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Cross-Correlation of Two Exponential Sequences

Compute and plot the cross-correlation of two 16-sample exponential sequences, xa=0.84n and xb=0.92n, with n0.

N = 16;
n = 0:N-1;

a = 0.84;
b = 0.92;

xa = a.^n;
xb = b.^n;

r = xcorr(xa,xb);

stem(-(N-1):(N-1),r)

Figure contains an axes object. The axes object contains an object of type stem.

Determine c analytically to check the correctness of the result. Use a larger sample rate to simulate a continuous situation. The cross-correlation function of the sequences xa(n)=an and xb(n)=bn for n0, with 0<a,b<1, is

cab(n)=1-(ab)N-|n|1-ab×{an,n>0, 1,n=0, b-n,n<0.

fs = 10;
nn = -(N-1):1/fs:(N-1);

cn = (1 - (a*b).^(N-abs(nn)))/(1 - a*b) .* ...
     (a.^nn.*(nn>0) + (nn==0) + b.^-(nn).*(nn<0));

Plot the sequences on the same figure.

hold on
plot(nn,cn)

xlabel('Lag')
legend('xcorr','Analytic')

Figure contains an axes object. The axes object contains 2 objects of type stem, line. These objects represent xcorr, Analytic.

Verify that switching the order of the operands reverses the sequence.

figure

stem(-(N-1):(N-1),xcorr(xb,xa))

hold on
stem(-(N-1):(N-1),fliplr(r),'--*')

xlabel('Lag')
legend('xcorr(x_b,x_a)','fliplr(xcorr(x_a,x_b))')

Figure contains an axes object. The axes object contains 2 objects of type stem. These objects represent xcorr(x_b,x_a), fliplr(xcorr(x_a,x_b)).

Generate the 20-sample exponential sequence xc=0.77n. Compute and plot its cross-correlations with xa and xb. Output the lags to make the plotting easier. xcorr appends zeros at the end of the shorter sequence to match the length of the longer one.

xc = 0.77.^(0:20-1);

[xca,la] = xcorr(xa,xc);
[xcb,lb] = xcorr(xb,xc);

figure

subplot(2,1,1)
stem(la,xca)
subplot(2,1,2)
stem(lb,xcb)
xlabel('Lag')

Figure contains 2 axes objects. Axes object 1 contains an object of type stem. Axes object 2 contains an object of type stem.

See Also

Functions