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Let’s assume that we have two audio signals, x(t) and y(t) affected by the noise as shown below. And we would like to cross-correlate these two signals and the cross-correlation plot is shown as below.

enter image description here

enter image description here

In this correlation plot there is a peak value around -11 msec. I am trying to understand how we interpret this peak in this plot? What does it mean? Please also explain me what do we obtain from the Fourier transform of the correlation function.

Thanks in advance!

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    $\begingroup$ This shows that $x(t)$ and $y(t)$ are very likely to be noisy versions of the same audio signal except that one of the signals (which one depends on what your definition of correlation is) is delayed by about 11 msec with respect to the other signal. That is, $x(t) = a(t) + n_1(t)$ while $y(t) = a(t-\tau) + n_2(t)$ for some $a(t)$. It also suggests that $a(t)$ is not a sustained musical note such as the A at 440 Hz that is sounded while the orchestra is tuning up before a performance. If it were, there would be multiple tall peaks and a broader spread of the crosscorrelation function. $\endgroup$ – Dilip Sarwate Jun 9 '14 at 12:51
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    $\begingroup$ @DilipSarwate Why not make that an answer? $\endgroup$ – Phonon Jun 10 '14 at 0:21
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I may answer things that you already know,but i will do my best anyway (i had similar problems once..). When you do the Correlation of the two sequences you simply shift the one relatively to the other (as you may already know) so each time you shift it you get lets say one "dot" for your plot over there,one single point.This points in a row construct the plot you are getting there.Now,in order to get each one of them you perform one operation,lets say better one calculation which is:the number of equal/same bits minus the number of non equal/same bits,after you perform that you shift again and calculate again....Now when it come to the point that two identical sequences are perfectly aligned,you can understand that the number of "non equal bits" will be zero (cause apparently will be matching perfectly and the result will be a massive number (compared to the rest) that will be YOUR CORRELATION PEAK,meaning that you have reached correlation.As i see there that is something similar to BOC for satellite coms that i have been using in the past,is that case that time you need to shift the waveform in order to reach to the peak is the time that you need to uses to calculate the distance in your GPS system! I hope that helps...

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    $\begingroup$ CDMA Systems Engineering Handbook has a detailed analysis for that in the gold codes section and not only,maybe you find it usefull. Unfortunatelly,i can not provide any info about the fourier analysis you are asking.... $\endgroup$ – Giwrgos Rizeakos Jun 10 '14 at 10:52
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Correlation across two signals is typically used to figure out the time shift between the signals. The correlation operation simply time-shifts the signal and calculates a value each time for each sample. When you see a peak this means the signal and the shift signal both align maximally. This is without mathematics.

Correlation mathematics multiplies a signal and the complex conjugate of another signal and this happens for each of the signal. Because of complex conjugation of the 2nd signal the final correlation value will produce a peak (-ve or +ve) when the product reach a maximum or minimum value.

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    $\begingroup$ Many thanks for all answers and sorry for my late reponse on this !!The explanations realy helped my understanding the issue. $\endgroup$ – tuner Apr 30 '15 at 13:41

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