Digital Signal Processing Reference
In-Depth Information
11.8.
A function
y
[
n
] has the unilateral
z
-transform
z
3
Y(z) =
+ 5z - 9
.
z
3
- 3z
2
(a)
Find the
z
-transform of
(b)
Find the
z
-transform of
(c)
Evaluate
y
[
n
] for
y
1
[n] = y[n - 3]u[n - 3].
y
2
[n] = y[n + 3]u[n].
n = 0
and 3,
y[n - 3]u[n - 3]
for
n = 3,
and
y[n + 3]u[n]
for
by expanding the appropriate
z
-transform into power series.
(d)
Are the values found in part (c) consistent? Explain why.
n = 0
f[n] = a
n
u[n],
11.9.
Given
find the
z
-transforms of the following:
(a)
f[
n
/7]
(b)
; verify your result by a power series expansion
f[n - 7]u[n - 7]
(c)
f[n + 3]u[n]
; verify your result by a power series expansion
b
2n
f[n]
(d)
; using two different procedures.
11.10. (a)
Given the following unilateral
z
-transforms, find the inverse
z
-transform of each
function:
0.5z
2
(z - 1)(z - 0.5)
(i)
X(z) =
0.5z
(z - 1)(z - 0.5)
(ii)
X(z) =
0.5
(z - 1)(z - 0.5)
(iii)
X(z) =
z
(iv)
X(z) =
z
2
- z + 1
(b)
Verify the partial-fraction expansions in part (a) using MATLAB.
(c)
Evaluate each
x
[
n
] in part (a) for the first three nonzero values.
(d)
Verify the results in part (c) by expanding each
X(z)
in part (a) into a power series
using long division.
(e)
Use the final-value property to evaluate for each function in part (a).
(f)
Check the results of part (e), using each
x
[
n
] found in part (a).
(g)
Use the initial-value property to evaluate
x
[0] for each function in part (a).
(h)
Check the results of part (g), using each
x
[
n
] found in part (a).
x[
q
]
11.11.
Consider the transforms from Problem 11.10.
0.5z
2
(z - 1)(z - 0.5)
;
X
1
(z) =
0.5z
(z - 1)(z - 0.5)
;
X
2
(z) =
0.5
(z - 1)(z - 0.5)
.
X
3
(z) =
