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Unformatted text preview: t o have a linear phase response. O n t he o ther
h and, if a recursive filter can be found t o d o t he j ob o f a nonrecursive filter, t he
r ecursive filter is of lower order; t hat is, i t is faster (with less processing delay) a nd
r equires less memory. I f processing delay is n ot c ritical, t he nonrecursive filter is t he
o bvious choice. T hey also have a n i mportant place in nonaudio applications, where
a l inear phase response is i mportant. We shall review t he c oncept of nonrecursive
s ystems briefly.
As discussed in Sec. 12.3, nonrecursive filters may b e viewed as recursive filters,
w here all the feedback or recursive coefficients are zero; t hat is , when
aO = a l = a2 = . .. = a nI =0 Consequently, t he t ransfer function of t he r esulting n thorder nonrecursive filter is (12.71a)
(12.71b) where h[k] T o p lot t he a mplitude a nd t he p hase response, we c an u se t he l ast 9 f unctions in E xample
C 12.5. 1 2.7 (12.69b) H[z] = E h[k]zk t. t. + b n_IZ n  1 + ... + bIZ + bo Hz=.:::.'.....::.~=~ W s=[O.l O .2];Wp=[O.045 O .4];Gp=2.1;Gs=20; t. 757 12.7 Nonrecursive F ilters b nk
= { O ::;k::;n 0 k >n (12.72) T he i mpulse response h[k] h as a finite w idth o f (n + 1) elements. Hence, these
filters are finite impulse response (FIR) filters. We shall use t he t erms nonrecursive
a nd F IR interchangeably. Similarly, t he t erms recursive a nd I IR (infinite impulse
response) will be used interchangeably in our future discussion.
T he impulse response h[k] c an b e expressed as h[k] = h[0]8[k] + h[1]8[k  1] + ... + h[n]8[k  n] (12.73) T he frequency response of t his filter is o btained from Eq. (12.71a) as H[e jwT ] = h[O] + h [l]e jwT
= E h[k]e ikwT + ... + h[n]e jnwT (12.74a)
(l2.74b) k=O Filter Realization T he nonrecursive ( FIR) filter in Eq. (12.69a) is a special case of a general
filter with all feedback (or recursive) coefficients zero. Therefore, t he r ealization o f
t his filter is t he s ame as t hat o f t he n thorder recursive filter w ith all t he feedback
connections omitted. Figure 12.7b shows a canonical realization of t his filter. I t is
easy t o verify from t his figure t hat for t he i nput 8[kJ, t he o utput is h[k] given in Eq.
(12.73). 12 Frequency Response a nd Digital Filters 758 T he filter in Fig. 12.7b is a t apped delay line with successive t aps a t u nit delay
( T seconds). Such a filter is known as a t ransversal filter. Tapped analog delays
a re i ntegrated circuits, which are available commercially. I n t hese circuits t he time
delay i s implemented by using charge transfer devices, which sample t he i nput signal
every T seconds ( unit delay) a nd t ransfer t he successive values of t he samples t o m
s torage cells. T he s tored signal a t t he k th t ap is t he i nput s ignal delayed by k time
u nits ( kT seconds). T he sampling interval can be varied electronically over a wide
range. Time delay can also be obtained by using shift registers. 759 12.7 Nonrecursive Filters
h[k] o 3 2 ""I I k •
2 0 4 I I (b) ( a) 1 2.71 Symmetry Conditions for Linear Phase Response
F ig. 1 2.17 C onsider a n n thorder finite impulse response (FIR) filt...
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 Spring '13
 Bayliss
 Signal Processing, The Land

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