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

T he h amming w indow f unction is given b y whk i n

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Unformatted text preview: frequency response H[eiwTJ of an ideallowpass filter is shown (shaded) in Fig. 12.21. In this case No = 7, No - 1 ~ 6 = 7' wo = 21f N oT 21f = 7T T he seven samples H r in Eq. (12.97) are Thus H r = H [eirwoTle-ir¥woT =H . hr e JNO'f' 1e - Jr&quot;l'IQ . &quot; '=' [ (12.97) R ecall t hat t he N o s amples H r a re t he uniform samples of t he p eriodic extension o f H [eiwTle-i¥wT. Hence, Eq. (12.97) applies t o s amples of t he frequency range from ~ w ~ T. T he r emaining samples a re o btained b y using t he c onjugate s ymmetry p roperty H r = H 'No-r. T he desired h [kl is t he I DFT o f H r; t hat is, The remaining three samples should be determined using the conjugate property of DFT, H r = H No - r ' t hat is H r = H 7 r . Thus - ° The desired h[kJ is t he IDFT of H r given by [Eq. {12.98)J k = 0, 1, 2, 3, . .. , 6 k = 0, 1, 2, . .. , N o - 1 a nd (12.98) r =O We may compute this IDFT by using the I FFT algorithm or by straightforward substitution of values of H r in the above equation as I II T 774 12 F requency R esponse a nd D igital F ilters r =i-l; H r(i)=H(i)*exp(-j*r*pi*(NO-l)/NO); 6 h[O] =~L Hr = ~[I + e -i6 ,,/7 + ei6&quot;/7] = ~ (I + 2 cos 6 ;) = - 0.1146 r =O h[l] =~&quot; 6 7~ H r ei2 ,!/ = ~[I + e - i4 ,,/7 + ei4&quot;/7] = ! r=O h[3] I 775 12.8 N onrecursive F ilter D esign ,,6or 6 7 7 ( I + 2 cos 411&quot;) 7 = 0.0792 I = 7 ~HreJ--r- = 7[1 + I + I] = 0.4285 r =O Similarly, we c an show t hat h[4] = 0.3209, h[5] = 0.0792, a nd z - 0.1146z 6 + 0.3209 + 0.4285 + 0.3209 + 0.0792 z2 z3 z4 0 h[6] = - 0.01146 O bserve t hat h[k] is symmetrical a bout i ts center point k = 3 a s e xpected. C ompare t hese values w ith t hose found by t he i mpulse invariance m ethod in T able 12.3 for a r ectangular window. Although t he two s ets o f values a re different, t hey a re c omparable. W hat i s t he difference i n t he two filters? T he i mpulse invariance filter optimizes t he d esign w ith r espect t o a ll frequencies. I t minimizes t he m ean s quared value o f t he difference between t he desired a nd t he realized frequency response. T he frequency sampling m ethod, i n c ontrast, realizes a filter whose frequency response matches exactly t o t he d esired frequency response a t No uniformly spaced frequencies. T he m ean squared error in t his design will generally b e h igher t han t hat in t he i mpulse invariance method. T he filter transfer function is H [z] = - 0.1146 + 0.0792 e nd k =O:6 h k=ifft(Hr); s ubplot(2,1,1); s tem(k,hk); x label('k');ylabel('h[k]'); M =512 h E=[hk z eros(1,M-7)] H E=fft(hE); s ubplot(2,1,2); r =O:M-l; W = r. * 2*pi/512 p lot(W,abs(HE»; x label('W');ylabel('F(W)');grid; An A lternate M ethod Using Frequency Sampling Filters W e n ow s how a n a lternative a pproach t o t he f requency s ampling m ethod, w hich u ses a n N o-order c omb f ilter i n c ascade w ith a p arallel b ank o f N o - 1 f irst-order f ilters. T his s tructure f orms a f requency s ampling f ilter. W e s tart w ith E qs. (6.55). T he t ransfer f...
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