Digital Sigal Procssig Brli C 4 Rfrcs:.. V. Oppi ad R. W. Scafr, Discrt-ti Sigal Procssig, 999.. uag t. al., Spo Laguag Procssig, Captrs 5, 6. J. R. Dllr t. al., Discrt-Ti Procssig of Spc Sigals, Captrs 4-6 4. J. W. Pico, Sigal odlig tciqus i spc rcogitio, procdigs of t IEEE, Sptbr 99, pp. 5-47
Digital Sigal Procssig Digital Sigal Discrt-ti sigal wit discrt aplitud [] Digital Sigal Procssig Maipulat digital sigals i a digital coputr 4 SP- Brli C
Two Mai pproacs to Digital Sigal Procssig Filtrig Sigal i [] Filtr plify or attuat so frqucy copots of [ ] Sigal out y [ ] Paratr Etractio Sigal i [] Paratr Etractio.g.:. Spctru Estiatio. Paratrs for Rcogitio Paratr out c c c c c c c c c L L L 4 SP- Brli C
[ ] ( φ ) Siusoid Sigals : aplitud ( 振幅 ) : agular frqucy ( 角頻率 ), : pas ( 相角 ) φ f : oralidfrqucy f f T Priod, rprstd by ubr of sapls [] [ ] ( ) (,,... ) E.g., spc sigals ca b dcoposd as sus of siusoids [] T 5 sapls 4 SP- Brli C 4
[ ] Siusoid Sigals (cot.) is priodic wit a priod of (sapls) [ ] [ ] ( ) ( φ ) ( ) φ (,,... ) Eapls (siusoid sigals) [] ( ) is priodic wit priod 8 / 4 [] ( / 8) is priodic wit priod 6 [] ( ) is ot priodic 4 SP- Brli C 5
4 SP- Brli C 6 Siusoid Sigals (cot.) [ ] ( ) ( ) 8 itgrs positiv ar ad 8 4 4 4 ) ( 4 4 4 / [ ] ( ) ( ) ( ) 6 ubrs positiv ar ad 6 8 8 8 8 8 8 / [ ] ( ) ( ) ( ) ( ) ( ) t ist! dos' itgrs positiv ar ad Q
Siusoid Sigals (cot.) Copl Epotial Sigal Us Eulr s rlatio to prss copl ubrs y φ ( φ si φ ) y y y y ( is a ral ubr ) Iagiary part I φ y si φ R ral part 4 SP- Brli C 7
Siusoid Sigals (cot.) Siusoid Sigal ral part [] ( φ ) R R { ( φ ) } { φ } 4 SP- Brli C 8
Siusoid Sigals (cot.) Su of two copl potial sigals wit sa frqucy ( φ ) ( φ ) ( ) φ φ φ ( φ ) W oly t ral part is idrd T su of siusoids of t sa frqucy is aotr siusoid of t sa frqucy ( φ ) ( φ ) ( φ ), ad ar ral ubrs 4 SP- Brli C 9
4 SP- Brli C Siusoid Sigals (cot.) Trigootric Idtitis si si ta φ φ φ φ φ ( ) ( ) ( ) ( ) si si si si φ φ φ φ φ φ φ φ φ φ
So Digital Sigals 4 SP- Brli C
4 SP- Brli C So Digital Sigals y sigal squc ca b rprstd as a su of sift ad scald uit ipuls squcs (sigals) [ ] [] [] [ ] δ scal/wigtd Ti-siftd uit ipuls squc [] [] [ ] [] [ ] [ ] [ ] [ ] [ ] [ ] [ ] [ ] [ ] [ ] [ ] [ ] [ ] () [ ] ( ) [ ] ( ) [] () [ ] ( ) [ ] () [ ] δ δ δ δ δ δ δ δ δ δ δ δ δ δ,,...,,,,...,
Digital Systs digital syst T is a syst tat, giv a iput sigal [], grats a output sigal y[] [ ] T { [ ] } y [] { } T y[ ] 4 SP- Brli C
Liar Proprtis of Digital Systs Liar cobiatio of iputs aps to liar cobiatio of outputs { a [ ] b [ ] } at { [ ] } bt { [ ] } T Ti-ivariat (Ti-sift) ti sift of i t iput by sapls giv a sift i t output by sapls y ± T ±, [ ] { [ ]} ti sift [ ] (if > ) rigt sift [ ] (if > ) lft sift sapls sapls 4 SP- Brli C 4
Proprtis of Digital Systs (cot.) Liar ti-ivariat (LTI) T syst output ca b prssd as a covolutio ( 迴旋積分 ) of t iput [] ad t ipuls rspos [] T syst ca b caractrid by t syst s ipuls rspos [], wic also is a sigal squc If t iput [] is ipuls δ [], t output is [] δ [ ] [ ] Digital Syst 4 SP- Brli C 5
Proprtis of Digital Systs (cot.) Liar ti-ivariat (LTI) Eplaatio: Ipuls rspos T δ [ ] [ ] Ti ivariat δ δ Digital Syst [ ] [ ] δ [ ] T [] [ ] T [ ] [ ] { } { [] } T [] δ [ ] scal [] T { δ [ ]} [][ ] [] [] Ti-siftd uit ipuls squc liar Ti-ivariat covolutio 4 SP- Brli C 6
Proprtis of Digital Systs (cot.) Liar ti-ivariat (LTI) Covolutio Eapl δ[] [ ] δ [ ] δ LgtL [ ] δ LTI [ ]? LTI [] [ ] [ ] [ ] LgtM - LgtLM- 9-6 6-4 4 - Su up y[] - 4-4 SP- Brli C 7
Proprtis of Digital Systs (cot.) Liar ti-ivariat (LTI) Covolutio: Graliatio Rflct [] about t origi ( [-]) Slid([-] [-] or [-(-)] ), ultiply it wit [] Su up [ ] y [] [][ ] [ ] [ ( )] [ ] Rflct Multiply [ ] slid Su up 4 SP- Brli C 8
[ ] - y [ ] [ ] [ ] Rflct [ ] y[ ], - [ ] - - [ ] [ ] [ ] [ 4] Covolutio - - y - y - - y 4 4 - - y [ ], [ ], [ ], [ ], 4 [][ ] Su up - y[ ] 4-4 SP- Brli C 9
Proprtis of Digital Systs (cot.) Liar ti-ivariat (LTI) Coutatio Covolutio is coutativ ad distributiv [] [ ] [ ] [ ] [ ]* [ ]* [ ] [] * [] * [] Distributio [ ] [ ] [ ] [ ] [ ]* ( [ ] [ ]) [] * [] [] * [] ipuls rspos as fiit duratio» Fiit-Ipuls Rspos (FIR) ipuls rspos as ifiit duratio» Ifiit-Ipuls Rspos (IIR) y [ ] [ ]* [ ] [] * [] [][ ] [][ ] 4 SP- Brli C
4 SP- Brli C Proprtis of Digital Systs (cot.) Prov covolutio is coutativ [] [ ] [ ] [ ] [ ] [ ] [ ] ( ) [] [] y * lt *
4 SP- Brli C Proprtis of Digital Systs (cot.) Liar ti-varyig Syst E.g., is a aplitud odulator [] [ ] y [] [ ] [] [ ] [] [] [ ] [ ] [ ] ( )? But suppos y y y y
Proprtis of Digital Systs (cot.) Boudd Iput ad Boudd Output (BIBO): stabl y [ ] B < [] B < LTI syst is BIBO if oly if [] is absolutly suabl y [ ] 4 SP- Brli C
Proprtis of Digital Systs (cot.) Causality syst is casual if for vry coic of, t output squc valu at idig dpds o oly t iput squc valu for y [ ] K M α y [ ] [ ] β [ ] β y[ ] - β α - - β α - - β M α - y ocausal FIR ca b ad causal by addig sufficit log dlay 4 SP- Brli C 4
Discrt-Ti Fourir Trasfor (DTFT) Frqucy Rspos ( ) Dfid as t discrt-ti Fourir Trasfor of ( ) is cotiuous ad is priodic wit priod [ ] proportioal to two tis of t saplig frqucy ( ) is a copl fuctio of ( ) ( ) ( ) r i ( ) ( ) agitud pas 4 SP- Brli C 5
4 SP- Brli C 6 Discrt-Ti Fourir Trasfor (cot.) Rprstatio of Squcs by Fourir Trasfor sufficit coditio for t istc of Fourir trasfor [] < [ ] ( ) ( ) [ ] d δ,, si ) ( ) ( ) ( ( ) [] [] ( ) [ ] [ ] [ ] [ ] [ ] d d d δ Fourir trasfor is ivrtibl: ) ( ( ) [] [] ( ) d absolutly suabl DTFT Ivrs DTFT
4 SP- Brli C 7 Discrt-Ti Fourir Trasfor (cot.) Covolutio Proprty ( ) [] [] [][ ] ( ) [ ] [ ] [] [ ] ( ) ( ) [] ( ) ( ) Y y ] [ ' ] [ ] [ ' ' ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) Y Y Y ' ' '
4 SP- Brli C 8 Discrt-Ti Fourir Trasfor (cot.) Parsval s Tor Dfi t autocorrlatio of sigal [] ( ) d powr spctru [ ] [] ( ) ] [ ] [ ] [ ] [ ] [ ] [ * * l l R l ( ) ( ) ( ) ( ) * S [] ( ) ( ) d d S R [ ] [ ] [ ] [ ] ( ) d R * St T total rgy of a sigal ca b giv i itr t ti or frqucy doai. ( l) l l y y * cougat copl : *
Discrt-Ti Fourir Trasfor (DTFT) LTI syst wit ipuls rspos Wat is t output for y [ ] si ( φ ) ( φ ) [] [] [] ( φ ) [] * [] y ( ( ) φ ) [] ( φ ) [] ( φ ) ( ) ( φ ) ( ) ( ) [ ] y [ ] [ ] ( φ ) Syst s frqucy rspos ( ) > aplify ( ) < attuat ( φ ) ( ) ( ) [ ] ( ) ( φ ) ( ) ( ) si ( φ ) ( ) [] ( ) ( φ ) ( ) y [ ] y 4 SP- Brli C 9
Discrt-Ti Fourir Trasfor (cot.) 4 SP- Brli C
Z-Trasfor -trasfor is a graliatio of (Discrt-Ti) Fourir trasfor [ ] ( ) [ ] -trasfor of is dfid as [ ] ( ) [ ] ( ) r Wr, a copl-variabl For Fourir trasfor I copl pla ( ) ( ) R -trasfor valuatd o t uit circl ( ) uit circl 4 SP- Brli C
Z-Trasfor (cot.) Fourir trasfor vs. -trasfor Fourir trasfor usd to plot t frqucy rspos of a filtr -trasfor usd to aaly or gral filtr caractristics,.g. stability I copl pla R ROC (Rgio of Covrg) Is t st of for wic -trasfor ists (covrgs) R R [] < absolutly suabl I gral, ROC is a rig-sapd rgio ad t Fourir trasfor ists if ROC icluds t uit circl ( ) 4 SP- Brli C
Z-Trasfor (cot.) LTI syst is dfid to b causal, if its ipuls rspos is a causal sigal, i.. [] for < Siilarly, ati-causal ca b dfid as y [ ] [ ]* [ ] [] * [] [][ ] Rigt-sidd squc [][ ] [] for > Lft-sidd squc LTI syst is dfid to b stabl, if for vry boudd iput it producs a boudd output cssary coditio: [ ] < Tat is Fourir trasfor ists, ad trfor -trasfor iclud t uit circl i its rgio of covrg 4 SP- Brli C
Rigt-Sidd Squc t uit cycl E.g., t potial sigal. Z-Trasfor (cot.) [] a u[], wr u[] a ( ) ( ) a a ROC is > I a R a If a < for for < Fourir tr asfor of [] ists if a < av a pol at a (Pol: -trasfor gos to ifiity) 4 SP- Brli C 4
Z-Trasfor (cot.) Lft-Sidd Squc E.g. [] a u[ ] (,-,-,...,- ). ( ) a u[ ] t uit cycl I a If a < a ROC is < a R ( ) a w a a a <, t a a [] dos' t ist, bcaus [] potially as a Fourir trasfor of will go 4 SP- Brli C 5
4 SP- Brli C 6 Z-Trasfor (cot.) Two-Sidd Squc E.g. [] [] [ ]. u u [] [ ],, < > u u R I t uit cycl [ ] uit circl t t iclud dos' bcaus t ist, dos' Fourir tr asfor of ROC ad is > < ROC ( )
Fiit-lgt Squc E.g.. 4 4 [] a, Z-Trasfor (cot.), otrs ( ) ( ) ( a ) a a a - a a a a... a ROC 4 is tir - pla cpt t uit cycl I ( ) a,,.., 4 If 8 R 7 pols at ro pol ad ro at is caclld a 4 SP- Brli C 7
Z-Trasfor (cot.) Proprtis of -trasfor []. If is rigt-sidd squc, i.. [ ], ad if ROC is t trior of so circl, t all fiit for wic > r will b i ROC If,ROC will iclud causal squc is rigt-sidd wit ROC is t trior of circl icludig []. If is lft-sidd squc, i.. [],, t ROC is t itrior of so circl, < If,ROC will iclud []. If is two-sidd squc, t ROC is a rig 4. T ROC ca t cotai ay pols 4 SP- Brli C 8
Suary of t Fourir ad -trasfors 4 SP- Brli C 9
LTI Systs i t Frqucy Doai Eapl : copl potial squc Syst ipuls rspos y [ ] [] [] [] [] [ ] - [] ( ) - Trfor, a copl potial iput to a LTI syst rsults i t sa copl potial at t output, but odifid by ( ) T copl potial is a igfuctio of a LTI syst, ad is t associatd igvalu T ( ) scalar { [ ] } ( ) [ ] ( ) ( ) : t t It is oft syst Fourir tr syst y [ ] [ ]* [ ] [] * [] [][ ] ipuls rfrrd frqucy asfor to rspos. as of t rspos. [][ ] 4 SP- Brli C 4
4 SP- Brli C 4 [] ( ) ( ) ( ) ( ) [ ] ( ) ( ) ( ) ( ) [ ] ( ) ( ) [ ] ) ( ) ( ) ( ) ( * y φ φ φ φ φ φ φ LTI Systs i t Frqucy Doai (cot.) Eapl : siusoidal squc Syst ipuls rspos [ ] ( ) φ w [] ( ) φ φ φ [ ] ( ) θ θ θ θ θ θ θ θ θ i i si si ( ) ( ) ( ) ( ) ( ) * * ( ) ( ) ( ) si si si * si * y y agitud rspos pas rspos
LTI Systs i t Frqucy Doai (cot.) Eapl : su of siusoidal squcs y K [] ( φ ) K [] ( ) [ ( )] φ agitud rspos siilar prssio is obtaid for a iput istig of a su of copl potials pas rspos 4 SP- Brli C 4
LTI Systs i t Frqucy Doai (cot.) Eapl 4: Covolutio Tor [] δ [ P ] [] a u[], a < DTFT DTFT [ ] [ ] ( ) ( ) ( ) δ P a ( ) Y P ( ) ( ) ( ) a P a δ P P as a oro valu w P 4 SP- Brli C 4
4 SP- Brli C 44 LTI Systs i t Frqucy Doai (cot.) Eapl 5: Widowig Tor [][] ( ) ( ) W w [] [ ] P δ [] otrwis,,...,,.46.54 w aig widow ( ) ( ) ( ) ( ) ( ) ( ) P W P P W P P W P P P W W Y δ δ δ as a oro valu w P
Diffrc Equatio Raliatio for a Digital Filtr T rlatio btw t output ad iput of a digital filtr ca b prssd by M [ ] β y[ ] y α y β Y [ ] [ ] [ ] liarity ad dlay proprtis M ( ) α Y ( ) β ( ) ratioal trasfr fuctio ( ) Y M ( ) ( ) α dlay proprty β [ ] ( ) [ ] ( ) - - - β M Causal: Rigtsidd, t ROC outsid t outost pol Stabl: T ROC icluds t uit circl Causal ad Stabl: all pols ust fall isid t uit circl (ot icludig ros) β β α α α - - - 4 SP- Brli C 45
Diffrc Equatio Raliatio for a Digital Filtr (cot.) 4 SP- Brli C 46
Magitud-Pas Rlatiosip Miiu pas syst: T -trasfor of a syst ipuls rspos squc ( a ratioal trasfr fuctio) as all ros as wll as pols isid t uit cycl Pols ad ros calld iiu pas copots Maiu pas: all ros (or pols) outsid t uit cycl ll-pass syst: Cosist a cascad of factor of t for -a * a ± Caractrid by a frqucy rspos wit uit (or flat) agitud for all frqucis Pols ad ros occur at cougat rciprocal locatios -a * a 4 SP- Brli C 47
Magitud-Pas Rlatiosip (cot.) y digital filtr ca b rprstd by t cascad of a iiu-pas syst ad a all-pass syst ( ) ( ) ( ) ( ) i Suppos tat as oly o ro ( a < ) a* outsid t uit circl. ca b prssd as : ( ) * ( ) ( )( a ) ( ) ()( ) ( ( ( ) * a a a ) wr : ( )( a ) * ( a ) is a ( a ) is also a all - pass iiu filtr. is a ap iiu pas filtr. pas filtr) 4 SP- Brli C 48
FIR Filtrs FIR (Fiit Ipuls Rspos) T ipuls rspos of a FIR filtr as fiit duratio av o doiator i t ratioal fuctio o fdbac i t diffrc quatio y [] M β [ r ] [] ( ) r β, Y r, M ( ) ( ) otrwis M β [ ] Ca b ipltd wit sipl a trai of dlay, ultipl, ad add opratios - - - β β β M ( ) y[ ] 4 SP- Brli C 49
First-Ordr FIR Filtrs spcial cas of FIR filtrs y[] [ ] α [ ] ( ) θ ( ) α ( si ) ( α ) ( α si ) α si ( ) arcta α α α < α ( ) α : pr-pasis filtr log ( ) 4 SP- Brli C 5
Discrt Fourir Trasfor (DFT) T Fourir trasfor of a discrt-ti squc is a cotiuous fuctio of frqucy W d to sapl t Fourir trasfor fily oug to b abl to rcovr t squc For a squc of fiit lgt, saplig yilds t w trasfor rfrrd to as discrt Fourir trasfor (DFT) ( ) [] [] ( ),, DFT, alysis Ivrs DFT, Sytsis 4 SP- Brli C 5
4 SP- Brli C 5 Discrt Fourir Trasfor (cot.) [ ] [ ] ( ) ( ) ( )( ) [] [] [ ] [] [] [ ], M M L M L M M L L
4 SP- Brli C 5 Discrt Fourir Trasfor (cot.) Ortogoality of Copl Epotials ( ) otrwis, if, -r r [] [] [] [] ( ) [] ( ) [] [] [] r r r r r [ ] [ ] [] r r
Discrt Fourir Trasfor (DFT) Parsval s tor [] ( ) Ergy dsity 4 SP- Brli C 54
alog Sigal to Digital Sigal alog Sigal Discrt-ti Sigal or Digital Sigal [] ( T ), T :saplig priod a t T Digital Sigal: Discrt-ti sigal wit discrt aplitud F s T saplig rat saplig priod5μs >saplig rat8 4 SP- Brli C 55
Cotiuous-Ti Sigal a () t s alog Sigal to Digital Sigal (cot.) () t δ ( t T ) Cotiuous-Ti to Discrt-Ti Covrsio Saplig switc a s s ( t ) Ipuls Trai To Squc ˆ [ ] ( ( T)) ()() t s t () t δ ( t T ) a a ( T ) δ ( t T ) []( δ t T ) Priodic Ipuls Trai ( t) ca b uiquly spcifid by [ ] Discrt-Ti Sigal a [ ] Digital Sigal a ( t ) Discrt-ti sigal wit discrt aplitud s () t δ ( t T) δ ( t ) δ () t dt, t -T -T T T T 4T 5T 6T 7T 8T 4 SP- Brli C 56
alog Sigal to Digital Sigal (cot.) cotiuous sigal sapld at diffrt priods a () t ( t ) a T a s ( t) ()() t s t () t δ ( t T ) a a ( T ) δ ( t T ) []( δ t T ) 4 SP- Brli C 57
alog Sigal to Digital Sigal (cot.) Spctra a ( Ω) S T ( Ω ) δ ( Ω Ω ) s Ω Ω T s F s (saplig frqucy) s s T ( Ω ) ( Ω ) S ( Ω ) ( Ω ) ( ( Ω Ω )) ig frqucy copots got supriposd o low frqucy copots a a aliasig distortio s T Ω < Ω s / R s ( Ω) a ( Ω) RΩ ( Ω) p ( Ω) s Ω a otrwis Low-pass filtr ( Ω) ca' t b rcovrd fro ( Ω) p Ω Ω < ( Ω Ω ) Q s Ω s > Ω s > Ω Ω T > Ω s T ( Ω Ω ) Q Ω s s < Ω 4 SP- Brli C 58 < Ω
alog Sigal to Digital Sigal (cot.) To avoid aliasig (ovrlappig, fold ovr) T saplig frqucy sould b gratr ta two tis of frqucy of t sigal to b sapld Ω > (yquist) saplig tor To rtruct t origial cotiuous sigal Filtrd wit a low pass filtr wit bad liit Covolvd i ti doai ( t ) sic Ω t s s Ω Ω a s () t ( T ) ( t T ) a a ( T ) sic Ω ( t T ) s 4 SP- Brli C 59