The Convolution Theorem with Application Examples¶
The convolution theorem is a fundamental property of the Fourier transform. It is often stated like
"Convolution in time domain equals multiplication in frequency domain"
or vice versa
"Multiplication in time equals convolution in the frequency domain"
In this notebook we will illustrate what that means by pictorial examples. First, let us state the first version of the theorem mathematically. Let x(t) and y(t) be two arbitrary signals. Then, we have
\mathcal{F}\{x(t)*y(t)\}(f)=\mathcal{F}\{x(t)\}(f)\cdot\mathcal{F}\{y(t)\}(f),where \mathcal{F}\{x(t)\}(f) denotes the Fourier transform of x(t), evaluated at the frequency f. In other words, we can say:
"The spectrum of the convolution two signals equals the multiplication of the spectra of both signals"
Let us first recap convolution (a more detailed description is given in another article on convolution): Given two signals x(t) and y(t), their convolution is defined by z(t)=x(t)*y(t)=\int_{-\infty}^{\infty}x(\tau)y(t-\tau)d\tau.
Literally, we take one signal, mirror it in time and shift it in time domain. Then we multiply this signal with the other signal and calculate the integral of the overlapping part. Let us now calculate the convolution of two arbitrary signals and look at the result in time and frequency domain. Here, ...