Autocorrelation

[Paul C]

In a plot of a cross-correlation function, lag (measured in samples) is on the x-axis. Is it correct that the y-axis is the output of correlation function for a given value of lag? If so, what are the units? Does a correlation function plot represent all values of the function given ‘tau’ for a single moment in time, t, of the original signal?

[Paul C]

Also, in choosing the peak in such a plot, why do we need to find exactly one? If the value of tau is greater than a single pitch period, could it not be that multiple peaks in the plot correspond to multiple moments of maximal self-similarity (i.e. multiple pitch periods aligning)?

[Simon]

Why do we need to find just one peak? Well, you are right to say the each peak corresponds to a similarity in the waveform with the shifted version of itself. But, perhaps your misunderstanding is that you are thinking of these as “moments” (in time). That’s not what they are. The horizontal axis in the autocorrelation plot is lag and not time (although the units of lag are the same as the units of time).

So, multiple peaks correspond to multiple lags, and therefore to multiple candidate values for the fundamental period (= 1/F0). Since F0 really does only have exactly one value (*) we do just want to find the one correct peak.

(*) although that’s not going to be true for creaky voice or any other type of irregular vocal fold vibration

[Simon]

Yes, you are correct that “the y-axis is the output of correlation function for a given value of lag”.

The units on the horizontal axis are simply time (measured in whatever units you would like to measure lag in, which is most commonly samples).

The units on the vertical axis are the square of the units on a waveform. However, we tend not to write units on the vertical axis of a waveform, because the scale is usually not calibrated. We simply label the axis with, say, amplitude. If we do put units, then they will be relative, such as the bits used to store each sample, or decibels relative to the maximum possible amplitude.

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