数字信号处理-基于计算机的方法(第四版)第九章答案

−α p / 20
9.2 We obtain the solutions by using Eqs. (9.3) and (9.4). € (a) α p = −20log10 1 − δ p = −20log10 (1 − 0.04) = 0.3546dB, €
(
)
α s = −20log10 (δ s ) = −20log10 (0.08) = 21.9382 dB.
are mapped onto a single point in the z -plane as z €in the figure below mapping is illustrated €
jΩ
3π T π T
⎛ 2 πk ⎞ j ⎜Ω o ± ⎟T T ⎠ = eσ oT e ⎝
= eσ oT e jΩ oT . The
HLP (e jω )
€
€ €
9.4 €
€
€
HHP (e jω )
€
1 + δp 1 – δp
1 – δp
1 + δp
δs –π – ωs – ωp 0 ωp ωs π
δs
ω
–π –(π – ω p ) –(π – ω s )
0
ω π π − ωs π − ω p
Therefore, the passband edge and the stopband edge of the highpass filter are given by ω p,HP = π − ω p , and ω s,HP = π − ω s , respectively. 9.5 Note that G( z) is a complex bandpass filter with a passband in the range 0 ≤ ω ≤ π . Its passband edges are at ω p,BP = ω o ± ω p ,and stopband edges at ω s,BP = ω o ± ω s . A real € coefficient bandpass transfer function can be generated according to GBP ( z) = H LP (e jω o z) + H LP (e – jω o z) which will have a passband € in the range 0 ≤ ω ≤ π €
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Chapter 9
9.1 We obtain the solutions by using Eq. (9.3) and Eq. (9.4). (a) δ p = 1 − 10 (b) δ p = 1 − 10
−α p / 20
= 1 − 10−0.24 / 20 = 0.0273, δ s = 10−α s / 20 = 10−49 / 20 = 0.0035. = 1 − 10−0.14 / 20 = 0.016,δ s = 10−α s / 20 = 10−68 / 20 = 0.000398.
∑δ (t − nT ). Thus, h p (t ) = ∑ ha (nT )δ (t − nT )..
n =−∞
∞
Now, H a ( s) =
€
∫ ha (t)e−st dt and
€ € ∞ ∞ ∞ − st − st H p ( s) = ∫ h p ( t )e dt = ∑ ∫ ha ( nT )δ ( t − nT )e dt = ∑ ha ( nT )e− snT . € −∞ n =−∞ −∞ n =−∞ € ∞ ∞ Comparing the above expression with G( z) = ∑ g[ n ]z− n = ∑ h ( nT ) z− n , we
HLP (e
jω
)
€
1 + δp 1 – δp
G(e ) 1 + δp
jω
1 – δp
δs –π – ωs – ωp 0 ωp ωs π
δs
ω
–π
ω 0 ωo π ωo − ωs ω − ω ω + ω ωo + ωs o p o p
∞
∞ n =−∞
9.6 (a) h p ( t ) = ha ( t )⋅ p( t ) where p( t ) = We also have, g[ n ] = ha ( nT ).
∞
∑
k =−∞ ∞
⎛ 2πkt ⎞ H a ⎜ s + j ⎟. As a result, we have ⎝ T ⎠
€
G( z) =
€
1 T
∑
k =−∞
⎛ 2πkt ⎞ H a ⎜ s + j ⎟ ⎝ T ⎠
ln z.
1 s= T
(7-1)
€
sT (b) The transformation from the s -plane to z -plane is given by z = e . If we express jω σ oT jΩ oT
k =−∞
∞ ∞ ⎛ ⎞ 1 1 − j (2 πkt /T ) ⎟ h p ( t ) = ⎜ e h ( t ) = ∑ ha (t)e− j (2 πkt /T ) . ⎜ T ∑ ⎟ a T ⎝ k =−∞ ⎠ k =−∞
. 1 ln z T
Hence,
€
1 H p ( s) = T
€ 9.3
€
G(e jω ) = H 2 (e jω ) = H (e jω ) .
2
Let δ p and δ s denote the passband and stopband ripples of H (e jω ) , respectively. Also, let
jω δ p,2 = 2 δ p, and δ s,2 denote the passband and stopband ripples of G(e ) , respectively. € Then δ p,2 = 1 − (1 − δ p ) 2 , and δ s,2 = ( δ s ) 2 . For a cascade of sections, € € M δ p,M = 1 − (1 − δ p ) M , and δ s,M = ( δ s ) . € €
∞ n =−∞ n =−∞
−∞
€
conclude that G( z) = H p ( s)
s=
We can also show that a Fourier series expansion of p(t) is given by € ∞ 1 − j (2 πkt /T ) p( t ) = . Therefore, ∑e € T
(d) For z = e jω = e jΩT , or equivalently, ω = ΩT . € € € 9.7 Assume ha ( t ) is causal. Now, ha ( t ) = ∫ H a ( s)e st ds. Hence, € € snT g[ n ] = ha ( nT ) = ∫ H a ( s) e ds. Therefore,
SOLUTIONS MANUAL
to accompany
Digital Signal Processing: A Computer-Based Approach
Fourth Edition
Sanjit K. Mitra
Prepared by Chowdary Adsumilli, John Berger, Marco Carli, Hsin-Han Ho, Rajeev Gandhi, Martin Gawecki, Chin Kaye Koh, Luca Lucchese, Mylene Queiroz de Farias, and Travis Smith
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and another passband in the range – π ≤ ω ≤ 0. However because of the overlap of the two spectra a simple formula for the bandedges cannot be derived.
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Im z
€
€
σ –π T
π – 3T
−1
1
Re z
s -plane
z -plane
Note that the strip of width 2π / T in the s-plane for values of s in the range π π − ≤ Ω ≤ is mapped into the entire z -plane, and so are the adjacent strips of width T T
s = σo + jΩo , then we can write z = re
€
=e
e
. Therefore,
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⎧ < 1, for σo < 1, ⎪ z = ⎨ = 1, for σo = 1, Or in other words, a point in the left-half -plane is mapped onto ⎪ ⎩ > 1, for σo > 1. a point inside the unit circle in the z -plane, a point in the right-half -plane is mapped onto a point outside the unit circle in the z -plane, and a point on the jω -axis in the splane is mapped onto a point on the unit circle in the z -plane. As a result, the mapping has the desirable properties enumerated in Section 9.1.3. € € 2πk € (c) However, all points in the s -plane defined by s = σ + j Ω ± j , k = 0, 1, 2, …, , a o o € T