Digital Signal Processing

Transcription

1 Digial Signal Processing by Andreas Spanias, Ph.D Professor and SenSIP Cener Co- direcor Deparmen Elecrical Engineering Arizona Sae Universiy Tempe AZ February 27 for Universiy of Cyprus 27 Copyrigh 27 Andreas Spanias I- Digial Signal Processing (DSP) Inroducion Digial Signal Processing (DSP) is a branch of signal processing ha emerged from he rapid developmen of VLSI echnology ha made feasible real-ime digial compuaion. DSP involves ime and ampliude quanizaion of signals and relies on he heory of discree-ime signals and sysems. DSP emerged as a field in he 96s. Early applicaions of off-line DSP include seismic daa analysis, voice processing research. 27 Copyrigh 27 Andreas Spanias I-2

2 Digial vs Analog Signal Processing Advanages of digial over analog signal processing: flexibiliy via programmable DSP operaions, sorage of signals wihou loss of fideliy, off-line processing, lower sensiiviy o hardware olerances, rich media daa processing capabiliies, opporuniies for encrypion in communicaions, Mulimode funcionaliy and opporuniies for sofware radio. -Disadvanages : Large bandwidh and CPU demands 27 Copyrigh 27 Andreas Spanias I-3 DSP Hisorical Perspecive Nyquis Theorem 92's. Saisical Time Series, PCM 94's. Digial Filering, FFT, Speech Analysis mid 96s (MIT, Bell Labs, IBM). Adapive Filers, Linear Predicion (Sanford, Bell Labs, Japan 96s). Digial Specral Esimaion, Speech Coding (97s). 27 Copyrigh 27 Andreas Spanias I-4

3 DSP Hisorical Perspecive (2) Firs Generaion DSP Chips (Inel microconroler, TI, AT&T, Moorola, Analog Devices (early 98s) Low-cos DSPs (lae 98s) Vocoder Sandards for civilian applicaions (lae 98s) Migraion of DSP echnologies in general purpose CPU/Conrollers "naive" DSP (99s) High Complexiy Rich Media Applicaions Low Power (Porable) Applicaions 27 Copyrigh 27 Andreas Spanias I-5 DSP Applicaions Miliary Applicaions (arge racking, radar, sonar, secure communicaions, sensors, imagery) Telecommunicaions (cellular, channel equalizaion, vocoders, sofware radioec) PC and Mulimedia Applicaions (audio/video on demand, sreaming daa applicaions, voice synhesis/recogniion) Enerainmen (digial audio/video compression, MPEG, CD, MD, DVD, MP3) Auomoive (Acive noise cancellaion, hands-free communicaions, navigaion-gps, IVHS) Manufacuring, insrumenaion, biomedical, oil exploraion, roboics Remoe sensing, securiy 27 Copyrigh 27 Andreas Spanias I-6

4 Communicaions and DSP DTMF (use of he FFT and digial oscillaors) Adapive echo cancellaion (Hands-free elephony, Speakerphones) Speech coding (speech coding in cellular phones) Modem (daa/compuer conneciviy) Sofware radio (muli-mode/muli sandard wireless communicaions) Channel esimaion (equalizaion) Anenna beamforming (space division muliple access - SDMA) CDMA (modulaing wih random sequences) 27 Copyrigh 27 Andreas Spanias I-7 Typical Digial Signal Processing Sysem Nowdays LPF and A/D inegraed DSP chip x() x`() sample x(nt) LPF & A/D Digial Signal Processor Anialiasing f s Reconsrucion y(nt) y() LPF y`() D/A Nowdays LPF and D/A inegraed Remarks: The diagram shows he sampling, processing, and reconsrucion of an analog signal. There are applicaions where processing sops a he digial signal processor, e.g., speech recogniion. 27 Copyrigh 27 Andreas Spanias I-8

5 x a ( ) nt Symbols and Noaion x( nt ) x( n) ;discree ime inpu y( n) ;discree ime oupu H (.) ; ransfer and frequency response funcions h(.) ;impulse response(sysemfuncion) n ;discree ime index Remarks: In general and unless oherwise saed lower case symbols will be used for ime-domain signals and upper case symbols will be used for ransform domain signals. Bold face or underlined face symbols will be Be generally used for vecors or marices. 27 Copyrigh 27 Andreas Spanias I-9 Coninuous vs Discree-ime x() Coninuous-ime (analog) Signal x(n) Discree-ime (digial) signal T 2T n x() Q x(n) Remarks: A coninuous-ime signal is convered o discree-ime using sampling and quanizaion. As a resul aliasing and quanizaion noise is inroduced. This noise can be conrolled by properly designing he quanizer and ani-aliasing filer. 27 Copyrigh 27 Andreas Spanias I-

6 Quanizaion Noise quanized waveform x q () quanizaion noise e q () sampling period T x a () analog waveform x q ( ) x ( ) e ( ) 27 Copyrigh 27 Andreas Spanias I- q Simples Quanizaion Scheme - Uniform PCM Performance in erms of Signal o Noise Raio (SNR) SNR PCM 6.2R b K where R b is he number of bis and he value of K depends on signal saisics. For elephone speech K = - 27 Copyrigh 27 Andreas Spanias I-2

7 Oversampling / or / Conversion Inegraed oversampling and -bi quanizaion Very compac and inexpensive circuiry (some low power applicaions as well) Lowers analog circui complexiy wih a modes increase in sofware (DSP MIPS) complexiy Uses conceps from mulirae signal processing and Dela Modulaion Will be described in he conex of mulirae signal processing 27 Copyrigh 27 Andreas Spanias I-3 Time vs Frequency Domain x() im e-dom ain X(f) frequency-domain f x() X(f) f Remarks: Slowly ime-varying signals end o have low-frequency conen while signals wih abrup changes in heir ampliudes have high frequency conen. The frequency conen of signals can be esimaed using Fourier echniques. 27 Copyrigh 27 Andreas Spanias I-4

8 Example: Time vs Frequency Domain Speech. Time domain speech segmen 5 fundamenal TAPE TIME: 84 frequency Forman Srucure Ampliude. Magniude (db) Time (ms) Frequency (KHz) Periodic waveform gives harmonic specra. Time domain speech segmen TAPE TIME: Ampliude. Magniude (db) Time (ms) Frequency (KHz) 27 Copyrigh 27 Andreas Spanias I-5 Review of Analog Signals and Sysems FREQUENCY DOMAIN ANALYSIS The Fourier series (measuring he specrum of periodic signals) The Fourier ransform (measuring he specrum of non-periodic signals and generally all signals) SAMPLING The Sampling heorem (how we conver o digial signals wihou losing informaion) FILTERS Coninuous-ime sysems (analog filers) Convoluion (how filering is done) 27 Copyrigh 27 Andreas Spanias I-6

9 Coninuous-ime Impulse Some Imporan Signals ( ) lim { /.. }.. Discree-ime Impulse (n).... n Think of signals as a sum of impulses. Impulses help in analyzing signals and filers 27 Copyrigh 27 Andreas Spanias I-7 Coninuous-ime uni sep Discree-ime uni sep Some Imporan Signals (2) u() {.. u (n).. n..} {... } 27 Copyrigh 27 Andreas Spanias I-8

10 The sinusoid Some Imporan Signals (3) Period T 2 sin( ) sin( ) T { } 2 f 2 T unis: (rad/s) f (Hz) T(s) Sinusoids are used in analyzing or synhesizing acousic and oher signals 27 Copyrigh 27 Andreas Spanias I-9 Some Imporan Signals (4) The sinc funcion sidelobes mainlobe { sin( ) sinc ( ) } 2 Sinc funcions ofen appear in signal and filer analysis paricularly when considering frequency domain behavior 27 Copyrigh 27 Andreas Spanias I-2

11 Random noise Some Imporan Signals (5) Encounered in communicaion sysems and oher applicaion Characerized by heir mean and variance 27 Copyrigh 27 Andreas Spanias I-2 Frequency-domain represenaions of signals In order o observe and analyze he specrum, he signal is usually represened in erms of oher basic ( basis ) signals. Basis signals or more precisely basis funcions are ypically chosen o be orhogonal. The mos common orhogonal basis funcion used in signal analysis is he sinusoid. This is mainly because of: he physical properies of a sinusoid, i.e., as an acousic one he fac ha sinusoids are eigenfuncions of linear sysems ( Sinusoid In Sinusoid Ou ) linear sysem 27 Copyrigh 27 Andreas Spanias I-22

12 x( ) Represening Periodic Signals wih Sinusoids Fourier series: Trigonomeric form: a a k cos( k o ) bk sin( k o ) k k Fourier series: Complex (magniude/phase) form: Preferred in engineering -->> x( ) k X k e jk o X k are complex F.S. coefficiens and provide specral magniude and phase info and e jk o cos( k ) o j sin( k ) o 27 Copyrigh 27 Andreas Spanias I-23 The Complex Fourier Series x( ) k X k e jk o Synhesis Expression X k T jk o x( ) e d Analysis Expression X T k are discree F.S. specral coefficiens where o 2 / T The magniude of F.S. coefficiens, X k, provides info on frequency conen. Phase of X k ofen provides info on evens in signal (e.g., beginning of a period p ec.) 27 Copyrigh 27 Andreas Spanias I-24

13 Use Sinusoids o synhesize a periodic pulse using he Fourier series (only one period shown) sinusoids sinusoid 2 sinusoids 5 sinusoids 3 sinusoids sinusoids 27 Copyrigh 27 Andreas Spanias I-25 Fourier Series Analysis Example Represening a Periodic Pulse Train as a Sum of Harmonic Sinusoids x() d T X k T d / 2 e d / 2 jk o d k o d d sinc T 2 Remarks: A periodic pulse signal has a discree F.S. specrum described by samples ha fall on a sinc (sinc(x)=sin(x)/x) funcion. As he period increases he F.S. componens become more dense in frequency and weaker in ampliude. If T goes o infiniy periodiciy is los and he F.S. vanishes. 27 Copyrigh 27 Andreas Spanias I-26

14 Fourier Series Example (2) Harmonic Specrum d T / 4 X k 2 o harmonics 3 o Fundamenal frequency o 27 Copyrigh 27 Andreas Spanias I-27 d/t=/5 Fourier Series Example (3) 4 d/t=/ 2 27 Copyrigh 27 Andreas Spanias I-28

15 Remarks on he Fourier Series F.S. represens periodic signals wih a sum of harmonic sinusoids he F.S.specrum is discree and F.S. componens correspond o ineger muliples of he fundamenal frequency periodic signals have a discree specrum a uniformly sampled specrum implies periodiciy in he ime domain A discree-ime F.S. is also available T If he F.S. vanishes F.S. can be used for specral analysis, filer design, and many oher applicaions 27 Copyrigh 27 Andreas Spanias I-29 Seleced F.S. Properies Lineariy (superposiion holds addiion&scaling in ime is addiion&scaling in frequency) if x( ) F. S. X k y( ) z ( ) x ( ) y ( ) F. S. Y k hen Z k X k Y k Parseval s Theorem (power is preserved from ime o frequency) o T 2 2 k X k 27 Copyrigh 27 Andreas Spanias I-3 T o x ( ) d

16 From he F.S. o he Coninuous Fourier Transform For non-periodic signals x( ) x( ) 2 [ lim { 2 lim { 2 x( ) e k k j T X k e / jk / d ] e x( ) e j } & jk d 2 d e o jk X ( ) e } Remarks: For non-periodic signals he F.S. vanishes. If he limi is aken hen we can derive he coninuous Fourier ransform. The las equaion is known as he inverse Fourier ransform. Noe ha is now a coninuous variable j d 27 Copyrigh 27 Andreas Spanias I-3 The Coninuous Fourier Transform (CFT) Equaions The Fourier ransform The inverse Fourier ransform X ( ) x ( ) e j d j x () X( ) e d 2 Analysis Expression Synhesis Expression A Fourier ransform pair is designaed by: x ( ) X ( ) Remarks: Boh ime and frequency are coninuous variables. The CFT can handle non-periodic signals as long as hey are inegrable. Periodic signals can be handled using he impulse and CFT properies. 27 Copyrigh 27 Andreas Spanias I-32

17 Fourier ransform of a ime-limied pulse Given he signal x X ( ) d / 2 e d / 2 j d d d sinc d 2 Remarks: Noe ha a ime-limied signal has a non-bandlimied CFT specrum. The sinc funcion has zero crossings a ineger muliples of 2/d. As he pulse widh increases he sinc funcion shrinks. In he limi, if T goes o infiniy (i.e., pulse becomes D.C. signal) he sinc funcion collapses o a uni impulse. 27 Copyrigh 27 Andreas Spanias I-33 Fourier ransform of a ime-limied pulse(con.) X ( ) d d sinc 2 pulse ime domain d 2 /d a ransform pair 4/d frequency domain sinc 27 Copyrigh 27 Andreas Spanias I-34

18 Remarks on he CFT Proper compuaion of he CFT specrum requires ha he signal is known everywhere in ime Specra of runcaed signals suffer from specral leakage and loss of resoluion A ime-limied signal has a non-band-limied CFT specrum A band-limied signal can no be ime-limied The forward and inverse CFT formulas are symmeric and herefore we observe dualiies in CFT ransform pairs and CFT properies Numerical compuaion of he CFT is done using he fas Fourier Transform (FFT) 27 Copyrigh 27 Andreas Spanias I-35 Symmery of he Fourier ransform if x ( ) X ( ) hen X ( ) 2 x ( ) (ime-limied) x() (non band-limied) X() (non ime-limied) X() (band-limied) 2 x(-) 27 Copyrigh 27 Andreas Spanias I-36

19 Seleced F.T properies - Lineariy if x ( ) X ( ) & y ( ) Y ( ) hen Example: x ( ) y ( ) X ( ) Y ( ) x ( ) y ( ) x ( ) y ( ) 27 Copyrigh 27 Andreas Spanias I-37 Seleced F.T properies - Scaling sreching a signal in ime implies compressing i in frequency compressing a signal in ime implies sreching i in frequency x ( a ) a X ( ) a Noe ha ime expansion implies frequency compression 27 Copyrigh 27 Andreas Spanias I-38

20 Seleced F.T Properies - Time Shif if x( ) X ( ) j hen ( ) x e X( ) -linear phase facor -has uni magniude - phase is linear across frequency Remarks: A ime shif inroduces linear phase in he frequency domain 27 Copyrigh 27 Andreas Spanias I-39 Seleced F.T Properies Frequency Shif MODULATION - VERY IMPORTANT IN WIRELESS COMMUNICATIONS If x( ) X ( ) hen e j x ) X( ) ( muliplicaion by sinusoid ranslaes he signal in frequency 27 Copyrigh 27 Andreas Spanias I-4

21 The Time-Domain Convoluion (Filering) Propery x ( ) X ( ) h ( ) H ( ) h( )* x( ) H ( ) X ( ) h ( ) * x ( ) h ( ) x ( ) d Muliplicaion in frequency is essenially a filering operaion * convoluion in ime is muliplicaion in frequency = Example: Convoluion of an exponenial wih a pulse DEMO 27 Copyrigh 27 Andreas Spanias I-4 Frequency Convoluion x ( ) w ( ) X ( ) W ( ) hen h( ) w( ) W ( )* X ( ) 2 If w() is ime limied hen his operaion runcaes Convoluion ends o have a spreading effec 27 Copyrigh 27 Andreas Spanias I-42

22 Imporan Fourier Transform Pairs ( ) 2 ( ) e j 2 ( ) 27 Copyrigh 27 Andreas Spanias I-43 Imporan Fourier Transform Pairs (2) cos( ) ( ( ) ( )) (- O) -O O 27 Copyrigh 27 Andreas Spanias I-44

23 Frequency Convoluion and Windowing Truncaion Truncaed Signals:.. w ( ) x ( ) x ( ) w ( ) Remarks: Truncaing an infinie-lengh signal is equivalen o muliplying i wih a finie-lengh window. Muliplying a sinusoid wih a recangular pulse resuls in a finie-lengh sinusoid. All real-life signals are finie lengh. 27 Copyrigh 27 Andreas Spanias I-45 s () cos( ) Truncaing a Cosine S( ) ( ( ) ( )) where 2 /T w () CFT T ; T w () ; oherwise CFT jt ( ) /2 T W Te sinc 2 s () s() w() w jt S /2 w( ) Te sinc T/2 +sinc T/2 CFT T T 27 Copyrigh 27 Andreas Spanias I-46

24 Truncaing in ime implies convoluion in frequency Truncaion wih a shor recangular window implies convoluion wih a wideband sinc Remark: Truncaion wih a wider recangular window implies convoluion wih a narrowband sinc. If he lengh of he recangular window becomes arbirarily long he sinc collapses o an impulse. Clearly, longer windows in ime imply improved specral resoluion and less specral leakage. 27 Copyrigh 27 Andreas Spanias I-47 Truncaing Signals wih Tapered Windows.... B 3dB 25dB B 27 Copyrigh 27 Andreas Spanias I-48

25 Truncaing Speech CFT Normalized frequency x rad/sec 27 Copyrigh 27 Andreas Spanias I-49 Truncaing Speech (apered window) CFT Normalized frequency x rad/sec 27 Copyrigh 27 Andreas Spanias I-5

26 Truncaing Signals wih Tapered Windows (2) Truncaion of a signal is ineviable in real life specral esimaion. Srong sidelobes conribue o specral leakage and specral smearing. The widh of he mailobe affecs he resoluion of specral esimaes. In choosing a window one is confroned wih he radeoff of he mainlobe widh and sidelobe level. Tapered windows have suppressed sidelobes relaive o a recangular window. However, he mainlobe widh of a apered window is wider han ha of a recangular window Truncaion is also involved in designing FIR digial filers from Fourier series componens. The sidelobe level affecs he rejecion and ripple characerisics while he mainobe widh affecs he ransiion characerisics. 27 Copyrigh 27 Andreas Spanias I-5 The Sampling Process A bandlimied signal ha has no specral componens a or above B can be uniquely represened by is sampled values spaced a uniform inervals ha are no more han /B seconds apar. T B or a signal ha is bandlimied o B mus be sampled a a rae of s where 2B s or f s B analog signal x = sampling digial signal 27 Copyrigh 27 Andreas Spanias I-52

27 Example: Audio - Bandwidh 2-32 Hz Basic Telephone Speech Inelligible Preserves Speaker Ideniy 5-7 Hz Wideband Speech AM- grade audio 5-5 Hz Near High Fideliy FM- grade Audio 2-2 Hz High- Fideliy CD/DAT Qualiy Voice 27 Copyrigh 27 Andreas Spanias Example: Sampling of Audio Signals Telephony Forma Wideband audio High-fideliy, CD Digial audio ape (DAT) Super audio CD (SACD) DVD audio (DVD-A) Bandwidh 3.2 khz 7 khz 2 khz 2 khz khz 96 khz Sampling frequency 8 khz 6 khz 44. khz 48 khz MHz 92 khz 27 Copyrigh 27 Andreas Spanias

28 The Mah Represenaion of Sampling Engineering represenaion s x s x Mahemaical represenaion x X x s s Remark: Muliplicaion wih he ideal swiching funcion resuls in a periodic specrum where he signal specrum is repeaed a ineger muliples of he sampling frequency. 27 Copyrigh 27 Andreas Spanias I-55 Sampling by muliplying wih impulses The swiching or sampling funcion s () ( nt) n s T 2T 3T analog signal x = sampling digial signal 27 Copyrigh 27 Andreas Spanias I-56

29 The Sampling Signal in he Frequency Domain I can be easily shown ha k nt ) k k ( s S T 2T 3T o 2 o 27 Copyrigh 27 Andreas Spanias I-57 Ideal Low-Pass filers and Bandlimied Signals Ideal Low-Pass Filer H B h B h Ideally Bandlimied Signal X B B 27 Copyrigh 27 Andreas Spanias I-58

30 Sampling and Periodic Specra x() X () () x s B B () X s /T T 2T 3T s B B s 2 s 27 Copyrigh 27 Andreas Spanias I-59 Signal Reconsrucion using an Ideal Filer () x s () X s T /T T 2T 3T s B B s 2 s x() X () B B 27 Copyrigh 27 Andreas Spanias I-6

31 Derivaion of he Sampling Theorem x () x()() s s where s () ( nt) n n x ( ) x ( nt ) ( nt ) s 2 X s ( ) X ( ) * k o T k X s ( ) X k o T and X s ( ) X ( ) * S ( ) k 27 Copyrigh 27 Andreas Spanias I-6 Signal Reconsrucion Analyically for s =2B h ( ) * x ( ) H ( ) X ( ) s j h ( ) H ( ) e d sinc ( B 2 x ( ) sinc( B)* x( nt) ( nt) n x ( ) x( nt )sinc( B( nt )) n Remark: Noe ha he reconsrucion filer inerpolaes beween he samples wih sinc funcions - hence he name inerpolaion filer. s ) 27 Copyrigh 27 Andreas Spanias I-62

32 ALIASING (UNDERSAMPLING) s <2B X s () B aliasing B s 2 s he signal can no be recovered perfecly even wih an ideal filer only a disored version of he signal can be recovered 27 Copyrigh 27 Andreas Spanias I-63 Oversampling s >>2B () X s s B B Guard bands s 2 s Oversampling relaxes he requiremens on anialiasing filers I is used in / (/ ) ) A-oA o-d D converers 27 Copyrigh 27 Andreas Spanias I-64

33 Non-ideal Consideraions of Sampling There are no ideal impulses in pracice - insead finie ampliude and finie duraion periodic pulses are used (no a big problem) All real-life life signals are no band-limied There are no ideal LPF Aliasing is always presen and can be viewed as noise in he signal Typically we use anialiasing filers ha limi aliasing noise several s of dbs under he useful signal energy Pracical Rule: Quanizaion noise reduces generally 6 db per added a bi of resoluion, e.g., a 6 bis we have approximaely 96 db SNR. Therefore we could buil an anialiasing filer ha keeps aliasing noise under he quanizaion noise by a leas 6 db. Bandpass signals can be sampled more efficienly 27 Copyrigh 27 Andreas Spanias I-65 Coninuous Linear Sysems In a linear sysem if we superimpose wo disinc inpu signals x () and x 2 () we ge an oupu ha consiss of he superposiion of he responses o each individual inpu, i.e., x () Linear Sysem y () x 2 () Linear Sysem y 2 () a x ()+ b x 2 () Linear Sysem a y ()+ b y 2 () 27 Copyrigh 27 Andreas Spanias I-66

34 R Example : RC Circui y() y y 2 ( ) ( ) x() R R x x 2 ( ) ( ) C c c Assuming zero iniial condiions x x ( ) d 2 ( ) d Noe ha: y ) y ( ) y ( ) ( 2 27 Copyrigh 27 Andreas Spanias I-67 Remarks on Coninuous Linear Sysems Analysis of coninuous linear sysems (CLS) relies on he heory of linear differenial equaions (LDE). The ransien and seady-sae responses of a CLS are obained from he homogeneous and paricular soluions respecively of he LDE. The oupu of a sysem can be obained from he convoluion inegral of is impulse response convolved wih he inpu The frequency response of a linear sysem is defined as he seadysae response o a specrum of sinusoids. The seady-sae oupu of a linear sysem due o a sinusoidal inpu is a sinusoid of he same frequency bu phase shifed and ampliude scaled. 27 Copyrigh 27 Andreas Spanias I-68

35 R Example x() i() C y() Differenial equaion: dy( ) d RC y ( ) RC x ( ) Transfer funcion: Hs () src Frequency response funcion: H ( ) jrc y ss if x( ) sin( ) hen sinusoid in sinusoid ou ( ) H ( ) sin( H ( )) 27 Copyrigh 27 Andreas Spanias I-69 Sin In / Sin Ou LTI sysem 27 Copyrigh 27 Andreas Spanias I-7

36 Coninuous Linear Sysems (Con.) x() h() y() The oupu is obained by convolving he inpu x() and he impulse response h() of he sysem, ha is: y( ) h( ) x( ) d h( ) * x( ) For a causal sysem and causal inpu: y ( ) h ( ) x ( ) d DEMO 27 Copyrigh 27 Andreas Spanias I-7 Example - Impulse Response Consider he circui below wih R=M, C=x -6 R C i() x() y() dh() d RC h () () RC The soluion: h () RC e RC e for > 27 Copyrigh 27 Andreas Spanias I-72..

37 Example - Convolve and obain an oupu Consider he RC wih impulse response h ( ) e u ( ) and he inpu x() u() u( ) y ( ) e d e for.... y ( ) e d e e ( ) for.. * = Copyrigh 27 Andreas Spanias I-73 Convoluion of Pulses x() h() y() * = Copyrigh 27 Andreas Spanias I-74