Y axis Component of Rxy in MATLAB

Question

I calculated the $R_{xy}$ by cross-correlating $x(n)$ and $y(n)$.

Now, in the MATLAB command window, I wish to construct the Y-axis component of $R_{xy}(n)$. I'm not sure how I'm going to do it. Is there a built-in function that I can use?

Answer 1

How are you?

So, if I understood appropriately, you computed the cross-correlation between $x(n)$ and $y(n)$ and you're trying to build the Y-axis for your plot. I'm assuming that you have mistook the X-axis for Y-axis since the cross-correlation $R_{xy}(n)$ is the Y-axis. In order words, you already have your Y-axis, it is $R_{xy}(n)$.

In order to define a X-axis for a cross-correlation between 2 signals, it is possible to initialize a vector with a length that is a sum of both $x(t)$ and $y(t)$ lengths. Similar with how convolution algorithm does. Bear in mind that your $R_{xy}(n)$ must have been previously initialized as a vector with the same length as your X_axis. You may do so in MATLAB with the following code

% ...Vectors x and y previously defined

N = length(x);
L = length(y);
Rxy = zeros(1,N+L);
X_axis = 0 : 1 : N + L - 1;

% compute cross-correlation...

The vector above is your X-axis given in the number of samples. If you have a sampling rate $f_s$ defined, you may turn X_axis into seconds by multiplying your X_axis by a factor of $1/f_s$ also known as the sampling period $T_s$.

X_axis = Ts * ( 0 : 1 : N + L - 1 );

We may compute the cross-correlation between x and y by the following:

for i = 1 : 1 : N
    for j = 1 : 1 : L
        Rxy( i + j ) = Rxy( i + j ) + x(i)*y( ( L + 1 ) - j );
    end
end
% Scale result with total number of points in order... 
% ...to have the appropriate magnitude
Rxy = ( 1 / ( N * L ) ) .* Rxy; 

The algorithm above slides signal $y$ across time unflipped, performing a summation of products with $x$. Thus performing a cross-correlation.

Hope I was able to clarify your issue.

Cheers!