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

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Unformatted text preview: 88a) - 0.02653z 6 + O.2387z 4 + 0 .5z 3 + 0.2387z 2 + 0.02653 E xample 1 2.11 D esign a d ifferentiator u sing a t enth-order n onrecursive f ilter. I n t his c ase n = 10 a nd No = 11. T he t ransfer f unction o f a n i deal d ifferentiator is Ha(s) = s . T herefore, (12.88b) H a(jw) = j w jwj:5 11" T 768 12 F requency R esponse a nd D igital F ilters a nd a ccording t o Eq. ( 12.81b)t "IT ~ J COS h[kJ = k71" j we jwkT dw 271" - "IT (12.89) kT T his i s a n oncausal f ilter. W e t runcate i t u sing a n l l-point w indow a nd t hen d elay i t b y N o -1 = 5 u nits t o m ake i t r ealizable. T he d esired i mpulse r esponse u sing a r ectangular 2 w indow is hR[kJ = h[k - 5J 0 ::; k::; 10 T able 12.4 shows t his i mpulse r esponse u nder t he r ectangular w indow. T he f requency r esponse is 10 12.8 N onrecursive F ilter D esign 769 T herefore, t he p hase c haracteristic o f HR[ejwTJ is i dentical t o t he d esired ( ideal) differentiator w ith a d elay o f 5 T s econds. F igure 1 2.20 s hows t he a mplitude r esponse IHI o f t he r ealized filter a nd t he IHal o f t he i deal ( desired) d ifferentiator. T he a mplitude r esponse IHR[ejwTlI, w hich is oscillatory, c an b e i mproved b y u sing a t apered w indow f unction t o t runcate h a(t) g radually. T he H amming w indow f unction wH[kJ [Eq. (12.87)J for n = 10 i s g iven by wH[kJ ={ 0 .54 + 0.46 c os ( ¥) Ikl ::; 5 o o therwise T able 12.4 shows t he ( delayed) H amming coefficients wH[kJ a nd t he c orresponding f ilter i mpulse r esponse hH[kJ = h[kJWH[kJ. N ote t hat hR[kJ a nd hH[kJ a re b oth a ntisymmetric jwT ], t he w indowed a bout k = 5. U sing t he H amming coefficients from T able 12.4, HH[e f ilter frequency r esponse, is given b y HH[~wTJ = ~ej(f-5WT) [1.8242 s in w T - HR[ejwTJ = L hR[kJe-ikTW 0.682 s in 2 wT + 0 .2652 s in 3 wT - 0.0839sin 4 wT + 0.032 s in 5wTJ k=O S ubstitution o f v alues o f hR[kJ f rom T able 12.4 i n t his e quation y ields _1 1 j2wT j3wT - e -j3wT) H R [e jwTJ - 7'e -j5wT [( e jwT - e - jwT) - 2" (e - e -j2wT) + 31" (e 1( - 4e j4wT - e - j4wT) = ~e-j5wT [2 s in + s e - e - j5wT)] 1 ( j5wT F igure 1 2.20 s hows t he a mplitude r esponse o f t he i deal, t he r ectangular-windowed, a nd t he H amming-windowed f ilter c haracteristics. T he p hase r esponse o f b oth t he r ealized filters a re i dentical, b ut w ith a d elay o f 5 T s econds. T he a mplitude r esponse o f t he w indowed ( Hamming) f ilter is p ractically t he s ame o ver a large b and ( two t hirds o f t he b and) a s t hat for t he i deal f ilter, a nd h ence is p referable t o t he u nwindowed ( or r ectangularwindowed) filter. w T - sin 2 wT + ~ s in 3 wT - ~ s in 4 wT + ~ s in 5wT] T ABLE 1 2.4 = ~ej( f -5wT) [2 s in w T - sin 2 wT + ~ s in 3 wT - ~ s in 4 wT + ~ s in 5wT] R ectangular W indow H ence IHR[ejwTJI = ~ 1 2sin w T - sin 2 wT+ ~sin 3 wT- ~sin 4 wT+ ~sin 5wTI k hR[kJ H amming W indow wH[kJ hH[kJ ( 12.90a) 1 /5T 0 .08 0 .016/T - 1/4T 0 0 .1678 - 0.04195/T a nd ( 12.90b) 2 1 /3T 0 .3978 0...
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