Convolution theorem

• categories: Signal processing, Theorem, Functional analysis

Statement

The convolution theorem states that the Convolution of two functions in the time domain is equivalent to the pointwise multiplication of their Fourier transform in the frequency domain, and vice versa. Formally:

1. Time Domain to Frequency Domain:
   For functions $f(t)$ and $g(t)$, the convolution $h(t)=(f\ast g)(t)$ satisfies:

   $$\mathcal{F}[f\ast g](\omega )=\mathcal{F}[f](\omega )\cdot \mathcal{F}[g](\omega ),$$

   where $\mathcal{F}[\cdot ]$ denotes the Fourier transform.

2. Frequency Domain to Time Domain:
   For functions $F(\omega )$ and $G(\omega )$, the convolution in the frequency domain satisfies:

   $${\mathcal{F}}^{-1}[F\cdot G](t)={\mathcal{F}}^{-1}[F](t)\ast {\mathcal{F}}^{-1}[G](t),$$

   where ${\mathcal{F}}^{-1}[\cdot ]$ is the inverse Fourier transform.