The cross-correlation sequence for two wide-sense stationary
random process, *x*(*n*) and *y*(*n*) is

$${R}_{xy}(m)=E\{x(n+m){y}^{*}(n)\},$$

where the asterisk denotes the complex conjugate and the expectation is over the ensemble of realizations that constitute the random processes.

Note that cross-correlation is not commutative, but a Hermitian (conjugate) symmetry property holds such that:

$${R}_{xy}(m)={R}_{yx}^{*}(-m).$$

The cross-covariance between *x*(*n*) and *y*(*n*) is:

$${C}_{xy}(m)=E\{(x(n+m)-{\mu}_{x})\text{\hspace{0.17em}}{(y(n)-{\mu}_{y})}^{*}\}={R}_{xy}(m)-{\mu}_{x}{\mu}_{y}{}^{*}.$$

For zero-mean wide-sense stationary random processes, the cross-correlation and cross-covariance are equivalent.

In practice, you must estimate these sequences, because it is
possible to access only a finite segment of the infinite-length random
processes. Further, it is often necessary to estimate ensemble moments
based on time averages because only a single realization of the random
processes are available. A common estimate based on *N* samples
of *x*(*n*) and *y*(*n*) is
the deterministic cross-correlation sequence (also called the time-ambiguity
function)

$${\widehat{R}}_{xy}(m)=\{\begin{array}{ll}{\displaystyle \sum _{n=0}^{N-m-1}x}(n+m){y}^{*}(n),\hfill & m\ge 0,\hfill \\ {\widehat{R}}_{yx}^{*}(-m),\hfill & m<0.\hfill \end{array}$$

where we assume for this discussion that *x*(*n*) and *y*(*n*) are
indexed from 0 to *N* – 1, and $${\widehat{R}}_{xy}(m)$$ from –(*N* – 1) to *N* – 1.

The functions `xcorr`

and `xcov`

estimate the cross-correlation and
cross-covariance sequences of random processes. They also handle autocorrelation
and autocovariance as special cases. The `xcorr`

function
evaluates the sum shown above with an efficient FFT-based algorithm,
given inputs *x*(*n*) and *y*(*n*) stored
in length *N* vectors `x`

and `y`

. Its operation is equivalent to convolution with one of the two subsequences reversed
in time.

For example:

x = [1 1 1 1 1]'; y = x; xyc = xcorr(x,y)

Notice that the resulting sequence length is one less than twice
the length of the input sequence. Thus, the *N*th
element is the correlation at lag 0. Also notice
the triangular pulse of the output that results when convolving two
square pulses.

The `xcov`

function estimates autocovariance
and cross-covariance sequences. This function has the same options
and evaluates the same sum as `xcorr`

, but first
removes the means of `x`

and `y`

.

An estimate of a quantity is *biased* if its expected value is
not equal to the quantity it estimates. The expected value of the
output of `xcorr`

is

$$E\{{\widehat{R}}_{xy}(m)\}=(N-\left|m\right|){R}_{xy}(m).$$

`xcorr`

provides the unbiased estimate, dividing
by *N* – |*m*| when you specify
an `'unbiased'`

flag after the input sequences.

```
xcorr(x,y,'unbiased')
```

Although this estimate is unbiased, the end points (near
–(*N* – 1)
and *N* – 1)
suffer from large variance because `xcorr`

computes
them using only a few data points. A possible trade-off is to simply
divide by *N* using the `'biased'`

flag:

```
xcorr(x,y,'biased')
```

With this scheme, only the sample of the correlation at zero
lag (the *N*th output element) is unbiased. This
estimate is often more desirable than the unbiased one because it
avoids random large variations at the end points of the correlation
sequence.

`xcorr`

provides one other normalization scheme.
The syntax

```
xcorr(x,y,'coeff')
```

divides the output by `norm(x)*norm(y)`

so
that, for autocorrelations, the sample at zero lag is 1.

For a multichannel signal, `xcorr`

and `xcov`

estimate
the autocorrelation and cross-correlation and covariance sequences
for all of the channels at once. If `S`

is an *M*-by-*N* signal
matrix representing *N* channels in its columns, `xcorr(S)`

returns
a (2*M* – 1)-by-*N*^{2} matrix
with the autocorrelations and cross-correlations of the channels of `S`

in
its *N*^{2} columns. If `S`

is
a three-channel signal

S = [s1 s2 s3]

then the result of `xcorr(S)`

is organized
as

R = [Rs1s1 Rs1s2 Rs1s3 Rs2s1 Rs2s2 Rs2s3 Rs3s1 Rs3s2 Rs3s3]

Two related functions, `cov`

and `corrcoef`

, are available in the standard MATLAB^{®} environment.
They estimate covariance and normalized covariance respectively between
the different channels at lag 0 and arrange them in a square matrix.

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