Signal Processing and Linear Systems-B.P.Lathi copy

730 12 frequency response a nd digital filters 1 731

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Unformatted text preview: p rocedure of finding H[zJ c an b e s ystematized for any n th-order s ystem. F irst we express a n n th-order a nalog t ransfer f unction H a(s) a s a s um o f p artial f ractions as n Ha(s) = " '"' - -·'~c ;=1 S - Ai (12.44) T hen t he c orresponding H[zJ is given b y n H [zJ=T c ·z L --' - . z eAt. T (12.45) i =l T his t ransfer function c an b e r eadily realized as a parallel combination of t he n firstorder s ystems if all t he n p oles o f H a (s) a re real. T he c omplex c onjugate poles, if any, m ust b e realized as a single second-order t erm. T able 12.1 lists several pairs of H a(s) a nd t heir c orresponding H [z J. Choosing the Sampling Interval T T he z -transform of this equation yields H[zJ = T Z(ha(kT)) (12.40) T his r esult yields t he desired transfer function H[zJ. L et us consider a first-order transfer function Ha(s) = _ c_ S -A (12.41a) I f Fh is t he h ighest frequency t o b e processed, t hen t he s ampling i nterval m ust b e no greater t han 1 j2Fh i n o rder t o avoid signal aliasing. However, in t he i mpulse invariance m ethod, t here is y et a nother c onsideration, which m ust a lso b e t aken i nto account. Consider a h ypothetical frequency response Ha{jw) (Fig. 12.lOa) t hat we wish t o realize using a digital filter, as illustrated in Fig. 12.8a. L et u s assume t hat we have a n e quivalent d igital filter t hat m eets t he t ime-domain equivalence criterion in E q. (12.37); t hat is , 1 ~ 0 .0...) Q ~ ~ <b'h OJ '6'0 ~ I h ~ I '" '" e+ i5 "d 00 u oj I 00 Q) ~ Q ~ Q) ;:; ~ C' Q) :.:: h It hl~ NI~ 'hI ~ '" +;:;-- ...... ~ I N~ '" '" h NI~ h I N '" h '""N'""'" '" h I '" h ~ 0 1!. 2n ITT ro- 00 0 '" 0:: » u ro- .c ~ 0 0. 2n T 0 T h h Q) 00 Q (a) I~I----------------- -Zl!; ::!.. 0 Q 737 12.5 Recursive F ilter Design: T he I mpulse Invariance m ethod u h ~ "0" I 00 '" U ~ ~ I N"N ' l- _~ -::n T h T '...... " ,----. "+ " ~~ -.:-.: ..., h I ~ e 00 ' --' 0 u .... .N ... r<l ~ ..., :.:: ' "C ~ ~ ~ h h I:Q ... " '".., .h ~I'" .h.. ''"""'" '"h'" " I'Q ... I h .h .., h ~ h[kJ = lim Tha(kT) T ....O oj ~ '" I n C hapter 5 (Fig. 5.6), we showed t hat t he Fourier transform of t he samples o f ha(t) consists of periodic repetition of Ha (jw) w ith period equal t o t he sampling frequency w. = 27r/T = 27rFs.t Also Ha(jw) is not generally bandlimited. Hence, aliasing among various repeating cycles cannot be prevented, as depicted in Fig. 12.10b. T he resulting spectrum will b e different from t he desired spectrum, especially a t higher frequencies. I f Ha(jw) were t o b e bandlimited; t hat is, if Ha(jw) = 0 for Iwl > WO, t hen t he overlap could be avoided if we select t he period 27r/T > 2wo· However, according t o t he Paley-Wiener criterion [Eq. (4.61)], every practical system frequency response is nonbandlimited, a nd t he cycle overlap is i...
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This note was uploaded on 04/14/2013 for the course ENG 350 taught by Professor Bayliss during the Spring '13 term at Northwestern.

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