What should be the definition of DFT in plain English from mathematical point of view?

Question

What should be the definition of DFT in plain English from mathematical point of view?

Signal Processing definition

DFT is a technique which converts a discrete signal from time domain to frequency domain representation.

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Mathematical definition

???

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I can think of the following:

Discrete Fourier Transform is a technique of converting a sequence of N complex numbers into a new sequence of N complex numbers.

But, its incomplete and not satisfactory.

Answer 1

The discrete Fourier transform is a linear operator $\mathbb{C}^N \to \mathbb{C}^N$ given by $$X(k) = \frac{1}{\sqrt{N}}\sum_{n=0}^{N-1} x(n) e^{-2i \pi nk/N}$$ This operator is orthonormal which means its inverse is its adjoint :

$$x(n) = \frac{1}{\sqrt{N}}\sum_{k=0}^{N-1} X(k) e^{2i \pi nk/N}$$ we have written the signal $x(.)$ as a sum of (complex) sinusoids

Answer 2

Not really a definition. DFT is an umbrella concept that to an algebraist looks something like:

A discrete Fourier transform on a finite Abelian group $G$ is the process of representing a function $f:G\to\Bbb{C}$ as a linear combination of the group $\hat{G}$ of characters $\chi:G\to\Bbb{C}$. Because $\hat{\hat{G}}=G$ in a natural way, we always also have an inverse transform.

• Some authors may restrict the scope to cyclic groups $G=\Bbb{Z}_N$, when the characters are of the form $\chi(\overline{n})=e^{2\pi i nk/N}$. Here $k=0,1,\ldots,N-1$ parametrizes the characters, and $\overline{n}\in G$ is an arbitrary element.
• When $G$ is an elementary 2-abelian group, many authors call it a Hadamard transformation instead of a DFT. Or a Walsh-Hadamard transformation.
• Any finite abelian group is a direct product of cyclic groups, and in such cases some authors call the resulting transformation a $k$-dimensional DFT ($k$= the number of component groups). Walsh-Hadamard transform is a special case of this where all the component groups are cyclic of order two.
• It is possible to replace $\Bbb{C}$ with a suitable field $K$ of positive characteristic. To get a nice theory we need $N=|G|\cdot 1_K$ to be invertible in $K$. The resulting DFT is used in many algorithms handling encoding and error correction with cyclic codes.

In all the above cases the DFT/IFT pair works because we have the orthogonality relations (familiar to some of you from representation theory of finite groups): $$ \sum_{x\in G}\chi(x)=\begin{cases}|G|,&\ \text{if $\chi$ is the trivial character, and}\\0,&\ \text{otherwise.}\end{cases} $$ As well as the dual result $$ \sum_{\chi\in \hat{G}}\chi(x)=\begin{cases}|G|,&\ \text{if $x$ is the neutral element, and}\\0,&\ \text{otherwise.}\end{cases} $$