A Fourier Transform Conventions

The generic Fourier transform of a function $f\ab(x)$ is defined as:

\[\begin{aligned} \tilde{f}\ab(\xi) &= \ab[\mathcal{F}\ab(f)]\ab(\xi) \\ &= \int_{\mathbb{R}} \dd{x} f\ab(x) e^{-2\pi i \xi x} \end{aligned}\]

and has the inverse transform:

\[\begin{aligned} f\ab(x) &= \ab[\mathcal{F}^{-1}\ab(\tilde{f})]\ab(x) \\ &= \int_{\mathbb{R}} \dd{\xi} \tilde{f}\ab(\xi) e^{2\pi i \xi x}. \end{aligned}\]

Unfortunately, there are many different conventions for what $\xi$ is leading to many different definitions of the Fourier transform. In these notes, we try to be self-consistent, and consistent with a broad community of physicists.

Let $x\mapsto x$ denote a spatial variable and let $\xi\mapsto k$ be the associated dual spatial frequency variable. The convention we use in these notes is that $k = 2\pi \xi$ such that a spatial domain signal $f\ab(x)$ has a Fourier transform $\tilde{f}\ab(k)$ with:

\[\begin{aligned} \tilde{f}\ab(k) = \int_{\mathbb{R}} \dd{x} f\ab(x) e^{-i k x}, \qquad f\ab(x) = \frac{1}{2\pi}\int_{\mathbb{R}} \dd{k} \tilde{f}\ab(k) e^{i k x}. \end{aligned}\]

The same convention is taken for the discrete Fourier transform (DFT) and its inverse (iDFT).

For the temporal transform, we use a different sign. The idea is that we want plane waves to have the form $e^{i(\vb*{k}\cdot\vb*{r} - \omega t)}$ in the time domain, which is the convention used in physics. This means that we let $x\mapsto t$ be our temporal variable and $\xi \mapsto \nu$ be the associated dual temporal frequency variable. We then further define the angular frequency $\omega = 2\pi \nu$ such that a time domain signal $f\ab(t)$ has a Fourier transform $\tilde{f}\ab(\omega)$ with:

\[\begin{aligned} \tilde{f}\ab(\omega) = \int_{\mathbb{R}} \dd{t} f\ab(t) e^{i \omega t}, \qquad f\ab(t) = \frac{1}{2\pi}\int_{\mathbb{R}} \dd{\omega} \tilde{f}\ab(\omega) e^{-i \omega t}. \end{aligned}\]

In the notes, we've dropped the tilde in the frequency domain functions, as we do not deal with their time domain counterparts in these notes for very long.