% This file shows how signals buried in noise can be detected using
% correlation (assuming the noise is random and uncorrelated).
%
% Here we use two noisy signals (20 Hz sinusoid with noise),
% sampled at t_s = 0.01 seconds for 2 seconds.
%
% Taken from Digital Signal Processing: A Modern Approach by Ashok Ambardar,
% Thompson Publishing, ISBN 0-534-40509-6, Copyright 2007
%
ts = 0.01;t = 0:ts:2;N1 = length(t);            % Set up time array
x = sin(2*20*pi*t);                             % Signal x
z1 = 2*randn(size(x));z2 = 2*randn(size(x));    % Random noise
x1 = x + z1;x2 = x + z2;                        % Noisy signals
cc = conv(x1,fliplr(x2));                       % cross-correlation of x1 and x2
cn = conv(z1,fliplr(z2));                       % cross-correlation of noise
N = length(cn);                                 % Length of corr
tc = -(N1-1):N1-1;f=(0:N-1)/N/ts;               % corr and FFT axis values
ccspec = abs(fft(cc))/N;                        % FFT of cross-correlation
cnspec = abs(fft(cn))/N;                        % FFT of noise
subplot(2,2,1),plot(tc,cc);ax=axis;             % Cross-correlation cc
subplot(2,2,2),plot(tc,cn);axis(ax);            % Noise correlation cn
subplot(2,2,3),plot(f,ccspec);ay=axis;          % spectrum of cc
subplot(2,2,4),plot(f,cnspec);axis(ay);         % spectrum of cn