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Autocorrelation Function of Exponential Sequence

Compute the autocorrelation function of a 28-sample exponential sequence, x=0.95n for n0.

a = 0.95;

N = 28;
n = 0:N-1;
lags = -(N-1):(N-1);

x = a.^n;
c = xcorr(x);

Determine c analytically to check the correctness of the result. Use a larger sample rate to simulate a continuous situation. The autocorrelation function of the sequence x(n)=an for n0, with |a|<1, is

c(n)=1-a2(N-|n|)1-a2×a|n|.

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

dd = (1-a.^(2*(N-abs(nn))))/(1-a^2).*a.^abs(nn);

Plot the sequences on the same figure.

stem(lags,c);
hold on
plot(nn,dd)
xlabel('Lag')
legend('xcorr','Analytic')
hold off

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

Repeat the calculation, but now find an unbiased estimate of the autocorrelation. Verify that the unbiased estimate is given by cu(n)=c(n)/(N-|n|).

cu = xcorr(x,'unbiased');

du = dd./(N-abs(nn));

stem(lags,cu);
hold on
plot(nn,du)
xlabel('Lag')
legend('xcorr','Analytic')
hold off

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

Repeat the calculation, but now find a biased estimate of the autocorrelation. Verify that the biased estimate is given by cb(n)=c(n)/N.

cb = xcorr(x,'biased');

db = dd/N;

stem(lags,cb);
hold on
plot(nn,db)
xlabel('Lag')
legend('xcorr','Analytic')
hold off

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

Find an estimate of the autocorrelation whose value at zero lag is unity.

cz = xcorr(x,'coeff');

dz = dd/max(dd);

stem(lags,cz);
hold on
plot(nn,dz)
xlabel('Lag')
legend('xcorr','Analytic')
hold off

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

See Also

Functions