Digital Signal Processing Reference
In-Depth Information
(a)
Without calculating the inverse transforms, state how
x
1
[n], x
2
[n],
and
x
3
[n]
are
related.
(b)
Verify the results of part (a) by finding the inverse transforms.
(c)
Verify the partial-fraction expansions in part (b) using MATLAB.
11.12.
Consider the following difference equation and excitation:
y[n] - 1.5y[n - 1] + 0.5y[n - 2] = x[n];
1,
n = 1
b
x[n] =
otherwise
.
0,
(a)
Find
y
[
n
], using the
z
-transform.
(b)
Check the partial-fraction expansions in part (a) using MATLAB.
(c)
Verify the results of part (a) for
n = 0, 1, 2, 3,
and 4, by solving the difference
equation by iteration.
(d)
Use MATLAB to check the results in part (c).
(e)
Verify the value of
y
[0] in part (a), using the initial-value property.
(f)
Will the final-value property give the correct value of If your answer is yes,
find the final value. Otherwise, state why the final-value property is not applicable.
11.13.
Consider the following difference equation and excitation:
y[
q
]?
y[n] - 0.75y[n - 1] + 0.125y[n - 2] = x[n];
1,
n = 0
b
x[n] =
otherwise
.
0,
(a)
Find
y
[
n
], using the
z
-transform.
(b)
Check the partial-fraction expansions in part (a) using MATLAB.
(c)
Verify the results of part (a) for
n = 0, 1, 2, 3,
and 4, by solving the difference
equation by iteration.
(d)
Use MATLAB to check the results in part (c).
(e)
Verify the value of
y
[0] in part (a), using the initial-value property.
(f)
y[
q
]?
Will the final-value property give the correct value of If your answer is yes,
find the final value. Otherwise, state why the final-value property is not applicable.
11.14. (a)
The difference equation
y[n] - 0.75y[n - 1] + 0.125y[n - 2] = x[n].
models an LTI system. Find the system transfer function.
(b)
Find the unit step response for the system of part (a).
(c)
Use MATLAB to check the partial-fraction expansion in part (b).
(d)
From part (b), give the values of
y
[0],
y
[1], and
y
[2].
(e)
Verify the results of part (d) by solving the difference equation iteratively.
(f)
Verify the results in part (e) using MATLAB.
11.15.
Consider the block diagram of a discrete-time system given in Figure P11.15.
(a)
Find the difference-equation model of this system.
(b)
Find the system transfer function.
