Normalizing correlation coefficients for sequences of unequal length without unnecessary zero padding

I have 2 questions:

1. Can some one explain why normalization{unbiased} of cross-corelation coefficents is carried out? I want to understand this at an intuitive level and not a mathematical one.

2. I have written my own code to carry out cross corelation between 2 sequences. The sequences are of unequal length. The code has been written in such a way that only the longer sequence has to be zero padded and that to only to the length of the smaller sequence-1.

How am I supposed to carry out normalization in this case?

Kindly help me figure this out.

Answer 1

If you are estimating the autocorrelation of a random process or cross correlation of two wide sense stationary processes then you are just obtaining the estimate of the true autocorrelation or cross-correlation. The normalizations used in xcorr() 1/N and 1/(N-abs(lag)) are just two ways of constructing the sample expectation (as an estimate of the true expectation, which you cannot compute).

Just like 1/N in computing the mean represents the expectation operator. 1/N in the sample mean results in an unbiased estimate of the mean, for the cross-correlation sequence, you have to take into account the lag to come up with an unbiased estimate. Note that in time-series analysis, the biased estimate is actually preferred in many cases.

If you have different length input vectors, then you should not scale the output.

Answer 2

Gentle bounce.

Answer 3

The issue that I just cannot wrap my head around id this:

Irrespective of whether I carry out biased normalization or unbiased normalization or no normalization whatsoever, the correlation coefficients still give me the same information.

As in, by looking at the correlation coefficients I can make out the nature of variability of the signals and the spectral bandwidth of the common signals.

So, how does normalization help at all?

If I scale to get a peak of 1 for autocorrelation, I get a warm fuzzy feeling when I get 0.76 for crosscorrelation at zero lag because I think I know how similar the functions are.

If I don't scale to get a peak of 1 for autocorrelation, I have to look at both autocorrelation curves to try to interpret the crosscorrelation curve.