Digital Signal Processing Reference
In-Depth Information
an LTID system can be computed using the convolution property of either the
DTFT or the z-transform. In addition, the frequency-domain approach offers
insight about the system characteristics, which is not readily available from the
time-domain approach. However, an important issue is to determine which of
the two transforms should be used to analyze the LTID system. Both approaches
have their own advantages. Depending upon the application under considera-
tion, the appropriate transform should be selected.
Example 13.19
Consider an LTID system represented by the unit impulse response
h
[
k
]
=
0
.
8
k
u
[
k
]. Calculate the overall output and steady state output of the LTID system
for the input sequence
x
[
k
]
=
cos(
π
k
/
3)
u
[
k
].
Solution
z-transform method
Using Table 13.1, the z-transforms of the impulse
response
h
[
k
] and the input
x
[
k
] are given by
1
1
−
0
.
8
z
−
1
H
(
z
)
=
and
−
1
cos(
π/
3)
1
−
2
z
−
1
cos(
π/
3)
+
z
−
2
−
1
1
−
z
−
1
+
z
−
2
.
1
−
z
1
−
0
.
5
z
X
(
z
)
=
=
Using the convolution property, the z-transform of the output response is given
by
−
1
1
−
0
.
5
z
Y
(
z
)
=
H
(
z
)
X
(
z
)
=
−
2
)
.
(1
−
0
.
8
z
−
1
)(1
−
z
−
1
+
z
By partial fraction expansion, the above expression becomes
−
1
Y
(
z
)
=
2
7
1
1
−
0
.
8
z
−
1
+
5
7
1
+
0
.
5
z
1
−
z
−
1
+
z
−
2
−
1
1
−
z
−
1
+
z
−
2
.
Taking the inverse z-transform, the output response is given by
−
1
=
2
7
1
1
−
0
.
8
z
−
1
+
5
7
1
−
0
.
5
z
+
5
7
z
1
−
z
−
1
+
z
−
2
y
[
k
]
=
2
7
0
.
8
k
u
[
k
]
+
5
7
π
k
3
10
7
π
k
3
cos
u
[
k
]
+
sin
√
u
[
k
]
3
u
[
k
]
π
k
3
0
.
287(0
.
8)
k
+
1
.
091 cos
−
0
.
857
r
=
where the superscript
r
indicates that the angle is expressed in radians.
The steady state output
y
ss
[
k
] is computed by neglecting the transient term
(0.8)
k
, which decays to zero with time. The steady state output response is,
Search WWH ::
Custom Search