Digital Signal Processing Reference
In-Depth Information
This expression has a Laplace transform denoted as
∞
X
∗
(s)
x(nT )e
−
snT
=
(2.11)
n
=
0
Now we use a frequency transformation
e
sT
z, (
where
z
is a complex variable),
and substituting it in expression (2.11), we get
=
∞
X
∗
(s)
e
sT
=
z
=
x(nT )z
−
n
n
=
0
Since
T
is a constant, we consider the samples
x(nT )
as a function of
n
and
obtain the
z
transform of
x(n)
as
∞
X
∗
(s)
e
sT
=
z
=
x(nT )z
−
n
n
=
0
∞
x(n)z
−
n
X(z)
=
(2.12)
n
=
0
Although the first definition of a discrete sequence given in (2.8) is devoid of
any signal concepts, soon concepts such as frequency response and time-domain
response are used in the analysis of discrete-time systems and signal processing.
Our derivation of the
z
transform starts with a continuous-time signal that is sam-
pled by impulse sampling and introduces the transformation
e
sT
z
to arrive at
the same definition. In Chapter 3, we will study the implication of this transfor-
mation in more detail and get a fundamental understanding of the relationship
between the frequency responses of the continuous-time systems and those of
the discrete-time systems. Note that we consider in this topic only the unilateral
z
transform as defined by (2.12), so we set the lower index in the infinite sum
as
n
=
=
0.
Example 2.3
Let us derive the
z
transform of a few familiar discrete-time sequences. Consider
the unit pulse
1
n
=
0
δ(n)
=
0
n
=
0
There is only term in the
z
transform of
δ(n)
, which is one when
n
=
0. Hence
Z
[
δ(n)
]
=
1.
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