Convolution Theorem

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DSP for Radio Astronomy

1 Fourier Transform

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.

2 Convolution Theorem

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:

2.1 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: