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

6 t o p lot t he a mplitude a nd t he p hase response

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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 non-audio 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 n-I =0 Consequently, t he t ransfer function of t he r esulting n th-order 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]z-k 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 n-k = { 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 th-order 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 2.7-1 Symmetry Conditions for Linear Phase Response F ig. 1 2.17 C onsider a n n th-order finite impulse response (FIR) filt...
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