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Unformatted text preview: M ethod
T he i mpulse invariance m ethod is h andicapped b y a liasing. Consequently this
m ethod c an b e used t o design filters where H aUw) b ecomes negligible b eyond some
frequency B Hz. T his c ondition r estricts t he p rocedure t o lowpass a nd b andpass
filters. T he i mpulse invariance m ethod c annot b e u sed for high pass o r b andstop
filters. Moreover, t o reduce aliasing effects t he s ampling r ate h as t o b e v ery high,
w hich m akes its implementation costly. I n g eneral, t he f requencydomain m ethod
d iscussed in t he n ext s ection is s uperior t o t his m ethod. 1 2.6 Recursive Filter Design by the FrequencyDomain Criterion:
T he Bilinear Transformation M ethod
T he b ilinear t ransformation m ethod discussed i n t his s ection is preferable to
t he i mpulse invariance m ethod i n filtering problems where t he g ains are c onstant
o ver c ertain b ands (piecewise c onstant a mplitude r esponse). T his c ondition e xists in
lowpass, b andpass, highpass, a nd b andstop filters. Moreover, t his m ethod requires = 2)
(T = S 15 (12.54) 2 (~) ::~:! e sT / 2 _ e  sT / 2
e sT/2 + e sT/2 E quation (12.53) now c an b e e xpressed as 8 +20 2;00 + 2. ( ST)5 + ...J For small T ( T  t 0 ), we c an i gnore t he h igherorder t erms i n t he i nfinite series o n
t he r ighthand s ide t o y ield
e ST / 2 _ e  ST / 2 )
sT
l im
=T~O ( e sT / 2 + e  sT / 2
2 Ha(s)=~
Answer: H[z] = z!~!iOT w ith T = ~ sT 2 t anh ( ST) == e /  e / = [ ST _
( ST)3
e sT / 2 + e  sT / 2
2
3
2
2 0.3142z
z  0.7304 Exercise E12.4 (12.53) T~O a conclusion which agrees with o.ur result in Eq. (12.50). To plot the amplitude and the
phase response, we can use t he last 8 functions in Example C12.1. 0
£:, 741 H[e sT = Ha (~ ::~: !) F rom t his result, i t follows t hat H[z] = Ha (
= 2 ZI) T z +1 Ha(s)ls=.q. ~:;:: (12.55a) (12.55b)
(12.55c) T herefore, we c an o btain H[z] from H a(s) b y u sing t he t ransformationt S=(~)::! (12.57) T his t ransformation is known as t he b ilinear t ransformation.
t There e xist o ther t ransformations, w hich c an b e u sed t o derive H[z] from H .{s). We s tart w ith
t he power series
e  sT = 1  sT + ~{8T)2  i{8T)3 + ...
I n t he l imit as T
yields > 0, all b ut t he first two t erms o n t he r ighthand side c an be ignored. T his T his r esults in a t ransformation 12 7 42 F requency R esponse a nd D igital F ilters 1 2.6 R ecursive F ilter d esign: T he B ilinear T ransformation M ethod 7 43 Choice o f T in Bilinear Transformation Method
B ecause o f t he a bsence o f a liasing i n t he b ilinear t ransformation m ethod, t he
v alue o f t he s ampling i nterval T c an b e m uch s maller c ompared t o t he i mpulse
i nvariance m ethod. B y a bsence o f a liasing w e m ean o nly t he k ind o f a liasing o bserved i n i mpulse i nvariance m ethod ( Fig. 1 2.lOb). T he s ignal a liasing, w hich l imits
t he h ighest u sable f requency, is s till p resent. T hus i f t he h ighest f requency t o b e
p rocessed i s Fh H z, t hen t o a void s ig...
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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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