Digital Signal Processing Reference
In-Depth Information
which reduces to
x
[
k
]
=
∞
x
[
m
]
δ
[
k
−
m
]
.
(10.5)
m
=−∞
Equation (10.5) provides an alternative representation of an arbitrary DT func-
tion using a linear combination of time-shifted DT impulses. In Eq. (10.5),
variable
m
denotes the dummy variable for the summation that disappears as
the summation is computed. Recall that a similar representation exists for the
CT functions and is given by Eq. (3.24).
10.3 Impulse respo
nse of a system
In Section 10.1, a constant-coefficient difference equation is used to specify the
input-output characteristics of an LTID system. An alternative representation of
an LTID system is obtained by specifying its impulse response. In this section,
we will formally define the impulse response and illustrate how the impulse
response of an LTID system can be derived directly from the difference equation
modeling the LTID system.
Definition 10.1
The impulse response h
[
k
]
of an LTID system is the output of
the system when a unit impulse
δ
[
k
]
is applied at the input of the LTID system.
Following the notation introduced in Eq. (2.1b), the impulse response can be
expressed as follows:
δ
[
k
]
→
h
[
k
]
,
(10.6)
with zero ancillary conditions.
Note that an LTID system satisfies the linearity and the time-shifting properties.
Therefore, if the input is a scaled and time-shifted impulse function
a
δ
[
k
−
k
0
],
the output, Eq. (10.6), of the DT system is also scaled by a factor of
a
and
time-shifted by
k
0
, i.e.
a
δ
[
k
−
k
0
]
→
ah
[
k
−
k
0
]
,
(10.7)
for any arbitrary constants
a
and
k
0
. Section 10.4 illustrates how Eq. (10.7) can
be generalized to calculate the output of LTID systems for any arbitrary input.
Example 10.3
Consider the LTID systems with the following input-output relationships:
(i)
y
[
k
]
=
x
[
k
−
1]
+
2
x
[
k
−
3];
(10
.
8)
(ii)
y
[
k
+
1]
−
0
.
4
y
[
k
]
=
x
[
k
]
.
(10
.
9)
Calculate the impulse responses for the two LTID systems. Also, determine
the output responses of the LTID systems when the input is given by
x
[
k
]
=
2
δ
[
k
]
+
3
δ
[
k
−
1].
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