Digital Signal Processing Homework 7 Solutions in progress

Transcription

1 Digital Signal Prossing Homwork 7 Solutions in progrss Du Wnsay 0 Novmbr 00 Problm 46 a, b, ) Fin th maximum valu of th magnitu of th frquny rspons ) Fin th pols an ros of H() f) Compar th minimum an maximum magnitu of th frquny rspons for F s 0a, F s a, an F s 0a Explain ths rsults at () t u() t a) For h whr a is a positiv ral-valu paramtr, th frquny rspons is H ( j) h skth of th magnitu of th frquny rspons shoul look a j lik th uppr right plot in th MALAB plot blow whr th paramtr a has bn arbitrarily hosn to b 5 th impuls rspons in ontinuous tim is shown in th uppr lft plot for th sam tim onst an b) For h [] n h ( n ) u[ n], th isrt tim frquny rspons is givn by H ( j ) h skth shoul look lik on of th thr right-han si a j MALAB plots blow hs plots show th sampl impuls rspons an th frquny rspons for th thr valus of rqust in part (f) ) h minimum magnitu of th frquny rspons will our at π an will b H ( jπ ) a ) h maximum magnitu of th frquny rspons will our at 0 an will b H ( j0 ) a ) h systm funtion is H hr will b a ro at 0 a a a an a pol at whih will b btwn 0 an f) hr was an rror in this problm statmnt h valus spifi wr mant to b th sampling frquny, not th sampling intrval As th problm is stat, numri valus an not b alulat h solution shows th orrt valus F s Minimum magnitu rspons Maximum magnitu rspons 0a 0/a tn 0 / a 0 / a 0 / a 0 / a tim onsts 0 / a 0 / a a /aon tim / a / a / a / a onst / a 58 / a

2 0a 0/a 0 tim onst 0/ a 0/ a / a 0/ a 0/ a / a h son row of th plot blow shows th first 0 sampls of th impuls rspons an th frquny rspons o th infinit uration sampl squn whn 0*tim onst Hr 00 for tau 0 h frquny rspons shows a vry narrow ban low-pass filtr with high attnuation at frqunis outsi th narrow low-pass filtr passban his orrspons to th wi filtr with h thir row shows th sam rsults for th tim onst hr is som, but not muh attnuation outsi th low-pass filtr passban Aftr about 5 sampls of th impuls rspons th sampl valus ar los to ro In th fourth row th sampling intrval is so larg that all offiints ar almost ro xpt for th first his filtr auss almost no hang in th input signal an th frquny rspons magnitu is almost ompltly onst

3 Us th bilinar transform to sign a first orr isrt tim Buttrworth filtr of orr with π h transfr funtion for a first orr analog Buttrworth filtr is () 4 s s H a) Fin th ontinuous utoff frquny as a funtion of b) Fin H() ) Fin th pols an ros of H() ) What is th minimum magnitu of j H? a) h utoff frquny will b π b) h systm funtion will b th ontinuous transfr funtion with th substitution s H ) his filtr has a ro at - an a pol at As th utoff frquny in th isrt tim spifiation inrass from 0 to pi/, th pol movs from to th ntr of th unit irl If th utoff frquny ontinus to inras, th pol ontinus to mov towar - ) sin thr is a ro at -, th minimum magnitu of th frquny rspons will b ro 3 Problm 7

4 π Using impuls invarian, Sin 0ms an, th ontinuous utoff 4 frquny in raians/son us for th prototyp filtr π π is / π ( 50) Problm 7 π Using th bilinar transform, Sin 0ms an, th ontinuous utoff frquny in raians/son us for th prototyp filtr π is 000() Following th xampls in th lass hanout omparing lowpass IIR filtr signs, us MALAB to gt th offiints of a Buttrworth filtr with th spifiations p π / 4, s π /, passban rippl05 an stopban rippl 60B a) What orr filtr is rommn to mt th spifiations? b) Us frq with a sampling frquny of 8000 H to alulat th frquny rspons Plot th magnitu of th frquny rspons as a funtion of frquny, an also plot 0*log(magnitu of th frquny rspons) Vrify that th spifiations wr mt Submit th plot with th homwork assignmnt ) Us -plan to plot th pols an ros Vrify that all ros our at - Submit th plot 6 Rpat parts (a) an (b) of th prvious problm for an llipti filtr ) Us -plan to plot th pols an ros Math th ros to th minimum valus in th frquny rspons Submit th plot 7 Fin th offiints of an FIR lowpass filtr of lngth 3 using b fir(30, 05, rtwin(3)); a) For a sampling frquny of 8000H, plot th magnitu of th frquny rspons Also us stm to plot th impuls rspons b) How muh iffrn is thr btwn th magnitu of th rspons at 0H an th maximum magnitu of th rspons?

5 ) What is th maximum valu of th frquny rspons aftr th first ro in th rspons? How muh rippl woul this b in B? 8 Crat a lngth 3 sampl osin with a frquny of 4000 H an a sampling frquny of 8000H a) Multiply this sampl osin an th filtr offiint vtor from th prvious problm using th * opration to prou a nw filtr of lngth 3 Plot th impuls rspons an th magnitu of th frquny rspons of th nw filtr an ompar to th rsults of th prvious problm Is th nw filtr low pass, ban pass, or high pass? b) Rpat (a) for a sampl osin with a frquny of 000 H ) Rpat (a) for a sampl osin with a frquny of 000 H