Periodic correlations for random sequences

This might be a very simple query, but I do just seem not to get over it. I have a random initialization of 2 x codes that I have been using for aperiodic correlations. Means I have random initialization of a nx2 matrix, each column of the matrix depicting a code as a vector of complex numbers. Now I want to use them for periodic correlations, say for 5 periods. However, when I repmat the codes and use xcorr, I get a sharp peak at central period, with monotonically reducing peaks at every period on either sides. The peak in centre is scaled by period with the remaining peaks reducing by a factor of 1. Does that mean that I need to scale down the xcorr by 1/N, N being the period? Also, I want to optimize the correlations of these sequences for a specific target; do i scale down the periodic correlations and then subject them to optimization?

Any guidance will be appreciated. Thanks

Answer 1

[M S Rashed]

So, yes, by replicating the sequences to as many cycles as we want to have, we will get periodic peaks of auto-correlations at every multiples of "N", where "N" is the number of cycles we have chosen for replicating the sequence or code set. %% n=100; %length of code K=2; % number of codes N=4; % Number of cycle ab = randn(n,K) + 1j*randn(n,K); % initialisation abc=repmat(ab, [N,1]); %Replicating the codes as many times as nu,mber of cycles "N" EE = xcorr(abc)/N; % find all possible auto and cross correlations %%

The highest peak will be at centre of the correlation length, i.e. at "N*n", where "N" is number of cycles, and "n" is length of the 2-code set we started with (the random initialization). And all other peaks away from this centre will be monotonically reducing in height as we move on each side towards the extremes of correlation length.

In order to normalize, we need to divide the correlations by "N", so that the central peak is normalized to "1", whereas all other peaks are at magnitude less than this.