Friday, October 20, 2017

Conceptually, how does a Fourier transform differ from an autocorrelation?


I realize the two are derived using different algorithms, and the units are different, but from a conceptual standpoint of the information they provide how do they differ?


I'm thinking here about the more or less specific (yet general) case of a "representative" block of samples from a time series, where the gross statistics of the data are assumed to vary slowly relative to the block size. (I gather that's roughly what's defined as a WSS process.)


Note that I've got an intuitive understanding of Fourier transform (at least the simple 1-dimensional real version) as a "spectrum". I haven't developed an intuitive concept for autocorrelation -- that's what I'm groping for.


Update: Rather than scatter things about, I'll put this here, since it's related to my desire to understand what an autocorrelation is doing...



The following (primitive) chart is of a brute-force autocorrelation of one of my signals. (The units are largely meaningless, and 256 bins are jammed into 150 print position.) The curious thing is the bifurcated tail. What would cause this? (It so happens that I see this twin tail at the peak of a snore -- otherwise the tail is kind of fuzzy, and the peak and slope are not nearly as pronounced.) Checking the numeric data shows that every other value is about 10x different from it's immediate neighbors.


I suppose it's some sort of artifact of the sampling, but it's not obvious to me what specifically that might be.


16893892.00 :           *                                                                                                                                           
12668632.00 : *
9500134.00 :
7124095.00 : *
5342317.50 :
4006173.25 : *
3004206.00 :
2252836.75 : *

1689389.25 : *
1266863.12 :
950013.38 : *
712409.50 : * *
534231.75 : * * **
400617.31 : * *
300420.59 : * * * * *
225283.67 : * *
168938.92 : * **
126686.32 : ** ** **

95001.34 : * * * * * *
71240.95 : * * * * * **
53423.18 : * * * *
40061.73 : * ** ** ** * **
30042.06 : * * * ** ***
22528.37 : * * *** ** ** *
16893.89 : * * * * * * * * **** ** **
12668.63 : * * * * ****** ****
9500.13 : * * * * * ** ****** ***** **
7124.10 : * * * * * *** ****** ***** ** *

5342.32 : ** * * * ** * *****
4006.17 : * ** ** ***
3004.21 : * * * ** *** *** *
2252.84 : * *** *** * *
1689.39 : * * **** ** *
1266.86 : * ** **** * *** **
950.01 : * * * * * * ** ** * *
712.41 : ** ** **** ***** ***** * * *
534.23 : * * * * *
400.62 : * *

300.42 :
225.28 : * * *
========= : ======================================================================================================================================================
: 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 4 4 4 4 4 4 4 4 4 4 4
: 0 0 1 2 2 3 4 4 5 5 6 7 7 8 9 9 0 0 1 2 2 3 4 4 5 5 6 7 7 8 9 9 0 0 1 2 2 3 4 4 5 5 6 7 7 8 9 9 0 0 1 2 2 3 4 4 5 5 6 7 7 8 9 9 0 0 1 2 2 3 4 4 5 5 6
: 3 9 5 1 8 4 0 6 3 9 5 1 8 4 0 6 3 9 5 1 8 4 0 6 3 9 5 1 8 4 0 6 3 9 5 1 8 4 0 6 3 9 5 1 8 4 0 6 3 9 5 1 8 4 0 6 3 9 5 1 8 4 0 6 3 9 5 1 8 4 0 6 3 9 5
: 1 4 6 9 1 4 6 9 1 4 6 9 1 4 6 9 1 4 6 9 1 4 6 9 1 4 6 9 1 4 6 9 1 4 6 9 1 4 6 9 1 4 6 9 1 4 6 9 1 4 6 9 1 4 6 9 1 4 6 9 1 4 6 9 1 4 6 9 1 4 6 9 1 4 6

Answer



Ok, in the context of one specific application: If you're trying to find the frequency of a waveform, you can calculate it similarly from the position of the peak in a Fourier transform or the peak of an autocorrelation. (And the autocorrelation can be calculated efficiently using the Fourier transform, so I don't know why everyone is naysaying "they're totally different and unrelated".)


The Fourier transform shows you all the individual frequency components of the signal. If the fundamental frequency happens to be the biggest, you can just pick it out of the spectrum and the position of that peak will be the fundamental frequency of the waveform. If the peaks are harmonically related, but missing the fundamental, there won't be a peak at the fundamental, and you'll have to use some special processing of the positions of the harmonics to find it.



The autocorrelation shows you periodicities of the entire waveform. The position of the peak will be the fundamental period of the waveform, which can easily be converted to frequency. If the fundamental is missing, the autocorrelation will still find it (GCD of the harmonics).


Note: If the waveform is not perfectly repetitive, the autocorrelation peak will be shifted and inaccurate. The sound of the human voice and bowed string instruments will work fine, for instance, while plucked string instruments will be slightly high, due to inharmonicity making the harmonics slightly high. On the other hand, the human-perceived pitch is slightly higher than the true fundamental, too, so one or the other method might be more appropriate, depending on what kind of signal you're processing and what you want to get out of it.


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