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

Such a realization requires t he use o f infinite

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Unformatted text preview: a c onstant ~. Generalize t his r esult t o show t hat a n L TID s ystem w ith t wo poles a t z = r e±j9 a nd t wo zeros a t z = ~ e±j9 ( r ::; 1) is a n allpass filter. I n o ther words, show t hat t he a mplitude response o f a s ystem w ith t he t ransfer function (z - !e j9 )(z - !e- j9 ) Z 2 _ (~ cos Ii)z + 1 H[z] = r r = r ~ ( z - re j9 )(z - re j9) z2 - (2r cos Ii)z + r 2 is c onstant w ith frequency. + v 2s + 1 A ssume t he filter t o b e b andlimited t o t he f requency where t he g ain d rops t o 1% o f i ts m aximum value. r ::; Ha(s)=~ s +wc ( c) S ynthesize a discrete-time i ntegrator u sing t he s tep i nvariant m ethod a nd c ompare i ts a mplitude r esponse w ith t hat o f t he i deal i ntegrator. 1 2.5-6 S ynthesize a discrete-time differentiator a nd i ntegrator, u sing t he r amp i nvariance m ethod. I n t his m ethod, for a given Ha(s), we design H[z] s uch t hat y (kT) in Fig. 12.8b is identical t o y[k] i n Fig. 12.8a when f (t) = t u(t). 1 2.5-7 I n a n i mpulse invariance design, show t hat i f Ha(s) is a t ransfer f unction o f a s table s ystem, t he c orresponding H[z] is also a t ransfer f unction o f a s table s ystem. 1 782 12 F requency R esponse a nd D igital F ilters A pole o f Ha(s) a t S = S i is transformed into a pole o f H[z] a t z = t hat if Re S i < 0, t hen JZiJ < l . 1 2.6-1 Zi =e S ". P roblems Show ( a) Design a digital differentiator, using t he b ilinear t ransformation. ( b) Show a realization o f t his filter. ( c) F ind a nd s ketch t he a mplitude response o f t his filter a nd c ompare i t w ith t hat o f t he ideal differentiator. ( d) I f t his filter is used primarily for processing audio signals (voice a nd music) u p t o 20 kHz, determine a s uitable value for T . 1 2.6-2 Design a digital lowpass B utterworth filter using th<;: b ilinear trans~ormation w ith p rewarping t o s atisfy t he following specifications: G p = - 2 d B, G s = - 11 d B, W p = 1007r r ad/s, a nd W s = 2007r r ad/s. T he h ighest significa,nt f requency is 250 Hz. I t is desirable t o o versatisfy (if possible) t he r equirement o f G s . H[z] = zn + 1 zn ( a) F ind t he i mpulse response o f t his filter for n = 6. ( b) F ind c anonical realization o f t his filter. ( c) F ind a nd s ketch t he a mplitude a nd p hase r esponse o f t his filter. 1 2.8-1 ( a) Design a nonrecursive ( FIR) filter w ith n = 14 t o a pproximate a n i deallowpass filter w ith a c utoff frequency a t 20 kHz. Use t he s ampling frequency F s = 200 kHz. ( b) M odify t he d esign in (a), using t he H amming window. 1 2.8-2 U sing t he i mpulse invariance m ethod, d esign a t enth-order (n = 10) nonrecursive ( FIR) filter t o a pproximate t he i deal b andpass c haracteristic depicted in Fig. P12.82. U seT=2x 1 0- 3 . R epeat P rob. 12.6-1 for a digital integrator. 1 2.6-3 1 2.6-4 R epeat P rob. 12.6-3 for a chebyshev filter. 1 2.6-5 Design a digital highpass B utterworth filter using th~ b ilinear t rans!ormation w ith p rewarping t o s atisfy t he following specifications: G p = - 2 d B, G s = - 10 d B, W p = 15071' r ad/s, a nd W s = 10071' r ad/s. T he h ighest signific...
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