PRACTICAL MATLAB® FOR ENGINEERS PRACTICAL MATLAB

534 Practical MATLAB® Applications for Engineers

Observe that the magnitude spectrum for f(n) and f(n − 32) are similar. The phase-error plot

shows variations that are proportional to the shift (32). Note that the error in time is large;

but in frequency, approximately 1000 Hz, the magnitude and phase errors are small.

Example 5.11

This example explores and verifi es some properties that relate the correlation with the

convolution process.

Let f 1 (n) = sin(2πn/16) and f 2 (n) = cos(2πn/16), over the range n = 0, 1, 2, 3, 4, ..., 16.

Create the script fi le conv_corr that returns the following:

a. fi gure(1): f 1 (n) versus n and f 2 (n) versus n, over the range 0 ≤ n ≤ 31 , using the stairs

command

b. fi gure(2), explore the conv and xcorr functions in time by obtaining and comparing

the following plots:

i. [corr = xcorr(f1n, f2n)] versus n

ii. [conv1 = conv(f1n, f2(−n))] versus n

iii. [conv2 = conv(f1(−n), f2n)] versus n

c. fi gure(3) verifi es that the maximum point occurs at the midrange, and this point

corresponds to R 11 (0) and R 22 (0), by obtaining the following plots:

i. [R 11 = aut_corr1 = xcorr(f1n, f1n)] versus n

ii. [R 1 = aut_xcorr(f1n)] versus n

iii. [R 22 = aut_corr2 = xcorr(f2n, f2n)] versus n

Observe that R 1 ≠ R 22 , but R 11 = R 22.

d. fi gure(4), normalized frequency analysis (verify that the two following plots are

identical)

i. abs[conj[fft(f1n)] * fft(f2n)] versus W

ii. abs[f ft(xcorr(f 1(n), f 2(n)))] versus W

e. fi gure(5), error analysis in time

i. [error1(n) = xcorr(f 1(n), f 2(n)) − conv(f1(n), f2(−n))] versus n and

ii. [error2(n) = abs(abs(xcorr(f1(n), f2(n))) − abs(conv(f1(−n), f 2(n))))] versus n

MATLAB Solution

% Script file: conv _ corr

n=0:1:31;

f1n=sin(2*pi*n./16);

f2n=cos(2*pi*n./16);

f1n _ rev=fliplr(f1n);

f2n _ rev=fliplr(f2n);

figure(1) % discrete time plots of f1(n) and f2(n)

subplot(2,1,1)

stairs(n,f1n), axis([0 33 -1.1 1.1]),

title(‘ Stair plot of f1(n) vs. n’);

ylabel(‘Amplitude ’);

subplot(2,1,2)

stairs(n,f2n); axis([0 33 -1.1 1.1]),

title(‘ Stair plot of f2(n) vs. n’);

xlabel(‘time index n’);ylabel(‘Amplitude’);