Introduction to Signal Processing
1
Copyright © 2005-2009 – Hayder Radha
V.
Discrete-Time System Analysis Using
The Z-Transform
The z-transform is the discrete time domain
counterpart of the Laplace transform. .
A.
The z-Transform
We define
[ ]
X z
as the direct z-transform of
[ ]
x n
.
The z-transform represents signals as the sum of
everlasting exponentials.
Introduction to Signal Processing
2
Copyright © 2005-2009 – Hayder Radha
[ ]
[ ]
n
n
X z
x n z
∞
−
=−∞
=
∑
where
z
is a
complex variable
.
The inverse z-transform is found using,
1
1
[ ]
[ ]
2
n
x n
X z z
dz
j
π
−
=
∫
v
Symbolic representations of the z-transform and its
inverse,
{
}
[ ]
[ ]
X z
Z
x n
=
and
{
}
1
[ ]
[ ]
x n
Z
X z
−
=
Introduction to Signal Processing
3
Copyright © 2005-2009 – Hayder Radha
Or
[ ]
[ ]
x n
X z
⇔
.
Note that
{
}
1
[ ]
[ ]
Z
Z
x n
x n
−
⎡
⎤
=
⎣
⎦
and
{
}
1
[ ]
[ ]
Z
Z
X z
X z
−
⎡
⎤
=
⎣
⎦
.
Introduction to Signal Processing
4
Copyright © 2005-2009 – Hayder Radha
Linearity of the z-Transform
Like its continuous-time counterpart the z-transform too
is linear, i.e.
1
1
[ ]
[ ]
x n
X
z
⇔
and
2
2
[ ]
[ ]
x
n
X
z
⇔
Then,
1
1
2
2
1
1
2
2
[ ]
[ ]
[ ]
[ ]
a x n
a x
n
a X
z
a X
z
+
⇔
+
.

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