Digital Signal Processing Reference
In-Depth Information
Do not confuse convolution with multiplication. The multiplication property
of the impulse function is given by
d[n]g[n - n
0
] = g[-n
0
]d[n]
and
d[n - n
0
]g[n] = g[n
0
]d[n - n
0
],
d[n], -
q
6 n 6
q
.
because is the only nonzero value of
From the convolution sum (10.16), we see that if
h
[
n
] is known, the system re-
sponse for any input
x
[
n
] can be calculated. Hence, the impulse response
h
[
n
] of a
discrete LTI system contains a complete
input-output description
of the system. We
now give two examples that illustrate the use of the convolution sum.
d[0]
A finite impulse response system
EXAMPLE 10.1
We consider the system depicted in Figure 10.3, in which the blocks labeled D are unit delays.
We can write the system difference equation directly from the figure:
y[n] = (x[n] + x[n - 1] + x[n - 2])
>
3.
(10.19)
This system averages the last three inputs. It is a moving-average filter, which has many ap-
plications. We find the impulse response
h
[
n
] for this system by applying the input
x[n] = d[n]:
y[n] = h[n] = (d[n] + d[n - 1] + d[n - 2])
>
3.
(10.20)
Thus,
h[0] = (d[n] + d[n - 1] + d[n - 2])
>
3 ƒ
n= 0
= (1 + 0 + 0)
>
3 = 1
>
3;
h[1] = (d[n] + d[n - 1] + d[n - 2])
>
3 ƒ
n= 1
= (0 + 1 + 0)
>
3 = 1
>
3;
h[2] = (d[n] + d[n - 1] + d[n - 2])
>
3 ƒ
n= 2
= (0 + 0 + 1)
>
3 = 1
>
3;
h[n] = 0, all other n.
This is a
finite impulse response
(FIR) system; that is, the impulse response contains a
finite number of nonzero terms. As an exercise, the reader should trace the signal
through the system in Figure 10.3 to verify
h
[
n
]. (Initially the numbers stored in
the two delays must be zero. Otherwise the output also includes an initial-condition response,
in addition to the impulse response, by superposition.)
x[n] = d[n]
x
[
n
]
y
[
n
]
1/3
x
[
n
1]
D
x
[
n
2]
D
Figure 10.3
Discrete system.
■
