Convolution Theorem

Fourier Transform

Here are the definitions we will use for the forward ($\mathcal{F}$) and inverse ($\mathcal{F}^{-1}$) Fourier transforms:

\begin{align} f(t)& = \mathcal{F}(\hat f(\omega )) = \frac1{2\pi }\int {\hat f(\omega )e^{i\omega t}d\omega }\\ \hat f(\omega )& = \mathcal{F}^{-1}(f(t)) = \int {f(t)e^{-i\omega t}dt} \end{align}\,\!

where $\omega \equiv 2\pi \nu$ 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:

\begin{align} \left[f*g\right](\tau )& \equiv \int {f(t)g(\tau -t)dt}\\ & = \frac1{(2\pi )^2}\int \! \! \! \int {\hat f(\omega _1)e^{i\omega _1 t}d\omega _1\, \hat g(\omega _2)e^{i\omega _2(\tau -t)}d\omega _2\, dt}\\ & = \frac1{(2\pi )^2}\int \! \! \! \int {\hat f(\omega _1)\hat g(\omega _2)e^{i(\omega _1-\omega _2) t} e^{i\omega _2\tau }d\omega _1\, d\omega _2\, dt}\\ & = \frac1{2\pi }\int {\hat f(\omega )\hat g(\omega ) e^{i\omega \tau }d\omega }. \end{align}\,\!

Renaming τ to be t (which we are totally free to do), we get a statement of the convolution theorem:

$f(t)*g(t) = \int {\hat f(\omega ) \hat g(\omega ) e^{i\omega t}d\omega } = \mathcal{F}^{-1}\! \! \left(\mathcal{F}(f)\cdot \mathcal{F}(g)\right). \,\!$

Convolution vs. Correlation

Correlation is very similar to convolution, and it is best defined through its equivalent “correlation theorem”:

$f(t)\star g(t) = \int {\hat f(\omega ) \hat g^*(\omega ) e^{i\omega t}d\omega } . \,\!$

The difference between correlation and convolution is that that when correlating two signals, the Fourier transform of the second function ($\hat g(\omega )$ in equation ) is conjugated before multiplying and integrating. Using that

\begin{align} g^*(-t)& =\frac1{2\pi }\int {\hat g^*(\omega )e^{-i\omega (-t)}d\omega }\\ & =\frac1{2\pi }\int {\hat g^*(\omega )e^{i\omega t}d\omega }, \end{align}\,\!

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):

$f(t)*g^*(-t) = f(t)\star g(t). \,\!$

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:

\begin{align} f(t-\tau )& =\frac1{2\pi }\int {\hat f(\omega )e^{i\omega (t-\tau )}d\omega }\\ & =\frac1{2\pi }\int {\hat f(\omega )e^{i\omega t}e^{-i\omega \tau }d\omega } , \end{align}\,\!

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:

\begin{align} f(t)\star f(t-\tau )& =\frac1{2\pi }\int {\hat f(\omega )\hat f^*(\omega )e^{-i\omega \tau }e^{i\omega t}d\omega }\\ & =\frac1{2\pi }\int {|f|^2e^{i\omega (t-\tau )}d\omega }\\ & =|f|^2\cdot \delta (t-\tau ). \end{align}\,\!