What can I add to this function to allow me to normalize the cross-correlation of 2 signals in the frequency domain?

I am in the process of refining the "xcorrFD" function to include the option to normalize the cross-correlation of 2 signals in the frequency domain, mirroring the process undertaken by MATLAB's built-in "xcorr" function. I've come across answers on how to do so in the time domain but am facing some difficulties integrating them into "xcorrFD". I'm hoping someone can determine what I need to add to the current content of the function to achieve this. For a more detailed description for how the function works, I've included the author's DL link just above what is written. Thank you for your help.

xcorrFD DL Link: https://www.mathworks.com/matlabcentral/fileexchange/63353-xcorrfd-x-y

function [lags,ck,td] = xcorrFD(x,y)

a = y'; b = x';

len = length(a);

c = [ zeros(1,len-1) a ];

d = [ b zeros(1,len-1) ];

% Compute FFTs

X1 = fft(c);

X2 = fft(d);

% Compute cross correlation

X = X1.*conj(X2);

ck = ifft((X));

[~,i] = max(ck);

td = i - len;

lags = [-(len-1):len-1];

Answer 1

[Sudarsanan A K]

Hello,

I understand that you are trying to modify the given implementation of "xcorrFD" function to include the normalization option in the frequency domain, and expecting the result obtained from MATLAB's built-in "xcorr" function.

The following example demonstrates how you can include the normalization (where the cross-correlations at zero lag are identically 1.0):

% Generate input signals
x = [1, 1, 1, 1];
y = [1, 1, 1, 1];

% Calculate the cross-correlation with normalization
[lags, ck_custom, td_custom] = xcorrFD(x, y, true); % see the function definition at the bottom of the script
% Calculate the cross-correlation using MATLAB's "xcorr" function with normalization
[ck_matlab, lags_matlab] = xcorr(x, y, 'normalized');

disp('Cross-correlation result:');

Cross-correlation result:

disp(ck_custom);

    0.2500
    0.5000
    0.7500
    1.0000
    0.7500
    0.5000
    0.2500

disp('Lags:');

Lags:

disp(lags);

    -3    -2    -1     0     1     2     3

% Plot the input signals and the cross-correlation result
figure
subplot(3,1,1);
stem(x);
xlabel('Sample index');
ylabel('Amplitude');
title('Signal x');

subplot(3,1,2);
stem(y);
xlabel('Sample index');
ylabel('Amplitude');
title('Signal y');

subplot(3,1,3);
stem(lags_matlab, ck_matlab, 'bo');
hold on
stem(lags, ck_custom, 'r*');
hold off
legend("xcorr - MATLAB built-in function", "xcorrFD - Custom function")
xlabel('Lag');
ylabel('Cross-correlation');
title('Cross-correlation of x and y');

%% xcorrFD function definition
function [lags, ck, td] = xcorrFD(x, y, normalize)
    a = y(:); % Ensure y is a column vector
    b = x(:); % Ensure x is a column vector
    len = length(a);
    c = [a; zeros(len-1, 1)]; % Zero-padding and flipping a
    d = [b; zeros(len-1, 1)]; % Zero-padding b
    % Compute FFTs
    X1 = fft(c);
    X2 = fft(d);
    % Compute cross correlation 
    X = X1.*conj(X2);
    ck = fftshift(ifft((X)));
    if normalize
        ck = ck / abs(ck(len)); % Normalize to ensure magnitude at zero lag is 1
    end
    [~, i] = max(ck);
    td = i - len;
    lags = -(len-1):len-1;
end

To know more about different scaling options in "xcorr" function, you can refer to the following MathWorks documentation:

https://mathworks.com/help/matlab/ref/xcorr.html#d126e1804395

I hope this helps you to achieve the normalization in the cross-correlation performed in the frequency domain.