Digital Signal Processing Reference
In-Depth Information
From (12.35), we see that the term in brackets can be evaluated as
N- 1
N,
n - l - m = 0
e
j2pk(n- l -m)/N
b
=
a
0,
otherwise
k= 0
and, therefore,
N- 1
N- 1
1
N
a
y[n] =
x[l]
a
h[m]Nd[n - l - m].
l = 0
m= 0
Because the impulse function is zero except when
m = n - l,
we can rewrite the
equation as
N- 1
y[n] =
a
x[l]h[n - l],
(12.45)
l = 0
which is clearly related to the equation for linear convolution (10.16). However, the
summation is over only one period rather than for all time. This equation represents
the process called
periodic convolution,
or
circular convolution
. In this topic, we
usually use the latter title.
The symbol
*
is used to signify the operation of circular convolution as de-
scribed by (12.45):
N- 1
y[n] = x[n]
*
h[n] =
a
x[l]h[n - l].
(12.46)
l = 0
In this development, we have established the discrete Fourier transform pair:
x[n]
*
h[n]
Î
"
X[k]H[k].
(12.47)
The process of (12.46) is called circular convolution because it is easily (if some-
times tediously) evaluated by using two concentric circles, as shown in Figure 12.22.
We can evaluate the circular convolution of (12.46) by writing the
N
values of
x[n]
,
x
[0]
x
[1]
x
[7]
h
[0]
h
[7]
h
[1]
x
[2]
h
[6]
h
[2]
x
[6]
h
[5]
h
[3]
h
[4]
x
[3]
x
[5]
x
[4]
Figure 12.22
Circular convolution.
