Digital Signal Processing Reference
In-Depth Information
(k)
It can be shown that the closed-loop transfer function is given by
H
c
(z)H
p
(z)
1 + H
c
(z)H
p
(z)
.
Y(z)
U(z)
= H(z) =
Verify your results in part (h) by showing that this equation is satisfied by the de-
rived transfer functions in parts (b) and (e).
13.10.
Consider the of Problem 13.2. Parts of this problem are repeated from Problem
13.2. Use those results if available.
a-filter
(a)
Write the state equations of the filter, with the state variable equal to the output
y
[
n
].
(b)
Use the results of part (a) to find the filter transfer function.
13.11.
Consider the filter of Figure P13.3. Parts of this problem are repeated from Prob-
lem 13.3. Use those results if available.
a- b
(a)
Write the state equations of the filter, with the state variables equal to the delay
outputs and the output equal to
y
[
n
].
(b)
Use the results of part (a) to find the filter-transfer function
(c)
H(z) = Y(z)/U(z).
Show that, with
b = 0,
the transfer function is that of the
a-filter
in Problem 13.10(b).
13.12.
Consider the system of Figure P13.6.
(a)
Write the state equations, with the outputs of the delays as the states.
(b)
Find the state-transition matrix.
(c)
Find the system output for and the initial states given by
(d)
Find the system unit step response, with using (13.32).
(e)
Verify the results of part (d), using the
z
-transform and the system-transfer function.
(f)
Find the system response, with the initial conditions given in part (c) and the input
in part (d).
(g)
Verify the results in part (f), using S
IMULINK
.
13.13.
Consider the system of Figure P13.7. Replace the gains of 1.5 and 2.3 with gains of zero
for this problem.
2]
T
.
u[n] = 0
x
(0) = [1
x
(0) =
0
,
(a)
Write the state equations, with the outputs of the delays as the states.
(b)
Find the state-transition matrix.
(c)
Find the system output for and the initial states given by
(d)
Find the system unit step response, with using the
z
-transform as in (13.32).
(e)
Verify the results of part (d), using the
z
-transform and the system-transfer function.
(f)
Find the system response, with the initial conditions given in part (c) and the input
in part (d).
(g)
Verify the results in part (f), using S
IMULINK
.
13.14. (a)
9]
T
.
u[n] = 0
x
(0) = [1
x
(0) =
0
,
Consider the system described by
01
00
0
1
B
R
B
R
x
[n + 1] =
x
[n] +
u[n];
y[n] = [1
0]
x
[n].
