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Unformatted text preview: tems. I t s hould
b e remembered, however, t hat w ith this criterion we a re ensuring only t hat t he
t Because T is a c onstant, s ome a uthors ignore t he f actor T , w hich yields a lternate c riterion
= h a(kT). I gnoring T m erely scales t he a mplitude response o f t he r esulting filter. h[kJ ,
7 34 12 F requency Response a nd D igital Filters d igital filter's response matches exactly t hat of t he d esired analog filter a t s ampling
i nstants. I f we w ant t he two responses t o m atch a t e very value o f t , we m ust have
T ~ O. T herefore 735 12.5 Recursive F ilter Design: T he I mpulse Invariance m ethod
cT c h [ k] (b) (12.37)
A Practical Difficulty k- t- B oth o f t hese c riteria for filter design require t he c ondition T ..... 0 for realizing a digital filter equivalent t o a given analog filter. However, this condition is
~mpos~ible .in practice because i t n ecessitates a n i nfinite sampling r ate, r esulting
m an mfimte d ata r ate. I n p ractice, we must choose a small b ut n onzero T t o
achieve a compromise between t he two conflicting requirements, namely closeness
of approximation a nd s ystem cost.
T his a pproximation, however, does not m ean t hat t he s ystem in Fig. 12.8a
is inferior t o t hat in Fig. 12.8b, because often H a(s) i tself is a n a pproximation
t o w hat we a re seeking. For example, in lowpass filter design we strive t o design
a s ystem w ith i deal lowpass characteristics. Failing t hat, however, we s ettle for
s ome a pproximation such as B utterworth lowpass t ransfer functions. I n fact, i t is
entirely possible t hat H[z], which is a n a pproximation t o Ha(s), m ay b e a b etter
a pproximation t o t he desired characteristics t han is H a(s) itself. Fig. 1 2.9 Procedure for t he impulse invariance method of filter design. T he i mpulse response h (t) o f t his filter is t he inverse Laplace t ransform o f Ha(s),
which in t his case is
T he c orresponding digital filter u nit s ample response h[kJ is given by Eq. (12.39) h[kJ = T ha(kT) = Tce kAT (12.42) Figures 12.9a a nd b show h a(t) a nd h[kJ. According t o E q. (12.40), H[zJ is T t imes
t he z -transform of h[kJ. T hus,
H [zJ=-z - eAT 1 2.5 Recursive Filter Design by the T ime-Domain Criterion:
T he Impulse Invariance M ethod T he t ime-domain design criterion for t he e quivalence o f t he s ystems in Figs.
12.8a a nd 12.8b is [see Eq. (12.31)J h[kJ = lim T ha(kT)
T~O (12.38) where h[kJ is t he u nit impulse response of H[z], ha(t) is t he u nit i mpulse response
of Ha(s), a nd T is t he s ampling interval in Fig. 12.8a.
As indicated earlier, i t is impractical t o l et T ..... O. I n p ractice, T is chosen
t o be small b ut nonzero. We have already discussed t he effect o f a liasing a nd t he
c onsequent distortion in t he frequency response caused by nonzero T . A ssuming
t hat we have selected a suitable value o f T , we c an i gnore t he c ondition T ~ 0, a nd
E q. (12.38) c an b e expressed as h[kJ = T ha(kT) (12.39) T he...
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