The cross-correlation of two complex functions 
and 
of a real variable 
, denoted 
is defined by
 |
(1)
|
where 
denotes convolution and 
is the complex conjugate
of 
.
Since convolution is defined by
 |
(2)
|
it follows that
=int_(-infty)^inftyf^_(-tau)g(t-tau)dtau.](/images/equations/Cross-Correlation/NumberedEquation3.svg) |
(3)
|
Letting 
,
,
so (3) is equivalent to
 |  |  |
(4)
|
 |  |  |
(5)
|
The cross-correlation satisfies the identity
 |
(6)
|
If 
or 
is even, then
 |
(7)
|
where 
again denotes convolution.
