VI. FIR digital filters

Transcription

1 Digital Sigal Procssig 6 Dcmbr 24, 29 VI. FIR digital filtrs (No chag i 27 syllabus). 27 Syllabus: Charactristics of FIR digital filtrs, Frqucy rspos, Dsig of FIR digital filtrs usig Widow tchiqus, Frqucy samplig tchiqu, Compariso of IIR ad FIR filtrs. Cotts: 6. FIR Rcapitulatio 6.2 Charactristics if FIR digital filtrs 6.3 Frqucy rspos 6.4 Dsig of FIR digital filtrs Th Fourir sris ad widowig mthod 6.5 Choosig btw FIR ad IIR filtrs 6.6 Rlatioship of th DFT to th z-trasform DSP-6 (FIR) of 82 Dr. Ravi Billa

2 FIR - Rcapitulatio Nomclatur With a i th liar costat cofficit diffrc quatio, a y() + a y( ) + + a N y( N) b x() + b x( ) + + b M x( M), a w hav, H(z) M i i N i b z i a z i i This rprsts a IIR filtr if at last o of a through a N is ozro, ad all th roots of th domiator ar ot cacld xactly by th roots of th umrator. I gral, thr ar M fiit zros ad N fiit pols. Thr is o rstrictio that M should b lss tha or gratr tha or qual to N. I most cass, spcially digital filtrs drivd from aalog dsigs, M N. Systms of this typ ar calld N th ordr systms. This is th cas with IIR filtr dsig. Wh M > N, th ordr of th systm is o logr uambiguous. I this cas, H(z) may b tak to b a N th ordr systm i cascad with a FIR filtr of ordr (M N). Wh N, as i th cas of a FIR filtr, accordig to our covtio th ordr is. Howvr, it is mor maigful i such a cas to focus o M ad call th filtr a FIR filtr of M stags or (M+) cofficits. 8 Exampl Th systm H(z) ( z ) ( z ) is a FIR filtr. Why (vrify)? A FIR filtr th has oly th b cofficits ad all th a cofficits (xcpt a which quals ) ar zro. A xampl is th thr-trm movig avrag filtr y() (/3) x() + (/3) x( ) + (/3)x( 2). I gral th diffrc quatio of a FIR filtr ca b writt M y() br x( r) b x() + b x( ) + + b M x( M) () r Thr ar (M + ) cofficits; som us oly M cofficits. This quatio dscribs a orcursiv implmtatio. Its impuls rspos h() is mad up of th cofficits {b r } {b, b,, b M } h() b, for M {b, b,, b M }, lswhr Equivaltly, th fiit lgth impuls rspos ca also b writt i th form of a wightd sum of fuctios as was do i Uit I for xampl, x( ) x( k) ( k) k M h() br ( r) b () + b ( ) + + b M ( M) r Th diffrc quatio () is also quivalt to a dirct covolutio of th iput ad th impuls rspos: M y() br x( r) b( r) x( r) r M r whr w hav writt b r as b(r), i.., th subscript i b r is writt as a idx i b(r). DSP-6 (FIR) 2 of 82 Dr. Ravi Billa

3 Th trasfr fuctio H(z) of th FIR filtr ca b obtaid ithr from th diffrc quatio or from th impuls rspos h(): M H(z) h( ) z b 2 b z b z... b M z M M b z b z... bm z bm M z M b M bm bm b z z z b b b... M z Th trasfr fuctio has M otrivial zros ad a M th ordr (trivial) pol at z. This is cosidrd as a all-zro systm. j W may obtai th frqucy rspos H( ) or H() of th FIR filtr ithr from H(z) as j H ( ) H( z) j z or, from th impuls rspos, h(), as th discrt-tim Fourir trasform (DTFT) of h(): j H ( ) h ) j ( M b 2 j b M j j2 b b2... Th ivrs DTFT of H(ω) is of cours th impuls rspos, giv by h() 2 H ( ) j d Th basic dsig problm is to dtrmi th impuls rspos h(), or, th cofficits b r, for r to M, rquird to achiv a dsird H(). Ths cofficits ar of cours th costats that appar i th umrator of th trasfr fuctio H(z). Th various trasformatios usd i IIR filtr dsig caot b usd hr sic thy usually yild IIR fuctios, i.., with both umrator ad domiator cofficits. b M jm 6.2 Charactristics of FIR digital filtrs Illustratio Th quatios of th thr-trm movig avrag filtr ar rpatd blow x( ) x( ) x( 2) y() 3 2 Y ( z) z z H(z) X ( z) 3 j j2 j j j j ( ) 2cos H ( ) j This is a crud low pass filtr with liar phas, H() ω. %Magitud ad phas rspos of 3-cofficit movig avrag filtr %Filtr cofficits: h() {/3, /3, /3} b3[/3, /3, /3], a[] w-pi: pi/256: pi; Hw3frqz(b3, a, w); subplot(2,, ), plot(w, abs(hw3)); lgd ('Magitud'); xlabl('frqucy \omga, rad/sampl'), ylabl('magitud of H(\omga)'); grid DSP-6 (FIR) 3 of 82 Dr. Ravi Billa

4 Magitud of H() Phas of H() subplot(2,, 2), plot(w, agl(hw3)); lgd ('Phas'); xlabl('frqucy \omga, rad/sampl'), ylabl('phas of H(\omga)'); grid Magitud Frqucy, rad/sampl 4 2 Phas Frqucy, rad/sampl Normalizd frqucy W dfi th r ω/π. As ω gos from to th variabl r gos from to. This corrspods to a frqucy rag of F s /2 to F s /2 Hz. I trms of th ormalizd frqucy th frqucy rspos of th thr-trm movig avrag filtr bcoms j r j2 r H(r) 3 %Magitud ad phas rspos of 3-cofficit movig avrag filtr %Filtr cofficits: h() {/3, /3, /3} subplot(2,,); fplot('abs((/3)*(+xp(-j*pi*r)+xp(-j*2*pi*r)))', [-, ], 'k'); lgd ('Magitud'); xlabl('normalizd frqucy, r'); ylabl('magitud of H(r)'); grid subplot(2,,2);fplot('agl((/3)*(+xp(-j*pi*r)+xp(-j*2*pi*r)))', [-, ], 'k'); lgd ('Phas'); xlabl('normalizd frqucy');ylabl('phas of H(r)'); grid DSP-6 (FIR) 4 of 82 Dr. Ravi Billa

5 Magitud of H(r) Phas of H(r) Magitud Normalizd frqucy, r 4 2 Phas Normalizd frqucy DSP-6 (FIR) 5 of 82 Dr. Ravi Billa

6 Magitud of H() Phas of H() W illustrat blow th charactristics of svral typs of FIR filtr. Th filtr lgth N may b a odd (prfrrd) or a v umbr. Furthr, w ar typically itrstd i liar phas. This rquirs th impuls rspos to hav ithr v or odd symmtry about its ctr. Exampl 6.2. Fid th frqucy rspos of th followig FIR filtrs A. h() {.25,.5,.25} Ev symmtry B. h() {.5,.3,.2} No symmtry C. h() {.25,.5,.25} No symmtry D. h() {.25,,.25} Odd symmtry Solutio (A) Th squc h() {.25,.5,.25} has v symmtry. %Magitud ad phas rspos of h() {.25,.5,.25} %Filtr cofficits Ev symmtry b3[.25,.5,.25], a[] w-pi: pi/256: pi; Hw3frqz(b3, a, w); subplot(2,, ), plot(w, abs(hw3)); lgd ('Magitud'); xlabl('frqucy \omga, rad/sampl'), ylabl('magitud of H(\omga)'); grid subplot(2,, 2), plot(w, agl(hw3)); lgd ('Phas -\omga'); xlabl('frqucy \omga, rad/sampl'), ylabl('phas of H(\omga)'); grid Magitud Frqucy, rad/sampl 4 2 Phas Frqucy, rad/sampl DSP-6 (FIR) 6 of 82 Dr. Ravi Billa

7 Phas of H() Magitud of H() (B) Th squc h() {.5,.3,.2}is ot symmtric. %Magitud ad phas rspos of h() {.5,.3,.2} %Filtr cofficits No symmtry b3[.5,.3,.2], a[] w-pi: pi/256: pi; Hw3frqz(b3, a, w); subplot(2,, ), plot(w, abs(hw3)); lgd ('Magitud'); xlabl('frqucy \omga, rad/sampl'), ylabl('magitud of H(\omga)'); grid subplot(2,, 2), plot(w, agl(hw3)); lgd ('Noliar Phas'); xlabl('frqucy \omga, rad/sampl'), ylabl('phas of H(\omga)'); grid.8 Magitud Frqucy, rad/sampl.5 Noliar Phas Frqucy, rad/sampl DSP-6 (FIR) 7 of 82 Dr. Ravi Billa

8 Magitud of H() Phas of H() (C) Th squc h() {.25,.5,.25} is ot symmtric. %Magitud ad phas rspos of h() {.25,.5,.25} %Filtr cofficits This is ot odd symmtry b3[.25,.5, -.25], a[] w-pi: pi/256: pi; Hw3frqz(b3, a, w); subplot(2,, ), plot(w, abs(hw3)); lgd ('Magitud'); xlabl('frqucy \omga, rad/sampl'), ylabl('magitud of H(\omga)'); grid subplot(2,, 2), plot(w, agl(hw3)); lgd ('Noliar Phas'); xlabl('frqucy \omga, rad/sampl'), ylabl('phas of H(\omga)'); grid.9.8 Magitud Frqucy, rad/sampl 4 2 Noliar Phas Frqucy, rad/sampl (D) Th squc h() {.25,,.25} has odd symmtry. %Magitud ad phas rspos of h() {.25,,.25} %Filtr cofficits This is odd symmtry b3[.25,, -.25], a[] w-pi: pi/256: pi; Hw3frqz(b3, a, w); subplot(2,, ), plot(w, abs(hw3)); lgd ('Magitud'); xlabl('frqucy \omga, rad/sampl'), ylabl('magitud of H(\omga)'); grid subplot(2,, 2), plot(w, agl(hw3)); lgd ('Phas -\omga + \pi/2 '); xlabl('frqucy \omga, rad/sampl'), ylabl('phas of H(\omga)'); grid DSP-6 (FIR) 8 of 82 Dr. Ravi Billa

9 Magitud of H() Phas of H() Magitud Frqucy, rad/sampl 2 Phas - + / Frqucy, rad/sampl 6.3 Frqucy rspos Ralizatio of liar phas FIR filtrs A importat spcial subst of FIR filtrs has a liar phas charactristic. Liar phas rsults if th impuls rspos is symmtric about its ctr. For a causal filtr whos impuls rspos bgis at ad ds at N, this symmtry is xprssd thus Ev: h() h(n ), Odd: h() h(n ), for,,, (N ) a total of N poits for,,, (N ) a total of N poits This symmtry allows th trasfr fuctio to b rwritt so that oly half th umbr of multiplicatios is rquird for th rsultig ralizatio. Liar phas phas ad dlay distortio Assum a low pass filtr with frqucy rspos j H( ) giv by j j k H ( ), ω < ω c, ω c < ω < π whr k is a itgr. This is a liar phas filtr with th slop of th phas curv i th pass j bad big k. Lt X ( ) rprst th Fourir trasform of a iput squc x(). Th th j j j j trasform of th output squc y() is giv by Y ( ) X ( ). H ( ). If X ( ) is tirly j withi th pass bad of H( ) th j j Y ( ) X ( ). j k DSP-6 (FIR) 9 of 82 Dr. Ravi Billa

10 j So th output sigal y() ca b obtaid as th ivrs F-trasform of Y( ) as y() x( k), a dlayd vrsio of x() H ω c ω c ω kω H ω Thus th liar phas filtr did ot altr th shap of th origial sigal, simply traslatd (dlayd) it by k sampls. If th phas rspos had ot b liar, th output sigal would hav b a distortd vrsio of x(). It ca b show that a causal IIR filtr caot produc a liar phas charactristic ad that oly spcial forms of causal FIR filtrs ca giv liar phas. Thorm If h() rprsts th impuls rspos of a discrt tim systm, a cssary ad sufficit coditio for liar phas is that h() hav a fiit duratio N, ad that it b symmtric about its midpoit. Exampl 6.3. (a) For th FIR filtr of lgth N 7 with impuls rspos h() lt h() h(n ). Show that th filtr has a liar phas charactristic. (b) Rpat for N 8. h() N Solutio (a) For N 7, th positiv symmtry rlatio h() h(n ) lads to h() h(6 ) which mas that h() h(6), h() h(5), ad h(2) h(4), as show i figur abov. DSP-6 (FIR) of 82 Dr. Ravi Billa

11 6 j H(z) h ( ) z ad H ( ) H ( z) j h( ) z j H ( ) h () + h + j ( ) + h(4) j4 + h(2) h(5) j3 j3 j2 { h () + h () + j + h( 4) + h(5) Sic h() h(6), tc., w ca writ j H ( ) j2 j3 j 3 { ()( 3 j h ) j3 { ()cos 3 j2 j5 + h(3) + h(6) h 2) j3 j6 j ( + h (3) + h(6) j3 } 6 j j 2 + ()( 2 j h j j ) + h(2)( ) + h (3) } 2h + 2h ()cos 2 + 2h (2)cos + h (3) } a(3) a(2) a() a() 3 j3 a( k) cos k, with a() h(3) ad a(k) 2h(3 k), k, 2, 3 k Th cofficits, i gral, ar giv by N N a() h ad a(k) 2 h k, for k, 2,..., (N )/2 2 2 j j H ( ) ± H( ) 3 j j H ( ) 3 j3 k a( k) cos k j j whr ± H ( ) a( k) cos k ad H( ) Θ(ω) 3ω. Th phas rspos is obviously k liar, with slop 3 (N )/2 which mas that th dlay is a itgr umbr of sampls. Slop 3 H(ω) or Θ(ω) (b) For N 8, th positiv symmtry rlatio h() h(n ) lads to h() h(7 ), which mas h() h(7), h() h(6), h(2) h(5), ad h(3) h(4) as show i figur blow. h() N DSP-6 (FIR) of 82 Dr. Ravi Billa

12 7 j H(z) h ( ) z ad H ( ) H ( z) j h( ) z j H ( ) h () + h + j ( ) + h(4) + h(4) Sic h() h(7), tc., w ca writ j H ( ) j4 + h(2) h(5) j2 j5 + + h(3) h(6) j3 j6 7 j7 / 2 j7 / 2 j5 / 2 j3 / 2 { h () + h () + h (2) + j / 2 + h(5) j7 / 2 j7 / 2 j7 / 2 { h()( ) j7 / 2 j3 / 2 + h(6) j5 / 2 j5 / 2 j5 / 2 + h()( ) + h(7) h (3) + h(7) j j7 j / 2 j7 / 2 j3 / 2 j3 / 2 j / 2 j / 2 + h(2)( ) + h(3)( )} h () cos 2h() cos 2h(2) cos 2h(3) cos b(4) b(3) b(2) b() With b(k) 2h((N/2) k), for k, 2,, N/2, w ca writ j j H ( ) ± H( ) 4 j j H ( ) 4 j7 / 2 k b( k) cos[( k / 2) ] j j j whr ± H ( ) b( k)cos[( k / 2) ] ad H( ) Θ(ω) 7ω/2. Th phas, H( ), k is clarly liar. Howvr, th slop of th phas curv is ( 7/2), which is ot a itgr. Th o-itgr dlay will caus th valus of th squc to b chagd, which, i som cass, may b udsirabl. } Slop 7/2 H(ω) or Θ(ω) DSP-6 (FIR) 2 of 82 Dr. Ravi Billa

13 Implmtatio For a causal filtr whos impuls rspos has v symmtry: h() h(n ), th trasfr fuctio H(z) ʓ{h()} v or odd, as follows. for,,, (N ) a total of N poits N h( ) z For v N Th diffrc quatio is drivd startig from H(z), ( N / 2) H(z) ( N ) h ( ) z z Sic Y(z) H(z) X(z), w ca writ ( N / 2) ( N h( ) z z ( N) h() z X ( z ) Y(z) ca b writt, dpdig o whthr N is X(z) h() z ) + ( N2) z X ( z) N N N h z z X ( z) 2 Takig th ivrs z-trasform of th abov w gt y() as ( ) x( ) x N h ( ) x( ) x N 2 y() h N N N h x x Th dlayd vrsios of x() ar addd i pairs ad th multiplid by cofficits h(.). This is show i figur blow for N 8. Not that thr ar a odd umbr ( 7) of dlay lmts. Thr ar N/2 4 multiplicatios ad (N/2) addrs (actually th umbr of twooprad additios is ). Figur for N 8 x() x( ) x( 2) x( 3) x( 4) x( 5) x( 6) z z z z z z z x( 7) h() h(7) + h() h(6) + + h(2) h(5) + y() + h(3) h(4) DSP-6 (FIR) 3 of 82 Dr. Ravi Billa

14 For odd N W d ot driv th quatios (thy would b cssary if w wr writig a computr program to automat it). For N 7, thr ar N 6 dlay lmts a v umbr of dlay lmts. Thr ar (N + )/2 (7 + )/2 4 multiplicatios ad 4 addrs (th umbr of two-oprad additios is 6). Figur for N 7 x() x( ) x( 2) x( 3) x( 4) x( 5) z z z z z z x( 6) + h() h(6) + + h() h(5) h(2) h(4) + y() h(3) Proprtis of FIR digital filtrs Th siusoidal stady stat trasfr fuctio of a digital filtr is priodic i th samplig frqucy. W hav j H ( ) H ( z) j j h( ) z i which h() rprsts th trms of th uit puls rspos. Th abov xprssio ca b dcomposd ito ral ad imagiary compots by writig j H ( ) h( ) cos j h( )si H R (ω) + j H I (ω) whr th ral ad imagiary parts of th trasfr fuctio ar giv by H R (ω) h( ) cos ad H I (ω) h( )si Ths xprssios for H R (ω) ad H I (ω) show that. H R (ω) is a v fuctio of frqucy ad H I (ω) is a odd fuctio of frqucy. 2. If h() is a v squc, th imagiary part of th trasfr fuctio, H I (ω), will b zro. (Th v squc, h(), multiplid by th odd squc si ω will yild a odd squc. A odd squc summd ovr symmtric limits yilds zro.) I this cas j H ( ) h( ) cos H R (ω) 3. Similarly, if h() is a odd squc, th ral part of th trasfr fuctio, H R (ω), will b zro H ( j ) j h( )si j H I (ω) DSP-6 (FIR) 4 of 82 Dr. Ravi Billa

15 Thus a v uit puls rspos yilds a ral-valud trasfr fuctio ad a odd uit puls rspos yilds o imagiary-valud trasfr fuctio. Rcall that a ral trasfr fuctio has a phas shift of or radias, whil a imagiary trasfr fuctio has a phas shift of / 2 radias as show i figurs blow. So, by makig th uit puls rspos ithr v or odd, w ca grat a trasfr fuctio that is ithr ral or imagiary. h() (Ev) h() (Odd) Θ(ω) Θ(ω) Θ(ω) π/2 π π π π π/2 Two typs of applicatios I dsigig digital filtrs w ar usually itrstd i o of th followig two situatios:. Filtrig W ar itrstd i th amplitud rspos of th filtr (.g., low pass, bad pass, tc.) without phas distortio. This is ralizd by usig a ral j valud trasfr fuctio, i.., H ( ) H R (ω), with H I (ω). 2. Filtrig plus quadratur phas shift Ths applicatios iclud itgrators, diffrtiators, ad Hilbrt trasform dvics. For all of ths th dsird j trasfr fuctio is imagiary, i.., H ( ) j H I (ω), with H R (ω) FIR Filtr Dsig Procdur. Dcid whthr H R (ω) or H I (ω) is to b st qual to zro. Typically, H d (ω) H R (ω) + j for filtrig, ad H d (ω) + j H I (ω) for itgrators, diffrtiators ad Hilbrt trasformrs 2. Expad H d (ω) ito Fourir sris h d (). This is th dsird impuls rspos. 3. Dcid o th lgth N of th impuls rspos duratio. Trucat th squc h d () to N sampls {h t (), (N )/2 to (N )/2}. Ev valus of N rsult i dlays of half-sampl priods; odd valus of N avoid this problm. 4. Apply widow fuctio {w(), (N )/2 to (N )/2} 5. Fid th trasfr fuctio H(z) z (N )/2 H t (z) ad th frqucy rspos H(ω). If ot satisfactory th valu of N may hav to b icrasd or a diffrt widow fuctio may b trid. Phas dlay ad group dlay If w cosidr a sigal that cosists of svral frqucy compots (such as a spch wavform or a modulatd sigal) th phas dlay of th filtr is DSP-6 (FIR) 5 of 82 Dr. Ravi Billa

16 th amout of tim dlay ach frqucy compot of th sigal suffrs i goig through th filtr. Mathmatically, th phas dlay τ p is giv by scat () τ p Th group dlay o th othr had is th avrag tim dlay th composit sigal suffrs at ach frqucy. Th group dlay τ g is giv by th slop (tagt) at ω d ( ) τ g d j whr Θ(ω) H( ) of th filtr. A oliar phas charactristic will caus phas distortio, which is udsirabl i may applicatios, for xampl, music, data trasmissio, vido ad biomdici. A filtr is said to hav a liar phas rspos if its phas rspos satisfis o of th followig rlatioships: Θ(ω) kω (A) or Θ(ω) β kω (B) whr k ad β ar costats. If a filtr satisfis quatio (A) its group dlay ad phas dlay ar th sam costat k. It ca b show that for coditio (A) to b satisfid th impuls rspos of th filtr must hav positiv symmtry (aka v symmtry or just symmtry). Th phas rspos i this cas is simply a fuctio of th filtr lgth N: h() h(n ),,, 2,, (N )/2 for N odd,, 2,, (N/2) for N v k (N )/2 If quatio (B) is satisfid th filtr will hav a costat group dlay oly. I this cas, th impuls rspos h() has gativ symmtry (aka odd symmtry or atisymmtry): h() h(n ) k (N )/2 β /2 DSP-6 (FIR) 6 of 82 Dr. Ravi Billa

17 Aalog filtr backgroud of phas ad group dlay Phas dlay at a giv frqucy is th slop of th scat li from dc to th particular frqucy ad is a sort of ovrall avrag dlay paramtr. Phas dlay is computd ovr th frqucy rag rprstig th major portio of th iput sigal spctrum ( to F i th figur blow). H(F) Tagt at F Slop group dlay at F Phas curv F Scat at F Slop phas dlay ovr to F F Th group dlay at a giv frqucy rprsts th slop of th tagt li at th particular frqucy ad rprsts a local or arrow rag (ighborhood of F i th figur) dlay paramtr. A cas of sigificac ivolvig both phas dlay ad group dlay is that of a arrow bad modulatd sigal. Wh a arrow bad modulatd sigal is passd through a filtr, th carrir is dlayd by a tim qual to th phas dlay, whil th vlop (or itlligc) is dlayd by a tim approximatly qual to th group dlay. Sic th itlligc (modulatig sigal) rprsts th dsird iformatio cotaid i such sigals, strog mphasis o good group dlay charactristics is oft mad i filtrs dsigd for procssig modulatd wavforms. DSP-6 (FIR) 7 of 82 Dr. Ravi Billa

18 Summary of symmtry Positiv symmtry (or just symmtry or v symmtry about th middl) is charactrizd by h() h(n ). Show that for positiv symmtry ( N ) / 2 j a) For N odd (Typ I): H ( ) j( N ) / 2 k a() a( k) cos k N N h & a(k) 2 h k, k 2 2 j j( N ) / 2 b) For N v (Typ II): H ( ) b( k)cos[( k / 2) ] b(k) N / 2 k N 2 h k 2 h() Typ I N is odd Ctr of Symmtry h() Typ II N is v Ctr of Symmtry 6 N 7 N DSP-6 (FIR) 8 of 82 Dr. Ravi Billa

19 Summary of symmtry, cot d Ngativ symmtry (or atisymmtry or odd symmtry about th middl) is charactrizd by h() h(n ). Show that for gativ symmtry N j ( N ) / 2 j a) For N odd (Typ III): H ( ) 2 2 k a() N j N / 2 a( k)si k N N h & a(k) 2 h k, k 2 2 j b) For N v (Typ IV): H ( ) 2 2 b( k)si[( k / 2) ] b(k) k N 2 h k 2 h() Typ III N is odd Ctr of Symmtry h() Typ IV N is v Ctr of Symmtry 6 N 7 N DSP-6 (FIR) 9 of 82 Dr. Ravi Billa

20 Magitud of H() Phas of H() Qualitativ atur of symmtry Typ I Positiv symmtry, N is odd. To illustrat tak N 5: ( N ) / 2 j H ( ) j( N ) / 2 k j2 a( k) cos k a( k)cos k 2 k j2 [a() + a() cos ω + a(2) cos 2ω] W hav to add up a(), ad th two cosi trms. It is clar that at ω all th cosi trms ar j at thir positiv pak, so that wh addd th rspos of H ( ) vs. ω would idicat a low pass filtr. Cosidr x( ) x( ) x( 2) x( 3) x( 4) y() 5 j j2 j3 j4 j H ( ) H( z) j z 5 j2 j j j2 j2 2cos 2cos j2 %Frqucy rspos of movig avrag filtr h() {.2,.2,.2,.2,.2} b5 [.2,.2,.2,.2,.2], a [] w-pi: pi/256: pi; Hw5frqz(b5, a, w); subplot(2,, ), plot(w, abs(hw5)); lgd ('Magitud'); titl ('Typ I, N is odd'); xlabl('frqucy \omga, rad/sampl'), ylabl('magitud of H(\omga)'); grid subplot(2,, 2), plot(w, agl(hw5)); lgd ('Phas'); xlabl('frqucy \omga, rad/sampl'), ylabl('phas of H(\omga)'); grid Typ I, N is odd Magitud Frqucy, rad/sampl 4 2 Phas Frqucy, rad/sampl DSP-6 (FIR) 2 of 82 Dr. Ravi Billa

21 Magitud of H() Phas of H() Typ II Positiv symmtry, N is v. Tak N 6: N / 2 j j( N ) / 2 j5 / 2 H ( ) b( k)cos[( k / 2) ] b( k)cos[( k / 2) ] k j5 / 2 [b() cos ω/2 + b(2) cos 3ω/2 + b(3) cos 5ω/2] At, corrspodig to half th samplig frqucy (maximum possibl frqucy), all th cosi trms will b zro. Thus this typ of filtr is usuitabl as a high-pass filtr. It should b ok as a low pass filtr. Cosidr x( ) x( ) x( 2) x( 3) x( 4) x( 5) y() 6 j j2 j3 j4 j5 j H ( ) H( z) j z 6 j(5/ 2) j 5/ 2) j(3/ 2) j(/ 2) 6 cos( / 2) cos(3 / 2) cos(5 / 2) 3 3 k ( j(/ 2) j(3/ 2) j(5/ 2) 2) j(5/ %Frqucy rspos of movig avrag filtr h() {/6, /6, /6, /6, /6, /6} b6 [/6, /6, /6, /6, /6, /6], a [] w-pi: pi/256: pi; Hw6frqz(b6, a, w); subplot(2,, ), plot(w, abs(hw6)); lgd ('Magitud'); titl ('Typ II, N is v'); xlabl('frqucy \omga, rad/sampl'), ylabl('magitud of H(\omga)'); grid subplot(2,, 2), plot(w, agl(hw6)); lgd ('Phas'); xlabl('frqucy \omga, rad/sampl'), ylabl('phas of H(\omga)'); grid Typ II, N is v Magitud Frqucy, rad/sampl 4 2 Phas Frqucy, rad/sampl DSP-6 (FIR) 2 of 82 Dr. Ravi Billa

22 Magitud of H() Phas of H() Typ III Ngativ symmtry, N is odd. This itroducs a 9 ( π/2) phas shift. Bcaus of th si trms H is always zro at ω ad at ω π/2 (half th samplig frqucy). Thrfor th filtr is usuitabl as a low pass or a high pass filtr. To illustrat tak N 5 ad h() {.2,.2,, -.2, -.2} N j ( N ) / 2 j H ( ) 2 2 k j2 j( / 2) a( k)si k 2 k a( k)si k 2 2 k N a() h h(2) 2 N a(k) 2 h k 2 h2 k, k 2 Etc. 5 j (5 ) / 2 a( k)si k %Frqucy rspos of Typ III filtr, h() {.2,.2,, -.2, -.2} b5 [.2,.2,, -.2, -.2], a [] w-pi: pi/256: pi; Hw5frqz(b5, a, w); subplot(2,, ), plot(w, abs(hw5)); lgd ('Magitud'); titl ('Typ III, N is odd'); xlabl('frqucy \omga, rad/sampl'), ylabl('magitud of H(\omga)'); grid subplot(2,, 2), plot(w, agl(hw5)); lgd ('Phas'); xlabl('frqucy \omga, rad/sampl'), ylabl('phas of H(\omga)'); grid.8.6 Typ III, N is odd Magitud Frqucy, rad/sampl 4 2 Phas Frqucy, rad/sampl DSP-6 (FIR) 22 of 82 Dr. Ravi Billa

23 Magitud of H() Phas of H() Typ IV Ngativ symmtry, N is v. This itroducs a 9 ( π/2) phas shift. Bcaus of th si trms H is always zro at ω. Thrfor th filtr is usuitabl as a low pass filtr. To illustrat tak N 6 ad h() {/6, /6, /6, -/6, -/6, -/6} N j N / 2 j H ( ) 2 2 b( k)si[( k / 2) ] 2 2 b( k)si[( k / 2) ] Etc. k 3 j3 j( / 2) b( k)si[( k / 2) ] k 6 j 6/ 2 k ad b(k) h3 k 2, k to 3 %Frqucy rspos of Typ IV filtr h() {/6, /6, /6, -/6, -/6, -/6} b6 [/6, /6, /6, -/6, -/6, -/6], a [] w-pi: pi/256: pi; Hw6frqz(b6, a, w); subplot(2,, ), plot(w, abs(hw6)); lgd ('Magitud'); titl ('Typ IV, N is v'); xlabl('frqucy \omga, rad/sampl'), ylabl('magitud of H(\omga)'); grid subplot(2,, 2), plot(w, agl(hw6)); lgd ('Phas'); xlabl('frqucy \omga, rad/sampl'), ylabl('phas of H(\omga)'); grid.8.6 Typ IV, N is v Magitud Frqucy, rad/sampl 4 2 Phas Frqucy, rad/sampl Typs III ad IV ar oft usd to dsig diffrtiators ad Hilbrt trasformrs bcaus of th 9 phas shift that ach o ca provid. DSP-6 (FIR) 23 of 82 Dr. Ravi Billa

24 Th phas dlay for Typ I ad II filtrs or group dlay for all four typs of filtrs is xprssibl i trms of th umbr of cofficits of th filtr ad so ca b corrctd to giv a zro phas or group dlay rspos. N Typs I ad II: τ p τ g Θ(ω)/ω T 2 d ( ) N Typs III ad IV: τ g T d 2 Magitud ( H(ω) ) rspos at ω rad. ω π rad. Typ I Max Low pass Typ II Zro OK as LP Not OK as HP filtr Typ III Zro Zro 9 Phas shift Typ IV Zro 9 Phas shift 6.4 Dsig of FIR digital filtrs Th Fourir sris ad widowig mthod This mthod of filtr dsig origiats with th obsrvatio that th siusoidal stady stat trasfr fuctio of a digital filtr is priodic i th samplig frqucy. Sic H() is a cotiuous ad priodic fuctio of w ca xpad it ito a Fourir sris. Th rsultig Fourir cofficits ar th impuls rspos, h(). Th major disadvatag is that o caot asily spcify i advac th xact valus for pass bad ad stop bad attuatio/rippl lvls, so it may b cssary to chck svral altrat dsigs to gt th rquird o. j Cosidr th idal low pass filtr with frqucy rspos Hd ( ) or H d () as show blow. Th subscript d mas that it is th dsird or idal filtr. H d (ω) H d (ω) 2π π ω c ω c π 2π ω Th impuls rspos is giv by j j h d () H d d 2 ( ) 2 c si c,, c, c j d DSP-6 (FIR) 24 of 82 Dr. Ravi Billa

25 This is a o-causal ifiit impuls rspos squc. It is mad a fiit impuls rspos squc by trucatig it symmtrically about ; it is mad causal by shiftig th trucatd squc to th right so that it starts at. Th shiftig rsults i a tim dlay ad w shall igor it for ow. Th trucatio rsults i th squc h t () whr th subscript t mas trucatio but w shall igor th subscript h() h d (), (N )/2 (N )/2, othrwis I gral h() ca b thought of as obtaid by multiplyig h d () with a widow fuctio w() as follows h() h d (). w() For th h() obtaid by simpl trucatio as abov th widow fuctio is a rctagular widow giv by w(), (N )/2 (N )/2, othrwis j j j Lt H ( ), H ( ) ad W( ) rprst th Fourir trasforms of h(), h d () ad w() d j rspctivly. Th th frqucy rspos H( ) of th rsultig filtr is th covolutio of j j H ( ) ad W ( ) giv by d j H ( ) 2 H d j j( ) j j ( ) W ( d H ( )* W( ) Th covolutio producs a smard vrsio of th idal low pass filtr. I othr words, j j j H( ) is a smard vrsio of H d ( ). I gral, th widr th mai lob of W ( ), th mor spradig or smarig, whras th arrowr th mai lob (largr N), th closr j j H( ) coms to H ( ). d For ay arbitrary widow th trasitio bad of th filtr is dtrmid by th width of th mai lob of th widow. Th sid lobs of th widow produc rippls i both pass bad ad stop bad. d DSP-6 (FIR) 25 of 82 Dr. Ravi Billa

26 I gral, w ar lft with a trad-off of makig N larg ough so that smarig is miimizd, yt th umbr of filtr cofficits ( N) is ot too larg for a rasoabl implmtatio. Som commoly usd widows ar th rctagular, Bartltt (triagular), Haig, Hammig, Blackma, ad Kaisr widows. j h d () H ( ) d ω j w() W ( ) ω j h() h d ().w() H ( ) W tur ow to filtrig applicatios. W ar itrstd i th amplitud rspos ω of th filtr (.g., low pass, bad pass, tc.) without phas distortio. This is ralizd by usig a ral j valud trasfr fuctio, i.., H ( ) H R (ω), with H I (ω). Exampl 6.4. [Dsig of 9-cofficit LP FIR filtr] Dsig a i-cofficit (or 9-poit or 9-tap) FIR digital filtr to approximat a idal low-pass filtr with a cut-off frqucy c.2. Th magitud rspos, H (), is giv blow. Tak (). d H d (ω) H d π π/5 π/5 π ω, rad/sampl Solutio Th impuls rspos of th dsird filtr is DSP-6 (FIR) 26 of 82 Dr. Ravi Billa

27 h d () 2 H.2 d j j ( ) d 2 j j.2 j.2 ( ) si.2 2 j 2 j.2 Sic h d () for < this is a ocausal filtr (also, it is ot BIBO stabl s Uit IV). Th rst of th dsig is aimd at comig up with a ocausal approximatio of th abov impuls rspos. For a rctagular widow of lgth 9, th corrspodig impuls rspos is obtaid by valuatig h d () for 4 4 o a calculator. I th MATLAB sgmt blow, which grats th h d () cofficits for 5 5, divisio by zro for causs NaN (Not a Numbr), whil all th othr cofficits ar corrct. I gratig th frqucy rspos H () w copy ad past all th cofficits xcpt for h d () which is trd by had. d(si.2 ) h d () d d( ) d cos.2 j d.2 Asid (MATLAB) Th sgmt blow grats ad stm-plots th h d () cofficits. %Calculat hd (si (.2*pi*)) / (pi*) ad stm plot -5: 5, hd (si(.2*pi*))./(pi*), stm(, hd) xlabl(''), ylabl('hd()'); grid; titl ('hd() (si (.2*pi*)) / (pi*)') -5 to 5 Warig: Divid by zro. hd -, -.38, -.63, -.64, -.4,, ,.46 -, ,.55.92, , ,.78, , , , ,, NaN , , , , , , DSP-6 (FIR) 27 of 82 Dr. Ravi Billa

28 hd() hd() (si (.2*pi*)) / (pi*) Not Th sgmt blow is usd to gt a quick look at th h d () cofficits ad thir symmtry. Th MATLAB problm with may b avoidd by rplacig with (.). This will affct th othr cofficits vry slightly (which is ot a srious problm as far as dmostratig th v symmtry of h d ()) but th accuracy of th cofficits is somwhat compromisd i th third or fourth sigificat digit. %Calculat hd (si (.2*pi*)) / (pi*) ad stm plot -4: 4, hd (si(.2*pi*(-.) ) )./(pi*(-.) ), stm(, hd), xlabl(''), ylabl('hd()'); grid; titl ('hd() (si (.2*pi*)) / (pi*)') hd.467,.9,.53,.87,.2,.87,.54,.,.468 DSP-6 (FIR) 28 of 82 Dr. Ravi Billa

29 hd() hd() (si (.2*pi*)) / (pi*) Not 2 As a altrativ to th abov, o could writ a custom program to calculat all cofficits xactly icludig h d (). Ed of Asid Th valus ar show i tabl blow: h t () {.47,.,.5,.87,.2,.87,.5,.,.47} By a rctagular widow of lgth 9 w ma that w rtai th abov 9 valus of h d (.) ad trucat th rst outsid th widow. Thus h t ( 4).47, h t ( 3).,, h t ().2,, h t (4).47 h d () DSP-6 (FIR) 29 of 82 Dr. Ravi Billa

30 Th trasfr fuctio of this filtr is 4 H t (z) h d ( ) z z +. z z z +.2 z z +.5 z +. 3 z +.47 z 4 Th causal filtr is th giv by dlayig th squc h t () by 4 sampls. That is, h() h t ( 4), ad th rsultig trasfr fuctio is H(z) z 4 H t (z) 8.47 ( z ) +. ( z z ) +.5 ( z z ) ( z z ) +.2 z W may obtai th frqucy rspos of this ralizabl (causal) filtr by sttig z j, j that is, H ( ) H( z) j. Bcaus of th trucatio th magitud H will oly b z approximatly qual to H d Gibbs phomo, s compariso of 9 cofficits vrsus cofficits blow. Furthr, bcaus of th dlay th phas H 4ω whras, as origially d H ( ) spcifid, H d. Th slop 4, showig that th filtr itroducs a dlay of 4 d sampls. 4 4ω H ω %Magitud ad phas rspos of 9-cofficit LP filtr %Filtr cofficits b9[.468,.9,.54,.87,.2,.87,.54,.9,.468], a[] w-pi: pi/256: pi; Hw9frqz(b9, a, w); subplot(2,, ), plot(w, abs(hw9)); lgd ('9 cofficits'); xlabl('frqucy \omga, rad/sampl'), ylabl('magitud of H(\omga)'); grid subplot(2,, 2), plot(w, agl(hw9)); lgd ('9 cofficits'); xlabl('frqucy \omga, rad/sampl'), ylabl('phas of H(\omga)'); grid DSP-6 (FIR) 3 of 82 Dr. Ravi Billa

31 Magitud of H() Phas of H() 9 cofficits Frqucy, rad/sampl cofficits Frqucy, rad/sampl DSP-6 (FIR) 3 of 82 Dr. Ravi Billa

32 Magitud of H() %Compariso of 9 cofficits vs. cofficits %Filtr cofficits b9[.468,.9,.54,.87,.2,.87,.54,.9,.468], a[] b [ , , ,.55.92, , , , , , , , , , , , ] w-pi: pi/256: pi; Hw9frqz(b9, a, w); Hwfrqz(b, a, w); plot(w, abs(hw9), w, abs(hw), 'k') lgd ('9 cofficits', ' cofficits'); titl('magitud Rspos'); xlabl('frqucy \omga, rad/sampl'), ylabl('magitud of H(\omga)'); grid.4.2 Magitud Rspos 9 cofficits cofficits Frqucy, rad/sampl Rctagular widow I this xampl th squc h d (), which xtds to ifiity o both sids, has b trucatd to 9 trms. This trucatio procss ca b thought of as multiplyig th DSP-6 (FIR) 32 of 82 Dr. Ravi Billa

33 ifiitly log squc by a widow fuctio calld th rctagular widow, w R (). Th figur blow shows both h d () ad w R () i th udlayd form, that is, symmtrically disposd about. Spcifyig th widow fuctio Th itrval ovr which th widow fuctio is dfid h d () w R () dpds o whthr w first dlay h d () ad th trucat it or th othr way aroud. If, istad of trucatig first ad th dlayig, w adopt th procdur of first dlayig h d () ad th trucatig it, th widow fuctio may b dfid ovr th itrval N, whr N is th umbr of trms rtaid. With this udrstadig th rctagular widow, w R (), is giv blow. w R (), N, lswhr If, howvr, w dfi w R () symmtrically about, as w do latr, w hav w R (), (N )/2 (N )/2, lswhr I this cas w hav implid that N is odd. DSP-6 (FIR) 33 of 82 Dr. Ravi Billa

34 Hammig widow As a altrativ to th rctagular widow w shall apply th Hammig widow, dfid ovr th itrval N, by w Ham () cos[ 2 /( N )], N, lswhr For N 9, th Hammig widow is giv by w Ham () cos( / 4), w Ham () {.8,.25,.54,.865,,.865,.54,.25,.8} w Ham () Imagi that w li up this squc alogsid th h d () giv arlir. (This mas w should imagi w Ham () is movd to th lft by 4 sampls). W th multiply th two squcs at ach poit to gt th widowd squc, h t (): h d () {.47,.,.5,.87,.2,.87,.5,.,.47} w() {.8,.25,.54,.865,,.865,.54,.25,.8} h t () {.382,.26,.85,.67,.2,.67,.85,.26,.382} Th trasfr fuctio is giv by H t (z).382 z z z z z +.85 z z z 4 Dlayig by 4 sampl priods w gt H(z) z 4 8 H t (z).382 ( z ) +.26 ( z z ) +.85 ( z z ) ( z z ) +.2 z 4 HW Writ th corrspodig diffrc quatio ad show th filtr structur. W compar blow th 9-tap Hammig widowd filtr to th filtr without th widow. %Compariso of o widow vs. Hammig widow %Ni-tap filtr cofficits, o widow b9[.468,.9,.54,.87,.2,.87,.54,.9,.468], % %Ni-tap, Hammig-widowd filtr cofficits DSP-6 (FIR) 34 of 82 Dr. Ravi Billa

35 Magitud of H() b9ham[.382,.26,.85,.67,.2,.67,.85,.26,.382] a[] w-pi: pi/256: pi; Hw9frqz(b9, a, w); Hw9Hamfrqz(b9Ham, a, w); plot(w, abs(hw9), w, abs(hw9ham), 'k') lgd ('9 cofficits, No widow', '9 cofficits, Hammig widow'); titl('magitud Rspos'); xlabl('frqucy \omga, rad/sampl'), ylabl('magitud of H(\omga)'); grid.4.2 Magitud Rspos 9 cofficits, No widow 9 cofficits, Hammig widow Frqucy, rad/sampl Exampl [Low pass filtr] [23] Dsig a low pass FIR filtr that approximats th followig frqucy rspos H(F), F Hz, lswhr i F F s /2 whr th samplig frqucy F s is 8 sps. Th impuls rspos duratio is to b limitd to 2.5 msc. Draw th filtr structur. Solutio Not th spcs ar giv i Hrtz. O th digital frqucy scal gos from to 2, with 2 corrspodig to th samplig frqucy of F s 8 Hz or to s 2 8 rad/sc. Basd o this Hz corrspods to (/8)2 /4. Or w may us th rlatio T to covrt th aalog frqucy Hz to th digital frqucy. Thus T 2F T 2 (/8) /4 Thus th spcificatios ar rstatd o th ω scal as DSP-6 (FIR) 35 of 82 Dr. Ravi Billa

36 j H ( ), /4 /4 d, lswhr i th rag [ to ] H(F) or H d (ω) Tak H d (ω) k k 4k 8k π/4 π 2π F ω W ca valuat th impuls rspos h d () 2 H d j j ( ) d 2 / 4 / 4. j d 2 j j j / 4 j / 4 ( ) si.25 2 j W d to dcid th umbr of cofficits, that is, th filtr lgth, dd. Th problm spcifis th impuls rspos duratio is to b limitd to 2.5 msc. At 8 h() / 4 / 4 2 N 2.5 msc (2 Poits) sampls/sc., th samplig priod T /8.25 msc. So th duratio of 2.5msc traslat to 2.5/.25 2 sampl priods. Arragig ths o a liar scal w s that th filtr lgth N 2 or w d 2 cofficits. Thrfor dtrmi th valus h d ( ) through h d (), h t () si.25, h t ().25 by L Hopital s rul %Calculat hd (si (.25*pi*)) / (pi*) ad stm plot -5: 5, hd (si(.25*pi*))./(pi*), stm(, hd) -5 to 5 Warig: Divid by zro. DSP-6 (FIR) 36 of 82 Dr. Ravi Billa

37 hd [ NaN ] ± ±2 ±3 ±4 ±5 ±6 ±7 ±8 ±9 ± h t () Th tak th z trasform H t (z) h t ( ) z. Th dtrmi th trasfr fuctio, H(z), of th ralizabl FIR filtr as H(z) z H t (z) from which th filtr structur ca b draw. Th frqucy rspos is compard blow for 2 ad cofficits. Th 2-tap filtr is ot that bad. %2-tap filtr cofficits b2[ ], a[] %-tap filtr cofficits b[ ], w-pi: pi/256: pi; Hw2frqz(b2, a, w); Hwfrqz(b, a, w); plot(w, abs(hw2), w, abs(hw), 'k') lgd ('2 cofficits', ' cofficits'); titl('magitud Rspos'); xlabl('frqucy \omga, rad/sampl'), ylabl('magitud of H(\omga)'); DSP-6 (FIR) 37 of 82 Dr. Ravi Billa

38 Magitud of H() Magitud Rspos 2 cofficits cofficits Frqucy, rad/sampl Exampl [Vry arrow bad pass filtr] [23] Dsig a bad pass FIR filtr that approximats th followig frqucy rspos: H(F), 6 F 2 Hz, lswhr i F F s /2 wh th samplig frqucy is 8 sps. Limit th duratio of impuls rspos to 2 msc. Draw th filtr structur. Solutio Th samplig frqucy F s 8 Hz corrspods to ω 2 rad. Thus 6 Hz corrspods to ω (6/8)2.4 rad., ad 2 Hz corrspods to ω (2/8)2.5 rad. H d (ω) Tak H d (ω) π.5π.4π.4π.5π π ω DSP-6 (FIR) 38 of 82 Dr. Ravi Billa

39 j H ( ),.5 ω.4 ad.4 ω.5 d h d () 2 2, lswhr i th rag [ to ].4.5 j j j j H d ( ) d 2 d d.5.4 j j2 j j j.5.4 j j j j j.5 j2 j.5 si.5 si. 4 j.4 j2 j.4 h() 6 N 2 msc (7 Poits) Filtr lgth N is dtrmid by th duratio of th impuls rspos. Two milliscods corrspods to.2 / (/8) 6 sampl priods. This mas that th filtr lgth N 7, ad thr will b 7 cofficits. Dtrmi th valus of h d () for 8 8, so that h t () { h d ( 8), h d ( 7),, h d (),, h d (8) }, ad H t (z) h t ( ) z 8 8 Dlay th impuls rspos by 8 sampl priods so that h() h t ( 8) ad H(z) z 8 H t (z) h d (). by L Hopital s rul %Calculat hd (si(.5*pi*) - si(.4*pi*)) /(pi*) ad stm plot -5: 5, hd (si(.5*pi*) - si(.4*pi*))./(pi*), stm(, hd) -5 to 5 Warig: Divid by zro. hd [ DSP-6 (FIR) 39 of 82 Dr. Ravi Billa

40 NaN ] ± ±2 ±3 ±4 ±5 ±6 ±7 ±8 h d () Th frqucy rsposs 7-tap ad -tap filtrs ar show blow. Th 7-tap filtr looks mor lik a low pass filtr! Owig to th vry arrow pass bad a vry larg umbr of cofficits is dd bfor th pass bad bcoms discribl. I gral FIR filtrs ar charactrizd by a larg umbr of cofficits compard to IIR filtrs. %Filtr cofficits b7[ ], a[] b[ ] w-pi: pi/256: pi; Hw7frqz(b7, a, w); Hwfrqz(b, a, w); subplot(2,, ), plot(w, abs(hw7)); lgd ('7 cofficits'); xlabl('frqucy \omga, rad/sampl'), ylabl('magitud of H(\omga)'); grid subplot(2,, 2), plot(w, abs(hw)); lgd (' cofficits'); xlabl('frqucy \omga, rad/sampl'), ylabl('magitud of H(\omga)'); grid DSP-6 (FIR) 4 of 82 Dr. Ravi Billa

41 Magitud of H() Magitud of H() 7 cofficits Frqucy, rad/sampl.8.6 cofficits Frqucy, rad/sampl Exampl [Bad pass filtr] [23] Dsig a bad pass filtr to pass frqucis i th rag to 2 rad/sc usig Haig widow with N 5. Draw th filtr structur ad plot its spctrum. Solutio Th Haig widow is also kow as th Ha widow. It is a raisd cosi, is vry similar to th Hammig widow, ad is giv by w Ha ().5 cos2 /( N ), N, lswhr Not I ordr to covrt th aalog frqucis to digital w d th samplig tim T. Th samplig frqucy F s (or th samplig tim T) is ot spcifid. W assum T sc or, what amouts to th sam, w assum that th frqucis giv ar actually digital, that is, to 2 rad/sampl istad of to 2 rad/sc. Howvr, th solutio blow assums that th frqucis ar giv corrctly, that is, that thy ar aalog, ad uss a samplig frqucy of s 4 rad / sc. Although thr is a spcializd vrsio of th samplig thorm for bad pass sigals w shall simply tak th samplig frqucy to b twic th highst frqucy which is 2 rad / sc. Thus w shall tak s 4 rad / sc. This th givs us a high pass filtr rathr a bad pass. Howvr, if w tak, say, s 8 rad / sc., w shall hav a bad pass filtr. W shall xt covrt th aalog frqucy spcs to digital (): s 4 rad / sc corrspods to 2 rad rad / sc corrspods to 2/4 /2 rad 2 rad / sc corrspods to (2/4) 2 rad DSP-6 (FIR) 4 of 82 Dr. Ravi Billa

42 Thus H d (), /2 ad /2, lswhr i [, ] H d (ω) Tak H d (ω) π π/2 π/2 π ω Evaluat impuls rspos h d () 2 2 H d j ( ) d j j / 2 + j j2 j 2 / 2 j j d d / 2 DSP-6 (FIR) 42 of 82 Dr. Ravi Billa j j / 2 j / 2 j2 j / 2 j j j / 2 j2 j / 2 h d ().5 by L Hopital s rul Dtrmi h d () for 2 2, so that h t () {h d ( 2), h d ( ), h d (), h d (), h d (2)} si( ) si( / 2) For N 5 th Haig widow is giv by w Ha ().5 cos2 / 5, 5.5 cos / 2, 4 Thus w() {,.5,,.5, }. Multiplyig h t () ad w Ha () poit by poit w gt hd ( ) hd () h t () h t () w Ha (),, hd (),, ad 2 2 H t (z) h t ( ) z 2 2 Dlay by 2 sampls to gt h() h t ( 2) ad H(z) z 2 H t (z). Now draw th dirct form structur j for H(z). Th spctrum is giv by H ( ) H( z) j. W compar blow filtrs lgths of 5 ad without th Haig widow. Grat filtr cofficits: %Calculat hd (si(pi*) - si(pi*/2)) /(pi*) ad stm plot -5: 5, hd (si(pi*) - si(pi*/2))./(pi*), stm(, hd) z

43 to 5 Warig: Divid by zro. hd [ NaN ] Grat frqucy rsposs: %5-tap filtr cofficits b5[ ], a[] %-tap filtr cofficits b[ ], w-pi: pi/256: pi; Hw5frqz(b5, a, w); Hwfrqz(b, a, w); plot(w, abs(hw5), w, abs(hw), 'k') lgd ('5 cofficits', ' cofficits'); titl('magitud Rspos'); xlabl('frqucy \omga, rad/sampl'), ylabl('magitud of H(\omga)'); grid DSP-6 (FIR) 43 of 82 Dr. Ravi Billa

44 Magitud of H() Magitud Rspos 5 cofficits cofficits Frqucy, rad/sampl W compar blow th frqucy rsposs of th 5-tap filtr with ad without th Haig widow. It ca b s that th Haig widow aggravats what is alrady a poor (short) filtr lgth. Thr may b mor to gai by icrasig th filtr lgth tha by widowig. Grat Haig widow: %Grat Haig widow w.5*( cos(pi*/2)) ad stm plot -2: 2, w.5*(-cos(pi*/2)), stm(, w) -2 to 2 w Grat frqucy rsposs: %5-tap filtr cofficits b5[ ], %Haig widow cofficits w [..5.5.], %Widowd cofficits b5ha b5.*w, a[] w-pi: pi/256: pi; Hw5frqz(b5, a, w); Hw5Hafrqz(b5Ha, a, w); plot(w, abs(hw5), w, abs(hw5ha), 'k') lgd ('5 cofficits, No widow', '5 cofficits, Haig widow '); DSP-6 (FIR) 44 of 82 Dr. Ravi Billa

45 Magitud of H() titl('magitud Rspos'); xlabl('frqucy \omga, rad/sampl'), ylabl('magitud of H(\omga)'); grid.4.2 Magitud Rspos 5 cofficits, No widow 5 cofficits, Haig widow Frqucy, rad/sampl DSP-6 (FIR) 45 of 82 Dr. Ravi Billa

46 Exampl [High pass filtr] [28] Dsig a high pass liar phas filtr with frqucy rspos j H ( ) d j3, c, lswhr i th rag [ to] Th umbr of filtr cofficits is N 7 ad c /4. Us (a) rctagular widow ad (b) Hammig widow. H d (ω) π π/4 π/4 π ω H d (ω) 3ω π π ω h d () 2 2 H d j ( ) d / 4 j( 3) ( 3) 2 j( 3) j2 ( 3) 2 d / 4 j / 4 + / 4 j3 j j3 j d d / 4 j( 3) d ( 3) j j( 3) / 4 j ( 3)/ 4 j ( 3) j ( 3) j ( 3)/ 4 j ( 3) j ( 3) j ( 3)/ 4 j ( 3)/ 4 ( 3) j2 si ( 3) si ( 3) / 4 ( 3) j2 DSP-6 (FIR) 46 of 82 Dr. Ravi Billa

47 This squc is ctrd at 3. Sic th filtr lgth is N 7 w ca calculat th 7 cofficits as {h d (), 6}. I othr words w trucat it outsid th itrval 6; morovr, thr is o d to right-shift th trucatd squc. I gral, o may ot kow th filtr lgth with crtaity ad thr is o spcial advatag i spcifyig th phas, H d (ω), as aythig but zro. W calculat cofficits si( ) si( / 4) of th squc ctrd about, that is, h d () : h d ().75 by L Hopital s rul % Grat hd (si(pi*) - si(pi*/4)) /(pi*) ad stm plot -5: 5, hd (si(pi*) - si(pi*/4))./(pi*), stm(, hd) 5 to 5 Warig: Divid by zro. hd [ NaN ] DSP-6 (FIR) 47 of 82 Dr. Ravi Billa

48 Magitud of H() Th 7-tap ad -tap filtr rsposs ar show blow %7-tap filtr cofficits b7[ ], a[] %-tap filtr cofficits b[ ], w-pi: pi/256: pi; Hw7frqz(b7, a, w); Hwfrqz(b, a, w); plot(w, abs(hw7), w, abs(hw), 'k') lgd ('7 cofficits', ' cofficits'); titl('magitud Rspos'); xlabl('frqucy \omga, rad/sampl'), ylabl('magitud of H(\omga)'); grid.4.2 Magitud Rspos 7 cofficits cofficits Frqucy, rad/sampl DSP-6 (FIR) 48 of 82 Dr. Ravi Billa

49 Exampl [Bad-stop filtr] Dsig a bad-stop liar phas filtr with th followig frqucy rspos. Th umbr of filtr cofficits is N 3. H d (ω) π 3π/4 π/4 π/4 3π/4 π ω h d () 2 2 H j d j 3 / 4 + j ( ) d 2 j / 4 + j / 4 3 / 4 j j j 3 / 4 d / 4 / 4 j d Exampl [Dsig of 9-cofficit arrow-bad LP FIR filtr] This is a rpat of Exampl 2. with th badwidth rducd to.2π. Th bad width is arrowr by a factor of from th bad width i that xampl. Th objctiv is to show that th arrowr th bad width th largr th umbr of cofficits dd to achiv th spcifid frqucy rspos. Dsig a i-cofficit (or 9-poit or 9-tap) FIR digital filtr to approximat a idal low-pass filtr with a cut-off frqucy c.2. Th magitud rspos, H (), is giv blow. Tak (). H d H d (ω) 3 / 4 d j π.2π π.2π ω Solutio Th impuls rspos of th dsird filtr is h d () 2 2 j j H d.2.2 j j ( ) d ( 2 j 2 j.2 j ) j d si.2 DSP-6 (FIR) 49 of 82 Dr. Ravi Billa

50 d(si.2 ) h d () d d( ) d.2 cos.2.2 Th MATLAB calculatio of cofficits follows (good for all xcpt whr w fill i th valu.2): % Grat hd (si (.2*pi*)) / (pi*) ad stm plot -5: 5, hd (si(.2*pi*))./(pi*), stm(, hd) xlabl(''), ylabl('hd()'); grid; titl ('hd() (si (.2*pi*)) / (pi*)') -5 to 5 Warig: Divid by zro. hd NaN Th followig MATLAB plot of cofficits is good for all xcpt at. DSP-6 (FIR) 5 of 82 Dr. Ravi Billa

51 hd() 2 x -3 hd() (si (.2*pi*)) / (pi*) MATLAB plots of magitud ad phas rspos of th 9-cofficit arrowr bad LP filtr follow: %Magitud ad phas rspos of 9-cofficit LP filtr %Filtr cofficits b9x7 [ ], a[] w-pi: pi/256: pi; Hw9x7frqz(b9x7, a, w); subplot(2,, ), plot(w, abs(hw9x7)); lgd ('9 cofficits'); xlabl('frqucy \omga, rad/sampl'), ylabl('magitud of H(\omga)'); grid subplot(2,, 2), plot(w, agl(hw9x7)); lgd ('9 cofficits'); xlabl('frqucy \omga, rad/sampl'), ylabl('phas of H(\omga)'); grid DSP-6 (FIR) 5 of 82 Dr. Ravi Billa

52 Magitud of H() Phas of H() 9 cofficits Frqucy, rad/sampl cofficits Frqucy, rad/sampl DSP-6 (FIR) 52 of 82 Dr. Ravi Billa

53 Magitud of H() Th MATLAB plots blow abl us to compar (oly visually) th magitud rsposs of th two 9-cofficit filtrs, th oly diffrc big that Ex has a bad width of.2π whil Ex 2 is vry arrow at.2π. % Magitud rsposs compard: Ex ad Ex 7 (9-cofficit LP filtrs) %Filtr cofficits b9x[.468,.9,.54,.87,.2,.87,.54,.9,.468], b9x7 [ ], a[] w-pi: pi/256: pi; Hw9xfrqz(b9x, a, w); Hw9x7frqz(b9x7, a, w); plot(w, abs(hw9x), w, abs(hw9x7), 'k') lgd ('Ex, 9 cofficits', 'Ex7, 9 cofficits'); xlabl('frqucy \omga, rad/sampl'), ylabl('magitud of H(\omga)'); grid From th plots it clar that th arrow-bad filtr of Exampl 6 is o whr ar ithr th bad width or th gai spcifid. I cotrast, th widr bad width filtr of Exampl is rlativly much closr to spcificatio..4.2 Ex, 9 cofficits Ex6, 9 cofficits Frqucy, rad/sampl DSP-6 (FIR) 53 of 82 Dr. Ravi Billa