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xcorr
Cross-correlation function estimate.
c = xcorr(x,y) c = xcorr(x) c = xcorr(x,y,'option') c = xcorr(x,y,maxlags) c = xcorr(x,y,maxlags,'option') [c,lags] = xcorr(x,y) [c,lags] = xcorr(x,y,maxlags) [c,lags] = xcorr(x,y,maxlags,'option')
xcorr estimates the cross-correlation sequence of random processes. Autocorrelation is handled as a special case.
The true cross-correlation sequence is
where xn and yn are stationary random processes, 
, and E {} is the expected value operator. xcorr must estimate the sequence because, in practice, access is available to only a finite segment of the infinite-length random process.
c = xcorr(x,y)
returns the cross-correlation sequence in a length 2N-1 vector, where x and y are length N vectors.
c = xcorr(x)
is the autocorrelation sequence for the vector x. Where x is an N-by-P matrix, c = xcorr(X) returns a matrix with 2N-1 rows whose P2 columns contain the cross-correlation sequences for all combinations of the columns of X.
By default, xcorr computes raw correlations with no normalization. For a length N vector:
The output vector c has elements given by c(m) = cxy(m-N), m=1,...,2N-1.
The correlation function requires normalization to estimate the function properly.
c = xcorr(x,y,'option')
specifies a scaling option, where 'option' is
biased, for biased estimates of the cross-correlation function:
unbiased, for unbiased estimates of the cross-correlation function:
coeff, to normalize the sequence so the autocorrelations at zero lag are identically 1.0
none, to use the raw, unscaled cross-correlations (default)
[c,lags] = xcorr(x,y)
where x and y are length N vectors, returns the cross-correlation sequence in a length 2N-1 vector and the output lags in the vector [-N+1:N-1]. That is, the maximum lag is N-1.
[c,lags] = xcorr(x,y,maxlags)
where x and y are length N vectors, returns the cross-correlation sequence in a length 2*maxlags+1 vector c. lags is a vector of the lag indices where c was estimated, that is, [-maxlags:maxlags].
[c,lags] = xcorr(x,maxlags)
returns the autocorrelation sequence over the lag range [-maxlags:maxlags].
[c,lags] = xcorr(X,maxlags)
where x is an M-by-P matrix, is a matrix with 2*maxlags+1 rows whose P2 columns contain the cross-correlation sequences for all combinations of the columns of X.
[c,lags] = xcorr(x,maxlags,'option')
and
[c,lags] = xcorr(x,y,maxlags,'option')
specifies both a maximum number of lags and a scaling option.
In all cases, xcorr gives an output such that the zeroth lag of the correlation vector is in the middle of the sequence, at element or row maxlags+1 or at N.
The second output lags is useful when plotting. For example, the estimated autocorrelation of zero-mean Gaussian white noise cww(m) can be displayed for -10 
m 10 using
ww = randn(1000,1); [c_ww,lags] = xcorr(ww,10,'coeff'); stem(lags,c_ww)Swapping the
x and y input arguments reverses (and conjugates) the output correlation sequence. For row vectors, the resulting sequences are reversed left to right; for column vectors, up and down. The following example illustrates this property (mat2str is used for a compact display of complex numbers):
x = [1,2i,3]; y = [4,5,6];
[c1,lags] = xcorr(x,y);
c1 = mat2str(c1,2), lags
c1 =
[12+i*4.2e-16 15-i*8 22-i*10 5-i*12 6-i*1.4e-15]
lags =
-2 -1 0 1 2
c2 = conj(fliplr(xcorr(y,x)));
c2 = mat2str(c2,2)
c2 =
[12+i*4.2e-16 15-i*8 22-i*10 5-i*12 6-i*1.4e-15]
For the case where input argument x is a matrix, the output columns are arranged so that extracting a row and rearranging it into a square array produces the cross-correlation matrix corresponding to the lag of the chosen row. For example, the cross-correlation at zero lag can be retrieved by
randn('seed',0)
X = randn(2,2);
[M,P] = size(X);
c = xcorr(X);
c0 = zeros(P); c0(:) = c(M,:) % Extract zero-lag row
c0 =
1.7500 0.3079
0.3079 0.1293
You can calculate the matrix of correlation coefficients that the MATLAB function corrcoef generates by substituting
c = xcov(X,'coef')in the last example. The function
xcov subtracts the mean and then calls xcorr.
Use fftshift to move the second half of the sequence starting at the zeroth lag to the front of the sequence. fftshift swaps the first and second halves of a sequence.
For more information on estimating covariance and correlation functions, see [1] and [2].
There must be at least one vector input argument; otherwise, xcorr gives the following error message:
1st arg must be a vector or matrix.The string
'option' must be the last argument; otherwise, xcorr gives the following error message:
Argument list not in correct order.If the second argument was entered as a scalar, it is taken to be
maxlag and no succeeding input can be a scalar. When the second argument is a vector, the first must also be a signal vector. The third argument, when present, must be a scalar or a string. If they are not, xcorr gives the appropriate error message(s):
3rd arg is maxlag, 2nd arg cannot be scalar. When b is a vector, a must be a vector. Maxlag must be a scalar.Normally the lengths of the vector inputs should be the same; if they are not, then the only allowable scaling option is
'none'. If it is not, xcorr gives the following error message:
OPTION must be 'none' for different length vectors A and B.
Cross-covariance function estimate (equal to mean-removed cross-correlation). |