数字信号处理-基于计算机的方法(第四版)答案 8-11章

€
€
€
⎡
⎢
transfer matrices for the three-pair: Type 2A: ⎢
z−2 z−1
z−2 −1 z−1
0⎤⎥ 1⎥, €
€
€
⎣⎢z−2 +1 z−2 0⎦⎥
⎡ ⎢ Type 2B: ⎢
z−2 z−1(z−1 −1)
z−1 +1 z−1
0⎤⎥
,
then
from
Eq.
(8.151a),
the
€
numerator coefficients of HN −1(z) are given by
pk'
=
p0dk+1 p0dN
− −
pk+1 pN
=
dN
dk+1 − dN dN2 −1
−k−1
,
€
and from Eq. (8.151b) the denominator coefficients of HN −1(z) are given by
⎡ ⎢
Problem 8.38 then reduces to the lattice structure employed in the Gray-Markel realization
procedure.
€
€
8.41 (a) Consider th€e realization of Ty€pe 1B allpas€s structure. From its transfer parameters
given in Eq. (8.50b) we arrive at Y1 = z−1X1 + (1+ z−1)X2 = z−1(X1 + X2) + X2, and Y2 = (1 − z−1)X1 − z−1X2 = X1 − z−1(X1 + X2). A realization of the two-pair based on these
dN' −k−1
=
pN −k−1dN p€0 dN
− dN −k−1 − pN
=
dk+1dN − dN −k−1 €dN2 −1
=
pk' ,
implying
Байду номын сангаас
HN −1(z)
is
an
allpass
transfer function of order N −1. Since here €pN = 1 and p0 = dN , the lattice structure of
1
Chapter 8 - Part 2
8.40 When HN (z) is an allpass transfer function of the form
HN
(z)
=
AN
(z)
=
dN + dN −1z−1 + +d1z−N +1 + z−N 1+d1z−1 + + dN −1z−N +1 + dN z−N
Substituting Eq. (4) in Eq. (1) we get
X
(z)
=
⎛⎜⎜1 ⎝
−
a
− 1
z−1 + − z−1
az−1
⎞ ⎟⎟W ⎠
(z).
€ Finally, from Eqs. (5) and (6) we arrive at
H(z)
=
Y (z) X(z)
€−(1 − a) + z−1 = 1 − (1 − a)z−1
Luca Lucchese, Mylene Queiroz de Farias, and Travis Smith
Copyright © 2011 by Sanjit K. Mitra. No part of this publication may be reproduced or distributed in any form or by any means, or stored in a database or retrieval system, without the prior written consent of Sanjit K. Mitra, including, but not limited to, in any network or other electronic Storage or transmission, or broadcast for distance learning.
Simple block diagram manipulation of the above structure leads to:
Finally, by delay sharing between adjacent allpass sections we arrive at the following equivalent realization requiring now 4 delays compared to 6 delays in the direct realization shown on the previous page.
€
Consider the case t13 = 0. Then the above equation reduces to t31t12t23 = z−4 −1. From
€ this equation a€nd t23t32 €= z−2, it follows that t32 = z−2, t23 = 1, t31t12 = z−4 −1
8.43 The structure of Figure P8.15 with internal variables labeled is shown below:
Y(z)
Its analysis yields (1): W (z) = X(z) − z−1U(z), (2): U(z) = aW (z) + z−1U(z), and (3):
(b) A cascade connection of 3 Type 1At first-order allpass networks is shown below which requires 8 delays.
3
By delay sharing between adjacent allpass sections we arrive at the following equivalent realization requiring now 4 delays.
two equations is as shown below which leads to the structure of Figure 8.24(b).
€
€
(b) From the transfer parameters of Type 1At allpass given in Eq. (8.50c) we obtain Y1 = z−1X1 + X2, and Y2 = (1 − z−2)X1 − z−1X2 = X1 − z−1(z−1X1 + X2) = X1 − z−1Y1. A realization of the two-pair based on these two equations is as shown on next page which
Y (z) = −W (z) + U(z).
From
Eq.
(2)
we
obtain
(4):
U(z)
=
a 1 − z−1
W
(z).
Substituting Eq.
€
(4) in Eq. (3)€we get (5):
Y
(z)
=
−⎛⎜⎜1 ⎝
− 1
a −
−€z−1 z−1
⎞ ⎟⎟W ⎠
(z).
= (z−1 −1)(z−1 +1)(z−2€+1). There are four €possible realizable sets of values of t21 and t31
satisfyin€g the last equation and t12t21 = z−1(z−2 −1). €These lead to 4 different realizable
⎤ ⎥ ⎥
=
⎡⎢⎢tt1211
t12 t 22
t13 t 23
⎤ ⎥ ⎥
⎡⎢⎢XX12
⎤ ⎥⎥,
and
X2
= −d1Y 2,
⎣⎢Y3 ⎦⎥ ⎣⎢t 31 t 32 t 33⎦⎥ ⎣⎢X3 ⎦⎥
X3
= d2Y 3.
From these equations, we get after some algebra €
A2(z) =
Y1 X1
=
N (z) , D(z)
where
D(z) N(z)
= =
1+ t11
d+1td212(t−11dt222t3−€3 +t12dt12d12)(−t2d32t3(2t1−1tt3232−t3t31)3,t3a1n)d
€
{ } +d1d2 t21(t12t33 − t13t32) + t31(t22t13 − t12t23) + t11€(t23t32 − t22t33) .
.
€
€
8.44
We realize
A2(z)
=
d1d2 + d1z−1 + z−2 1+ +d1z−1 + d1d2z−2
in the form of a constrained three-pair as
indicated below:
Y2
€
4
From
the
above
figure,
we
have
⎡⎢⎢YY12
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,
€
transfer function with N(z) we get t11 = z−2, t12t21 = z−1(z−2 −1), t13t31 = 0, and
t32(t11t23 − t21t13) + t31(t22t13 − t12t23) = 1. Substituting the a€ppropriate transfer
€ €
Comparing the denominator of the desired allpass transfer function with D(z) we get t22 = z−1, t33 = 0, t23t32 = z−2. Next, comparing the numerator of the desired allpass
leads to the structure of Figure 8.24(c).
€
€
(c) From the transfer parameters of Type 1At allpass given in Eq. (8.50d) we obtain
Y1 = z−1X1+(1 − z−1)X2 = z−1(X1 − X2) + X2, and Y2 = (1+ z−1)X1 − z−1X2 = X1 + z−1(X1 − X2). A realization of the two-pair based on these two equations is as
shown below which leads to the structure of Figure 8.24(d).
€
€
€
2
8.42 (a) A cascade connection of 3 Type 1A first-order allpass networks is shown below:
€
parameters from the previous equations into the last equation we simplify it to
t13t21t32 + t3€1t12t23 = z−4€−1.
€
Since t13t31 = 0, either t13 = 0, or t31 = 0. (Both cannot be simultaneously equal to zero, as this will violate the condition t13t21t32 + t31t12t23 = z−4 −1.