A signal’s autocorrelation is a cosine sum over its sinusoidal partials.

Evidence

Fourier analysis is a way of building up a waveform from much simpler bits. There are many ways of building up complex waveforms from simple basic shapes, like rectangles, but Fourier analysis was developed (by a French mathematician called Jean Baptiste Joseph Fourier in 1807) to solve the problem of heat diffusion in a metal bar! His idea was very simple. He built up the more complicated shapes of the functions, or waveforms, that he was studying by using the humble sine wave. He found that by adding together sine waves with different frequencies, phases and amplitudes, he could very accurately approximate the functions he was interested in. He found that using sine waves as a basis for approximation was a very useful thing to do. For a start, different sine wave frequencies were independent from or, more strictly, orthogonal to each other and so could be manipulated, or used in equations, independently of each other. — David M. Howard, Jamie Angus, Acoustics and Psychoacoustics, 440 (#expands, via linked claim)

Autocorrelation is a check for repetition or self-similarity. Does the signal, which might look quite unruly, tend in fact to repeat patterns at regular intervals? For example, if there is a large positive fluctuation in s(t) at some time t, does there tend to be another a time τ later? Or, is it anticorrelated at time τ: most often s(t + τ) has the opposite sign as does s(t). The product s(t)s(t + τ) would be most often negative and 〈s(r)s(t + τ)〉 would be negative for that value of the time delay τ. If s(t) and s(t + τ) are uncorrelated, so that s(t)s(t + τ) is as likely positive as negative, the average and thus c(τ) will be zero at time τ. — Eric J. Heller, Why You Hear What You Hear- An…, 369 (#expands, via linked claim)

With the preceding discussion as pretext, the so-called Wiener-Khinchin theorem will come as little surprise: it states that the power spectrum and the autocorrelation are obtainable from one another; specifically, the power spectrum is the Fourier cosine transform of the autocorrelation function. — Eric J. Heller, Why You Hear What You Hear- An…, 410 (#expands, via linked claim)

Categorization

• XREF: connects to: The power spectrum is the Fourier cosine transform of the autocorrelation — Eric J. Heller, Why You Hear What You Hear- An…, 410; Autocorrelation measures periodic self-similarity in a signal across time delays. — Eric J. Heller, Why You Hear What You Hear- An…, 369; Any waveform can be approximated by summing orthogonal sine waves — David M. Howard, Jamie Angus, Acoustics and Psychoacoustics, 440 (+3 more)

Broader Connections

• The power spectrum is the Fourier cosine transform of the autocorrelation #expands (0.80) — Eric J. Heller, Why You Hear What You Hear- An…, 410
• Autocorrelation measures periodic self-similarity in a signal across time delays. #expands (0.80) — Eric J. Heller, Why You Hear What You Hear- An…, 369
• Any waveform can be approximated by summing orthogonal sine waves #expands (0.80) — David M. Howard, Jamie Angus, Acoustics and Psychoacoustics, 440
• Frequency, phase, and impulse responses are interchangeable transforms of each other #expands (0.75)