Conventions

In the following we details the conventions and notations used in this package, particulary as found in the Theory page.

Notation

We will denote vectors/lists in boldface and use \(\v{a}[n]\) to denote the \(n\)th element of \(\v{a}\), where the argument in square brackets is always interpreted modulo \(N\). Where helpful, we will use the notation \((f(n))_n\) to represent the vector \((f(1), f(2),\dots,f(N))\) where the length \(N\) is implicit and easily deduced from context, and the notation \(f(\v{x})\equiv(f(\v{x}[n]))_n.\)

We use the floor function, which takes a real number \(x\) and returns \(\lfloor x \rfloor\), the greatest integer that is less than or equal to \(x\).

Fourier transform conventions

We choose the ordinary-frequency convention for the continuous Fourier transform (CFT) and its inverse

\[ \begin{align}\label{eqn:FT} F(k) &= \hat{\mathcal{F}}_x[f(x)](k) = \left. \int_{-\infty}^\infty f(x)\exp(-i 2\pi \xi x) \d x \right|_{\xi = k}, \\ f(x) &= \hat{\mathcal{F}}^{-1}_k[F(k)](x) = \left. \int_{-\infty}^\infty F(k)\exp(i 2\pi k \xi) \d k \right|_{\xi = x}. \end{align} \]

The fast Fourier transform (FFT) is a fast, mathematically equivalent way of performing the discrete Fourier transform (DFT). Please be aware that we casually use these terms interchangeably. We use the usual definition of the FFT

\[ \begin{align} \v{A}[m] &= \Big(\text{fft}(\v{a})\Big)[m] = \sum_{n=0}^{N-1} \v{a}[n] \exp\left( -i 2\pi \frac{mn}{N} \right), \\ \v{a}[m] &= \Big(\text{ifft}(\v{A})\Big)[m] = \frac{1}{N} \sum_{n=0}^{N-1} \v{A}[n] \exp\left( +i2\pi \frac{mn}{N} \right). \end{align} \]