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Cross-correlation of Two Moving Average Processes

This example shows how to find and plot the cross-correlation sequence between two moving average processes. The example compares the sample cross-correlation with the theoretical cross-correlation. Filter an N(0,1) white noise input with two different moving average filters. Plot the sample and theoretical cross-correlation sequences.

Create an N(0,1) white noise sequence. Set the random number generator to the default settings for reproducible results. Create two moving average filters. One filter has impulse response δ(n)+δ(n-1). The other filter has impulse response, δ(n)-δ(n-1).

rng default;
w = randn(100,1);
x = filter([1 1],1,w);
y = filter([1 -1],1,w);

Obtain the sample cross-correlation sequence up to lag 20. Plot the sample cross-correlation along with the theoretical cross-correlation.

[xc,lags] = xcorr(x,y,20,'biased');
stem(lags,xc,'markerfacecolor',[0 0 1]);
Xc = zeros(size(xc));
Xc(20) = -1;
Xc(22) = 1;
hold on;
stem(lags,Xc,'color',[1 0 0],'marker','none','linewidth',2);
q = legend('Sample Cross-correlation', ...
           'Theoretical Cross-correlation', ...
           'Location','NorthWest');
set(q,'FontSize',9,'Box','off')

The theoretical cross-correlation is -1 at lag -1, 1 at lag 1, and zero at all other lags. The sample cross-correlation sequence approximates the theoretical cross-correlation.

As expected, there is not perfect agreement between the theoretical cross-correlation and sample cross-correlation. The sample cross-correlation does accurately represent both the sign and magnitude of the theoretical cross-correlation sequence values at lag -1 and lag 1.

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