Discrete-Time Signal Processing

Transcription

1 Discrt-Tim Signal Procssing Hnry D. Pfistr March 3, 07 Th Discrt-Tim Fourir Transform. Dfinition Th discrt-tim Fourir transform DTFT) maps an apriodic discrt-tim signal x[n] to th frquncy-domain function X jω ) F {x[n]}. Likwis, w writ x[n] F { X jω ) }. F Th DTFT pair x[n] X jω ) satisfis: x[n] X jω ) jωn dω π π X jω ) x[n] jωn synthsis quation) analysis quation) Rmarks: Not that jω+π)n jωn jπn jωn for any intgr n. So, in th synthsis quation, rprsnts intgration ovr any π intrval [a, a + π) of lngth π. On th othr hand, x[n] is assumd to b apriodic, so th summation in th analysis quation is ovr all n. Thus, th DTFT X jω ) is always priodic with priod π. Th frquncy rspons of a DT LTI systm with unit impuls rspons h[n] H jω ) h[n] jωn is prcisly th DTFT of h[n], so th rspons y[n] that corrsponds to th input x[n] jωn is givn by y[n] H jω ) jωn.

2 Convrgnc. If x[n] is absolutly summabl, i.., x[n] < thn X jω ) is wll-dfind i.., finit) for all Ω R. Thus, thr is a uniqu X jω ) for ach absolutly summabl x[n]. Invrsion. If X jω ) is th DTFT of an absolutly summabl DT signal x[n], thn X jω ) jωn dω ) x[m] jωm jωn dω π π π π m x[m] jωn m) dω π m π }{{} πδ[n m] x[m]δ[n m] m x[n]. Thus, thr is a on-to-on corrspondnc btwn an absolutly summabl x[n] and its DTFT X jω ). Priodic Signals. For a priodic DT signal, x[n] x[n + N], lt a k b th discrt-tim Fourir sris cofficints a k N N n0 x[n] jkω 0n, whr Ω 0 π/n. Thn, th DTFT consists is a sum of Dirac dlta functions:. DTFT Exampls X jω ) π k a k δω kω 0 ). Thr ar a fw important DTFT pairs that ar usd rgularly. Exampl. Considr th discrt-tim rctangular puls x[n] M δ[n k]. k0

3 Th DTFT of this signal is givn by X jω ) M k0 x[n] jωn jωn jωm+) jω jωm+)/ jωm+)/ ) jωm+)/ jω/ jω/ ) jω/ sinωm + )/) jωm/. sinω/) Th first trm is th discrt-tim analog of th sinc function and th scond trm is th phas shift associatd with th fact that th puls is not cntrd about n 0. If M is vn, w can advanc th pul by M/ sampls to rmov this trm. Exampl. An idal discrt-tim low-pass filtr passing Ω < Ω c π has th DTFT frquncy-rspons { H jω if Ω < Ω c ) 0 if Ω c < Ω π. Thus, th invrs DTFT implis that h[n] H jω ) jωn dω π π Ωc jωn dω π Ω c [ ] jωn Ω c πjn Ω c πn sinω cn). This impuls rspons is not absolutly summabl du to th discontinuity in H jω ). To undrstand th DTFT in this cas, lt H M jω ) b th frquncy rspons of th truncatd impuls rspons { h[n] if n M h M [n] 0 if n > M. Du to th Gibb s phnomnon, H M jω ) dos not convrg uniformly to H jω ). But, sinc h[n] <, th frquncy rspons instad convrgs in th man-squar sns π lim H jω ) H M jω ) dω 0. M π 3

4 .3 Proprtis of th DTFT Th canonical proprtis of th DTFT ar vry similar to th canonical proprtis of th F continuous-tim Fourir transform CTFS). For x[n] X jω F ) and y[n] Y jω ), w obsrv that: ) Linarity: ax[n] + by[n] ) Tim shift: x[n n 0 ] 3) Frquncy shift: jω 0n x[n] 4) Conjugation: x [n] 5) Tim flip: x[ n] 6) Convolution: x[n] y[n] 7) Multiplication: x[n]y[n] 8) Symmtry: F ax jω ) + by jω ) F jωn 0 X jω ) F X jω Ω 0) ) F X jω ) F X jω) F X jω )Y jω ) F X jω ) Y jω ) π π Xjθ )Y jω θ) )dθ x[n] ral R{X jω )} vn, Im{X jω )} odd, X jω ) vn, X jω ) odd x[n] ral & vn X jω ) ral & vn x[n] ral & odd X jω ) pur imaginary & odd 9) Parsval s rlation: x[n] X jω ) dω π π.4 Frquncy-Domain Charactrization of DT LTI Systms Th tim-domain charactrization of DT LTI systms is givn by y[n] x[n] h[n], whr h[n] is th unit impuls rspons of th systm. Taking th DTFT of this quation givs th frquncy-domain charactrization of DT LTI systms, Y jω ) X jω )H jω ). Th DTFT, H jω ), of h[n] is calld th frquncy rspons of th systm. 4

5 .5 Spctral Dnsity For an nrgy-typ signal x[n], th nrgy spctral dnsity is dfind to b R xx jω ) X jω ). This notation is usd bcaus R xx jω ) is th DTFT of th autocorrlation signal r xx [l]. To s this, w first not that r xx [l] x[n] y[n] for y[n] x [ n]. Thn, th convolution thorm implis that R xx jω ) F {x[n]} F {x [ n]} X jω )X jω ) X jω ). For a powr-typ signal.g., a priodic signal), on considrs a squnc in M) of normalizd nrgy spctral dnsitis for th windowd signals x M [n] x[n]w[n] with w[n] u[n + M] u[n M + ]. Using this, th powr spctral dnsity is dfind to b R xx jω ) lim XM jω ) M M + lim M M + F {x M[n]} F {x M[ n]} M+min0,l) lim F x[n]x [n l] M M + n M+max0,l) M+min0,l) F lim x[n]x [n l] M M + n M+max0,l) { } M F lim x[n]x [n l] M M + F {r xx [l]}, n M whr r xx [l] is th tim-avrag autocorrlation function of x[n]. Th oprational maning of ths dfinitions is that th total nrgy or powr) signal nrgy containd in th frquncy band [Ω, Ω ] is givn by E [Ω,Ω ] π whr th π scal factor is chosn so that E [ π,π] π π π Ω Ω R xx jω )dω π R xx jω )dω, π π X jω ) dω x[n] follows from Parsval s rlation. If th signal is ral, thn conjugat symmtry implis that R xx jω ) R xx jω ). Thus, on typically assums 0 Ω Ω π and says th total nrgy in th positiv frquncy band [Ω, Ω ] is E E [ Ω, Ω ] + E [Ω,Ω ] E [Ω,Ω ]. Thus, on must b carful to distinguish btwn ths two convntions. 5

6 .6 Th Discrt Fourir Transform Th DTFT maps a discrt-tim signal x[n] to a continuous but priodic) frquncy domain X jω ). Th discrt Fourir transform DFT) of lngth-n maps a discrt-tim squnc x[n] supportd on 0 n N ) to a discrt st of frquncis X[k] supportd on 0 k N ) using th rul X[k] N n0 x[n] πjkn/n. It is asy to vrify that th invrs DFT is givn by x[n] N N k0 X[k] πjkn/n. This DFT is closly rlatd to computing valus of th DTFT on a rgularly spacd grid. For xampl, lt Ω k πk and obsrv that N X[k] X πjk/n ) X jω k ). If th DT signal x[n] is priodic, thn this transform ariss naturally bcaus th DTFT consists of Dirac dlta functions supportd on a discrt st of harmonic frquncis Ω k πk N for k 0,,..., N. For ral signals, conjugat symmtry i.., X[k] X [N k]) implis that th signal information is containd compltly in th first N + )/ valus of X[k]. Th output X[k 0 ] is calld th DFT for frquncy bin k 0 bcaus th DFT can b sn to transform N tim-domain sampls into N frquncy-domain sampls. Morovr, this transform is unitary xcpt for an ovrall scal factor) and prsrvs th nrgy in th sns that E x N n0 N n0 N N x[n] N N i0 N N ) X[i] πjin/n N X [i]x[k] N i0 k0 N N N n0 X [i]x[k]δ[k i] i0 k0 N X[k]. N k0 N k0 πjk i)n/n X[k] πjkn/n ) Thus, th DFT dcomposs th signal nrgy E x into N frquncy bins that xpos th signal s nrgy distribution ovr frquncy. 6

7 For a short duration signal x[n] supportd on 0 n M ), on can comput th DFT on a dnsr st of frquncis by zro padding th signal up to lngth N > M. In practic, this computation can b don quickly by zro padding th squnc up to a powr of lngth N m and applying th fast Fourir transform FFT). On can also us th DFT to numrically approximat th nrgy or powr) spctral dnsity of a DT signal x[n]. Using a lngth-n DFT, th ida is to approximat th intgrals in th prvious sction. For th nrgy spctral dnsity, on gts E [Ω,Ω ] N Ω N/π) k Ω N/π) X[k] and an additional scal factor of /N is usd for th powr spctral dnsity. FIR Filtr Dsign via th Window Mthod As w saw in th last sction, th impuls rspons of an idal DT low-pass filtr can b computd in closd form. This filtr cannot b implmntd, howvr, bcaus its impuls rspons xtnds to n. To fix this, on can truncat impuls rspons to gt h M [n] { sinω πn cn) if n M 0 if n > M. To undrstand th ffct of this truncation, w rwrit this as h M [n] πn sinω cn)v M [n], whr v M [n] M k M δ[n k]. From this, w obsrv that th frquncy rspons is givn by H M jω ) H jω ) V M jω ). From th DTFT xampls, it is asy to s that V M jω ) sinωm + )/). sinω/) Th portion of this function satisfying Ω Ω 0 π/m + ) is calld th main lob of th window rspons. Th sction from Ω 0 Ω Ω 0 is calld th first sidlob of th window rspons. This figur shows th rspons for two M valus. 7

8 80 60 M 0 M 40 VM jω ) Ω In th window mthod of filtr dsign, th main lob of th window smars th cutoff frquncy into a transition band whos width approximatly quals th width of th window s main lob. Likwis, th hight of th first sid lob typically limits th stop-band rjction. Choosing th window function to optimiz this trad-off is known as th window mthod of filtr dsign. If dlay and complxity ar not a problm, on can always choos a long nough window so that th rror is quit small. To achiv bttr stop-band rjction, on nds to choos a smoothr window function. For xampl, th Hann window function w[n] πn )) + cos M for n M, M +,..., M) provids rasonabl prformanc. For th Hann window, this filtr dsign mthod is implmntd by th MATLAB function hfir*m,wc/pi,hann*m+)). 3 From Discrt Tim to Continuous Tim Th procss of filling in th signal wavform btwn sampl points is calld intrpolation. From a purly discrt-tim prspctiv, th procss of intrpolation consists of dsigning a filtr that rsults in a non-intgr tim dlay. Exampl 3. An idal dlay of n 0 sampls maps jωn to jωn n 0). Thus, it can b rprsntd by an LTI systm whos DTFT frquncy rspons is H jπk+ω)) jωn 0, 8

9 for Ω π, π] and intgr k. From this, th invrs DTFT implis that h[n] H jω ) jωn dω π π π π π πjn n 0 ) jωn 0 jωn dω sinπn n 0)). πn n 0 ) [ jωn n 0 ) ] π Sinc y[n] x[n] h[n] implis that y[0] is qual to th signal valu at tim t n 0, th gnral intrpolation formula is givn by sinπn t)) xt) x[n]. πn t) In practic, on nds to window this filtr to mak its duration finit. Th sam intrpolation formula can also b drivd as th rsult of filtring th CT signal xt) k x[k]δt k) with th idal CT low-pass filtr ht) π π 4 Sampl Rat Convrsion π jωt sinπt). πt To achiv sampl rat convrsion by a rational factor R L/M, th dirct approach is to upsampl by L, filtr, and thn downsampl by M. Th complxity of this approach can b quit high and thr ar a varity of ways to rduc th complxity. Exampl 4 Upsampling by L). Considr th signal { x[n/l] if L divids n y[n] 0 othrwis, whr y[n] is formd by insrting L zros btwn ach sampl of x[n]. Th DTFT of y[n] is givn by Y jω ) y[n] jωn X jlω ). y[nl] jωnl x[n] LjΩn π 9

10 Sinc X jω ) is priodic with priod π, th DTFT of Y jω ) is priodic with priod π/l and it contains L aliasd copis or imags) of X jω ). Passing y[n] through an idal low-pass filtr with cutoff frquncy Ω c π/l lavs only a singl low-pass vrsion and rmovs th xtra imags. Th rsult is a signal with th sam frquncy contnt as x[n] but sampld L tims fastr. Exampl 5 Downsampling by M). Considr th signal y[n] x[nm] whr y[n] is formd by taking only vry M-th sampl. In th frquncy domain, this opration is bst undrstood as th combination of two stps. First, w lt v[n] k δ[n km] and dfin z[n] x[n]v[n]. Thn, w lt y[n] z[nm]. Sinc th DTFT tabl shows that V jω ) π δ Ω πk ), M M k th ffct of th first stp in th frquncy domain is givn by Z jω ) X jω ) V jω ) M M k0 X jω πk/m)). From this, w s that th DTFT Z jω ) is priodic with priod π/m. Nxt, w s that th DTFT of y[n] z[nm] is givn by Y jω ) y[n] jωn Thus, w find that Y jω ) M Z jω/m ). M k0 z[nm] jωnm/m z[n] jω/m)n X jω/m πk/m)). Finally, if X jω ) 0 for π M Ω π, thn Y jω ) M XjΩ/M ). 5 Th Z-Transform 5. Introduction Th Z-transform is a gnralization of th discrt-tim Fourir transform DTFT) that can b applid to a largr class of signals. It is also th discrt-tim analog of th Laplac 0

11 transform. For a DT signal x[n], th Z-transform Xz) Z {x[n]} is dfind by Xz) x[n]z n. Th rgion of convrgnc ROC) for a signal x[n] is th st of complx) z-valus for which th sum is absolutly convrgnt { } R {x[n]} z C x[n]z n <. Z Th rlationship btwn x[n] and Xz) is dnotd by x[n] Xz). Ignoring th ROC, th Z-transform can b sn as a gnralization of th DTFT bcaus choosing z jw rcovrs th dfinition of th DTFT. Lt x[n] b a finit-duration signal whr x[n] 0 for n < n and n > n. Thn, Xz) n nn x[n]z n and z n Xz ) is a polynomial. Considr th cas of x[n] {, 3,, 4, 7}, whr th undrlin dnots th location associatd with tim n 0. In this cas, w hav Xz) z + 3z + + 4z + 7z and th ROC is 0 < z <. From this, w s that th Z-transform simply uss th powr of th indtrminat, z, to kp track of th tim indx associatd with a signal valu. This rprsntation is particularly usful for convolution bcaus ) Z {h[n] x[n]} h[k]x[n k] z n k k h[k]z k Hz)Xz). x[n k]z n+k Exampl 6. Lt h[n] {, 0, }and obsrv that Hz) z z. Thus, Z {h[n] x[n]} Hz)Xz). z z ) z + 3z + + 4z + 7z ) z 3 + 3z + z z ) z z + 4z + 7z 3) z 3 + 3z + z + + 5z 4z 7z 3. Thus, it is asy to s that h[n] x[n] {, 3,,, 5, 4, 7}.

12 5. Th Rgion of Convrgnc To undrstand th importanc of th ROC, w considr two simpl xampls. Exampl 7. Considr th signal x[n] a n u[n] whos Z-transform is givn by Xz) a n z n n0 a n u[n]z n, if z > a. az Th last stp follows from th gomtric sum formula and th ROC is z > a. Exampl 8. Considr th signal x[n] a n u[ n ] whos Z-transform is givn by Xz) a z a n u[ n ]z n a n z n a n z n n0 a z, if z < a a z, if z < a. az Although th signals in ths two xampls ar quit diffrnt, th only diffrnc btwn thir Z-transforms is th ROC. Thus, if on ignors th ROC, thn thr ar multipl signals with th sam Z-transform. Exampl 9. Considr th signal x[n] a n u[n] + b n u[ n ]. By linarity, w can comput th Z-transform using th two prvious xampls and w gt Xz) az, if a < z < b. bz In this cas, th ROC is mpty if b a and is a donut shapd rgion othrwis. Dfinition 0. A signal x[n] is right-sidd if thr is an n 0 such that x[n] 0 for all n < n 0. For this n 0, it follows that th shiftd signal y[n] x[n + n 0 ] is causal. Similarly, a signal is lft-sidd if thr is an n 0 such that x[n] 0 for all n > n 0. Any signal that is nithr right-sidd nor lft-sidd is calld two-sidd.

13 Th ROC of a right-sidd squnc is of th form z > r for som r [0, ]. Similarly, th ROC of a lft-sidd squnc is of th form z < r for som r [0, ]. In gnral, th ROC of a two-sidd squnc is of th form r < z < r for som r, r [0, ]. This matchs xactly what w saw in th prvious xampls. 5.3 Block Diagrams 5.4 Proprtis of th Z-Transform Th following proprtis of th Z-transform ar analogous to similar proprtis of th DTFT: ) Linarity: ax[n] + by[n] ) Tim shift: x[n n 0 ] 3) Frquncy shift: a n x[n] 4) Conjugation: x [n] 5) Tim flip: x[ n] 6) Convolution: x[n] y[n] 7) Diffrntiation: n x[n] Z axz) + by z), ROC givn by intrsction Z z n 0 Xz), ROC unchangd xcpt possibly {0, } Z Xa z), nw ROC is a r < z < a r Z X z ), ROC unchangd Z X z ), nw ROC /r < z < /r Z Xz)Y z), ROC includs intrsction Z z d Xz), ROC unchangd dz In gnral, th invrs of th Z-transform can b computd using a contour intgral. In practic, on typically uss partial fraction xpansion to obtain a sum of simpl trms and thn tabl lookup is usd to invrt ach trm. 6 Filtr Implmntation 6. Block Diagrams x[n] b 0 + v[n] v[n] + y[n] z z b a Considr th systm y[n] b 0 x[n] + b x[n ] a y[n ] and notic that it can b implmntd as th cascad of th block diagrams shown abov: 3

14 v[n] b 0 x[n] + b x[n ] y[n] a y[n ] + v[n]. Taking th Z-transform of ths systms, on gts Ths can b rwrittn as V z) b 0 Xz) + b Xz)z Y z) a Y z)z V z). V z) b 0 + b z )Xz) V z) a 0 + a z )Y z), with a 0. Thus, th Z-transform of th impuls rspons is givn by Hz) Y z) Xz) b 0 + b z Bz) a 0 + a z Az) and it is a rational function of z. Currntly, this systm rquirs two dlays. Can it b implmntd with only on dlay? 6. Rational Z-Transforms Th signals with rational Z-transforms, Hz) Bz) Az) b[0] + b[]z + + b[m]z M a[0] + a[]z + + a[n]z N, ar a vry important class of signals. If w factor th numrator and dnominator, thn w can rwrit this as Hz) b[0] M a[0] zn M k z z k) N k z p k), whr z,..., z M ar th zros of Bz) and p,..., p N ar th zros of Az) i.., th pols of /Az)). It is worth noting that, up to an ovrall constant, th magnitud of th frquncy rspons H jω ) b[0] a[0] M k jω z k N k jω p k has a graphical intrprtation as th product of th distancs from jω to th zros dividd by th product of th distancs from jω to th pols. Th pol-zro rprsntation of a Z-transform, Hz), is particularly usful whn th signal h[n] is th impuls rspons of a filtr. This is bcaus th zros idntify th signals.g., of th form x[n] zk n ) that ar mappd to th zro signal and th pols idntify th signals.g., of th form x[n] p n k ) that xprinc infinit gain. 4

15 Exampl. Considr x[n] a n u[n] shows that has zro at z 0 and a pol at p a. Z Xz) az. Th pol-zro dcomposition Xz) z 0) z a) Exampl. Considr a finit impuls rspons FIR) filtr dfind by y[n] M b[k]x[n k]. k0 Thus, th output satisfis Y z) k0 y[n]z n M b[k]z k k0 Bz)Xz), M b[k]x[n k]z n x[n k]z n+k whr Bz) z M M k z z k) is a filtr with M zros and M trivial pols at z 0. Exampl 3. Considr an all-pol filtr dfind by y[n] x[n] N a[k]y[n k]. k Similarly, th output satisfis Y z) Xz)/Az) whr /Az) / z ) N N k z p k) is a filtr with N pols and N trivial zros at z 0. Exampl 4. Considr th linar constant-cofficint diffrnc quation LCCDE) dfind by N M a[k]y[n k] b[k]x[n k]. k0 In this cas, th Z-transform Az)Y z) Bz)Xz) implis that Hz) Y z)/xz) Bz)/Az) has a rational Z-transform. W not that solving an LCCDE using th Z- transform implicitly assums that th systm was initially at rst. k0 5

16 Exampl 5. Dtrmin th impuls rspons of Th Z-transform implis that y[n] y[n ] + x[n]. Hz) Y z) Xz) z. Basd on our prvious calculations, this implis that ) n h[n] u[n]. 6.3 Undrstanding Filtrs via Pol-Zro Plots A filtr, h[n], with a rational Z-transform, Hz), is compltly dfind by th locations of its pols and zros in th complx plan. A pol-zro plot shows ths locations in th complx plan with zros rprsntd by circls and pols ar rprsntd by x s. If thr is a zro at som location z, thn th filtr maps th input signal, x[n] z n, mappd to th zro signal. If thr is a pol at som point z, thn th filtr maps th input signal, x[n] z n, to an unboundd output and th ROC dos not includ z. Sinc y[n] Hz)x[n] for x[n] z n, w s that th unit circl corrsponds to z and x[n] jωn for som ω [ π, π]. For z insid th unit circl i.., z < ), th signal x[n] is dcaying xponntially in n whil, z outsid th unit circl i.., z > ), th signal x[n] is growing xponntially in n. Rcall that a filtr, h[n], is boundd-input boundd-output BIBO) stabl iff h[n]. From th dfinition of th ROC, it follows that h[n] is BIBO stabl iff only th ROC of its Z-transform, Hz), includs z. Thus, for a right-sidd BIBO-stabl filtr, this implis th ROC must includ z and th Z-transform must hav all its pols insid th unit circl. 6.4 First-Ordr FIR Filtrs Considr a first-ordr FIR filtr dfind by th input-output rlationship y[n] b 0 x[n] + b x[n ], whr b 0, b ar ral cofficints. It has a zro at z b /b 0 and a pol at z 0. Th magnitud of th frquncy rspons satisfis H jω ) b 0 + b jω) b 0 + b jω) b 0 + b 0 b cosω) + b. ) 6

17 To gt a low-pass filtr, on can rquir unity gain at DC i.., H jω) if Ω 0) and zro gain at Nyquist i.., H jω) 0 if Ω π). Ths quations imply that Thus, w find that b 0 b and b 0 + b b 0 b 0. ) HLP jω + cosω). To gt a high-pass filtr, on can rquir unity gain at Nyquist i.., H jω) if Ω π) and zro gain at DC i.., H jω) 0 if Ω 0). Ths quations imply that Thus, w find that b 0, b, and 6.5 First-Ordr IIR Filtrs b 0 b b 0 + b 0. ) HHP jω cosω). Considr th first-ordr FIR filtr dfind by th input-output rlationship y[n] b 0 x[n] + b x[n ] a y[n ], whr b 0, b, a ar ral cofficints. It has a zro at z b /b 0 and a pol at z a. From ), w s that th magnitud of th frquncy rspons satisfis ) H jω b 0 + b 0 b cosω) + b. + a cosω) + a To gt a low-pass filtr, on can rquir unity gain at DC i.., H jω) if Ω 0) and zro gain at Nyquist i.., H jω) 0 if Ω π). Ths quations imply that b 0 + b + a b 0 b a 0. Thus, w find that b 0 b +a and th magnitud of th frquncy rspons satisfis H ) ) LP jω + a + cosω). ) + a cosω) + a 7

18 To dtrmin a, a standard approach is to dfin th cutoff frquncy Ω 0 whr HLP jω 0 ). Using th quadratic formula to solv for a, on can show that a sinω 0). cosω 0 ) To vrify this, on can start with ), us th abov formula for a and substitut Ω Ω 0. Th following figur shows an xampl with Ω 0 tan 3/4) 0.π whr a implis b 0 b Im{z} 0 H jω ) R{z} Normalizd Frquncy f Ω/π To dsign a high-pass filtr, a similar argumnt shows that b 0 a, b a, and 6.6 All-Pass Filtrs a sinω 0). cosω 0 ) Anothr typ of filtr is an all-pass filtr. By dfinition, an all-pass filtr must satisfy H jω ) for all ral Ω. For a filtr with a rational Z-transform, a sufficint condition is that N M and Bz) z N A z ) bcaus this implis that ) H jω B jω) B jω) B jω) A jω ) A jω ) A jω ) jωn A ) jω jωn A jω). A jω ) A jω ) Th gnral form of a first-ordr all-pass filtr is givn by y[n] a x[n] x[n ] a y[n ], 8

19 whr a is a complx cofficint. Highr ordr all-pass filtrs can b implmntd by cascading first ordr sctions. Th Z-transform is givn by H AP z) a + z + a z and w obsrv that th filtr has a zro at z /a and a pol at z a. Now, w can comput th phas rspons, H AP jω ), using H AP jω ) arg a + jω ) + a jω ) ) arg a + jω ) + a jω ) ) arg R{a } + jω + a jω) atan a ) sinω), R{a } + a + ) cosω) ). Thus, th phas rangs from 0 at Ω 0 to π at Ω π. Th following figur shows an xampl pol-zro diagram and phas rspons for th cas of a 0.5. Im{z} 0 HAP jω ) dgrs) R{z} Normalizd Frquncy f Ω/π All-pass filtrs can also b usd to dsign low-pass and high-pass) filtrs. For th low-pass cas, th ida is to implmnt a filtr with Z-transform Hz) + H AP z)). Frquncis whr H AP jω ) 0 will hav a magnitud rspons clos to whil frquncis whr H AP jω ) ±π will hav a magnitud rspons clos to 0. Basd on this, th cutoff frquncy Ω 0 of an all-pass filtr is typically dfind by H AP jω 0 ) π/. In audio procssing, thr is a common block, calld a low-pass) shlving filtr, that has th Z-transform H S z) + H 0 + H AP z)). 9

20 In this cas, frquncis whr H AP jω ) 0 will hav a magnitud rspons clos to + H 0 whil frquncis whr H AP jω ) ±π will hav a magnitud rspons clos to. Choosing H 0 > givs a filtr that amplifis low frquncis. Choosing H 0 givs a filtr that attntuats low frquncis. 6.7 Notch Filtrs Notch filtrs ar band-stop filtrs with a vry narrow stop band. For xampl, analog notch filtrs ar commonly usd to rmov th 60 Hz hum from audio amplifirs that can lak into audio systms from th powr supply. In trms of th pol-zro diagram, on can dsign a digital FIR notch filtr that maps x[n] cosω 0 n + φ) to zro by placing filtr zros at z ±jω Im{z} 0 H jω ) R{z} Normalizd Frquncy f Ω/π For xampl, th first plot in th abov figur shows th pol-zro diagram of th filtr H z) z z j/ )z j/ ) cos )z + z, with zros at ±j/ and two pols at 0. Th frquncy rspons of this filtr is givn by H jω ) jω jω cos ) + jω ) cosω) cos ))jω. Th scond plot shows th magnitud of this rspons. Notic that th rspons at Ω is zro. Th main problm with FIR notch filtrs is that th rspons is vry flat in th pass band. This problm can b rctifid by dsigning an IIR notch filtr that includs pols that ar clos to th zros.g., at z r ±jω 0 for som r < ). For th input signal x[n] cosωn+φ) whr Ω is not too clos to Ω 0, ths pols cancl th ffct of th zros. 0

21 0 Im{z} 0 H jω ) R{z} Normalizd Frquncy f Ω/π Th first plot in th abov figur show th pol-zro plot of th filtr H z) z z j/ )z j/ ) z z 3 4 j/ )z 3 4 j/ ) cos )z + z 3 cos )z + 9, 6 z with zros at ±j/ and pols at 3 4 ±j/. Th magnitud-squard frquncy rspons of this filtr is givn by H jω ) cosω) cos )) jω 3 cos ) jω jω ) 3 cos )jω jω ) 4 cosω) cos )) cos ) cos 8 ) cosω) + cosω) Th scond plot in th abov figur shows th magnitud of th frquncy rspons. Notic that th rspons at Ω is still zro but matching th zros with pols maks th ovrall rspons much flattr. On can also dsign digital IIR notch filtrs by first dsigning an optimal analog notch filtr and thn transforming it into th digital domain via th bilinar transform. In this class, w will not covr this mthod. But, that is th approach takn by th MATLAB function iirnotch. 6.8 Gnral Scond-Ordr IIR Filtrs Considr th scond-ordr FIR filtr dfind by th input-output rlationship y[n] b 0 x[n] + b x[n ] + b x[n ] a y[n ] a y[n ], whr b 0, b, b, a, a ar ral cofficints. Such a filtr will hav two pols and two zros. Sinc th cofficints ar ral, th zros and pols) will ithr b ral or will appar in

22 complx conjugat pairs. A littl algbra shows that th magnitud of th frquncy rspons satisfis H jω ) b 0 + b + b + b b 0 + b ) cosω) + b 0 b cosω) + a + a + a + a ) cosω) + a cosω). Howvr, th dirct dsign of digital IIR filtrs with ordr gratr than on can b somwhat cumbrsom. Instad, on typically starts with th Laplac transform H c s) of an optimal continuous-tim filtr.g., Chbyshv, Buttrworth, Elliptical) and maps it to th Z-transform Hz) of a digital filtr using th bilinar transform s T z )/ + z ). Thus, th Z-transform of th digital filtr is givn by ) z Hz) H c, T + z whr T is a paramtr with units sconds pr sampl that can b viwd as th sampling rat associatd with this convrsion. A ky proprty of this transform is that it prsrvs th ordr of th filtr. Stting z jω and s jω, w s that jω T z )/ + z ) j tanω/)/t. Thus, th discrt-tim filtr xprincs th frquncy-domain rmapping Ω tan T ω/) of th continuous-tim filtr and a littl work shows that H jω) H c j ) ) Ω T tan. }{{} ω For xampl, considr th scond-ordr Buttrworth low-pass filtr with Laplac transform ωc H c s). s + sω c + ω c If w choos th cutoff frquncy to b ω c T Ω c and apply th bilinar transform, thn w gt Hz) T Ω T c) ) z +z + z Ω T +z T c + Ω T c) Ω c 4 ) z +z + z Ω +z c + Ω c Ω + z ) c 4 z ) + 8 z )Ω c + + z ) Ω c Ω + z + z c 4 + Ω c 8 + Ω c ) + Ω c 8)z + 4 Ω c 8 + Ω c )z. For sufficintly small T, this is a digital low-pass filtr whos cutoff frquncy is clos to Ω c. Us Matlab to plot th frquncy rspons of this filtr for Ω c 0.π.