Why does cross correlation need minimum number of 5000 samples for correct time delay estimation?

Question

1.wav

2.wav

I have two audio signals where x[n] is a farend signal and x1[n] is our nearend signal.

A part of the farend signal is coming into the nearend signal with a certain delay (echo.) I want to estimate that delay (echo delay estimation) for which I've used the 'cross correlation' xcorr function of MATLAB. When I take 5000 samples of both the nearend and farend signals and apply xcorr, it gives an accurate delay estimation in terms of index which in our case was 266.

Then I decreased the number of samples from 5000 to 4500 and used the same wav files for delay estimation using xcorr and it gave a delay of 167.

Why does the xcorr function of MATLAB only work accurately at 5000 or above samples? When we decrease the number of samples below 5000, why doesn't it give us an accurate delay estimation? Is there any particular reason behind this? Both of the audio wav files used, the graphs, and the MATLAB code are shared below.

Note:

Please consider the value of X in the graph for time delay.

Correct delay estimation code:

clear all
close all

[x, fs] = audioread('D:1.wav');
[x1, ~] = audioread('D:2.wav');

x = x(1:5000);
x1 = x1(1:5000);


[a,b] = xcorr(x,x1);
[~, index] = max(a);
delay = b(index);

Wrong delay estimation code:

clear all
close all

[x, fs] = audioread('D:1.wav');
[x1, ~] = audioread('D:2.wav');

x = x(1:4500);
x1 = x1(1:4500);


[a,b] = xcorr(x,x1);
[~, index] = max(a);
delay = b(index);

Answer 1

I had a look at the two .wav files.

The most likely cause of the "error" is that the file 1.wav has about 70 milliseconds (~3340 samples) of noise at the beginning.

Even 2.wav has a very long period of just noise at the beginning. It is just noise for about the first 30 milliseconds (about 1440 samples.)

With less than 5000 samples, you aren't comparing the two signals. You are just comparing the noise in the signals. 5000 samples gets you (just barely) enough of the signal for the correlation to be (approximately) correct.

Xcorr works correctly for your task, but you have to give it signals to compare rather than uncorrelated noise.

Answer 2

Two things that might help, though I can't be sure without those files.

You need to be finding the max of the absolute value of the cross-correlation because the phase of the correlation output in this context is irrelevant.

You may also benefit from using the 'normalized' scaleopt argument to xcorr considering that's the way MATLAB's documentation recommends estimating delays.

Edit: Your issue stems that when you are analyzing larger portions of the data, your correlation function changes because the two signals are not matching. I have a notebook to demonstrate the problem here: https://colab.research.google.com/drive/17eBFrUPt93NmgWvZCVDmkBwEsegfQAvw?usp=sharing

Go to File->"Open in playground mode" if you want to mess around with it.