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Unformatted text preview: p rocedure of finding H[zJ c an b e s ystematized for any n thorder s ystem. F irst
we express a n n thorder 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 secondorder 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 firstorder 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 imedomain 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 PaleyWiener 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.
 Spring '13
 Bayliss
 Signal Processing, The Land

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