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 discretetime 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 2.56 S ynthesize a discretetime 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 2.57 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 2.61 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 2.62 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 2.81 ( 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 2.82 U sing t he i mpulse invariance m ethod, d esign a t enthorder (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. 12.61 for a digital integrator. 1 2.63 1 2.64 R epeat P rob. 12.63 for a chebyshev filter.
1 2.65 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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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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