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Please help understanding the DSP usage of Autocorrelation & Cross-correlation

It seems this is strongly linked to calculating phase offsets, frequency offset for carrier recovery, symbol timing errors, recovery algorithms, what else?

What can I get from Cross-correlation? Started list below, please help

  • Biased version can give difference in constant phase offset between two signals, useful to rotate one set of symbols back into place.
  • Biased version can give The difference in power between the two which is useful if I want to get them to the same amplitude.

**What can I get from Auto-correlation? Started list below, please help **

  • I think that the magnitude or absolute of unbiased and unnormalised with no frequency offset or error can give me the bandwidth or Symbol rate of the signal on the Y axis where there is the maximum spike.
  • The frequency offset can be obtained from the imaginary part of the autocorrelation, how this is implemented to correct the offset in carrier recovery is still alluding me.
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    $\begingroup$ For the autocorrelation, this post gives some answers. $\endgroup$
    – Peter K. ♦
    Commented May 30, 2022 at 16:24
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    $\begingroup$ As you learn the details, keep the big picture in mind: correlation is a measure of the similarity between two signals. $\endgroup$
    – MBaz
    Commented May 30, 2022 at 16:30
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    $\begingroup$ I'd recommend moving your lists to answers, and making each answer a community wiki. $\endgroup$
    – Peter K. ♦
    Commented May 31, 2022 at 14:58

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In audio, I have used autocorrelation, or something like it, to accurately determine the period and fundamental frequency of a quasi-periodic audio signal; a musical note. Pitch detection

I have used cross-correlation to measure the difference, in time, that a single acoustic signal impinges upon two different microphones. From that inter-aural distance, we can compute an angular position of that single sound source is relative to the line connecting the two microphones. Localization

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In addition to the uses robert bristow-johnson (very well) provided, one can use cross-correlation to detect a (known) signal in noise. For more information on this use you can search for "matched-filter" approaches in Detection Theory. A personal preference of a related textbook is "Fundamentals of Statistical Signal Processing: Detection Theory" by Steven M. Kay, but of course you can search in your preferred sources.

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    $\begingroup$ Oh yeah. Matched filters. Also for binary or M-ary communication (which is really your first example). You have a constellation of signals all associated with a message containing some bits (the simplest is, of course, binary which is a constellation of two given signals, one for a 0 and another for 1. But you could have a constellation of 16 signals in a 4 $\times$ 4 grid or something like that. I think we call those "symbols" or similar. Given a noisy received signal you cross-correlate each symbol in the constellation and see which one the received signal resonates best with. $\endgroup$
    – robert bristow-johnson
    Commented Jun 26, 2022 at 18:09
  • $\begingroup$ Exactly the examples I had in mind, thanks for the extra info :). $\endgroup$
    – ZaellixA
    Commented Jun 27, 2022 at 9:04

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