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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.

user366312
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2 Answers2

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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 $$ \sum_{\chi\in \hat{G}}\chi(x)=\begin{cases}|G|,&\ \text{if $x$ is the neutral element, and}\\0,&\ \text{otherwise.}\end{cases} $$

Jyrki Lahtonen
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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

reuns
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