Introduction to Digital Signal and System Analysis - Tutorsindia

Introduction to Digital Signal and System Analysis
Z Domain Analysis
For the delayed unit impulse,
∞
−n
−n
X ( z)
= ∑d
[ n −1]
z = z = z
n=
0
n=
1
−1
n
For more general cases of shifting by
0 samples,
∞
X ( z)
= ∑d [ n − n
n=
0
0
]
z
−n
= z
−n
n=
n0
= z
−n0
For shifted signal x[n], i.e. delayed by n samples, the z-transform
0
∞
∑
n=
0
x
[ n − n0
] u[
n − n0
] z
−n
=
X ( z)
z
−n0
5.4 Transfer function
The transfer function describes the input-output relationship, or the transmissibility between input and output, in the
z-domain. Applying the z-transform to the output of a system, the relationship between the z-transforms of input and
output can be found:
Y ( z)
=
=
=
∞
∑
r=−∞
∞
∑
r=−∞
∞
∑
n=
0
x(
r)
z
x(
r)
z
y[
n]
z
∞
−r
−r
∑
n=
0
∞
∑
−n
m=−r
=
∞
∑
n=
0
h[
n − r]
z
h[
m]
z
r=−∞
−m
∞
∑
x[
r]
h[
n − r]
z
−(
n−r)
= X ( z)
H ( z)
−n
Therefore,
H ( z)
=
Y ( z)
X ( z)
(5.3)
i.e., the transfer function can be obtained from the z-transforms of input and output.
Alternatively, the transfer function H (z)
can be obtained by applying z-transform directly to the impulse response h[n]
. The relationships of input ( x [n]
and X (z)
), output ( y [n]
and Y (z)
) and system function ( h [n]
and H (z)
the time and z domains are depicted in Figure 5.2.
) in
63
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