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Unformatted text preview: al case of recursive filters, we e xpect t he p erformance of recursive
filters t o b e s uperior. This s tatement is t rue i n t he s ense t hat a given a mplitude
r esponse can b e achieved by a recursive filter of a n o rder smaller t han t hat r equired
f or t he c orresponding nonrecursive filter. However, nonrecursive filters have t he
a dvantage t hat i t c an realize a n a rbitrarily s haped a mplitude response a nd a l inear
p hase c haracteristic. Recursive filters are good for a piecewise constant a mplitude
r esponse a nd t hey c an realize linear phase only approximately. To realize a linear
p hase c haracteristic in nonrecursive filters, t he impulse response h[k] m ust b e e ither
s ymmetric o r antisymmetric a bout i ts c enter point. Problems 779
y [ k] f~ •
Q.4 M itra, S .K., Digital Signal processing: A Computer-Based Approach, M cGrawHill, New York, 1998. Find the amplitude and phase response of the digital filters depicted in Fig. PI2.1-1.
1 2.1-2 Find the amplitude and the phase response of t he filters shown in Fig. PI2.1-2.
Hint: Express H[ejf!J as e - j2 .S f! Ha[ejf!J.
For an LTID system specified by the equation + IJ - 0.5y[kJ = f [k 1· 3 }--------. (c) F ig. P 12.1-1. (a) f [k] y [k] f [k] y [k] F ig. P 12.1-2. 1 2.1-1 y[k . ( b) Problems 1 2.1-3 y [k ...•....... (a) References
1. l .. !/ 1 2.1-5 (a) A digital filter has the sampling interval T = 50 p.s. Determine the highest + IJ + 0.8f[kJ ( a) Find the amplitude and the phase response.
(b) Find the system response y[kJ for the input f[kJ = cos (0.5k - 1 2.1-4 For an asymptotically stable LTID system, show t hat the steady-state response to input ejf!ku[kJ is H[ejf!Jejf!ku[kJ. T he steady-state response is t hat p art of the response
which does not decay with time and persists forever.
Hint: Follow the procedure parallel to t hat used for continuous-time systems in Sec.
7.1-1. ~). ( b) 4
7 80 12 F requency R esponse a nd D igital F ilters frequency t hat c an b e processed by this filter w ithout a liasing.
( b) I f t he h ighest frequency t o b e p rocessed is 50 kHz, d etermine t he m inimum v alue
o f t he s ampling frequency F ., a nd t he m aximum value o f t he s ampling i nterval T
t hat c an b e u sed.
1 2.2-1 781 P roblems
1 2.4-1 ( a) A lowpass digital filter w ith a s ampling i nterval T = 50 I's h as a c utoff frequency
10 kHz. I f t he value o f T i n this filter is changed t o 251's, d etermine t he new cutoff
frequency of t he filter. R epeat t he p roblem if T is changed t o 1 001's. 1 2.5-1 ( a) Using t he i mpulse invariance criterion, design a digital filter t o realize a n a nalog
filter w ith t ransfer f unction Pole-zero configurations o f c ertain filters are shown in Fig. P 12.2-l. S ketch r oughly
t he a mplitude r esponse o f t hese filters. H S_
7 s + 20
a( ) - 2(s2 + 7s+ 10) Assume t he filter t o b e b andlimited t o t he f requency where t he g ain drops t o a bout
1% o f i ts m aximum value.
( b) Show a canonical a nd a p arallel realization o f t he filter.
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