Digital Signal Processing Reference
In-Depth Information
x
[
n
]
x
1
[
n
]
3
2
1
3
2
1
•••
•••
•••
•••
3
2
1 0
(a)
1
2
3
n
3
2
1 0
(b)
1
2
3
n
x
0
[
n
]
x
1
[
n
]
3
2
1
3
2
1
•••
•••
•••
•••
3
2
1 0
(c)
1
2
3
n
3
2
1 0
(d)
1
2
3
n
Figure 10.1
Representation of a signal
with discrete-time impulse functions.
These three signals are also shown in Figure 10.1. The only nonzero values of
x
[
n
]
are contained in these three signals; hence, we can express the signal
x
[
n
] as
x[n] = x
-1
[n] + x
0
[n] + x
1
[n]
= x[-1]d[n + 1] + x[0]d[n] + x[1]d[n - 1]
(10.6)
1
=
a
x[k]d[n - k].
k=-1
Next, we generalize this development, using the term
x[k],
n = k
x[k]d[n - k] =
b
n Z k
.
(10.7)
0,
The summation of terms for all
k
yields the general signal
x
[
n
]:
q
x[n] =
a
x[k]d[n - k].
(10.8)
k=-
q
This relation is useful in the developments in the sections that follow.
The function is called either the unit sample function or the unit impulse
function. We use the term
impulse function
to emphasize the symmetry of the rela-
tions between discrete-time impulse functions and systems and continuous-time im-
pulse functions and systems.
d[n]
10.2
CONVOLUTION FOR DISCRETE-TIME SYSTEMS
An equation relating the output of a discrete LTI system to its input will now be de-
veloped. Consider the system shown in Figure 10.2. A unit impulse function
d[n]
is
