What is the difference between normalized peak of correlation, versus peak of correlation divided by its average?

Question

Given a template, and a signal, the question arises as to how similar the signal is to the template.

Traditionally a simple correlation approach is used, whereby the template and a signal are cross-correlated, and then the entire result normalized by the product of both their norms. This gives a cross-correlation function that can range from -1 to 1, and the degree of similarity is given as the score of the peak therein.

• How does this compare to taking the value of that peak, and dividing by the mean or average of the cross-correlation function?
• What am I measuring here instead?

Attached is a diagram as my example.

In order to get the best measure of their similarity, I am wondering if I should look at:

1. Just the peak of the normalized cross-correlation as shown here?

2. Take peak but divide by the average of the cross-correlation plot?

3. My templates are going to be periodic square waves with some duty cycle as you can see - so should I not also somehow exploit the other two peaks we see here?

   • What would give the best measure of similarity in this case?

Thanks!

EDIT For Dilip:

I plotted the cross-correlation squared VS a cross-correlation that isnt squared, and it certainly does 'sharpen' the main peak over the others, but I am confused as to what calculation I should be using to determine similarity...

What I am trying to figure out is:

1. Can/Should I use the other secondary peaks in my calculations of similarity?

2. We have a squared cross-correlation plot now, and it certainly sharpens the main peak, but how does this help in determining final similarity?

Thanks again.

EDIT For Dilip:

The smaller peaks don't really help in similarity calculations; it is the main peak that matters. But the smaller peaks do provide support for the conjecture that the signal is a noisy version of the template. "

• Thanks Dilip, I am a little confused by that statement - if the smaller peaks do in fact provide support that the signal is a noisy version of the template, then doesnt that also aid in a measure of similarity?

What am I confused about is whether I should simply use the peak of the normalized cross-correlation function as my one and final measure of similarity and 'not care' about what the rest of the cross-corr function does/looks like, OR, should I take the peak value and some_other_metric of the cross-cor into account as well.

• If only the peak matters, then how/why would squaring the function help, since it just magnifies the main peak relative to the smaller ones? (More noise immunity?)

• Long and short: Should I care about the peak of the cross-correlation function only as my final measure of similarity, or should I also take the entire cross-correlation plot into account as well? (Hence my thought about looking at its mean).

Thanks again,

P.S. Time delay in this case is not a problem, in that, it is 'not cared' about for this application. P.P.S. I do not have control over the template.

Answer 1

To add some perspective, you could go back to the probabilistic interpretation of correlation. Remember that the correlation is used because it measures the degree of similarity under some linear generative model of the signal knowing the template (additive gaussian noise). The autocorrelation gives a measure of the log-Probability.

To go back to your question, there are a number of free parameters, notably the variance of noise, and this relates to your choice of normalisation.