Digital Signal Processing

Transcription

1 Digital Sigal Processig EC5 SUBJECT CODE : EC5 IA ARKS : 5 O. OF LECTURE HRS/WEEK : 4 EXA HOURS : 3 TOTAL O. OF LECTURE HRS. : 5 EXA ARKS : UIT - PART - A DISCRETE FOURIER TRASFORS DFT: FREQUECY DOAI SAPLIG AD RECOSTRUCTIO OF DISCRETE TIE SIGALS. DFT AS A LIEAR TRASFORATIO, ITS RELATIOSHIP WITH OTHER TRASFORS. UIT - 7 HOURS PROPERTIES OF DFT, ULTIPLICATIO OF TWO DFTS- THE CIRCULAR COVOLUTIO, ADDITIOAL DFT PROPERTIES, USE OF DFT I LIEAR FILTERIG, OVERLAP-SAVE AD OVERLAP-ADD ETHOD. UIT HOURS FAST-FOURIER-TRASFOR FFT ALGORITHS: DIRECT COPUTATIO OF DFT, EED FOR EFFICIET COPUTATIO OF THE DFT FFT ALGORITHS. UIT HOURS RADIX- FFT ALGORITH FOR THE COPUTATIO OF DFT AD IDFT DECIATIO-I- TIE AD DECIATIO-I-FREQUECY ALGORITHS. GOERTZEL ALGORITH, AD CHIRP-Z TRASFOR UIT - 5 PART - B 6 HOURS IIR FILTER DESIG: CHARACTERISTICS OF COOLY USED AALOG FILTERS BUTTERWORTH AD CHEBYSHEVE FILTERS, AALOG TO AALOG FREQUECY TRASFORATIOS. UIT HOURS FIR FILTER DESIG: ITRODUCTIO TO FIR FILTERS, DESIG OF FIR FILTERS USIG - RECTAGULAR, HAIG, BARTLET AD KAISER WIDOWS, FIR FILTER DESIG USIG FREQUECY SAPLIG TECHIQUE 6 HOURS SJBIT/ECE Page

2 Digital Sigal Processig EC5 UIT - 7 DESIG OF IIR FILTERS FRO AALOG FILTERS BUTTERWORTH AD CHEBYSHEV - IPULSE IVARIACE ETHOD. APPIG OF TRASFER FUCTIOS: APPROXIATIO OF DERIVATIVE BACKWARD DIFFERECE AD BILIEAR TRASFORATIO ETHOD, ATCHED Z TRASFORS, VERIFICATIO FOR STABILITY AD LIEARITY DURIG APPIG UIT HOURS IPLEETATIO OF DISCRETE-TIE SYSTES: STRUCTURES FOR IIR AD FIR SYSTES- DIRECT FOR I AD DIRECT FOR II SYSTES, CASCADE, LATTICE AD PARALLEL REALIZATIO. 6 HOURS TEXT BOOK: DIGITAL SIGAL PROCESSIG PRICIPLES ALGORITHS & APPLICATIOS, PROAKIS & OALAKIS, PEARSO EDUCATIO, 4 TH EDITIO, EW DELHI, 7. REFERECE BOOKS:. DISCRETE TIE SIGAL PROCESSIG, OPPEHEI & SCHAFFER, PHI, 3.. DIGITAL SIGAL PROCESSIG, S. K. ITRA, TATA C-GRAW HILL, D EDITIO, DIGITAL SIGAL PROCESSIG, LEE TA: ELSIVIER PUBLICATIOS, 7 SJBIT/ECE Page

3 Digital Sigal Processig EC5 IDEX SHEET SL.O TOPIC PAGE O. I Uit-: Discrete Fourier Trasforms -. Frequecy Domai Samplig. Recostructio of Discrete time sigals.3 DFT as a Liear Trasform.4 DFT relatioship with other trasforms II UIT - : Properties of DFT 3-5. ultiplicatio of two DFTs Circular covolutio. Additioal DFT properties.3 Use of DFT i Liear Filterig.4 Overlap save ad Overlap Add ethod.5 Solutio to Problems III UIT 3 : Fast Fourier Trasform Algorithms Direct computatio of DFT 3. eed for Efficiet computatio of DFT 3. FFT algorithms IV UIT 4 : Radix- FFT Algorithms for DFT ad IDFT Decimatio I Time Algorithms 4. Decimatio-i-Frequecy Algorithms 4.3 Goertel Algorithms 4.4 Chirp--Trasforms V UIT 5 : IIR Filter Desig Characteristics of commoly used Aalog Filters 5. Butterworth ad Chebyshev Filters 5.3 Aalog to Aalog Frequecy Trasforms 5.4 Solutio to problems VI UIT 6 : FIR Filter Desig Itroductio to FIR filters 6. Desig of FIR Filters usig Rectagular ad Hammig widow 6.3 Desig of FIR Filters usig Bartlet ad Hammig widow 6.4 F IR filter desig usig frequecy samplig techique VII UIT 7 : Desig of IIR Filters from Aalog Filters Impulse Ivariace ethod 7. appig of Trasfer Fuctios 7.3 Approximatio of derivative Bacward Differece ad Biliear Trasforms method 7.4 atched Z trasforms 7.5 Verificatio for stability ad Liearity durig mappig 7.6 Solutio to problems SJBIT/ECE Page 3

4 Digital Sigal Processig EC5 VIII UIT 8 : Implemetatio of Discrete time systems Structure of IIR ad FIR systems 8. Direct form I ad direct form II Systems 8.3 Cascade Realiatio 8.4 Lattice Realiatio 8.5 Parallel Realiatio 8.6 Solutio to problems SJBIT/ECE Page 4

5 Digital Sigal Processig EC5 UIT DISCRETE FOURIER TRASFORS DFT COTETS:-. FREQUECY DOAI SAPLIG. RECOSTRUCTIO OF DISCRETE TIE SIGALS 3. DFT AS A LIEAR TRASFORATIO 4. DFT RELATIOSHIP WITH OTHER TRASFORS. DFT RELATIOSHIP WITH FOURIER SERIES DFT RELATIOSHIP WITH Z-TRASFORS RECOEDED READIGS. DIGITAL SIGAL PROCESSIG PRICIPLES ALGORITHS & APPLICATIOS, PROAKIS & OALAKIS, PEARSO EDUCATIO, 4 TH EDITIO, EW DELHI, 7.. DISCRETE TIE SIGAL PROCESSIG, OPPEHEI & SCHAFFER, PHI, DIGITAL SIGAL PROCESSIG, S. K. ITRA, TATA C-GRAW HILL, D EDITIO, 4. UIT SJBIT/ECE Page 5

6 Digital Sigal Processig EC5 Discrete Fourier Trasform. Itroductio: Before we itroduce the DFT we cosider the samplig of the Fourier trasform of a aperiodic discrete-time sequece. Thus we establish the relatio betwee the sampled Fourier trasform ad the DFT.A discrete time system may be described by the covolutio sum, the Fourier represetatio ad the trasform as see i the previous chapter. If the sigal is periodic i the time domai DTFS represetatio ca be used, i the frequecy domai the spectrum is discrete ad periodic. If the sigal is o-periodic or of fiite duratio the frequecy domai represetatio is periodic ad cotiuous this is ot coveiet to implemet o the computer. Exploitig the periodicity property of DTFS represetatio the fiite duratio sequece ca also be represeted i the frequecy domai, which is referred to as Discrete Fourier Trasform DFT. DFT is a importat mathematical tool which ca be used for the software implemetatio of certai digital sigal processig algorithms.dft gives a method to trasform a give sequece to frequecy domai ad to represet the spectrum of the sequece usig oly frequecy values, where is a iteger that taes values, K=,,,..-. The advatages of DFT are:. It is computatioally coveiet.. The DFT of a fiite legth sequece maes the frequecy domai aalysis much simpler tha cotiuous Fourier trasform techique.. FREQUECY DOAI SAPLIG AD RECOSTRUCTIO OF DISCRETE TIE SIGALS: Cosider a aperiodic discrete time sigal x with Fourier trasform, a aperiodic fiite eergy sigal has cotiuous spectra. For a aperiodic sigal x[] the spectrum is: X w x e jw. Suppose we sample X[w] periodically i frequecy at a samplig of w radias betwee successive samples. We ow that DTFT is periodic with, therefore oly samples i the SJBIT/ECE Page 6

7 Digital Sigal Processig EC5 fudametal frequecy rage will be ecessary. For coveiece we tae equidistat samples i the iterval <=w<. The spacig betwee samples will be below i Fig... X[w] w as show w Fig. Frequecy Domai Samplig Let us first cosider selectio of, or the umber of samples i the frequecy domai. If we evaluate equatio at w X j / x e,,,...,.. We ca divide the summatio i ito ifiite umber of summatios where each sum cotais terms. X... x e j / x e j / x e j / l l l x e j / If we the chage the idex i the summatio from to -l ad iterchage the order of summatios we get: l x l e j / for,,,...,..3 SJBIT/ECE Page 7

8 Digital Sigal Processig EC5 Deote the quatity iside the bracet as x p []. This is the sigal that is a repeatig versio of x[] every samples. Sice it is a periodic sigal it ca be represeted by the Fourier series. x p c e j /,,,..., With FS coefficiets: j / c xp e,,,...,.4 Comparig the expressios i equatios.4 ad.3 we coclude the followig: c X,,...,..5 Therefore it is possible to write the expressio x p [] as below: j / xp X e,,...,..6 The above formula shows the recostructio of the periodic sigal x p [] from the samples of the spectrum X[w]. But it does ot say if X[w] or x[] ca be recovered from the samples. Let us have a loo at that: Sice x p [] is the periodic extesio of x[] it is clear that x[] ca be recovered from x p [] if there is o aliasig i the time domai. That is if x[] is time-limited to less tha the period of x p [].This is depicted i Fig.. below: SJBIT/ECE Page 8

9 Digital Sigal Processig EC5 x[] L x p [] >=L o aliasig L x p [] <L Aliasig Fig.. Sigal Recostructio Hece we coclude: The spectrum of a aperiodic discrete-time sigal with fiite duratio L ca be exactly recovered from its samples at frequecies w if >= L. We compute x p [] for =,,..., - usig equatio.6 The X[w] ca be computed usig equatio...3 Discrete Fourier Trasform: The DTFT represetatio for a fiite duratio sequece is -jω X jω = x = - jω X =/π X jω e dω, Where ω π/ SJBIT/ECE Page 9

10 Digital Sigal Processig EC5 π Where x is a fiite duratio sequece, Xjω is periodic with period π.it is coveiet sample Xjω with a samplig frequecy equal a iteger multiple of its period =m that is taig uiformly spaced samples betwee ad π. Let ω = π/, - as -jπ/ Therefore Xjω = x = Sice Xjω is sampled for oe period ad there are samples Xjω ca be expressed - -jπ/ X = Xjω ω=π/ x - =.4 atrix relatio of DFT The DFT expressio ca be expressed as [X] = [x] [W] Where [X] = [X, X,..] T [x] is the traspose of the iput sequece. W is a x matrix W = w w w3...w - w w4 w6 w-....w -- ex; 4 pt DFT of the sequece,,,3 X X -j - j X = - - X3 j - -j Solvig the matrix XK = 6, -+j, -, --j.5 Relatioship of Fourier Trasforms with other trasforms SJBIT/ECE Page

11 Digital Sigal Processig EC5.5. Relatioship of Fourier trasform with cotiuous time sigal: Suppose that x a t is a cotiuous-time periodic sigal with fudametal period T p = /F.The sigal ca be expressed i Fourier series as Where {c} are the Fourier coefficiets. If we sample x a t at a uiform rate Fs = /T p = /T, we obtai discrete time sequece Thus {c } is the aliasig versio of {c }.5. Relatioship of Fourier trasform with -trasform Let us cosider a sequece x havig the -trasform With ROC that icludes uit circle. If X is sampled at the equally spaced poits o the uit circle Z = e jπ/ for K=,,,..- we obtai The above expressio is idetical to Fourier trasform Xω evaluated at equally spaced frequecies ω = π/ for K=,,,..-. SJBIT/ECE Page

12 Digital Sigal Processig EC5 If the sequece x has a fiite duratio of legth or less. The sequece ca be recovered from its -poit DFT. Cosequetly X ca be expressed as a fuctio of DFT as Fourier trasform of a cotiuous time sigal ca be obtaied from DFT as SJBIT/ECE Page

13 Digital Sigal Processig EC5 Recommeded Questios with solutios Questio The first five poits of the 8-poit DFT of a real valued sequece are {.5,.5-j.38,,.5-j.58, }. Determie the remaiig three poits As: Sice x is real, the real part of the DFT is eve, imagiary part odd. Thus the remaiig poits are {.5+j.58,,,.5+j.38}. Questio Compute the eight-poit DFT circular covolutio for the followig sequeces. x = si 3π/8 As: Questio 3 Compute the eight-poit DFT circular covolutio for the followig sequece X 3 = cos 3π/8 Questio 4 SJBIT/ECE Page 3

14 Digital Sigal Processig EC5 Defie DFT. Establish a relatio betwee the Fourier series coefficiets of a cotiuous time sigal ad DFT Solutio The DTFT represetatio for a fiite duratio sequece is X jω = x -jω = - X =/π X jω e jω dω, Where ω π/ π Where x is a fiite duratio sequece, Xjω is periodic with period π.it is coveiet sample Xjω with a samplig frequecy equal a iteger multiple of its period =m that is taig uiformly spaced samples betwee ad π. Let ω = π/, Therefore Xjω = x -jπ/ = Sice Xjω is sampled for oe period ad there are samples Xjω ca be expressed as - X = Xjω ω=π/ x -jπ/ - = Questio 5 Solutio:- SJBIT/ECE Page 4

15 Digital Sigal Processig EC5 Questio 6 Fid the 4-poit DFT of sequece x = 6+ siπ/, =,, - Solutio :- Questio 7 Solutio SJBIT/ECE Page 5

16 Digital Sigal Processig EC5 Questio 8 Solutio SJBIT/ECE Page 6

17 Digital Sigal Processig EC5 UIT PROPERTIES OF DISCRETE FOURIER TRASFORS DFT COTETS:-. ULTIPLICATIO OF TWO DFT S- THE CIRCULAR COVOLUTIO,. ADDITIOAL DFT PROPERTIES 3. USE OF DFT I LIEAR FILTERIG 4. OVERLAP-SAVE AD OVERLAP-ADD ETHOD. RECOEDED READIGS. DIGITAL SIGAL PROCESSIG PRICIPLES ALGORITHS & APPLICATIOS, PROAKIS & OALAKIS, PEARSO EDUCATIO, 4 TH EDITIO, EW DELHI, 7.. DISCRETE TIE SIGAL PROCESSIG, OPPEHEI & SCHAFFER, PHI, DIGITAL SIGAL PROCESSIG, S. K. ITRA, TATA C-GRAW HILL, D EDITIO, 4. SJBIT/ECE Page 7

18 Digital Sigal Processig EC5 Uit Properties of DFT. Properties:- The DFT ad IDFT for a -poit sequece x are give as I this sectio we discuss about the importat properties of the DFT. These properties are helpful i the applicatio of the DFT to practical problems. Periodicity:-.. Liearity: If SJBIT/ECE Page 8

19 Digital Sigal Processig EC5 The A x + b x a X + b X..3 Circular shift: I liear shift, whe a sequece is shifted the sequece gets exteded. I circular shift the umber of elemets i a sequece remais the same. Give a sequece x the shifted versio x -m idicates a shift of m. With DFTs the sequeces are defied for to -. If x = x, x, x, x 3 X - = x 3, x, x.x X - = x, x 3, x, x..4 Time shift: If x X m The x -m W X..5 Frequecy shift If x X +o W x X+o - Cosider x = x W = - + o X+o= \ x W = o = x W W X+o o x W..6 Symmetry: SJBIT/ECE Page 9

20 Digital Sigal Processig EC5 For a real sequece, if x X X-K = X* For a complex sequece DFTx* = X*-K If x the X Real ad eve Real ad odd Odd ad imagiary Eve ad imagiary real ad eve imagiary ad odd real odd imagiary ad eve. Covolutio theorem; Circular covolutio i time domai correspods to multiplicatio of the DFTs If y = x h the Y = X H Ex let x =,,, ad h =,,, The y = x h Y = 9,,9,8 pt DFTs of real sequeces ca be foud usig a sigle DFT If g & h are two sequeces the let x = g +j h G = ½ X + X* H = /j XK +X* pt DFT of a real sequece usig a sigle pt DFT Let x be a real sequece of legth with y ad g deotig its pt DFT Let y = x ad g+ X = Y + W G Usig DFT to fid IDFT The DFT expressio ca be used to fid IDFT SJBIT/ECE Page

21 Digital Sigal Processig EC5 X = / [DFTX*]*.3 Digital filterig usig DFT I a LTI system the system respose is got by covolutig the iput with the impulse respose. I the frequecy domai their respective spectra are multiplied. These spectra are cotiuous ad hece caot be used for computatios. The product of DFT s is equivalet to the circular covolutio of the correspodig time domai sequeces. Circular covolutio caot be used to determie the output of a liear filter to a give iput sequece. I this case a frequecy domai methodology equivalet to liear covolutio is required. Liear covolutio ca be implemeted usig circular covolutio by taig the legth of the covolutio as >= +- where ad are the legths of the sequeces..3. Overlap ad add I order to covolve a short duratio sequece with a log duratio sequece x,x is split ito blocs of legth x ad h are ero padded to legth L+-. circular covolutio is performed to each bloc the the results are added. These data blocs may be represeted as The IDFT yields data blocs of legth that are free of aliasig sice the sie of the DFTs ad IDFT is = L+ - ad the sequeces are icreased to -poits by appedig eros to each bloc. Sice each bloc is termiated with - eros, the last - poits from each output bloc must be overlapped ad added to the first - poits of the succeedig SJBIT/ECE Page

22 Digital Sigal Processig EC5 bloc. Hece this method is called the overlap method. This overlappig ad addig yields the output sequeces give below..3. Overlap ad save method I this method x is divided ito blocs of legth with a overlap of - samples. The first bloc is ero padded with - eros at the begiig. H is also ero padded to legth. Circular covolutio of each bloc is performed usig the legth DFT.The output sigal is obtaied after discardig the first - samples the fial result is obtaied by addig the itermediate results. SJBIT/ECE Page

23 Digital Sigal Processig EC5 I this method the sie of the I/P data blocs is = L+- ad the sie of the DFts ad IDFTs are of legth. Each data bloc cosists of the last - data poits of the previous data bloc followed by L ew data poits to form a data sequece of legth = L+-. A - poit DFT is computed from each data bloc. The impulse respose of the FIR filter is icreased i legth by appedig L- eros ad a -poit DFT of the sequece is computed oce ad stored. The multiplicatio of two -poit DFTs {H} ad {Xm} for the mth bloc of data yields Sice the data record is of the legth, the first - poits of Ym are corrupted by aliasig ad must be discarded. The last L poits of Ym are exactly the same as the result from liear covolutio ad as a cosequece we get SJBIT/ECE Page 3

24 Digital Sigal Processig EC5 SJBIT/ECE Page 4

25 Digital Sigal Processig EC5 Recommeded Questios with solutios Questio State ad Prove the Time shiftig Property of DFT Solutio The DFT ad IDFT for a -poit sequece x are give as Time shift: If x X m The x -m W X Questio State ad Prove the: i Circular covolutio property of DFT; ii DFT of Real ad eve sequece. Solutio i Covolutio theorem Circular covolutio i time domai correspods to multiplicatio of the DFTs If y = x h the Y = X H Ex let x =,,, ad h =,,, The y = x h Y = 9,,9,8 pt DFTs of real sequeces ca be foud usig a sigle DFT If g & h are two sequeces the let x = g +j h G = ½ X + X* H = /j XK +X* SJBIT/ECE Page 5

26 Digital Sigal Processig EC5 pt DFT of a real sequece usig a sigle pt DFT Let x be a real sequece of legth with y ad g deotig its pt DFT Let y = x ad g+ X = Y + W G Usig DFT to fid IDFT The DFT expressio ca be used to fid IDFT X = / [DFTX*]* iidft of Real ad eve sequece. For a real sequece, if x X X -K = X* For a complex sequece DFTx* = X*-K If x the X Real ad eve real ad eve Real ad odd imagiary ad odd Odd ad imagiary real odd Eve ad imagiary imagiary ad eve Questio 3 Distiguish betwee circular ad liear covolutio Solutio Circular covolutio is used for periodic ad fiite sigals while liear covolutio is used for aperiodic ad ifiite sigals. I liear covolutio we covolved oe sigal with aother sigal where as i circular covolutio the same covolutio is doe but i circular patter depedig upo the samples of the sigal 3 Shifts are liear i liear i liear covolutio, whereas it is circular i circular covolutio. Questio 4 SJBIT/ECE Page 6

27 Digital Sigal Processig EC5 Solutioa Solutiob Solutioc Solutiod SJBIT/ECE Page 7

28 Digital Sigal Processig EC5 Questio 5 Solutio Questio 6 Solutio SJBIT/ECE Page 8

29 Digital Sigal Processig EC5 SJBIT/ECE Page 9

30 Digital Sigal Processig EC5 UIT 3 FAST FOURIER TRASFORS FFT ALOGORITHS COTETS:-. FAST-FOURIER-TRASFOR FFT ALGORITHS. DIRECT COPUTATIO OF DFT, 3. EED FOR EFFICIET COPUTATIO OF THE DFT FFT ALGORITHS. RECOEDED READIGS. DIGITAL SIGAL PROCESSIG PRICIPLES ALGORITHS & APPLICATIOS, PROAKIS & OALAKIS, PEARSO EDUCATIO, 4 TH EDITIO, EW DELHI, 7.. DISCRETE TIE SIGAL PROCESSIG, OPPEHEI & SCHAFFER, PHI, DIGITAL SIGAL PROCESSIG, S. K. ITRA, TATA C-GRAW HILL, D EDITIO, 4. SJBIT/ECE Page 3

31 Digital Sigal Processig EC5 UIT 3 FAST-FOURIER-TRASFOR FFT ALGORITHS 3. Direct Computatio of DFT The problem: Give sigal samples: x[],..., x[ - ] some of which may be ero, develop a procedure to compute for =,..., - where We would lie the procedure to be fast, simple, ad accurate. Fast is the most importat, so we will sacrifice simplicity for speed, hopefully with miimal loss of accuracy 3. eed for efficiet computatio of DFT FFT Algorithms Let us start with the simple way. Assume that has bee precompiled ad stored i a table for the of iterest. How big should the table be? is periodic i m with period, so we just eed to tabulate the values: Possibly eve less sice Si is just Cos shifted by a quarter periods, so we could save just Cos whe is a multiple of 4. Why tabulate? To avoid repeated fuctio calls to Cos ad si whe computig the DFT. ow we ca compute each X[] directly form the formula as follows For each value of, there are complex multiplicatios, ad - complex additios. There are values of, so the total umber of complex operatios is SJBIT/ECE Page 3

32 Digital Sigal Processig EC5 Complex multiplies require 4 real multiplies ad real additios, whereas complex additios require just real additios complex multiplies are the primary cocer. icreases rapidly with, so how ca we reduce the amout of computatio? By exploitig the followig properties of W: The first ad third properties hold for eve, i.e., whe is oe of the prime factors of. There are related properties for other prime factors of. Divide ad coquer approach We have see i the precedig sectios that the DFT is a very computatioally itesive operatio. I 965, Cooley ad Tuey published a algorithm that could be used to compute the DFT much more efficietly. Various forms of their algorithm, which came to be ow as the Fast Fourier Trasform FFT, had actually bee developed much earlier by other mathematicias eve datig bac to Gauss. It was their paper, however, which stimulated a revolutio i the field of sigal processig. It is importat to eep i mid at the outset that the FFT is ot a ew trasform. It is simply a very efficiet way to compute a existig trasform, amely the DFT. As we saw, a straight forward implemetatio of the DFT ca be computatioally expesive because the umber of multiplies grows as the square of the iput legth i.e. for a poit DFT. The FFT reduces this computatio usig two simple but importat cocepts. The first cocept, ow as divide-ad-coquer, splits the problem ito two smaller problems. The secod cocept, ow as recursio, applies this divide-ad-coquer method repeatedly util the problem is solved. SJBIT/ECE Page 3

33 Digital Sigal Processig EC5 Recommeded Questios with solutios Questio Solutio:- Questio Solutio:- SJBIT/ECE Page 33

34 Digital Sigal Processig EC5 Questio 3 Solutio:- Questio 4 SJBIT/ECE Page 34

35 Digital Sigal Processig EC5 Solutio:- a b SJBIT/ECE Page 35

36 Digital Sigal Processig EC5 UIT 4 FAST FOURIER TRASFORS FFT ALOGORITHS COTETS:-. RADIX- FFT ALGORITH FOR THE COPUTATIO OF DFT AD IDFT. DECIATIO-I-TIE AD DECIATIO-I-FREQUECY ALGORITHS. 3. GOERTZEL ALGORITH, 4. CHIRP-Z TRASFOR RECOEDED READIGS. DIGITAL SIGAL PROCESSIG PRICIPLES ALGORITHS & APPLICATIOS, PROAKIS & OALAKIS, PEARSO EDUCATIO, 4 TH EDITIO, EW DELHI, 7.. DISCRETE TIE SIGAL PROCESSIG, OPPEHEI & SCHAFFER, PHI, DIGITAL SIGAL PROCESSIG, S. K. ITRA, TATA C-GRAW HILL, D EDITIO, 4. SJBIT/ECE Page 36

37 Digital Sigal Processig EC5 UIT 4 RADIX- FFT ALGORITH FOR THE COPUTATIO OF DFT AD IDFT 4. Itroductio: Stadard frequecy aalysis requires trasformig time-domai sigal to frequecy domai ad studyig Spectrum of the sigal. This is doe through DFT computatio. -poit DFT computatio results i frequecy compoets. We ow that DFT computatio through FFT requires / log complex multiplicatios ad log additios. I certai applicatios ot all frequecy compoets eed to be computed a applicatio will be discussed. If the desired umber of values of the DFT is less tha log tha direct computatio of the desired values is more efficiet that FFT based computatio. 4. Radix- FFT Useful whe is a power of : = r v for itegers r ad v. r is called the radix, which comes from the Lati word meaig.a root, ad has the same origis as the word radish. Whe is a power of r =, this is called radix-, ad the atural.divide ad coquer approach. is to split the sequece ito two sequeces of legth =. This is a very clever tric that goes bac may years. 4.. Decimatio i time Fig 4. First step i Decimatio-i-time domai Algorithm SJBIT/ECE Page 37

38 Digital Sigal Processig EC5 SJBIT/ECE Page 38

39 Digital Sigal Processig EC5 4.. Decimatio-i-frequecy Domai Aother importat radix- FFT algorithm, called decimatio-i-frequecy algorithm is obtaied by usig divide-ad-coquer approach with the choice of = ad L= /.This choice of data implies a colum-wise storage of the iput data sequece. To derive the algorithm, we begi by splittig the DFT formula ito two summatios, oe of which ivolves the sum over the first / data poits ad the secod sum ivolves the last / data poits. Thus we obtai ow, let us split X ito the eve ad odd-umbered samples. Thus we obtai SJBIT/ECE Page 39

40 Digital Sigal Processig EC5 Fig 4. Shufflig of Data ad Bit reversal The computatio of the sequeces g ad g ad subsequet use of these sequeces to compute the /-poit DFTs depicted i fig we observe that the basic computatio i this figure ivolves the butterfly operatio. SJBIT/ECE Page 4

41 Digital Sigal Processig EC5 The computatio procedure ca be repeated through decimatio of the /-poit DFTs, X ad X+. The etire process ivolves v = log of decimatio, where each stage ivolves / butterflies of the type show i figure 4.3. Fig 4.3 First step i Decimatio-i-time domai Algorithm SJBIT/ECE Page 4

42 Digital Sigal Processig EC5 Fig 4.4 =8 poit Decimatio-i-frequecy domai Algorithm 4. Example: DTF Dual Toe ulti frequecy This is ow as touch-toe/speed/electroic dialig, pressig of each butto geerates a uique set of two-toe sigals, called DTF sigals. These sigals are processed at exchage to idetify the umber pressed by determiig the two associated toe frequecies. Seve frequecies are used to code the decimal digits ad two special characters 4x3 array SJBIT/ECE Page 4

43 Digital Sigal Processig EC5 I this applicatio frequecy aalysis requires determiatio of possible seve eight DTF fudametal toes ad their respective secod harmoics.for a 8 H samplig freq, the best value of the DFT legth to detect the eight fudametal DTF toes has bee foud to be 5.ot all 5 freq compoets are eeded here, istead oly those correspodig to ey frequecies are required. FFT algorithm is ot effective ad efficiet i this applicatio. The direct computatio of the DFT which is more effective i this applicatio is formulated as a liear filterig operatio o the iput data sequece. This algorithm is ow as Goertel Algorithm This algorithm exploits periodicity property of the phase factor. Cosider the DFT defiitio X x W Sice W is equal to, multiplyig both sides of the equatio by this results i; X W m x m W m m x m W m This is i the form of a covolutio y x h y x m W m m 3 h W u 4 Where y is the out put of a filter which has impulse respose of h ad iput x. The output of the filter at = yields the value of the DFT at the freq ω = π/ The filter has frequecy respose give by H W 6 The above form of filter respose shows it has a pole o the uit circle at the frequecy ω = π/. Etire DFT ca be computed by passig the bloc of iput data ito a parallel ba of sigle-pole filters resoators SJBIT/ECE Page 43

44 Digital Sigal Processig EC5 The above form of filter respose shows it has a pole o the uit circle at the frequecy ω = π/. Etire DFT ca be computed by passig the bloc of iput data ito a parallel ba of sigle-pole filters resoators.3 Differece Equatio implemetatio of filter: From the frequecy respose of the filter eq 6 we ca write the followig differece equatio relatig iput ad output; Y H The desired X output W is X = y for =,, -. The phase factor appearig i the differece y equatio W y ca be xcomputed yoce ad stored. 7 The form show i eq 7 requires complex multiplicatios which ca be avoided doig suitable modificatios divide ad multiply by W. The frequecy respose of the filter ca be alteratively expressed as W H cos / 8 This is secod order realiatio of the filter observe the deomiator ow is a secod-order expressio. The direct form realiatio of the above is give by v y cos / v v W v v v x v 9 The recursive relatio i 9 is iterated for =,,, but the equatio i is computed oly oce at time =. Each iteratio requires oe real multiplicatio ad two additios. Thus, for a real iput sequece x this algorithm requires + real multiplicatios to yield X ad X- this is due to symmetry. Goig through the Goertel SJBIT/ECE Page 44

45 Digital Sigal Processig EC5 algorithm it is clear that this algorithm is useful oly whe out of DFT values eed to be computed where log, Otherwise, the FFT algorithm is more efficiet method. The utility of the algorithm completely depeds o the applicatio ad umber of frequecy compoets we are looig for. 4.. Chirp - Trasform 4.. Itroductio: Computatio of DFT is equivalet to samples of the -trasform of a fiite-legth sequece at equally spaced poits aroud the uit circle. The spacig betwee the samples is give by π/. The efficiet computatio of DFT through FFT requires to be a highly composite umber which is a costrait. ay a times we may eed samples of -trasform o cotours other tha uit circle or we my require dese set of frequecy samples over a small regio of uit circle. To uderstad these let us loo i to the followig situatios:. Obtai samples of -trasform o a circle of radius a which is cocetric to uit circle The possible solutio is to multiply the iput sequece by a -. 8 samples eeded betwee frequecies ω = -π/8 to +π/8 from a 8 poit sequece From the give specificatios we see that the spacig betwee the frequecy samples is π/5 or π/4. I order to achieve this freq resolutio we tae 4- poit FFT of the give 8-poit seq by appedig the sequece with 896 eros. Sice we eed oly 8 frequecies out of 4 there will be big wastage of computatios i this scheme. For the above two problems Chirp -trasform is the alterative. Chirp - trasform is defied as: X x,,... L SJBIT/ECE Page 45

46 Digital Sigal Processig EC5 Where is a geeralied cotour. Z is the set of poits i the -plae fallig o a arc which begis at some poit ad spirals either i toward the origi or out away from the origi such that the poits { }are defied as, ote that, j j re Re,,... L a. if R < the poits fall o a cotour that spirals toward the origi b. If R > the cotour spirals away from the origi c. If R = the cotour is a circular arc of radius d.if r = ad R = the cotour is a arc of the uit circle. Additioally this cotour allows oe to compute the freq cotet of the sequece x at dese set of L frequecies i the rage covered by the arc without havig to compute a large DFT i.e., a DFT of the sequece x padded with may eros to obtai the desired resolutio i freq. e. If r = R = ad θ = Φ =π/ ad L = the cotour is the etire uit circle similar to the stadard DFT. These coditios are show i the followig diagram. SJBIT/ECE Page 46

47 Digital Sigal Processig EC5 Substitutig the value of i the expressio of X X x x r e j W 3 where W R e j Expressig computatio of X as liear filterig operatio: By substitutio of we ca express X as 5 X Where W / y y / h,,... L 6 h W / g j / x re W y g h 7 both g ad h are complex valued sequeces 4..3 Why it is called Chirp -trasform? If R =, the sequece h has the form of complex expoetial with argumet ω = Φ / = Φ /. The quatity Φ / represets the freq of the complex expoetial SJBIT/ECE Page 47

48 Digital Sigal Processig EC5 sigal, which icreases liearly with time. Such sigals are used i radar systems are called chirp sigals. Hece the ame chirp -trasform How to Evaluate liear covolutio of eq 7. Ca be doe efficietly with FFT. The two sequeces ivolved are g ad h. g is fiite legth seq of legth ad h is of ifiite duratio, but fortuately oly a portio of h is required to compute L values of X, hece FFT could be still be used. 3. Sice covolutio is via FFT, it is circular covolutio of the -poit seq g with a - poit sectio of h where > 4. The cocepts used i overlap save method ca be used 5. While circular covolutio is used to compute liear covolutio of two sequeces we ow the iitial - poits cotai aliasig ad the remaiig poits are idetical to the result that would be obtaied from a liear covolutio of h ad g, I view of this the DFT sie selected is = L+- which would yield L valid poits ad - poits corrupted by aliasig. The sectio of h cosidered is for - L- yieldig total legth as defied 6. The portio of h ca be defied i may ways, oe such way is, h = h-+ =,, Compute H ad G to obtai Y = GKH 8. Applicatio of IDFT will give y, for SJBIT/ECE Page 48

49 Digital Sigal Processig EC5 =,, -. The startig - are discarded ad desired values are y for - - which correspods to the rage L- i.e., y= y +- =,,,..L- 9. Alteratively h ca be defied as h h h L L. Compute Y = GKH, The desired values of y are i the rage L- i.e., y = y =,,.L-. Fially, the complex values X are computed by dividig y by h For =,, L- 4.3 Computatioal complexity I geeral the computatioal complexity of CZT is of the order of log complex multiplicatios. This should be compared with.l which is required for direct evaluatio. If L is small direct evaluatio is more efficiet otherwise if L is large the CZT is more efficiet Advatages of CZT a. ot ecessary to have =L b.either or L eed to be highly composite c.the samples of Z trasform are tae o a more geeral cotour that icludes the uit circle as a special case. 4.4 Example to uderstad utility of CZT algorithm i freq aalysis ref: DSP by Oppeheim Schaffer CZT is used i this applicatio to sharpe the resoaces by evaluatig the -trasform off the uit circle. Sigal to be aalyed is a sythetic speech sigal geerated by excitig a five-pole system with a periodic impulse trai. The system was simulated to correspod to a samplig freq. of H. The poles are located at ceter freqs of 7,9,3,35 & 45 H with badwidth of 3, 5, 6,87 & 4 H respectively. L SJBIT/ECE Page 49

50 Digital Sigal Processig EC5 Solutio: Observe the pole-ero plots ad correspodig magitude frequecy respose for differet choices of w. The followig observatios are i order: The first two spectra correspod to spiral cotours outside the uit circle with a resultig broadeig of the resoace peas w = correspods to evaluatig -trasform o the uit circle The last two choices correspod to spiral cotours which spirals iside the uit circle ad close to the pole locatios resultig i a sharpeig of resoace peas. SJBIT/ECE Page 5

51 Digital Sigal Processig EC5 4.5 Implemetatio of CZT i hardware to compute the DFT sigals The bloc schematic of the CZT hardware is show i dow figure. DFT computatio requires r =R =, θ = Φ = π/ ad L =. The cosie ad sie sequeces i h eeded for pre multiplicatio ad post multiplicatio are usually stored i a RO. If oly magitude of DFT is desired, the post multiplicatios are uecessary, I this case X = y =,,.- SJBIT/ECE Page 5

52 Digital Sigal Processig EC5 Recommeded Questios with solutios Questio Solutio:- SJBIT/ECE Page 5

53 Digital Sigal Processig EC5 Questio Solutio :- There are real, o trial multiplicatios Figure 4. DIF Algorithm for =6 SJBIT/ECE Page 53

54 Digital Sigal Processig EC5 Questio 3 Solutio:- Questio 4 Solutio:- SJBIT/ECE Page 54

55 Digital Sigal Processig EC5 Questio 5 Solutio:- Questio 6 Solutio:- This ca be viewed as the covolutio of the -legth sequece x with implulse respose of a liear filter SJBIT/ECE Page 55

56 Digital Sigal Processig EC5 SJBIT/ECE Page 56

57 Digital Sigal Processig EC5 UIT 5 IIR FILTER DESIG COTETS:-. IIR FILTER DESIG:. CHARACTERISTICS OF COOLY USED AALOG FILTERS 3. BUTTERWORTH AD CHEBYSHEVE FILTERS, 4. AALOG TO AALOG FREQUECY TRASFORATIOS. RECOEDED READIGS. DIGITAL SIGAL PROCESSIG PRICIPLES ALGORITHS & APPLICATIOS, PROAKIS & OALAKIS, PEARSO EDUCATIO, 4 TH EDITIO, EW DELHI, 7.. DISCRETE TIE SIGAL PROCESSIG, OPPEHEI & SCHAFFER, PHI, DIGITAL SIGAL PROCESSIG, S. K. ITRA, TATA C-GRAW HILL, D EDITIO, 4. SJBIT/ECE Page 57

58 Digital Sigal Processig EC5 5. Itroductio Uit 5 Desig of IIR Filters A digital filter is a liear shift-ivariat discrete-time system that is realied usig fiite precisio arithmetic. The desig of digital filters ivolves three basic steps: The specificatio of the desired properties of the system. The approximatio of these specificatios usig a causal discrete-time system. The realiatio of these specificatios usig fiite precisio arithmetic. These three steps are idepedet; here we focus our attetio o the secod step. The desired digital filter is to be used to filter a digital sigal that is derived from a aalog sigal by meas of periodic samplig. The specificatios for both aalog ad digital filters are ofte give i the frequecy domai, as for example i the desig of low pass, high pass, bad pass ad bad elimiatio filters. Give the samplig rate, it is straight forward to covert from frequecy specificatios o a aalog filter to frequecy specificatios o the correspodig digital filter, the aalog frequecies beig i terms of Hert ad digital frequecies beig i terms of radia frequecy or agle aroud the uit circle with the poit Z=- correspodig to half the samplig frequecy. The least cofusig poit of view toward digital filter desig is to cosider the filter as beig specified i terms of agle aroud the uit circle rather tha i terms of aalog frequecies. SJBIT/ECE Page 58

59 Digital Sigal Processig EC5 Figure 5.: Tolerace limits for approximatio of ideal low-pass filter A separate problem is that of determiig a appropriate set of specificatios o the digital filter. I the case of a low pass filter, for example, the specificatios ofte tae the form of a tolerace scheme, as show i Fig. 5.. ay of the filters used i practice are specified by such a tolerace scheme, with o costraits o the phase respose other tha those imposed by stability ad causality requiremets; i.e., the poles of the system fuctio must lie iside the uit circle. Give a set of specificatios i the form of Fig. 5., the ext step is to ad a discrete time liear system whose frequecy respose falls withi the prescribed toleraces. At this poit the filter desig problem becomes a problem i approximatio. I the case of ifiite impulse respose IIR filters, we must approximate the desired frequecy respose by a ratioal fuctio, while i the fiite impulse respose FIR filters case we are cocered with polyomial approximatio. 5. Desig of IIR Filters from Aalog Filters: SJBIT/ECE Page 59

60 Digital Sigal Processig EC5 The traditioal approach to the desig of IIR digital filters ivolves the trasformatio of a aalog filter ito a digital filter meetig prescribed specificatios. This is a reasoable approach because: The art of aalog filter desig is highly advaced ad sice useful results ca be achieved, it is advatageous to utilie the desig procedures already developed for aalog filters. ay useful aalog desig methods have relatively simple closed-form desig formulas. Therefore, digital filter desig methods based o aalog desig formulas are rather simple to implemet. A aalog system ca be described by the differetial equatio Ad the correspodig ratioal fuctio is The correspodig descriptio for digital filters has the form ad the ratioal fuctio I trasformig a aalog filter to a digital filter we must therefore obtai either H or h iverse Z-trasform of H i.e., impulse respose from the aalog filter desig. I such trasformatios, we wat the imagiary axis of the S-plae to map ito the it circle of the Z-plae, a stable aalog filter should be trasformed to a stable digital filter. That is, if the aalog filter has poles oly i the left-half of S-plae, the the digital filter must have poles oly iside the uit circle. These costraits are basic to all the techiques discussed here. SJBIT/ECE Page 6

61 Digital Sigal Processig EC5 5. Characteristics of Commoly Used Aalog Filters: From the previous discussio it is clear that, IIT digital filters ca be obtaied by begiig with a aalog filter. Thus the desig of a digital filter is reduced to desigig a appropriate aalog filter ad the performig the coversio from Has to H. Aalog filter desig is a well - developed field, may approximatio techiques, vi., Butterworth, Chebyshev, Elliptic, etc., have bee developed for the desig of aalog low pass filters. Our discussio is limited to low pass filters, sice, frequecy trasformatio ca be applied to trasform a desiged low pass filter ito a desired high pass, bad pass ad bad stop filters. 5.. Butterworth Filters: Low pass Butterworth filters are all - pole filters with mootoic frequecy respose i both pass bad ad stop bad, characteried by the magitude - squared frequecy respose Where, is the order of the filter, Ώc is the -3dB frequecy, i.e., cutoff frequecy, Ώp is the pass bad edge frequecy ad = /+ε is the bad edge value of HaΏ. Sice the product Has Ha-s ad evaluated at s = jώ is simply equal to HaΏ, it follows that The poles of HasHa-s occur o a circle of radius Ώc at equally spaced poits. From Eq. 5.9, we fid the pole positios as the solutio of Ad hece, the poles i the left half of the s-plae are SJBIT/ECE Page 6

62 Digital Sigal Processig EC5 ote that, there are o poles o the imagiary axis of s-plae, ad for odd there will be a pole o real axis of s-plae, for eve there are o poles eve o real axis of s-plae. Also ote that all the poles are havig cojugate symmetry. Thus the desig methodology to desig a Butterworth low pass filter with δ atteuatio at a specified frequecy Ώs is Fid, Where by defiitio, δ = / +δ. Thus the Butterworth filter is completely characteried by the parameters, δ, ε ad the ratio Ώs/Ώp or Ώc.The, from Eq. 5.3 fid the pole positios S; =,,,..-. Fially the aalog filter is give by 5.. Chebyshev Filters: There are two types of Chebyshev filters. Type I Chebyshev filters are all-pole filters that exhibit equiripple behavior i the pass bad ad a mootoic characteristic i the stop bad. O the other had, type II Chebyshev filters cotai both poles ad eros ad exhibit a mootoic behavior i the pass bad ad a equiripple behavior i the stop bad. The eros of this class of filters lie o the imagiary axis i the s-plae. The magitude squared of the frequecy respose characteristic of type I Chebyshev filter is give as Where ε is a parameter of the filter related to the ripple i the pass bad as show i Fig. 5.7, ad T is the th order Chebyshev polyomial defied as The Chebyshev polyomials ca be geerated by the recursive equatio SJBIT/ECE Page 6

63 Digital Sigal Processig EC5 Where Tx = ad Tx = x. At the bad edge frequecy Ώ= Ώp, we have Or equivaletly Figure 5.: Type I Chebysehev filter characteristic Where δ is the value of the pass bad ripple. The poles of Type I Chebyshev filter lie o a ellipse i the s-plae with major axis Ad mior axis Where β is related to ε accordig to the equatio SJBIT/ECE Page 63

64 Digital Sigal Processig EC5 The agular positios of the left half s-plae poles are give by The the positios of the left half s-plae poles are give by Where ζ = r Cos φ ad Ώ = r Siφ. The order of the filter is obtaied from Where, by defiitio δ = / +δ. Fially, the Type I Chebyshev filter is give by A Type II Chebyshev filter cotais ero as well as poles. The magitude squared respose is give as Where T x is the -order Chebyshev polyomial. The eros are located o the imagiary axis at the poits ad the left-half s-plae poles are give SJBIT/ECE Page 64

65 Digital Sigal Processig EC5 Where ad Fially, the Type II Chebyshev filter is give by The other approximatio techiques are elliptic equiripple i both passbad ad stopbad ad Bessel mootoic i both passbad ad stopbad. 5.3 Aalog to Aalog Frequecy Trasforms Frequecy trasforms are used to trasform lowpass prototype filter to other filters lie highpass or badpass or badstop filters. Oe possibility is to perform frequecy trasform i the aalog domai ad the covert the aalog filter ito a correspodig digital filter by a mappig of the s-plae ito -plae. A alterative approach is to covert the aalog lowpass filter ito a lowpass digital filter ad the to trasform the lowpass digital filter ito the desired digital filter by a digital trasformatio. Suppose we have a lowpass filter with pass edge Ω P ad if we wat covert that ito aother lowpass filter with pass bad edge Ω P the the trasformatio used is To covert low pass filter ito highpass filter the trasformatio used is SJBIT/ECE Page 65

66 Digital Sigal Processig EC5 Thus we obtai The filter fuctio is SJBIT/ECE Page 66

67 Digital Sigal Processig EC5 Recommeded Questios with aswers Questio I Desig a digital filter to satisfy the followig characteristics. -3dB cutoff frequecy of :5_ rad. agitude dow at least 5dB at :75_ rad. ootoic stop bad ad pass bad Usig Impulse ivariat techique Approximatio of derivatives Biliear trasformatio techique Figure 5.8: Frequecy respose plot of the example Solutio:- a Impulse Ivariat Techique From the give digital domai frequecy, _d the correspodig aalog domai frequecies. Where T is the samplig period ad /T is the samplig frequecy ad it always correspods to Π radias i the digital domai. I this problem, let us assume T = sec. The Ώc = :5Π ad Ώs = :75Π Let us fid the order of the desired filter usig SJBIT/ECE Page 67

68 Digital Sigal Processig EC5 Where δ is the gai at the stop bad edge frequecy ωs. Order of filter =5. The the 5 poles o the Butterworth circle of radius Ώc = :5 Π are give by The the filter trasfer fuctio i the aalog domai is where A 's are partial fractios coefficiets of Has. Fially, the trasfer fuctio of the digital filter is SJBIT/ECE Page 68

69 Digital Sigal Processig EC5 b c For the biliear trasformatio techique, we eed to pre-warp the digital frequecies ito correspodig aalog frequecies. The the order of the filter The pole locatios o the Butterworth circle with radius Ώc = are The the filter trasfer fuctio i the aalog domai is Fially, the trasfer fuctio of the digital filter is SJBIT/ECE Page 69

70 Digital Sigal Processig EC5 Questio Desig a digital filter usig impulse ivariat techique to satisfy followig characteristics i Equiripple i pass bad ad mootoic i stop bad ii -3dB ripple with pass bad edge frequecy at :5П radias. iii agitude dow at least 5dB at :75 П radias. Solutio: Assumig T=, Ώ= :5 П ad s = :75 П The order of desired filter is SJBIT/ECE Page 7

71 Digital Sigal Processig EC5 SJBIT/ECE Page 7

72 Digital Sigal Processig EC5 SJBIT/ECE Page 7

73 Digital Sigal Processig EC5 Questio 3 Solutio:- For the desig specificatios we have SJBIT/ECE Page 73

74 Digital Sigal Processig EC5 Questio 4 Solutio:- SJBIT/ECE Page 74

75 Digital Sigal Processig EC5 UIT 6 FIR FILTER DESIG COTETS:-. ITRODUCTIO TO FIR FILTERS,. DESIG OF FIR FILTERS USIG RECTAGULAR HAIG BARTLET KAISER WIDOWS, 3. FIR FILTER DESIG USIG FREQUECY SAPLIG TECHIQUE RECOEDED READIGS 4. DIGITAL SIGAL PROCESSIG PRICIPLES ALGORITHS & APPLICATIOS, PROAKIS & OALAKIS, PEARSO EDUCATIO, 4 TH EDITIO, EW DELHI, DISCRETE TIE SIGAL PROCESSIG, OPPEHEI & SCHAFFER, PHI, DIGITAL SIGAL PROCESSIG, S. K. ITRA, TATA C-GRAW HILL, D EDITIO, 4. SJBIT/ECE Page 75

76 Digital Sigal Processig EC5 UIT 6 Desig of FIR Filters 6. Itroductio: Two importat classes of digital filters based o impulse respose type are Fiite Impulse Respose FIR Ifiite Impulse Respose IIR The filter ca be expressed i two importat forms as: System fuctio represetatio; H b Differece Equatio represetatio; a a y b x Each of this form allows various methods of implemetatio. The eq ca be viewed as a computatioal procedure a algorithm for determiig the output sequece y of the system from the iput sequece x. Differet realiatios are possible with differet arragemets of eq The major issues cosidered while desigig a digital filters are : Realiability causal or o causal Stability filter output will ot saturate Sharp Cutoff Characteristics Order of the filter eed to be miimum this leads to less delay Geeralied procedure havig sigle procedure for all ids of filters Liear phase characteristics SJBIT/ECE Page 76

77 Digital Sigal Processig EC5 The factors cosidered with filter implemetatio are, a. It must be a simple desig b. There must be modularity i the implemetatio so that ay order filter ca be obtaied with lower order modules. c. Desigs must be as geeral as possible. Havig differet desig procedures for differet types of filters high pass, low pass, is cumbersome ad complex. d. Cost of implemetatio must be as low as possible e. The choice of Software/Hardware realiatio 6. Features of IIR: The importat features of this class of filters ca be listed as: Out put is a fuctio of past o/p, preset ad past i/p s It is recursive i ature It has at least oe Pole i geeral poles ad eros Sharp cutoff chas. is achievable with miimum order Difficult to have liear phase chas over full rage of freq. Typical desig procedure is aalog desig the coversio from aalog to digital 6.3 Features of FIR : The mai features of FIR filter are, They are iheretly stable Filters with liear phase characteristics ca be desiged Simple implemetatio both recursive ad orecursive structures possible Free of limit cycle oscillatios whe implemeted o a fiite-word legth digital system 6.3. Disadvatages: Sharp cutoff at the cost of higher order Higher order leadig to more delay, more memory ad higher cost of implemetatio 6.4 Importace of Liear Phase: The group delay is defied as d g d which is egative differetial of phase fuctio. oliear phase results i differet frequecies experiecig differet delay ad arrivig at differet time at the receiver. This creates problems with speech processig ad data SJBIT/ECE Page 77

78 Digital Sigal Processig EC5 commuicatio applicatios. Havig liear phase esures costat group delay for all frequecies. The further discussios are focused o FIR filter. 6.5 Examples of simple FIR filterig operatios:.uity Gai Filter y=x. Costat gai filter y=kx 3. Uit delay filter y=x- 4.Two - term Differece filter y = x-x- 5. Two-term average filter y =.5x+x- 6. Three-term average filter 3-poit movig average filter y = /3[x+x-+x-] 7. Cetral Differece filter y= /[ x x-] Whe we say Order of the filter it is the umber of previous iputs used to compute the curret output ad Filter coefficiets are the umbers associated with each of the terms x, x-,.. etc The table below shows order ad filter coefficiets of above simple filter types: SJBIT/ECE Page 78

79 Digital Sigal Processig EC5 Ex. order a a a - - K HP - - 5LP / / - 6LP /3 /3 /3 7HP / -/ 6.6 Desig of FIR filters: The sectio to follow will discuss o desig of FIR filter. Sice liear phase ca be achieved with FIR filter we will discuss the coditios required to achieve this Symmetric ad Atisymmetric FIR filters givig out Liear Phase characteristics: Symmetry i filter impulse respose will esure liear phase A FIR filter of legth with i/p x & o/p y is described by the differece equatio: y= b x + b x-+.+b - x-- = b x - Alteratively. it ca be expressed i covolutio form y h x - i.e b = h, =,,..- Filter is also characteried by SJBIT/ECE Page 79

80 Digital Sigal Processig EC5 SJBIT/ECE Page 8 h H -3 polyomial of degree - 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= ± h-- =,,.- -4 Icorporatig this symmetry & ati symmetry coditio i eq 3 we ca show liear phase chas of FIR filters... h h h h h H If is odd h h h h h h h H h h h h h h Applyig symmetry coditios for odd h h h h h h h h h h

81 Digital Sigal Processig EC5 SJBIT/ECE Page 8 / / 3 / / } { } { h H eve for similarly h h H 6.6. Frequecy respose: If the system impulse respose has symmetry property i.e.,h=h-- ad is odd j r j j e H e e H where cos 3 j r j r j r e H if e H if h h e H I case of eve the phase respose remais the same with magitude respose expressed as cos j r h e H If the impulse respose satisfies ati symmetry property i.e., h=-h--the for odd we will have 3 si.,. j r h e H h e i h h If is eve the,

82 Digital Sigal Processig EC5 H e r j h si I both cases the phase respose is give by 3 / / if if H H r r e e j j Which clearly shows presece of Liear Phase characteristics Commets o filter coefficiets: The umber of filter coefficiets that specify the frequecy respose is +/ whe is odd ad / whe is eve i case of symmetric coditios I case of impulse respose atisymmetric we have h-/= so that there are -/ filter coefficiets whe is odd ad / coefficiets whe is eve Choice of Symmetric ad atisymmetric uit sample respose Whe we have a choice betwee differet symmetric properties, the particular oe is piced up based o applicatio for which the filter is used. The followig poits give a isight to this issue. If h=-h-- ad is odd, H r w implies that Hr= & H r π=, cosequetly ot suited for lowpass ad highpass filter. This coditio is suited i Bad Pass filter desig. Similarly if is eve H r = hece ot used for low pass filter Symmetry coditio h=h-- yields a liear-phase FIR filter with o ero respose at w = if desired. Looig at these poits, atisymmetric properties are ot geerally preferred. SJBIT/ECE Page 8

83 Digital Sigal Processig EC Zeros of Liear Phase FIR Filters: Cosider the filter system fuctio H o h Expadig this equatio H sice h h the H H H h h h h h for Liear ; h [ h [ i. e., h h phase we eed h h This shows that if = is a ero the = - is also a ero ;... h ]... h h h H h;... h h h h] The differet possibilities:. If = the = - = is also a ero implyig it is oe ero. If the ero is real ad < the we have pair of eros 3. If ero is complex ad =the ad we agai have pair of complex eros. 4. If ero is complex ad the ad we have two pairs of complex eros SJBIT/ECE Page 83

84 Digital Sigal Processig EC5 The plot above shows distributio of eros for a Liear phase FIR filter. As it ca be see there is patter i distributio of these eros. 6.7 ethods of desigig FIR filters: The stadard methods of desigig FIR filter ca be listed as:. Fourier series based method. Widow based method 3. Frequecy samplig method 6.7. Desig of Liear Phase FIR filter based o Fourier Series method: otivatio: Sice the desired freq respose H d e jω is a periodic fuctio i ω with period π, it ca be expressed as Fourier series expasio H e where h d d j h d h e H d d are e j j e fourier series coefficiets j d This expasio results i impulse respose coefficiets which are ifiite i duratio ad o causal. It ca be made fiite duratio by trucatig the ifiite legth. The liear phase ca be obtaied by itroducig symmetric property i the filter impulse respose, i.e., h = h-. It ca be made causal by itroducig sufficiet delay depeds o filter legth 6.7. Stepwise procedure:. From the desired freq respose usig iverse FT relatio obtai h d. Trucate the ifiite legth of the impulse respose to fiite legth with assumig odd h h d for otherwise / / 3. Itroduce h = h- for liear phase characteristics 4. Write the expressio for H; this is o-causal realiatio 5. To obtai causal realiatio H = --/ H SJBIT/ECE Page 84

85 Digital Sigal Processig EC5 Exercise Problems Problem : Desig a ideal badpass filter with a frequecy respose: j 3 Hd e for 4 4 otherwise Fid the values of h for = ad plot the frequecy respose. h d trucatig to samples we have h For = the value of h is separately evaluated from the basic itegratio h =.5 Other values of h are evaluated from h expressio h=h-= h=h-=-.383 h3=h-3= h4=h-4= h5=h-5= H e si 4 The trasfer fuctio of the filter is 3 d / 4 / 4 e 3 si 4 j j e d j d 3 / 4 / 4 e j d h d for otherwise 5 SJBIT/ECE Page 85

86 Digital Sigal Processig EC5 H ' H ' h h'3 h'5 h h' h' [.5 / the filter coeff are [ h { the trasfer fuctio of h' h'9 }] the realiable filter is ].383 h' 7 h'8 h'4 h'6 The magitude respose ca be expressed as H e H e j / comparig this exp with j 5 a cos [ h 5 h cos ] We have a=h a=h= a=h= a3=h3= a4=h4= a5=h5= The magitude respose fuctio is He jω = cos ω which ca plotted for various values of ω ω i degrees =[ ]; SJBIT/ECE Page 86

87 Digital Sigal Processig EC5 He jω i dbs= [ ]; Problem : Desig a ideal lowpass filter with a freq respose H d e j for for Fid the values of h for =. Fid H. Plot the magitude respose From the freq respose we ca determie h d, / si j h d e d ad / Trucatig h d to samples h = / h=h-=.383 h=h-= h3=h-3=-.6 SJBIT/ECE Page 87

88 Digital Sigal Processig EC5 h4=h-4= h5=h-5=.6366 The realiable filter ca be obtaied by shiftig h by 5 samples to right h =h-5 h = [.6366,, -.6,,.383,.5,.383,, -.6,,.6366]; H ' Usig the result of magitude respose for odd ad symmetry H e r j [ h 3 h cos ] j H e [ cosw.cos3w.7cos5w] r Problem 3 : Desig a ideal bad reject filter with a frequecy respose: j H e d for ad 3 otherwise 3 Fid the values of h for = ad plot the frequecy respose As:h= [ ]; 6.8 Widow based Liear Phase FIR filter desig The other importat method of desigig FIR filter is by maig use of widows. The arbitrary trucatio of impulse respose obtaied through iverse Fourier relatio ca lead to distortios i the fial frequecy respose.the arbitrary trucatio is equivalet to multiplyig ifiite legth fuctio with fiite legth rectagular widow, i.e., h = h d w where w = for = ±-/ The above multiplicatio i time domai correspods to covolutio i freq domai, i.e., SJBIT/ECE Page 88

89 Digital Sigal Processig EC5 H e jω = H d e jω * We jω where We jω is the FT of widow fuctio w. The FT of w is give by W e j si / si / The whole process of multiplyig h by a widow fuctio ad its effect i freq domai are show i below set of figures. SJBIT/ECE Page 89

90 Digital Sigal Processig EC5 Suppose the filter to be desiged is Low pass filter the the covolutio of ideal filter freq respose ad widow fuctio freq respose results i distortio i the resultat filter freq respose. The ideal sharp cutoff chars are lost ad presece of rigig effect is see at the bad edges which is referred to Gibbs Pheomea. This is due to mai lobe width ad side lobes of the widow fuctio freq respose.the mai lobe width itroduces trasitio bad ad side lobes results i ripplig characters i pass bad ad stop bad. Smaller the mai lobe width smaller will be the trasitio bad. The ripples will be of low amplitude if the pea of the first side lobe is far below the mai lobe pea How to reduce the distortios?. Icrease legth of the widow - as icreases the mai lob width becomes arrower, hece the trasitio bad width is decreased -With icrease i legth the side lobe width is decreased but height of each side lobe icreases i such a maer that the area uder each sidelobe remais ivariat to chages i. Thus ripples ad rigig effect i pass-bad ad stop-bad are ot chaged.. Choose widows which tapers off slowly rather tha edig abruptly - Slow taperig reduces rigig ad ripples but geerally icreases trasitio width sice mai lobe width of these id of widows are larger. SJBIT/ECE Page 9

91 Digital Sigal Processig EC What is ideal widow characteristics? Widow havig very small mai lobe width with most of the eergy cotaied with it i.e.,ideal widow freq respose must be impulsive.widow desig is a mathematical problem, more complex the widow lesser are the distortios. Rectagular widow is oe of the simplest widow i terms of computatioal complexity. Widows better tha rectagular widow are, Hammig, Haig, Blacma, Bartlett, Traigular,Kaiser. The differet widow fuctios are discussed i the followig setio Rectagular widow: The mathematical descriptio is give by, w r for Haig widows: It is defied mathematically by, w ha.5 cos for SJBIT/ECE Page 9

92 Digital Sigal Processig EC Hammig widows: This widow fuctio is give by, w ham.54.46cos for Blacma widows: This widow fuctio is give by, 4 w bl.4.5cos.8cos for SJBIT/ECE Page 9

93 Digital Sigal Processig EC Bartlett Triagular widows: The mathematical descriptio is give by, w bart for Kaiser widows: The mathematical descriptio is give by, w I I for SJBIT/ECE Page 93

94 Digital Sigal Processig EC5 Type of widow Appr. Trasitio width of the mai lobe Pea sidelobe db Rectagular 4π/ -3 Bartlett 8π/ -7 Haig 8π/ -3 Hammig 8π/ -43 Blacma π/ -58 Looig at the above table we observe filters which are mathematically simple do ot offer best characteristics. Amog the widow fuctios discussed Kaiser is the most complex oe i terms of fuctioal descriptio whereas it is the oe which offers maximum flexibility i the desig Procedure for desigig liear-phase FIR filters usig widows:. Obtai h d from the desired freq respose usig iverse FT relatio. Trucate the ifiite legth of the impulse respose to fiite legth with SJBIT/ECE Page 94

95 Digital Sigal Processig EC5 assumig odd choosig proper widow h h w where d w is the widow fuctio defied for / / 3. Itroduce h = h- for liear phase characteristics 4. Write the expressio for H; this is o-causal realiatio 5. To obtai causal realiatio H = --/ H Exercise Problems Prob : Desig a ideal highpass filter with a frequecy respose: H d j e for 4 4 usig a haig widow with = ad plot the frequecy respose. h d [ / 4 e j d / 4 e j d ] SJBIT/ECE Page 95

96 Digital Sigal Processig EC5 h h d d [si [ / 4 d si ] 4 / 4 d ] for ad h d = h d -=-.5 h d = h d -= -.59 h d 3 = h d -3= -.75 h d 4 = h d -4= h d 5 = h d -5 =.45 The hammig widow fuctio is give by w h.5.5cos otherwise for w h.5.5cos w h = w h = w h -=.945 w h = w h -=.655 w h 3= w h -3=.345 w h 4= w h -4=.945 w h 5= w h -5= h= w h h d h=[ ] h' h 5 H' Usig the equatio SJBIT/ECE Page 96

97 Digital Sigal Processig EC5 H e r jw [ h 3 h cos H e r jw.75 4 h cos 5 The magitude respose is give by, Hre jω = cosω -.8 cosω -.5cos3ω ω i degrees = [ ] He jω i dbs = [ ] SJBIT/ECE Page 97

98 Digital Sigal Processig EC5 Prob : Desig a filter with a frequecy respose: H d e j e j3 for usig a Haig widow with = 7 Sol: The freq resp is havig a term e jω-/ which gives h symmetrical about = -/ = 3 i.e we get a causal sequece. h d / 4 e / 4 j3 e j d this gives h h h h d d d 3 si h d h d d h d The Haig widow fuctio values are give by w h = w h 6 = w h = w h 5 =.5 w h = w h 4 =.75 w h 3= h=h d w h h=[ ] SJBIT/ECE Page 98

99 Digital Sigal Processig EC5 6.9 Desig of Liear Phase FIR filters usig Frequecy Samplig method: 6.9. otivatio: We ow that DFT of a fiite duratio DT sequece is obtaied by samplig FT of the sequece the DFT samples ca be used i recostructig origial time domai samples if frequecy domai samplig was doe correctly. The samples of FT of h i.e., H are sufficiet to recover h. Sice the desiged filter has to be realiable the h has to be real, hece eve symmetry properties for mag respose H ad odd symmetry properties for phase respose ca be applied. Also, symmetry for h is applied to obtai liear phase chas. Fro DFT relatioship we have h H e j / for,,... H h e j / for,,... Also we ow H = H =e jπ/ The system fuctio H is give by H h Substitutig for h from IDFT relatioship H H j / e SJBIT/ECE Page 99

100 Digital Sigal Processig EC5 SJBIT/ECE Page Sice H is obtaied by samplig He jω hece the method is called Frequecy Samplig Techique. Sice the impulse respose samples or coefficiets of the filter has to be real for filter to be realiable with simple arithmetic operatios, properties of DFT of real sequece ca be used. The followig properties of DFT for real sequeces are useful: H* = H- H = H- - magitude respose is eve θ = - θ- Phase respose is odd / j e H h ca be rewritte as for odd / / / / / j j j e H e H H h e H H h Usig substitutio = r or r = - i the secod substitutio with r goig from ow - / to as goes from to -/ / / / * / / / / * / / / / / * / / / / / Re j j j j j j j j j e H H h e H e H H h e H e H H h e H e H H h e H e H H h Similarly for eve we have / / Re j e H H h

101 Digital Sigal Processig EC5 Usig the symmetry property h= h -- we ca obtai Liear phase FIR filters usig the frequecy samplig techique. Exercise problems Prob : 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. The desired respose ca be expressed as H d e j e with 7 j for otherwise ad c / c H d e j e j 8 for for / / Selectig for,, H j H d e 7 H H e e j j 7 for for for for 7 4 / The rage for ca be adjusted to be a iteger such as ad SJBIT/ECE Page

102 Digital Sigal Processig EC5 The freq respose is give by H e j 8 7 for for Usig these value of H we obtai h from the equatio h H i.e., h 7 h H / Re H e Re e cos j6 /7 8 7 e j / j /7 for,,...6 Eve though varies from to 6 sice we cosidered ω varyig betwee ad π/ oly values from to 8 are cosidered While fidig h we observe symmetry i h such that varyig to 7 ad 9 to 6 have same set of h 6. Desig of FIR Differetiator Differetiators are widely used i Digital ad Aalog systems wheever a derivative of the sigal is eeded. Ideal differetiator has pure liear magitude respose i the freq rage π to +π. The typical frequecy respose characteristics is as show i the below figure. SJBIT/ECE Page

103 Digital Sigal Processig EC5 Problem : Desig a Ideal Differetiator usig a rectagular widow ad bhammig widow with legth of the system = 7. Solutio: As see from differetiator frequecy chars. It is defied as He jω = jω betwee π to +π j cos hd j e d ad The h d is a add fuctio with h d =-h d - ad h d = a rectagular widow h=h d w r h=-h-=hd=- h=-h-=hd=.5 h3=-h-3=hd3=-.33 h =h-3 for causal system thus, H ' Also from the equatio H e r j 3 / h si For =7 ad h as foud above we obtai this as H j r e.66si3 si si j j H e jh e j.66si3 r si si b Hammig widow h=h d w h where w h is give by SJBIT/ECE Page 3

104 Digital Sigal Processig EC5 w h.54.46cos otherwise / / For the preset problem w h.54.46cos The widow fuctio coefficiets are give by for =-3 to +3 Wh= [ ] Thus h = h-5 = [.67, -.55,.77,, -.77,.55, -.67] Similar to the earlier case of rectagular widow we ca differetiator as j j H e jh e j.534si3.3si.54si r write the freq respose of We observe With 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 6. Desig of FIR Hilbert trasformer: Hilbert trasformers are used to obtai phase shift of 9 degree. They are also called j operators. They are typically required i quadrature sigal processig. The Hilbert trasformer SJBIT/ECE Page 4

105 Digital Sigal Processig EC5 is very useful whe out of phase compoet or imagiary part eed to be geerated from available real compoet of the sigal. Problem 3: Desig a ideal Hilbert trasformer usig a rectagular widow ad b Blacma Widow with = Solutio: As see from freq chars it is defied as H d e j j j The impulse respose is give by h d [ je j d je d ] cos At = it is hd = ad hd is a odd fuctio j except a Rectagular widow h = h d w r = h d for -5 5 h =h-5 h= [-.7,, -.,, -.636,,.636,,.,,.7] SJBIT/ECE Page 5

106 Digital Sigal Processig EC5 H e r H e j j 4 j H h si r e j 5 j{.54si 5.44si 3.7si } b Blacma Widow widow fuctio is defied as w b.4.5cos.8cos 5 5 otherwise 5 5 W b = [,.4,.,.59,.849,,.849,.59,.,.4,] for -5 5 h = h-5 = [,, -.44,, -.545,,.545,,.44,, ] H e j j[.848si 3.8si ] SJBIT/ECE Page 6

107 Digital Sigal Processig EC5 Recommeded questios with solutio Questio Solutio:- b agitude plot SJBIT/ECE Page 7

108 Digital Sigal Processig EC5 Phase plot c Hammig widow d Bartlett widow SJBIT/ECE Page 8

109 Digital Sigal Processig EC5 Questio Solutio:- SJBIT/ECE Page 9

110 Digital Sigal Processig EC5 SJBIT/ECE Page

111 Digital Sigal Processig EC5 Questio 3 Solutio:- agitude ad phase respose SJBIT/ECE Page

112 Digital Sigal Processig EC5 Questio 4 Solu tio:- SJBIT/ECE Page

113 Digital Sigal Processig EC5 UIT 7 Desig of IIR Filters from Aalog Filters COTETS:-. DESIG OF IIR FILTERS FRO AALOG FILTERS BUTTERWORTH AD CHEBYSHEV. IPULSE IVARIACE ETHOD 3. APPIG OF TRASFER FUCTIOS 4. APPROXIATIO OF DERIVATIVE BACKWARD DIFFERECE AD BILIEAR TRASFORATIO ETHOD, ATCHED Z TRASFORS 5. VERIFICATIO FOR STABILITY AD LIEARITY DURIG APPIG RECOEDED READIGS:-. DIGITAL SIGAL PROCESSIG PRICIPLES ALGORITHS & APPLICATIOS, PROAKIS & OALAKIS, PEARSO EDUCATIO, 4 TH EDITIO, EW DELHI, 7.. DISCRETE TIE SIGAL PROCESSIG, OPPEHEI & SCHAFFER, PHI, DIGITAL SIGAL PROCESSIG, S. K. ITRA, TATA C-GRAW HILL, D EDITIO, 4. SJBIT/ECE Page 3

114 Digital Sigal Processig EC5 UIT - 7 DESIG OF IIR FILTERS FRO AALOG FILTERS BUTTERWORTH AD CHEBYSHEV 7. Itroductio A digital filter is a liear shift-ivariat discrete-time system that is realied usig fiite precisio arithmetic. The desig of digital filters ivolves three basic steps: The specificatio of the desired properties of the system. The approximatio of these specificatios usig a causal discrete-time system. The realiatio of these specificatios usig _ite precisio arithmetic. These three steps are idepedet; here we focus our attetio o the secod step. The desired digital filter is to be used to filter a digital sigal that is derived from a aalog sigal by meas of periodic samplig. The speci_catios for both aalog ad digital filters are ofte give i the frequecy domai, as for example i the desig of low pass, high pass, bad pass ad bad elimiatio filters. Give the samplig rate, it is straight forward to covert from frequecy specificatios o a aalog _lter to frequecy speci_catios o the correspodig digital filter, the aalog frequecies beig i terms of Hert ad digital frequecies beig i terms of radia frequecy or agle aroud the uit circle with the poit Z=- correspodig to half the samplig frequecy. The least cofusig poit of view toward digital filter desig is to cosider the filter as beig specified i terms of agle aroud the uit circle rather tha i terms of aalog frequecies. Figure 7.: Tolerace limits for approximatio of ideal low-pass filter SJBIT/ECE Page 4

115 Digital Sigal Processig EC5 A separate problem is that of determiig a appropriate set of specificatios o the digital filter. I the case of a low pass filter, for example, the specificatios ofte tae the form of a tolerace scheme, as show i Fig. 4. ay of the filters used i practice are specified by such a tolerace scheme, with o costraits o the phase respose other tha those imposed by stability ad causality requiremets; i.e., the poles of the system fuctio must lie iside the uit circle. Give a set of specificatios i the form of Fig. 7., the ext step is to ad a discrete time liear system whose frequecy respose falls withi the prescribed toleraces. At this poit the filter desig problem becomes a problem i approximatio. I the case of ifiite impulse respose IIR filters, we must approximate the desired frequecy respose by a ratioal fuctio, while i the fiite impulse respose FIR filters case we are cocered with polyomial approximatio. 7. Desig of IIR Filters from Aalog Filters: The traditioal approach to the desig of IIR digital filters ivolves the trasformatio of a aalog filter ito a digital filter meetig prescribed specificatios. This is a reasoable approach because: The art of aalog filter desig is highly advaced ad sice useful results ca be achieved, it is advatageous to utilie the desig procedures already developed for aalog filters. ay useful aalog desig methods have relatively simple closed-form desig formulas. Therefore, digital filter desig methods based o aalog desig formulas are rather simple to implemet. A aalog system ca be described by the differetial equatio Ad the correspodig ratioal fuctio is SJBIT/ECE Page 5

116 Digital Sigal Processig EC The correspodig descriptio for digital filters has the form ad the ratioal fuctio I trasformig a aalog filter to a digital filter we must therefore obtai either Hor h iverse Z-trasform of H i.e., impulse respose from the aalog filter desig. I such trasformatios, we wat the imagiary axis of the S-plae to map ito the fiite circle of the Z-plae, a stable aalog filter should be trasformed to a stable digital filter. That is, if the aalog filter has poles oly i the left-half of S-plae, the the digital filter must have poles oly iside the uit circle. These costraits are basic to all the techiques discussed 7.3 IIR Filter Desig by Impulse Ivariace: This techique of trasformig a aalog filter desig to a digital filter desig correspods to choosig the uit-sample respose of the digital filter as equally spaced samples of the impulse respose of the aalog filter. That is, Where T is the samplig period. Because of uiform samplig, we have Or SJBIT/ECE Page 6

117 Digital Sigal Processig EC5 Figure 7.: appig of s-plae ito -plae Where s = jω ad Ω=ω/T, is the frequecy i aalog domai ad ω is the frequecy i digital domai. From the relatioship Z = e ST it is see that strips of width π/t i the S-plae map ito the etire Z-plae as show i Fig. 7.. The left half of each S-plae strip maps ito iterior of the uit circle, the right half of each S-plae strip maps ito the exterior of the uit circle, ad the imagiary axis of legth π/t of S-plae maps o to oce roud the uit circle of Z-plae. Each horiotal strip of the S-plae is overlaid oto the Z-plae to form the digital filter fuctio from aalog filter fuctio. The frequecy respose of the digital filter is related to the frequecy respose of the Figure 7.3: Illustratio of the effects of aliasig i the impulse ivariace techique SJBIT/ECE Page 7

118 Digital Sigal Processig EC5 aalog filter as From the discussio of the samplig theorem it is clear that if ad oly if The Ufortuately, ay practical aalog filter will ot be bad limited, ad cosequetly there is iterferece betwee successive terms i Eq. 7.8 as illustrated i Fig Because of the aliasig that occurs i the samplig process, the frequecy respose of the resultig digital filter will ot be idetical to the origial aalog frequecy respose. To get the filter desig procedure, let us cosider the system fuctio of the aalog filter expressed i terms of a partial-fractio expasio The correspodig impulse respose is Ad the uit-sample respose of the digital filter is the The system fuctio of the digital filter H is give by SJBIT/ECE Page 8

119 Digital Sigal Processig EC5 I comparig Eqs. 7.9 ad 7. we observe that a pole at s=s i the S-plae trasforms to a pole at exp st i the Z-plae. It is importat to recogie that the impulse ivariat desig procedure does ot correspod to a mappig of the S-plae to the Z-plae. 7.4 IIR Filter Desig By Approximatio Of Derivatives: A secod approach to desig of a digital filter is to approximate the derivatives i Eq. 4. by fiite differeces. If the samples are closer together, the approximatio to the derivative would be icreasigly accurate. For example, suppose that the first derivative is approximated by the first bacward differece Where y=yt. Approximatio to higher-order derivatives are obtaied by repeated applicatio of Eq. 7.3; i.e., For coveiece we defie Applyig Eqs. 7.3, 7.4 ad 7.5 to 7., we obtai Where y = yat ad x = xat. We ote that the operatio [ ] is a liear shiftivariat operator ad that [ ] ca be viewed as a cascade of operators [ ]. I particular Ad SJBIT/ECE Page 9

120 Digital Sigal Processig EC5 Thus taig the Z-trasform of each side i Eq. 7.6, we obtai Comparig Eq. 7.7 to 7., we observe that the digital trasfer fuctio ca be obtaied directly from the aalog trasfer fuctio by meas of a substitutio of variables So that, this techique does ideed truly correspod to a mappig of the S-plae to the Z- plae, accordig to Eq To ivestigate the properties of this mappig, we must express as a fuctio of s, obtaiig Substitutig s = jω, i.e., imagiary axis i S-plae Which correspods to a circle whose ceter is at =/ ad radius is /, as show i Fig It is easily verified that the left half of the S-plae maps ito the iside of the small circle ad the right half of the S-plae maps oto the outside of the small circle. Therefore, although the SJBIT/ECE Page

121 Digital Sigal Processig EC5 requiremet of mappig the jω-axis to the uit circle is ot satisfied, this mappig does satisfy the stability coditio. Figure 4.4: appig of s-plae to -plae correspodig to first bacward-differece approximatio to the derivative I cotrast to the impulse ivariace techique, decreasig the samplig period T, theoretically produces a better filter sice the spectrum teds to be cocetrated i a very small regio of the uit circle. These two procedures are highly usatisfactory for aythig but low pass filters. A alterative approximatio to the derivative is a forward differece ad it provides a mappig ito the ustable digital filters. 7.5 IIR Filter Desig By The Biliear Trasformatio: I the previous sectio a digital filter was derived by approximatig derivatives by differeces. A alterative procedure is based o itegratig the differetial equatio ad the usig a umerical approximatio to the itegral. Cosider the first - order equatio Where y a t is the first derivative of yat. The correspodig aalog system fuctio is We ca write yat as a itegral of y a t, as i SJBIT/ECE Page

122 Digital Sigal Processig EC5 I particular, if t = T ad t = - T, If this itegral is approximated by a trapeoidal rule, we ca write However, from Eq. 7., Substitutig ito Eq. 4. we obtai Where y = yt ad x = xt. Taig the Z-trasform ad solvig for H gives From Eq. 7. it is clear that H is obtaied from Has by the substitutio That is, This ca be show to hold i geeral sice a th - order differetial equatio of the form of Eq. 7. ca be writte as a set of first-order equatios of the form of Eq. 7.. Solvig Eq. 7.3 for gives SJBIT/ECE Page

123 Digital Sigal Processig EC The ivertible trasformatio of Eq. 7.3 is recogied as a biliear trasformatio. To see that this mappig has the property that the imagiary axis i the s-plae maps oto the uit circle i the -plae, cosider = e jω, the from Eq. 7.3, s is give by Figure 7.5: appig of aalog frequecy axis oto the uit circle usig the biliear Trasformatio Thus for o the uit circle, ζ = ad Ω ad ω are related by T Ω/ = ta ω/ or ω = ta - T Ω/ SJBIT/ECE Page 3

124 Digital Sigal Processig EC5 This relatioship is plotted i Fig. 7.5, ad it is referred as frequecy warpig. From the _gure it is clear that the positive ad egative imagiary axis of the s-plae are mapped, respectively, ito the upper ad lower halves of the uit circle i the -plae. I additio to the fact that the imagiary axis i the s-plae maps ito the uit circle i the -plae, the left half of the s-plae maps to the iside of the uit circle ad the right half of the s-plae maps to the outside of the uit circle, as show i Fig Thus we see that the use of the biliear trasformatio yields stable digital filter from aalog filter. Also this trasformatio avoids the problem of aliasig ecoutered with the use of impulse ivariace, because it maps the etire imagiary axis i the s-plae oto the uit circle i the -plae. The price paid for this, however, is the itroductio of a distortio i the frequecy axis. Figure 4.6: appig of the s-plae ito the -plae usig the biliear trasformatio 7.6 The atched-z Trasform: Aother method for covertig a aalog filter ito a equivalet digital filter is to map the poles ad eros of Has directly ito poles ad eros i the -plae. For aalog filter the correspodig digital filter is SJBIT/ECE Page 4

125 Digital Sigal Processig EC5 Where T is the samplig iterval. Thus each factor of the form s-a i Has is mapped ito the factor - e at -. Recommeded questios with solutio Questio SJBIT/ECE Page 5

126 Digital Sigal Processig EC5 Questio Questio 3 SJBIT/ECE Page 6

127 Digital Sigal Processig EC5 Questio 4 Questio 5 SJBIT/ECE Page 7

128 Digital Sigal Processig EC5 SJBIT/ECE Page 8

129 Digital Sigal Processig EC5 Questio 6 SJBIT/ECE Page 9

130 Digital Sigal Processig EC5 Questio 7 SJBIT/ECE Page 3

131 Digital Sigal Processig EC5 SJBIT/ECE Page 3

132 Digital Sigal Processig EC5 SJBIT/ECE Page 3

133 Digital Sigal Processig EC5 UIT 8 Implemetatio of Discrete time systems COTETS:-. IPLEETATIO OF DISCRETE-TIE SYSTES. STRUCTURES FOR IIR AD FIR SYSTES 3. DIRECT FOR I AD DIRECT FOR II SYSTES, 4. CASCADE, LATTICE AD PARALLEL REALIZATIO. RECOEDED READIGS:-. DIGITAL SIGAL PROCESSIG PRICIPLES ALGORITHS & APPLICATIOS, PROAKIS & OALAKIS, PEARSO EDUCATIO, 4 TH EDITIO, EW DELHI, 7.. DISCRETE TIE SIGAL PROCESSIG, OPPEHEI & SCHAFFER, PHI, DIGITAL SIGAL PROCESSIG, S. K. ITRA, TATA C-GRAW HILL, D EDITIO, 4. SJBIT/ECE Page 33

134 Digital Sigal Processig EC5 UIT 8 Implemetatio of Discrete-Time Systems 8. Itroductio The two importat forms of expressig system leadig to differet realiatios of FIR & IIR filters are a Differece equatio form y a y b Ratio of polyomials H Z b a Z Z b x The followig factors ifluece choice of a specific realiatio, Computatioal complexity emory requiremets Fiite-word-legth Pipelie / parallel processig 8.. Computatio Complexity This is do with umber of arithmetic operatios i.e. multiplicatio, additio & divisios. If the realiatio ca have less of these the it will be less complex computatioally. I the recet processors the fetch time from memory & umber of times a compariso betwee two umbers is performed per output sample is also cosidered ad foud to be importat from the poit of view of computatioal complexity. 8.. emory requiremets This is basically umber of memory locatios required to store the system parameters, past iputs, past outputs, ad ay itermediate computed values. Ay realiatio requirig less of these is preferred Fiite-word-legth effects These effects refer to the quatiatio effects that are iheret i ay digital implemetatio of the system, either i hardware or i software. o computig system has ifiite precisio. With fiite precisio there is boud to be errors. These effects are basically to do with trucatio & roudig-off of samples. The extet of this effect varies with type of arithmetic usedfixed or floatig. The serious issue is that the effects have ifluece o system characteristics. A structure which is less sesitive to this effect eed to be chose. SJBIT/ECE Page 34

135 Digital Sigal Processig EC Pipelie / Parallel Processig This is to do with suitability of the structure for pipeliig & parallel processig. The parallel processig ca be i software or hardware. Loger pipeliig mae the system more efficiet. 8. Structure for FIR Systems: FIR system is described by, y b x Or equivaletly, the system fuctio H Z b Z Where we ca idetify h b otherwise Differet FIR Structures used i practice are,. Direct form. Cascade form 3. Frequecy-samplig realiatio 4. Lattice realiatio 8.. Direct Form Structure Covolutio formula is used to express FIR system give by, y h x It is o recursive i structure As ca be see from the above implemetatio it requires - memory locatios for storig the - previous iputs It requires computatioally multiplicatios ad - 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 SJBIT/ECE Page 35

136 Digital Sigal Processig EC5 Prob: Realie the followig system fuctio usig miimum umber of multiplicatio H Z Z Z Z Z Z We recogie h,,,,, is eve = 6, ad we observe h = h-- h = h5- i.e h = h5 h = h4 h = h3 Direct form structure for Liear phase FIR ca be realied Exercise: Realie the followig usig system fuctio usig miimum umber of multiplicatio. H Z Z Z Z Z Z Z Z m=9 h,,,,,,, odd symmetry h = -h--; h = -h8-; hm-/ = h4 = h = -h8; h = -h7; h = -h6; h3 = -h5 SJBIT/ECE Page 36

137 Digital Sigal Processig EC5 8.. Cascade Form Structure The system fuctio HZ is factored ito product of secod order FIR system K H Z H Z Where H Z b bz b Z =,,.. K ad K = iteger part of + / The filter parameter b may be equally distributed amog the K filter sectio, such that b = b b. b 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 i } are real valued. SJBIT/ECE Page 37

138 Digital Sigal Processig EC5 I case of liear phase FIR filter, the symmetry i h implies that the eros of H also exhibit a form of symmetry. If ad * are pair of complex cojugate eros the / ad /* are also a pair complex cojugate eros. Thus simplified fourth order sectios are formed. This is show below, H C C * / / * C C C 3 4 Problem: Realie the differece equatio y x.5x.5x.75x 3 x i cascade form. 4 Y X { Sol: H H H.5.9 H H _ Frequecy samplig realiatio: We ca express system fuctio H i terms of DFT samples H which is give by H H W SJBIT/ECE Page 38

139 Digital Sigal Processig EC5 This form ca be realied with cascade of FIR ad IIR structures. The term - - is realied H as FIR ad the term as IIR structure. W 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. 8.4 Lattice structures 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 ba 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 = whe m = Y/ X = + a - SJBIT/ECE Page 39

140 Digital Sigal Processig EC5 SJBIT/ECE Page 4 y= x+ a x- f is ow as upper chael output ad ras lower chael output. f = r =x The outputs are, y f the a if b r f r a r f f If m= r f y x a x a x y a a X Y Substitutig a ad b i ] si ] ] [ x x x x x x x y x r f ce r f r f r f r f y We recogie

141 Digital Sigal Processig EC5 a a Solvig the above equatio we get a a ad a 4 Equatio 3 meas that, the lattice structure for a secod-order filter is simply a cascade of two first-order filters with ad as defied i eq 4 Similar to above, a th order FIR filter ca be implemeted by lattice structures with stages 8.4. Direct Form I to lattice structure For m =, -,.., do a m m a m m am i am m am m i i i m m SJBIT/ECE Page 4

142 Digital Sigal Processig EC5 The above expressio fails if m =. This is a idicatio that there isa ero o the uit circle. If m =, factor out this root from A ad the recursive formula ca be applied for reduced order system. for m ad m a & a for m a Thus a & i a a a a a [ a a ] a a 8.4. Lattice to direct form I For m =,,.- a a a m m m m i a m m i am m am m i i m Problem: Give FIR filter Give a, 3 Usig the recursive equatio for m =, -,,, here = therefore m =, if m= a 3 if m= a also, whe m= ad i= a a a Hece a 3 H Z Z 3 Z obtai lattice structure for the same a 3 3 SJBIT/ECE Page 4

143 Digital Sigal Processig EC5 Recommeded questios with solutio Problem: Cosider a FIR lattice filter with co-efficiets,, 3. Determie the FIR 3 4 filter co-efficiet for the direct form structure 3 H Z a a Z a Z a 3 Z a a a a 3 for m=, i= a a a a = a [ ] a 3 = for m=3, i= a 3 a a33 a = 3 = 3 = = 3 4 for m=3 & i= a 3 a a33 a = = SJBIT/ECE Page 43

144 Digital Sigal Processig EC5 3 = 6 a 3, 3 a 3, 4 a 3, a Structures for IIR Filters The IIR filters are represeted by system fuctio; HZ = b a ad correspodig differece equatio give by, y a y b x Differet realiatios for IIR filters are,. Direct form-i. Direct form-ii 3. Cascade form 4. Parallel form 5. Lattice form 8.5. 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 feedbac path. The umber of delays depeds o presece of most previous iput ad output samples i the differece equatio. SJBIT/ECE Page 44

145 Digital Sigal Processig EC5 SJBIT/ECE Page Direct form-ii The give trasfer fuctio H ca be expressed as,. V Y X V X Y H where V is a itermediate term. We idetify, a X V all poles b V Y all eros The correspodig differece equatios are, v a x v v b v y

146 Digital Sigal Processig EC5 This realiatio requires ++! multiplicatios, + additio ad the maximum of {, } memory locatio Cascade Form The trasfer fuctio of a system ca be expressed as, SJBIT/ECE Page 46

147 Digital Sigal Processig EC5 H H H... H Where H Z could be first order or secod order sectio realied i Direct form II form i.e., b bz b Z H Z az a Z where K is the iteger part of +/ Similar to FIR cascade realiatio, the parameter b ca be distributed equally amog the filter sectio B that b = b b..b. The secod order sectios are required to realie sectio which has complex-cojugate poles with real co-efficiets. Pairig the two complexcojugate 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 arithmetic, the various realiatios may differ sigificatly whe implemeted with fiite precisio arithmetic Parallel form structure I the expressio of trasfer fuctio, if we ca express system fuctio A H Z C C H Z p Z Where {p } are the poles, {A } are the coefficiets i the partial fractio expasio, ad the costat C is defied as C b a, The system realiatio of above form is show below. b bz Where H Z a Z a Z SJBIT/ECE Page 47

148 Digital Sigal Processig EC5 SJBIT/ECE Page 48 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. Problem Determie the idirect form-i ii Direct form-ii iii Cascade & ivparallel form realiatio of the system fuctio 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 H

149 Digital Sigal Processig EC5 SJBIT/ECE Page 49 Cascade Form H = H H Where H H

150 Digital Sigal Processig EC5 Parallel Form H = H + H H Problem: 3 Obtai the direct form I, direct form-ii Cascade ad parallel form realiatio for the followig system, y= -. y-+.y-+3x+3.6 x-+.6 x- Solutio: The Direct form realiatio is doe directly from the give i/p o/p equatio, show i below diagram Direct form II realiatio Taig ZT o both sides ad fidig H H Y X SJBIT/ECE Page 5

151 Digital Sigal Processig EC5 Cascade form realiatio The trasformer fuctio ca be expressed as: 3.6 H.5.4 which ca be re writte as 3.6 where H ad H.5.4 Parallel Form realiatio The trasfer fuctio ca be expressed as H = C + H + H where H & H is give by, H SJBIT/ECE Page 5

152 Digital Sigal Processig EC5 SJBIT/ECE Page Lattice Structure for IIR System: Cosider a All-pole system with system fuctio. Z A Z a Z H The correspodig differece equatio for this IIR system is, x y a y OR y a y x For = y a y x Which ca realied as, We observe f x g f f y y x a y y g f g For =, the y a y a x y

153 Digital Sigal Processig EC5 This output ca be obtaied from a two-stage lattice filter as show i below fig f x f f g g f g f f g g f g y f f f g g f f g g y y y y y g g x x Similarly g y y y We observe a ; a ; a -stage IIR filter realied i lattice structure is, f x f m f m m g m m=, -,--- g m m f m g m m=, -,--- SJBIT/ECE Page 53

154 Digital Sigal Processig EC5 SJBIT/ECE Page 54 g f y 8.6. Coversio from lattice structure to direct form: ; m m m a m a m a m a a a m m m m Coversio from direct form to lattice structure m a m a m m m a m a m a a a m m m m m 8.6. Lattice Ladder Structure: A geeral IIR filter cotaiig both poles ad eros ca be realied usig a all pole lattice as the basic buildig bloc. If, Z a Z b Z A Z B Z H Where A lattice structure ca be costructed by first realiig a all-pole lattice co-efficiets m m, for the deomiator A Z, ad the addig a ladder part for =. The output of the ladder part ca be expressed as a weighted liear combiatio of {g m }. ow the output is give by m m m g C y Where {C m } are called the ladder co-efficiet ad ca be obtaied usig the recursive relatio, ; m i i i m m m i C a b C m=, -,.

155 Digital Sigal Processig EC5 Problem:4 Covert the followig pole-ero IIR filter ito a lattice ladder structure, 3 Z Z Z H Z 3 Z Z Z Solutio: Give b Z Z Z Z A Z 4 Z 8 Z 3 Z ; a3 4 ; a3 8 ; a33 3 a33 3 Ad a Usig the equatio am am m am m am a m m for m=3, = a3 a33 a3 a a 3 for m=3, & = a a3 a33 a3 a for m=, & = a a a a a SJBIT/ECE Page