Digital Signal Processing Reference
In-Depth Information
It is interesting to note the following pattern. In the expressions for each
value of the output
y(n)
above, we have
x(
0
), x(
1
), x(
2
)...
and
h(n), h(n
−
1
),
2
)...
multiplied term by term in order and the products are added, while
the indices of the two samples in each product always add to
n
.
Convolution is a fundamental operation carried out by digital signal processors
in hardware and in the processing of digital signals by software. The design of
digital signal processors and the software to implement the convolution sum
have been developed to provide us with very efficient and powerful tools. We
will discuss this subject again in Section 2.5, after we learn the theory and
application of
z
transforms.
h(n
−
2.2
z
TRANSFORM THEORY
2.2.1 Definition
In many textbooks, the
z
transform of a sequence
x(n)
is simply defined as
∞
x(n)z
−
n
[
x(n)
]
=
X(z)
=
(2.8)
Z
n
=−∞
and the inverse
z
transform defined as
1
2
πj
Z
−
1
[
X(z)
]
=
X(z)z
n
−
1
dz
x(n)
=
(2.9)
C
Equation (2.8) represents the (double-sided or) bilateral
z
transform of a sequence
x(n)
defined for
−∞
∞
. The inverse
z
transform given in (2.9) is obtained
by an integration in the complex
z
plane, and this integration in the
z
plane is
beyond the scope of this topic.
We prefer to consider signals that are of interest in digital signal processing
and hence consider a sequence obtained by sampling a continuous-time signal
x(t)
with a constant sampling period
T
(where
T
is the sampling period), and
generate a sequence of numbers
x(nT )
. Remember that according to the sifting
theorem, we have
x(t)δ(t)
<n<
x(
0
)δ(t)
. We use this result to carry out a proce-
dure called
impulse sampling
by multiplying
x(t)
with an impulse train
p(t)
=
=
n
=
0
δ(t
nT )
. Consequently we consider a sequence of delayed impulse func-
tions weighted by the strength equal to the numerical values of the signal instead
of a sequence of numbers. By doing so, we express the discrete sequence as a
function of the continuous variable
t
, which allows us to treat signal processing
mathematically. The product is denoted as
−
∞
x
∗
(t)
=
x(t)δ(t
−
nT )
n
=
0
∞
=
x(nT )δ(t
−
nT )
(2.10)
n
=
0
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