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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 Nopoint window, t hen d elay t he t runcated h[k] by N o21 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 7point
r ectangular window a nd Fig. 12.19d shows t he t runcated h[k] delayed by N o21 = 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]zk
k=O = __ 37r
=z 3 ( + ~z2 + ~z3 + ~z4 _
1f 2 k ~z6 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]eiWkT
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
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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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