Let
be a periodic sequence, then the autocorrelation
of the sequence, sometimes called the periodic autocorrelation (Zwillinger 1995,
p. 223), is the sequence
(1)
|
where
denotes the complex conjugate and the final
subscript is understood to be taken modulo
.
Similarly, for a periodic array with
and
, the autocorrelation is the
-dimensional matrix given by
(2)
|
where the final subscripts are understood to be taken modulo and
, respectively.
For a complex function , the autocorrelation is defined by
(3)
| |||
(4)
|
where
denotes cross-correlation and
is the complex conjugate
(Bracewell 1965, pp. 40-41).
Note that the notation is sometimes used for
and that the quantity
(5)
|
is sometimes also known as the autocorrelation of a continuous real function (Papoulis 1962, p. 241).
The autocorrelation discards phase information, returning only the power, and is therefore an irreversible operation.
There is also a somewhat surprising and extremely important relationship between the autocorrelation and the Fourier transform
known as the Wiener-Khinchin theorem.
Let ,
and
denote the complex conjugate of
, then the Fourier transform
of the absolute square of
is given by
(6)
|
is maximum
at the origin; in other words,
(7)
|
To see this, let
be a real number. Then
(8)
|
(9)
|
(10)
|
Define
(11)
| |||
(12)
|
Then plugging into above, we have . This quadratic
equation does not have any real root,
so
,
i.e.,
.
It follows that
(13)
|
with the equality at .
This proves that
is maximum at the origin.