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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 2.53 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 2.54 F ig. P I2.21. 1 2.22 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 zeroinput 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 eroinput response is a d iscretetime s inusoid o f
f requency n = w T c orresponding t o 10 kHz.
( b) C hoose Ha(s) w hose zeroinput 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 2.55 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 3dB b andwidth o f t his filter when a = 0.2.
1 2.23 12.24 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 irstorder 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 firstorder 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.
 Spring '13
 Bayliss
 Signal Processing, The Land

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