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

# 9 t he second filter w ith transfer function h2z is a

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Unformatted text preview: 2 U sing t he i mpulse invariance criterion, design a digital filter t o r ealize t he s econdorder a nalog B utterworth filter w ith t ransfer f unction 1 Ha(s) = - - = - S2 ( a) (b) 1 2.5-3 D esign a digital i ntegrator u sing t he i mpulse invariance m ethod. F ind a nd s ketch t he a mplitude r esponse, a nd c ompare i t w ith t hat o f t he ideal i ntegrator. I f t his i ntegrator is used p rimarily for i ntegrating a udio s ignals (whose b andwidth is 20 k Hz), d etermine a s uitable value for T . 1 2.5-4 F ig. P I2.2-1. 1 2.2-2 A n o scillator by definition is a source ( no i nput) w hich g enerates a s inusoid o f a c ertain f requency woo T herefore a n o scillator is a s ystem w hose zero-input response is a sinusoid o f t he d esired frequency. F ind t he t ransfer f unction o f a d igital oscillator t o o scillate a t 10 kHz, by two m ethods: ( a) C hoose H[z] d irectly so t hat i ts z ero-input response is a d iscrete-time s inusoid o f f requency n = w T c orresponding t o 10 kHz. ( b) C hoose Ha(s) w hose zero-input response is a n a nalog sinusoid o f 10 kHz. Now use t he i mpulse invariance m ethod t o d etermine H[z]. I n b oth m ethods s elect T s o t hat t here a re 10 samples in each cycle o f t he s inusoid. ( c) S how a canonical realization o f t he o scillator. 1 2.5-5 A v ariant o f t he i mpulse invariance m ethod is t he s tep i nvariance m ethod o f d igital filter synthesis. I n t his m ethod, for a given Ha(s), we design H[z] i n F ig. 12.8a s um t hat y (kT) i n Fig. 12.8b is identical t o y[k] i n Fig. 12.8a when f (t) = u (t). ( a) S how t hat i n general ( a) Realize a digital filter whose t ransfer f unction is given by H[z] = K z + 1 z -a ( b) Sketch t he a mplitude r esponse o f t his filter, assuming lal < l . ( c) T he a mplitude r esponse of t his lowpass filter is m aximum a t n = O. T he 3 -dB b andwidth is t he frequency where t he a mplitude r esponse d rops t o 0.707 ( or 1 /v2) t imes its m aximum value. D etermine t he 3-dB b andwidth o f t his filter when a = 0.2. 1 2.2-3 12.2-4 Design a digital n otch filter t o r eject frequency 5000 H z completely, a nd t o h ave a s harp recovery o n e ither s ide o f 5000 Hz t o a g ain o f u nity. T he h ighest frequency t o b e processed is 20 kHz (Fh = 2 0,000). H int: See E xample 12.3. T he zeros s hould b e a t e±jwT for w c orresponding t o 5000 Hz, a nd t he poles a re a t ae±jwT w ith a < l . Leave your answer in t erms o f a. R ealize this filter using t he c anonical form. F ind t he a mplitude r esponse o f t he filter. Show t hat a f irst-order LTID s ystem w ith a pole a t z = r a nd a zero a t z = ~ ( r ::; 1) is a n allpass filter. I n o ther words, show t hat t he a mplitude r esponse IH[ejoJl o f a s ystem w ith t he t ransfer f unction z -! H[z] = _ _ r z -r r ::; 1 ( b) U sing t his m ethod, d esign H[z] w hen is c onstant w ith frequency. T his is a first-order allpass filter. Hint: Show t hat t he r atio o f t he d istances o f any p oint o n t he u nit circle from t he zero ( at z = ~) a nd t he p ole ( at z = r ) is...
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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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