Here are the definitions we will use for the forward () and inverse () Fourier transforms:
where is the angular frequency coordinate that is the Fourier complement of time , and a top-hat is generally used to denote Fourier-domain quantities.
The convolution is a useful operation with applications ranging from photo editing (blurring) to crystallography to astronomy.
Renaming to be (which we are totally free to do), we get a statement of the convolution theorem:
Convolution vs. Correlation
Correlation is very similar to convolution, and it is best defined through its equivalent “correlation theorem”:
The difference between correlation and convolution is that that when correlating two signals, the Fourier transform of the second function ( in equation None) is conjugated before multiplying and integrating. Using that
we can show that correlating and is equivalent to convolving with a conjugated, time-reversed version of :
Although this relation between convolution and correlation is often mentioned in the literature, I don’t personally find it very intuitively illuminating. I much prefer the “correlation theorem” in equation (None), because when it is combined with the expression of a time-shifted signal in Fourier domain:
it shows that correlating a flat-spectrum signal with a time-shifted version of itself yields a measure of the power of the signal at the delay corresponding to the time shift: