ELEC9721: Digital Signal Processing Theory and Applications

Transcription

1 ELEC97: Digital Sigal Pocssig Thoy ad Applicatios Tutoial ad solutios Not: som of th solutios may hav som typos. Q a Show that oth digital filts giv low hav th sam magitud spos: i [] [ ] m m i i i x c y ii [] [ ] m m i i i m x c y y[] output; x[] iput; c i cofficits As: m m i i c i m m i i c i Comput th db adwidth of th followig filts, ; < < a a a a a Which filt has a small db adwidth? As:. filt a a a a a d c c c c <

2 c Fid th magitud spos fo th systm fuctio ad commt o you sult Daw th caoic aliatio of th systm. As: allpass filt si si 6 6 x y - - Caoic alisatio - Q a Dtmi th magitud ad phas spos of th multipath chal y[] x[] x[-m]. At what fqucis? y x x-m X X -M -M M - sim φ ta M si M M si M M

3 Fo M φ ta ta si si φ If M / - - wh M - M i. /M -/ x[] Dtmi ad sktch th magitud ad phas spos of th systm show low y[] x[] y[] 8 8, φ

4 c Assumig that th digital filt G is to alid usig th cascad stuctu, daw a suital lock diagam ad dvlop th diffc quatios. 6 G G x p q y /6 6 6 q q y p p q p x x p φ

5 d Dtmi th fqucy spos of th ladd filt show low: X a a As : a Dtmi th fqucy spos of th lattic filt x[] y[] a -a - T T As a : a f Show that th filt stuctu show low has a lia phas chaactistic quatio giv y: φ -. x[] y[] 5

6 y x x- x- x- x φ N N 5 N [ N 5 si si N ] N N N si si g Dtmi th tasf fuctio of th systm show low. Chck th staility of th systm wh.8 ad. x[] si - y[] - - si As: si,.6 stal _ filt. Q X si P R - - P X si - - -si 6

7 7 Q P R ad R Q - Q P Q - - Fom & Q{- - } X si - si X Q - Fom th lock diagam: - si - Q - Fom & si si si si X X Q a Cosid th followig causal IIR tasf fuctio:.5 9 Is a stal fuctio? If it is ot stal, fid a stal tasf fuctio G such that G. Is th ay tasf fuctio havig th sam magitud spos as? As: ustal, { } filt Allpass A A G A G,.5 9 Aalys th digital stuctu giv low ad dtmi its tasf si - - Isid th staility tiagl Stal systm -

8 fuctio X i Is this a caoic stuctu? ii What should th valu of th multipli cofficit K so that has a uity gai at? iii What should th valu of th multipli cofficit K so that has a uity gai at? iv Is th a diffc tw ths two valus of K? If ot, why ot? X K - - α - β Q a A fist-od digital filt is dscid y a k β α K As: α β i dlays ocaoic ii& iii β α K α β iv o all pass filt a, < a, < assum i Dtmi k, so that th maximum valu of is qual to. ii Comput th -db adwidth of th filt. a k 5 c As: iii Daw a caoic aliatio of th systm fuctio. A two-pol lowpass filt has th systm fuctio 8

9 9 k Dtmi th valus of k ad such that th fqucy spos satisfy th coditios ; As: k.6,., k k k k k k k. ad k -..6 c A thid od FIR filt has a tasf fuctio G giv y 5 G Fom G, dtmi th tasf fuctio of a FIR filt whos magitud spos is idtical to that of G ad has a maximum phas spos. k -

10 5 G Max phas filt: 5 K P To gt th valu of K, G P G 5 K P 5 Thus, sustitutig ito th aov quatio 5 5 K K Max phas filt: 5 P Q5 Fo th systm show low, a Expss y [] i tms of y [-], y [-], ad x[]; do th sam fo y []. Assum A ; B si ; y - - ; y - si-. If x[], show that : y [] ad y [] si c Calculat fo aitay A ad B th systm fuctio X ad X d Tak ad daw th pols-os plot fo ad. Tak ad x[]δ[]. Calculat th impuls spos h [] fo -. 5

11 A x[] - B y [] -B A - y [] a y x -B y - Ay - y By -Ay - If x y Ay - By -, y y - - si y - si si, y y - si y, y y - si y - si si... y y By - Ay -, y si y - y - si - si, y si y y si, y si y y si si si... y si c X B - A - B - A - -A - B - B B X B A X A A B A X A A X A B A A A A B A A A B A X A A B B A A A X A B A

12 d W hav, B A B A, pols at pols at ± ± ± X if xδ X tasfom of u u y A B A B X os: ;, pols: ±, ±

13 Q6 a Otai a paalll aliatio fo th followig.5 Implmt th paalll aliatio of which you hav otaid. Otai a paalll aliatio fo th followig tasf fuctio. c A digital filt is pstd y Q7 i Dos this tasf fuctio pst a FIR o a IIR filt? ii Wit a diffc quatio fo usig th dict fom. iii Implmt a paalll aliatio of. a Th tasf fuctio of a disct-tim systm has pols at.5,. ±. ad os at - ad. i Sktch th pol-o diagam fo th systm ii Div th systm tasf fuctio fom th pol-o diagam. iii Dvlop th diffc quatio. iv Daw th lock diagam of th disct systm. As: A otch filt is giv y.8 Dtmi th fqucy spos at dc, ad f s samplig fqucy. Sktch th fqucy spos i th itval As:.8.8

14 ai. -.. ii X iii y x- x-.7 y- -.5y-.5y- iv x y

15 f s f s,.8.8 dc.8.8 si f s,.8 si.8 si.8.8 Im R.8.8 f Q8 A digital filt is show low. x[] a a a - - y[] a Dtmi th systm fuctio fo th aov stuctu. With a a ; a ;.5 ad -.75, Dtmi th pol-o patt of ad idicat if th systm is stal o ot. 5

16 Q9 Lt dot th disct-tim Foui tasfom of th squc. a Dtmi. Lt y[] dot a fiit-duatio squc of lgth ;, ad,. Th poit DFT of y[] is dotd y [k] cospods to qually spacd sampls of. Dtmi [k] ad y[]. As: Sic,, 6

17 Compaig th aov quatio with th DFT quatio, Thus, Q A spch sigal is sampld with a samplig piod of 5ms. A fam of 56 sampls is slctd ad a 56 poit DFT is computd. What is th spacig tw th DFT valus i As:.5 7

18 Q Comput th N-poit DFT, [k] of th squc h[]. Show that wh N th valu of /5 h [] 5 othwis Q A 5 k siusoidal sigal is sampld at k ad 8 sampls a collctd ad usd to comput th 8-poit DFT of th sigal. What is th tim duatio i scods of th collctd sampls? At what DFT idics do w xpct to s ay paks i th spctum? Duatio Num of sampls x Samplig piod 8 * /5.56 s Pak occus at 5k c, pak will occu at DFT idics 6 ad 8-6. Q Cosid th fiit lgth squc: x[] δ[].δ[-] Wit th quatio fo th N-poit DFT of x[] fo N 5 As:. 8

19 Q Lt a causal lia tim ivaiat disct-tim systm chaactisd y a al impuls spos h[] with a DTFT of. Cosid th systm of th figu show low, wh x[] is a fiit lgth squc. Dtmi th fqucy spos of th ovall systm G i tms of, ad show that it has a o-phas spos. h[] x[] y[] h[] Timvsal Timvsal Timvsal h[] x[] y[] h v u y DTFT x * h v x * h { h h }* x G G { } X * a a a c th quivalt tasf fuctio phas. h[] Timvsal v u G is al ad has o phas ad o 9

20 Q5 Fo th systm i th figu low, sktch th output y[] wh th iput x[] is δ[] ad is a idal lowpass filt as follows: < As: si ω [ ] si y d d w si [ ] [ ] si w w w y x[] ω[] - y[] y[] y

21 Q6 Th fqucy spos of a idal difftiato is giv y τ This spos is piodic with piod. Th quatity τ is th dlay of th systm i sampls. a Sktch th magitud ad phas sposs of th systm fo -. magitud > < ag τ ag τ Phas -τ < < ag{ } τ τ τ τ, wh > < / τ ag -/ τ

22 Fid th impuls spos h[] of th systm as a sum of a si ad i fuctio. k k d k Not: k si[ τ ] τ h As: τ τ