Digital Signal Processing Reference
In-Depth Information
3.
The classical method of solving a difference equation
4.
The analytical solution using the
z
transform.
We should point out that methods 1-3 require that the DT system be modeled by a
single-input, single-output equation. If we are given a large number of difference
equations describing the DT system, then methods 1-3 are not suitable for finding
the output response in the time domain. Method 4, using the
z
transform, is the
only powerful and general method to solve such a problem, and hence it will be
treated in greater detail and illustrated by several examples in this chapter. Given
a model in the
z
-transform domain, we will show how to derive the recursive
algorithm and the unit impulse response
h(n)
so that the convolution sum can be
applied. So the
z
-transform method is used most often for time-domain analysis,
and the frequency-domain analysis is closely related to this method, as will be
discussed in the next chapter.
2.1.3 Convolution Sum
In the discussion above, we have assumed that the unit impulse response of a
discrete-time system when it is excited by a unit impulse function
δ(n)
,exists
(or is known), and we denote it as
h(n)
. Instead of using the recursive algo-
rithm to find the response due to any input, let us represent the input sig-
nal
x(n)
not by its values in a sequence
}
but as
the values of impulse function at the corresponding instants of time. In other
words, we consider the sequence of impulse functions
x(
0
)δ(n)
,
x(
1
)δ(n
{
x(
0
)
,
x(
1
)
,
x(
2
)
,
x(
3
),...
−
1
)
,
x(
2
)δ(n
−
2
),...
as the input—and not the sequence of values
{
x(
0
)
,
x(
1
)
,
x(
2
)
,
x(
3
),...
}
. The difference between the values of the samples as a sequence
of numbers and the sequence of impulse functions described above should be
clearly understood. The first operation is simple sampling operation, whereas the
second is known as
impulse sampling
, which is a mathematical way to repre-
sent the same data, and we represent the second sequence in a compact form:
x(n)
=
k
=
0
x(k)δ(n
k)
. The mathematical way of representing impulse sam-
pling is a powerful tool that is used to analyze the performance of discrete-time
systems, and the values of the impulse functions at the output are obtained by
analytical methods. These values are identified as the numerical values of the
output signal.
Since
h(n)
is the response due to the input
δ(n)
,wehave
x(
0
)h(n)
as the
response due to
x(
0
)δ(n)
because we have assumed that the system is linear.
Assuming that the system is time-invariant as well as linear, we get the output
due to an input
x(
1
)δ(n
−
−
1
)
to be
x(
1
)h(n
−
1
)
. In general, the output due to
an input
x(k)δ(n
−
k)
is given by
x(k)h(n
−
k)
. Adding the responses due to all
=
k
=
0
x(k)δ(n
the impulses in
x(n)
−
k)
, we get the total output as the sum
∞
y(n)
=
x(k)h(n
−
k)
(2.5)
k
=
0
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