# Convolution Theorem

### From AstroBaki

### [edit] Short Topical Videos

- The Convolution Theorem by Aaron Parsons
- Introduction to the Convolution by Khan Academy

### [edit] Reference Material

- Convolution Integral (fourier-series.com, Brent Locher)
- Digital Convolution (fourier-series.com, Brent Locher)

## Contents |

# Convolution Theorem

## 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 *t*, and a top-hat is generally used to denote Fourier-domain quantities.

## Convolution Theorem

The *convolution* is a useful operation with applications ranging from photo editing to crystallography to astronomy. In words, the convolution of two functions *f*,*g* is what you get when you smooth one function (*f*) by another (*g*). Note that the order of *f* and *g* does not matter, though people often call the latter the “kernel”. Smoothing *f* by *g* means that you slide *g* along *f*, and at each step along the way, you sum up all of the parts of *f* with weights drawn from the value of *g* at the point you slid it to. In essence, you are blurring *f* by *g*.

Mathematically, this is described as:

Renaming τ to be *t* (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 ) is conjugated before multiplying and integrating. Using that

we can show that correlating *f*(*t*) and *g*(*t*) is equivalent to convolving *f*(*t*) with a conjugated, time-reversed version of *g*(*t*):

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 (), 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: