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

128 1 time domain equivalence method o f fir filter

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Unformatted text preview: w and the latter is a function of t. Clearly, H[ei wT ) is a t runcated Fourier series for t he gate function. As we increase n , H[e iWT ) converges t o the gate function in the manner depicted in Fig. 3.11. Regardless of the value of n , however, H[eiwT ) exhibits oscillatory behavior because of t he Gibbs phenomenon. (12.86) 766 12 F requency R esponse a nd D igital F ilters 767 N onrecursive F ilter D esign 12.8 F or t he H amming f ilter T ABLE 1 2.3 j H H [ejwTjj ( 12.88c) 0 .49 c os w T - 0.01696 c os 3 wT I n e ither f ilter, t he p hase r esponse is a l inear f unction o f w w ith s lope - 3T, i ndicating t ime d elay o f 3 T. N ote t hat b oth hR[k] a nd hh[k] a re s ymmetric a bout k = 3. • H amming W indow R ectangular W indow =~+ k hR[k] wH[k] hH[k] 0 -1/311" 0.08 - 0.00848 C omputer E xample C 12.9 U sing M ATLAB, f ind t he f requency r esponse o f t he l owpass filter i n E xample 1 2.10 for 9 8th-order filter. P lot t he f requency r esponse for r ectangular a nd H amming w indow 0 0.31 0 filters. 2 1/11" 0.77 0.245 3 1 /2 4 1/11" 0.77 0.245 5 0 0.31 0 6 -1/311" 0.08 - 0.00848 o N O=99; m =(NO-l)/2; k =O:NO-l; h l= ( 1/2) * sinc( ( k-m) / 2); n uml=hl; d enl=[I, z eros(I,NO-l)]; W = -pi:pi/l00:pi; H l=freqz(numl,denl,W); m agl=abs(Hl); p hasel=180 / pi*unwrap( a ngle(Hl»; f or i =I:NO k =i-l; h 2(i)=(1 / 2)*sinc( ( k-m) / 2) * ( O.54+0.46*cos(pi*(k-m) / m»; e nd n um2=h2; d en2=[I, z eros(I,NO-l)]; W = -pi:pi/l00:pi; H 2=freqz(num2,den2,W); m ag2=abs(H2); p hase2=180/pi*unwrap(angle(H2»; s ubplot(2,1,1); p lot(W , magl,W , mag2);grid; s ubplot(2,1,2); p lot(W , phasel,W , phase2);grid 0 .5 t ion s uch a s a H amming w indow, t he o scillatory b ehavior c an b e e liminated a t t he c ost o f i ncreasing t he t ransition b and ( from p assband t o s topband). T he H amming w indow f unction is given b y wH[k] = { I n o ur c ase No :5 jkj :5 0.54+0.46COS U:~l) - No -1 2 o o therwise = n + 1 = 7. wH[k] = { N o -1 2 (12.87) Hence, 0.54 + 0 .46 c os ("f) - 3 :5 jkj :5 3 o o therwise T able 12.3 also shows t he ( delayed) H amming w indow coefficients wH[k] a nd t he c orresponding i mpulse r esponse hH[k] = h[kjwH[k]. T he f requency r esponse o f t he H amming w indow filter is = e- a = e- H H[ejwTj j3wT j3wT [! + 0 .49 cos w T - + 0 .245 (e jwT + e- jWT )- 0.00848 (e j3wT + e- j3WT )] 0 6 . E xercise E 12.6 I f we were t o use n = 8 filter in Example 12.12, show t hat t he filter transfer function for t he b artlett ( triangular) window is 0.01696 c os 3wT] H[z] = W ith t he coefficients hR[k] ( or hH[k]) i n T able 12.3, t he d esired filter c an b e r ealized by u sing s ix d elay e lements, as d epicted i n F ig. 12.7. A ccording t o E qs. (12.86), we h ave a nd L HR[e . JW T = { -3wT 1 I"-3wT w hen w hen ! + ~ cos w T - -j; cos 3 wT ~ 0 ! + ~ cos w T - -j; cos 3wT < 0 z6 Observe t hat in this case t he filter order is reduced by 2 because t he two end-points have a zero 'V value for t he B artlett window. • ( 12....
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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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