Introduction to Digital Signal and System Analysis - Tutorsindia

Introduction to Digital Signal and System Analysis
Z Domain Analysis
X ( z)
=
∞
∑
n=
0
x[
n]
z
= 1+
0.8z
1
=
1−
0.8z
−1
−1
−n
+
= 1+
0.8z
−1
+ 0.8
−1
2
−1
( 0.8z
) + ( 0.8z
)
z
=
z − 0.8
2
z
−2
3
3
+ 0.8 z
+ ...
−3
+ ...
1
b) Reconstruct the signal corresponding to the z-transform: X ( z)
= z + 1.2
−1
( = 1 z
−1⎛
1 ⎞
X z)
= = z
⎜
⎟
−1
z + 1.2 1 + 1.2z
⎝1
− (1.2z
−1
) ⎠
Using the series summation formula in Eq.(3.13) (where 1.2z −1 < 1 is required.),
X(
z)
= z
= z
−1
−1
−1
−1
2
−1
3
{ 1+
( −1.2
z ) + ( −1.2
z ) + ( −1.2
z ) + ...}
−1.2z
−2
_1.44z
−3
−1.728z
−4
+ ...
Therefore, the reconstructed original signal can be obtained as
x[n]=[0 1 -1.2 1.44 -1.728 ... ].
↑
The signal is shown in Figure 5.1(b).
(a)
1
x[n]
0.5
0
-2 -1 0 1 2 3 4 5 6 7 8 9
n
(b)
5
x[n]
0
-5
-2 -1 0 1 2 3 4 5 6 7 8 9
n
Figure 5.1 Signals in the examples
61
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