How to compute the cross correlation of two fuctions

Question

I am trying to implement Matlab's cross correlation function but I can't seem to replicate the results obtained when using Matlab's built in function "xcorr".

Documentation for xcorr can be found HERE.

Looking at the Matlab documentation it seems as though the cross correlation is implemented as

$$ R_{xy}[m] = \left.\begin{cases} \sum_{n=0}^{N - m - 1} x[n + m] y[n] & m \geq 0\\ \\ R_{yx}[-m] & m < 0\\ \end{cases} \right\} $$

Or equivalently as

$$ R_{xy}[m] = \left.\begin{cases} \sum_{n=1}^{N - m} x[n + m] y[n] & m \geq 0\\ \\ \sum_{n=1}^{N + m} y[n - m] x[n] & m < 0\\ \end{cases} \right\} $$

if x and y have indexes starting at 1.The result will then be C, where

$$ C(m) = R_{xy}(m - N) \hspace{1cm} \text{for} \hspace{1cm} m = 1, 2,...,2N - 1. $$

The implementation of this formula in Matlab is shown below:

function [result] = myXCorr(A , B)
%Implementation of the Cross Coorelation Function

N = size(A,2);
M = size(B,2);
result = zeros(1, N + M - 1 );
len = size(result,2);

for m = 1 : len  
   arg = (m - N); 

   if(arg < 0)
       negativeCondition = 1;
       limit = N + arg;
   else
       negativeCondition = 0;
       limit = N - arg;
   end

   for n = 1:limit
           if(negativeCondition == 0)
            result(m) = result(m) + A(arg + n) + B(n);
           else
            result(m) = result(m) + A(n) + B(n - arg);
           end
   end
end

end

This implementation does not produce the same results as Matlab and I can't seem to find out what the error is.The plots of the autocorrelation's are shown below.

A = 1:1:10;
B = 1:1:10;

myResult = myXCorr(A,B);
mResult = xcorr(A,B);

lag = -9:1:9;

plot(lag, myResult);
plot(lag, mResult);

Autocorrelation using my function:

Autocorrelation using "xcorr":

Does anyone know what 'm doing wrong here? Any help will be appreciated.

Answer 1

In the loop where you compute your result vector, you should be multiplying the A and B values instead of adding them up.

Answer 2

for n = 1:limit
      if (negativeCondition==0)
        result(m) = result(m) + A(n) * B(n - arg);
      else
        result(m) = result(m) + A(arg + n) * B(n);
      end
end