Digital Signal Processing Reference
In-Depth Information
Let
m =-n;
then
N- 1
x¿[m]e
-j2pk(-m)/N
5
x¿[-n]
6
=
a
= X¿[-k]
k= 0
and
5
R
xy
[p]
6
= X¿[-k]Y¿[k].
If the discrete-time sequence is real valued, as sequences consisting of sample
values of physical signals always are, then it can be shown that
x[n]
X¿[-k] = X¿
*
[k].
Therefore, the correlation function can be derived from
R
xy
[p] =
t
5
X¿[-k]Y¿[k]
6
*
[k]Y¿[k]
=
t
5
X¿
6
.
(12.50)
We must take note of the fact that
R
yx
Z R
xy
.
Reversing the order of the sig-
nals in the correlation equation gives
R
yx
[p] =
q
n=-
q
y[n]x[p - (-n)] = x[n]
*
y[-n].
(12.51)
If the calculation is by the DFT method, then
*
[k]
R
yx
[p] =
t
5
X¿[k]Y¿[-k]
6
=
t
5
X¿[k]Y¿
6
.
(12.52)
Cross-correlation with the DFT
EXAMPLE 12.23
The cross-correlation of the two discrete-time sequences given in Example 12.21 will be cal-
culated by the DFT method.
Zero padding is used to extend the two sequences so that
x¿
1
[n] = [1, 2, 3, 4, 0, 0, 0]
and x¿
2
[n] = [0, 1, 2, 3, 0, 0, 0].
We compute the DFT of each sequence, using an FFT algorithm to get
X¿
1
[k] = [10, -2.0245 - j6.2240, 0.3460 + j2.4791, 0.1784 - j2.4220,
0.1784 - j2.4220, 0.3460 - j2.4791, -2.0245 + j6.2240];
X¿
2
[k] = [6, -2.5245 - j4.0333, -0.1540 + j2.2383, -0.3216 - j1.7950,
-0.3216 + j1.7950, -0.1540 - j2.2383, -2.5245 + j4.0333].
