xcorr
Cross-correlation
Syntax
c = xcorr(x,y) c = xcorr(x)
c = xcorr(x,y,'option') c = xcorr(x,'option') c = xcorr(x,y,maxlags) c = xcorr(x,maxlags)
c = xcorr(x,y,maxlags,'option') c = xcorr(x,maxlags,'option') [c,lags] = xcorr(...)
Description
xcorr estimates the cross-correlation sequence of a random process. Autocorrelation is handled as a special case. The true cross-correlation sequence is
where xn and yn are jointly stationary random processes, , and E {·} is the expected value operator. xcorr must estimate the sequence because, in practice, only a finite segment of one realization of the infinite-length random process is available.
c = xcorr(x,y) returns the cross-correlation sequence in a length 2*N-1 vector, where x and y are length N vectors (N>1). If x and y are not the same length, the shorter vector is zero-padded to the length of the longer vector.
Note The maximum allowable vector length for inputs to xcorr is 2^20. If you need to process longer sequences, see dfilt.fftfir. By default, xcorr computes raw correlations with no normalization.
The output vector c has elements given by c(m) = Rxy(m-N), m=1, ..., 2N-1. In general, the correlation function requires normalization to produce an accurate estimate (see below).
c = xcorr(x) is the autocorrelation sequence for the vector x. If x is an N-by-P matrix, c is a matrix with 2N-1 rows whose P2 columns contain the cross-correlation sequences for all combinations of the columns of x. For more information on matrix processing with xcorr, see Multiple Channels.
xcorr produces correlations identically equal to 1.0 at zero lag only when you perform an autocorrelation and only when you set the 'coeff' option. For example,
x=0:0.01:10; X = sin(x);
[r,lags]=xcorr(X,'coeff'); max(r)
ans = 1
c = xcorr(x,y,'option') specifies a normalization option for the cross-correlation, where 'option' is
?
'biased': Biased estimate of the cross-correlation function
?
'unbiased': Unbiased estimate of the cross-correlation function
'coeff': Normalizes the sequence so the autocorrelations at zero lag are identically 1.0.
? 'none', to use the raw, unscaled cross-correlations (default)
?
See [1] for more information on the properties of biased and unbiased correlation estimates.
c = xcorr(x,'option') specifies one of the above normalization options for the autocorrelation.
c = xcorr(x,y,maxlags) returns the cross-correlation sequence over the lag range [-maxlags:maxlags]. Output c has length 2*maxlags+1. c = xcorr(x,maxlags) returns the autocorrelation sequence over the lag range [-maxlags:maxlags]. Output c has length 2*maxlags+1. If x is an N-by-P matrix, c is a matrix with 2*maxlags+1 rows whose P2 columns contain the autocorrelation sequences for all combinations of the columns of x. c = xcorr(x,y,maxlags,'option') specifies both a maximum number of lags and a scaling option for the cross-correlation.
c = xcorr(x,maxlags,'option') specifies both a maximum number of lags and a scaling option for the autocorrelation.
[c,lags] = xcorr(...) returns a vector of the lag indices at which c was estimated, with the range [-maxlags:maxlags]. When maxlags is not specified, the range of lags is [-N+1:N-1].
In all cases, the cross-correlation or autocorrelation computed by xcorr has the zeroth lag in the middle of the sequence, at element or row maxlags+1 (element or row N if maxlags is not specified).
Examples
The second output, lags, is useful for plotting the cross-correlation or autocorrelation. 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):