 0s ad s. At first the sigal xt is sampled at regular itervals to produce a sequece x is of ifiite precisio. Each sample x is expressed i terms of a fiite umber of bits give the sequece xq. The differece sigal e xq -x is called A/D coversio oise. 6 MARKS, Explai about Limit Cycle Oscillatios. Limit Cycle Oscillatios: A limit cycle, sometimes referred to as a multiplier roudoff limit cycle, is a low-level oscillatio that ca exist i a otherwise stable filter as a result of the oliearity associated with roudig or trucatig iteral filter calculatios []. Limit cycles require recursio to exist ad do ot occur i orecursive FIR filters. As a example of a limit cycle, cosider the secod-order filter realied by 7 5 y Qr{ ^y 8y x

261 where Q r {} represets quatiatio by roudig. This is stable filter with poles at ± j Cosider the implemetatio of this filter with 4-b 3-b ad a sig bit two s complemet fixed-poit arithmetic, ero iitial coditios y y 0, ad a iput sequece x S, where S is the uit impulse or uit sample. The followig sequece is obtaied;

262 Notice that while the iput is ero except for the first sample, the output oscillates with amplitude /8 ad period 6. Limit cycles are primarily of cocer i fixed-poit recursive filters. As log as floatig-poit Notesegie.com filters Powered are by realied techoscript.com as the parallel or cascade coectio of first- ad secod-order subfilters, limit cycles will geerally ot be a problem sice limit cycles are practically ot observable i first- ad secod-order systems implemeted with 3-b floatig-poit arithmetic []. It has bee show that such systems must have a extremely small margi of stability for limit cycles to exist at aythig other tha uderflow levels, which are at a amplitude of less tha 0 38 []. There are at least three ways of dealig with limit cycles whe fixedpoit arithmetic is used. Oe is to determie a boud o the maximum limit cycle amplitude, expressed as a itegral umber of quatiatio steps [3]. It is the possible to choose a word legth that makes the limit cycle amplitude acceptably low. Alterately, limit cycles ca be preveted by radomly roudig calculatios up or dow [4]. However, this approach is complicated to implemet. The third approach is to properly choose the filter realiatio structure ad the quatie the filter calculatios usig magitude trucatio [5,6]. This approach has the disadvatage of producig more roudoff oise tha trucatio or roudig [see ]. 5,Explai about Overflow Oscillatios. With fixed-poit arithmetic it is possible for filter calculatios to overflow. This happes whe two umbers of the same sig add to give a value havig magitude greater tha oe. Sice umbers with magitude greater tha oe are ot represetable, the result overflows. For example, the two s complemet umbers 0.0 5/8 ad /8 add to give.00 which is the two s complemet represetatio of -7/8. The overflow characteristic of two s complemet arithmetic ca be represeted as R{} where X - X> For the example just cosidered, R{X} R{9/8} X 7/8. - < X < 3.7 A overflow oscillatio, sometimes X also referred X <- to as a adder overflow limit cycle, is a high- level oscillatio that ca exist i a otherwise stable fixed-poit filter due to the gross oliearity associated with the overflow of iteral filter calculatios [7]. Like limit cycles, overflow oscillatios require recursio to exist ad do ot occur i orecursive FIR filters. Overflow oscillatios also do ot occur with floatig-poit arithmetic due to the virtual impossibility of overflow. [Type text]

263 As a example of a overflow oscillatio, oce agai cosider the filter of 3.69 with 4-b fixed-poit y4 two s Qr R complemet Qr arithmetic ad with the two s 3.74 complemet overflow characteristic of 3.7: 75 y Qr\R 8y - - 8y - x 3.7 I this case we apply the iput 35 x -4 & - ^& - 5 5,, , 0, 6 8 s to scale the filter calculatios so as to reder overflow impossible. However, this may uacceptably restrict the filter dyamic rage. Aother method is to force completed sums-of- products to saturate at ±, rather tha overflowig [8,9]. It is importat to saturate oly the completed sum, sice itermediate overflows i two s complemet arithmetic do ot affect the accuracy of the fial result. Most fixed-poit digital sigal processors provide for automatic saturatio of completed sums if their saturatio arithmetic feature is eabled. Yet aother way to avoid overflow oscillatios is to use a filter structure for which ay iteral filter trasiet is guarateed to decay to ero [0]. Such structures are desirable ayway, sice they ted to have low roudoff oise ad be isesitive to coefficiet quatiatio [].

264 6, Explai about Coefficiet Quatiatio Error. y4 Qr R Qr 3.74 Coefficiet Quatiatio Error: Re Z FIGURE: Realiable pole locatios for the differece equatio of The sparseess of realiable pole locatios ear ± will result i a large coefficiet quatiatio error for poles i this regio.

265 Figure3.4 gives a alterative structure to 3.77 for realiig the trasfer fuctio of Notice that quatiig the coefficiets of this structure correspods to quatiig X r ad Xi. As show i Fig.3.5 from [5], this results i a uiform grid of realiable pole locatios. Therefore, large coefficiet quatiatio errors are avoided for all pole locatios. It is well established that filter structures with low roudoff oise ted to be robust to coefficiet quatiatio, ad visa versa []- [4]. For this reaso, the uiform grid structure of Fig.3.4 is also popular because of its low roudoff oise. Likewise, the low-oise realiatios of [7]- [0] ca be expected to be relatively isesitive to coefficiet quatiatio, ad digital wave filters ad lattice filters that are derived from low-sesitivity aalog structures ted to have ot oly low coefficiet sesitivity, but also low roudoff oise [5,6]. It is well kow that i a high-order polyomial with clustered roots, the root locatio is a very sesitive fuctio of the polyomial coefficiets. Therefore, filter poles ad eros ca be much more accurately cotrolled if higher order filters are realied by breakig them up ito the parallel or cascade coectio of first- ad secod-order subfilters. Oe exceptio to this rule is the case of liear-phase FIR filters i which the symmetry of the polyomial coefficiets ad the spacig of the filter eros aroud the uit circle usually permits a acceptable direct realiatio usig the covolutio summatio. Give a filter structure it is ecessary to assig the ideal pole ad ero locatios to the realiable locatios. This is geerally doe by simplyroudig or trucatigthe filter coefficiets to the available umber of bits, or by assigig the ideal pole ad ero locatios to the earest realiable locatios. A more complicated alterative is to cosider the origial filter desig problem as a problem i discrete FIGURE 3.4: Alterate realiatio structure.

266 optimiatio, ad choose the realiable pole ad ero locatios that give the best approximatio to the desired filter respose [7]- [30]..6 Realiatio Cosideratios Liear-phase FIR digital filters ca geerally be implemeted with acceptable coefficiet quatiatio sesitivity usig the direct covolutio sum method. Whe implemeted i this way o a digital sigal processor, fixed-poit arithmetic is ot oly acceptable but may actually be preferable to floatig-poit arithmetic. Virtually all fixed-poit digital sigal processors accumulate a sum of products i a doublelegth accumulator. This meas that oly a sigle quatiatio is ecessary to compute a output. Floatigpoit arithmetic, o the other had, requires a quatiatio after every multiply ad after every add i the covolutio summatio. With 3-b floatig-poit arithmetic these quatiatios itroduce a small eough error to be isigificat for may applicatios. Whe realiig IIR filters, either a parallel or cascade coectio of first- ad secod-order subfilters is almost always preferable to a high-order direct-form realiatio. With the availability of very low-cost floatig-poit digital sigal processors, like the Texas Istrumets TMS30C3, it is highly recommeded that floatig-poit arithmetic be used for IIR filters. Floatig-poit arithmetic simultaeously elimiates most cocers regardig scalig, limit cycles, ad overflow oscillatios. Regardless of the arithmetic employed, a low roudoff oise structure should be used for the secod- order sectios. Good choices are give i [] ad [0]. Recall that realiatios with low fixed-poit roudoff oise also have low floatigpoit roudoff oise. The use of a low roudoff oise structure for the secod-order sectios also teds to give a realiatio with low coefficiet quatiatio sesitivity. First-order sectios are ot as critical i determiig the roudoff oise ad coefficiet sesitivity of a realiatio, ad so ca geerally be implemeted with a simple direct form structure. UNIT V APPLICATIONS MARKS. What is the eed for multirate sigal processig? I real time data commuicatio we may require more tha oe samplig rate for processig data i such a cases we go for multi-rate sigal processig which icrease ad/or decrease the samplig rate.. Give some examples of multirate digital systems. Decimator ad iterpolator FIGURE: Realiable pole locatios for the alterate realiatio structure.

267 3. Write the iput output relatioship for a decimator. Fy Fx/D 4. Write the iput output relatioship for a iterpolator. Fy IFx 5. What is meat by aliasig? The origial shape of the sigal is lost due to uder samplig. This is called aliasig. 6. How ca aliasig be avoided? Placig a LPF before dow samplig. 7. How ca samplig rate be coverted by a factor I/D. Cascade coectio of iterpolator ad decimator. 8. What is meat by sub-bad codig? It is a efficiet codig techique by allocatig lesser bits for high frequecy sigals ad more bits for low frequecy sigals. 9. What is meat by up samplig? Icreasig the samplig rate. 0. What is meat by dow samplig? Decreasig the samplig rate.. What is meat by decimator? Dow samplig ad a ati-aliasig filter.. What is meat by iterpolator? A ati-imagig filters ad Up samplig. 3. What is meat by samplig rate coversio? Chagig oe samplig rate to other samplig rate is called samplig rate coversio. 4. What are the sectios of QMF. Aalysis sectio ad sythesis sectio. 5. Defie mea. MxE[x]itg xpxx, dx 6. Defie variace. ZxE[{xmx}] 7. Defie cross correlatio of radom process. R xy.m itxy*px,ymx,,y,mdxdy. 8. Defie DTFT of cross correlatio Txye jw x rxyl e jwl 9. What is the cutoff frequecy of Decimator? Pi/M where M is the dow samplig factor 0. What is the cutoff frequecy of Iterpolator? Pi/L where L is the UP samplig factor.. What is the differece i efficiet trasversal structure? Number of delayed multiplicatios are reduced.

268 . What is the shape of the white oise spectrum? Flat frequecy spectrum. 6 MARKS, Explai about Decimatio ad Iterpolatio. Defiitio. Give a iteger D, we defie the dowsamplig operator S ijd,show i Figure by the followig relatioship: >{»] - " *[«>] The operator S IW decreases the samplig frequecy by a factor of D, by keepig oe sample out of D samples. Figure Upsamplig jct] Figure Dowsamplig Example Let x [...,,,3,4,5,6,... ], the j[] S / x[] is give by?[«] [.3,5 f 7 ] bamplmg Kate Coversio by a Ratioal t* actor the problem of desigig a algorithm for resamplig a digital sigal x[] from the origial rate F x i H ito a rate F v UDF f with L ad D itegers. For example, we have a sigal at telephoe quality, F x 8 kh, ad we wat to resample it at radio quality, F y kh. I this case, clearly L ad D 4. First cosider two particular cases, iterpolatio ad decimatio, where we upsample ad dowsample by a iteger factor without creatig aliasig or image frequecies. Decimatio by a Iteger Factor D We have see i the previous sectio that simple dowsamplig decreases the samplig frequecy by a factor D. I this operatio, all frequecies of Xtu DTFT{jr[]} above 7tID cause aliasig, ad therefore they have to be filtered out before dowsamplig the sigal. Thus we have the scheme i Figure where the sigal is dowsampled by a factor D, without aliasig.

269 Iterpolatio by a Iteger Factor L As i the case of decimatio, upsamplig by a factor L aloe itroduces artifacts i the frequecy domai as image frequecies. Fortuately, as see i the previous sectio, all image frequecies are outside the iterval [-ttil, ttil], ad thev ca be elimiated by filterig the sigal after dowsamplig. This is show i Figure i

270

271

272 , Explai about Multistage Implemetatio of Digital Filters. I Multistage Implemetatio of Digital filters I some applicatios we wat to desig filters where the badwidth is just a small fractio of the overall samplig rate. For example, suppose we wat to desig a lowpass filter with badwidth of the order of a few hert ad a samplig frequecy of the order of several kilohert. This filter would require a very sharp trasitio regio i the digital frequecy a>, thus requirig a high-complcxity filter. [Type text] Example < As a example of applicatio, suppose you wat to desig a Filter with the fallowig specificatios: Passbad F p 450 H

273 Stopbad F s 500 H Samplig frequecy F s ~96 kh Notice that the stopbad is several orders of magitude smaller tha the samplig frequecy. This leads to a filter with a very short trasitio regio of high complexity. I Speech sigals From prehistory to the ew media of the future, speech has bee ad will be a primary form of commuicatio betwee humas. Nevertheless, there ofte occur coditios uder which we measure ad the trasform the speech to aother form, speech sigal, i order to ehace our ability to commuicate. The speech sigal is exteded, through techological media such as telephoy, movies, radio, televisio, ad ow Iteret. This tred reflects the primacy of speech commuicatio i huma psychology. Speech will become the ext major tred i the persoal computer market i the ear future. Speech sigal processig The topic of speech sigal processig ca be loosely defied as the maipulatio of sampled speech sigals by a digital processor to obtai a ew sigal with some desired properties. Speech sigal processig is a diverse field that relies o kowledge of laguage at the levels of Sigal processig Acoustics P Phoetics ^^^Laguageidepedet Phoology ^^ Morphology i^^^ Sytax ^, Laguag e-depedet Sematics \%X Pragmatics if,ffl^ 7 layers for describig speech From Speech to Speech Sigal, i terms of Digital Sigal Processig FO pitch - amplitude loudess At-* Acoustic ad perceptual features {traits - fudam etal freouecy

274 - spectrum timber It is based o the fact that - Most of eergy betwee 0 H to about 7KH, - Huma ear sesitive to eergy betwee 50 H ad 4KH I terms of acoustic or perceptual, above features are cosidered. From Speech to Speech Sigal, i terms of Phoetics Speech productio, the digital model of Speech Sigal will be discussed i Chapter. Motivatio of covertig speech to digital sigals:

275 pitch, Speech codig, A PC ad SBC fie structure Adaptive predictive codig APC is a techique used for speech Typic al Voice d speec h codig, that is data compressio of spccch sigals APC assumes that the iput speech sigal is repetitive with a period sigificatly loger tha the average frequecy cotet. Two predictors arc used i APC. The high frequecy compoets up to 4 kh are estimated usig a 'spectral or 'format prcdictor ad the low frequecy compoets H by a pitch or fie structure prcdictor see figure 7.4. The spcctral estimator may he of order 4 ad the pitch estimator about order 0. The lowfrequecy compoets of the spccch sigal are due to the movemet of the togue, chi ad spectral evelope, formats Figure 7.4 Ecoder for adaptive, predictive codig of speech sigals. The decoder is maily a mirrored versio of the ecoder The high-frequecy compoets origiate from the vocal chords ad the oise-like souds like i s produced i the frot of the mouth. The output sigal ytogether with the predictor parameters, obtaied adaptively i the ecoder, are trasmitted to the decoder, where the spcech sigal is recostructed. The decoder has the same structure as the ecoder but the predictors arc ot adaptive ad arc ivoked i the reverse order. The predictio parameters are adapted for blocks of data correspodig to for istace 0 ms time periods. A PC' is used for codig spcech at 9.6 ad 6 kbits/s. The algorithm works well i oisy eviromets, but ufortuately the quality of the processed speech is ot as good as for other methods like CELP described below. 3, Explai about Subbad Codig. Aother codig method is sub-bad codig SBC see Figure 7.5