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

76 thus the phase response in this case is lhe jwt

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Unformatted text preview: e) filter frequency response is d epicted i n Fig. 12.19a o n b oth w and !1 s cales. Recall t hat t he d igital frequency range is f rom - 7r t o 1f only. We wish t o design a n i deal lowpass filter o f c utoff frequency W e = f r r ad/s. T he frequency response has a period o f 27r o n !1 scale, a nd 27r / T o n W scale. R ather t han s ubstitute T = 12.5 X 1 0- 6 , i t is c onvenient t o leave T a s a n u nknown in o ur c omputations a nd s ubstitute t he value only i n t he end. Thus, we s hall u se t he r adian c utoff frequency W e = 7r / 2T. T he impulse response o f t he d esired i deallowpass (zero phase) filter in Fig. 12.19a is (Table 4.1, P air 18) h a(t) = 765 Nonrecursive F ilter Design G;) (a) 0 - 1t ::1! T JL 2T 0 : :lL - 1t E- 1t 2T 1t T 2 O)_ T il_ h[k] '. (12.82) (b) a nd a ccording t o t he i mpulse invariance criterion [Eq. (12.39)] h[k] = T ha(kT) = ~ sine (~~) = ~ sine C 2k) -.... (12.83) Figure 12.19b shows h[k]. T o m ake t his filter realizable, we need t o t runcate i t using a s uitable No-point window, t hen d elay t he t runcated h[k] by N o2-1 u nits. I n t he p resent e xample, N o = 7. F igure 12.19c shows t he i mpulse response t runcated b y a 7-point r ectangular window a nd Fig. 12.19d shows t he t runcated h[k] delayed by N o2-1 = 3 u nits. N ote t hat t he n oncausal filter in this case is m ade r ealizable a t t he c ost o f a d elay of t = 3 T seconds. T his c onstant d elay o f is w hat p roduces a linear p hase c haracteristic. T he r ectangular windowed, c ausal filter impulse response hR[k] is t he t runcated h[k] in Fig. 12.19d delayed by 3 T. ni 3)] 1 . [7r(k hR[k] = h [k - 3] = 2smc - - 2 - 0$k$6 k- i:}t . - {; -3 -5 -4 / i/ 1./~2 i "" T ""","" ' ['] I I I\.\i···. 0 -I (c) 3 ..l 4 5 6 7 k- (12.84) T he v alues o f t he coefficient h R[k] a re s hown in Table 12.3. Also ~[k] .. .. ........\ ' Truncated and delayed h[k] ( d) 6 H[z] = L h[k]z-k k=O = __ 37r =z -3 ( + ~z-2 + ~z-3 + ~z-4 _ 1f 2 k- ~z-6 7r 37r Fig. 1 2.19 13 1 1 1 -1 1 - 3) - 37r z + :;z+2++:;z - 37rz (12.85) H ence, t he frequency response H R[e iwT ] is 6 HR[e iWT ] = L hR[k]e-iWkT k=O = e= e- i3wT [~+ ~ ( e iWT + e - iwT ) _ 3~ i3wT [2 ~ cos w T - ~ cos 3 wT] ~+ 7r 31f = e - i3wT [1 + :2cos 2; (w) 8 0,000 (e i3wT + e - i3WT )] 2 (80~~00) ] - 31f cos N onrecursive m ethod o f l owpass filter design. T he t erm e - i3wT is a linear phase representing t he d elay o f 3 T s econds. T he m agnitude o f t he b racketed t erm, d epicted i n Fig. 12.19a by a solid curve, e xhibits oscillatory b ehavior which decays r ather slowly over t he s topband. A lthough i ncreasing n ( the s ystem o rder) improves t he f requency response, i ts o scillatory n ature p ersists ( Gibbs phenomenon).t I n s ome filtering applications, t he o scillatory c haracteristic ( which decays slowly a s l /w) i n t he s topband m ay n ot b e a cceptable. B y u sing a t apered w indow functEq. (12.86) is identical to the first three terms in Eq. (3.61) except t hat the former is a function of...
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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.

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