Convolution and Correlation – Some Case Studies - Digital Communication: Principles and System Modelling - page 226

between the signals, while zero correlation coefficient signifies that the two signals
are not at all similar. A negative number indicates a negative correlation, meaning
that one signal tends to decrease as the other increases.
Correlation between two different sequences is the cross-correlation and cor-
relation between two identical sequences is the auto-correlation. Correlation can
be viewed as convolution, by reversing one signal. Instead of considering signal
y[k] being indexed from 0 to some positive limit K, we will fold it around the
0th sample, and consider that it starts at − K. For example, if y[n] = [5,3,1,6],

= = = =

we interpret this as y[0] 5; y[1] 3; y[2] 1, and y[3] 6. If we fold this

− = − = − = =

signal around, we would have y[ 3] 6, y[ 2] 1, y[ 1] 3, and y[0] 5.

When we convolve y with x, we would sum x [ n ] × y [ n − k ]. Using the folded ver-

sion of y, the only thing to change is the index, e.g., x [ n ] × y [ n − ( − k ) ]orsimply

x [ n ] k ].

As the number of data samples affects the cross-correlation measure, so the cross-
correlation value must be divided by the number of data samples.

× y [ n +

N − 1

1
N

S xy [ k ] = x [ n ] y [ n + k ] (B.11)

n = 0

The function below finds the cross-correlation between two signals. The total
program is given in http://extras.springer.com .

correlate_apu.m
% Measure of similarity between two signals.
%
% Usage: [rho] = correlate_apu(k, x, y);
% inputs: k = shift (integer between 0 and length(y)-1)
%x,y = the signals to correlate
% output: rho = the correlation coefficient -1..0..1
%
% Note: This function assumes length(x) = length(y).
% Apurba Das, CDAC Kolkata

function [rho] = correlate_apu(k, x, orig_y)
% Shift y by k units
y = [orig_y(k+1:length(orig_y)), orig_y(1:k)];
N = length(x);
sxx = x ∗ x. - sum(x) ∗ sum(x)/N;
syy = y ∗ y. - sum(y) ∗ sum(y)/N;
sxy = x ∗ y. - sum(x) ∗ sum(y)/N;
rho = sxy / sqrt(sxx ∗ syy);