Wikipedia is great at obfuscating simple ideas in complex math. The "Efficient computation" explains the idea well, but it could be made even simpler. The amplitude squaring step drops the phase there.
The phase is dropped but isn't there more going on? Comparing the signal to itself at various lags? I don't see how just dropping the phase accomplishes that.
Not really. ACF is defined as a convolution of signal X with itself: XX. But FFT turns a convolution into a dot product: FFT[XX] = FFT[X]·FFT[X], or just |FFT[X]|². But what is this really? If X is a sum of A·cos(2πwt+φ) waves, then FFT[X] is a set of A·exp(iφ) complex numbers. What does |FFT[X]|² do? It turns those complex numbers into A². Inversing this FFT gives a sum of A²·cos(2πwt) waves, so in effect ACF has dropped the phases and squared amplitudes. This is also why ACF have this bright vertical line - this is cos(x) functions piling up together.
hereforphone|4 years ago
gbh444g|4 years ago