%%More indepth discussions about cross-correlation function
close all
clear all
clc
format compact

SimpleSignals
pause
close (1)
close (2)
close (3)
close (4)
close (5)

% s1: boxcar, s2: spike, s3: triangle, s4: simple wavelet, s5:wavelet+noise

%%%%%
ns4=length(s4) %21
[cs4,tlag4]=xcov(s4,'unbiased');
n04=find(tlag4==0);
cs40=cs4(n04);
ns5=length(s5) %100
[cs5,tlag5]=xcov(s5,'unbiased');
n05=find(tlag5==0);
cs50=cs5(n05);

% cross-correlation between s5 and s4
s4c=zeros(1,length(s5));
s4c(1,1:ns4)=s4(1:ns4);
[cov_ec,tlag] = xcov(s5, s4c, 75, 'unbiased');
nop=find(tlag==0);
Rcov_ec=cov_ec./cov_ec(nop);

figure(1)
subplot(2,2,2);
plot(s5,'-b')
title('inputs')
subplot(2,2,4);
plot(s4c,'-b')
subplot(2,2,[1 3]);
plot(tlag,Rcov_ec, '-b')
title('output')

%%%The cross-covariance is the cross-correlation function of
  %  two sequences with their means removed:
  %       C(m) = E[(A(n+m)-MA)*conj(B(n)-MB)] 
  %  where MA and MB are the means of A and B respectively.
  
  %In this problem, A is the wavelet in noise, and B is the simple wavelet.
  % According to the formulation in the program, one would anticpate 
  % maximum cross correlation starting around m=50 to m=70 (about the
  % length of the wavelet) since the wavelet is embedded in the noise
  % about 50 lagged-time in A 'after' the simple wavelet is presented in
  % B. Also, through this way, one can identify the location of the wavelet
  % embedded in the noise. These are indeed observed.