# discrete Fourier transform

## 1 Definition

Given a sequence $\{f_{n}\}_{n=0}^{N-1}$ of $N$ complex numbers  , the DFT is defined as a sequence $\{F_{k}\}_{k=0}^{N-1}$ of $N$ complex numbers according to the equation

 $F_{k}=\frac{1}{N^{(1-p)/2}}\sum_{n=0}^{N-1}f_{n}\,W_{N}^{-k,n}\quad\quad k=0,1% ,2,\dots,N-1$

where $W_{N}^{k,n}=e^{2\pi iqkn/N}$ are the basis functions, $p$ is an arbitrary real number and $q$ is either $1$ or $-1$.

Similarly, we can reconstruct $\{f_{n}\}_{n=0}^{N-1}$ from $\{F_{k}\}_{k=0}^{N-1}$ via the inverse discrete Fourier transform (IDFT):

 $f_{n}=\frac{1}{N^{(1+p)/2}}\sum_{k=0}^{N-1}F_{k}\,W_{N}^{k,n}\quad\quad n=0,1,% 2,\dots,N-1$

## 2 Explanation

Generically, if we have some vector $\mathbf{v}=(v_{1},v_{2},\dots,v_{n})$ that we wish to project on to a set of unit basis vectors ${\mathbf{\hat{B}}_{1},\mathbf{\hat{B}}_{2},\dots,\mathbf{\hat{B}}_{m}}$, we find that the $k$th component   of the projection $\mathbf{v}^{\prime}$ is given by

 $v_{k}^{\prime}=\langle\mathbf{v},\mathbf{\hat{B}}_{k}\rangle$

where $\langle\mathbf{u},\mathbf{v}\rangle$ is an inner product  . Using the standard dot product  of complex vectors, we have

 $v_{k}^{\prime}=\sum_{j=0}^{m}v_{j}\bar{B}_{k,j}$

(where $\bar{z}$ represents the complex conjugate  of $z$).

We may use a procedure to project a function onto basis functions. Here, the components of our vectors are the functions sampled at certain time intervals.

The idea of the Fourier transform  is to project our discrete function $\{f_{n}\}$ onto a basis made up of sinusoidal functions of varying frequency.

 $e^{ix}=\cos x+i\sin x$

so we can construct a sinusoidal function of any frequency $k$:

 $W_{N}^{k,n}=e^{2\pi iqkn/N}=\cos(2\pi iqkn/N)+i\sin(2\pi iqkn/N)$

By Nyquist’s theorem, any frequency components of $\{f_{n}\}$ above the Nyquist frequency $\frac{N}{2}$ will be aliased into lower frequencies, so we will only concern ourselves with the frequency components $\frac{-N}{2}\leq k\leq\frac{N}{2}$.

Now we substitute these results into the standard formula for projection to reveal that

 $F_{k}=\sum_{n=0}^{N-1}f_{n}\,W_{N}^{-k,n}\quad\quad k=0,1,2,\dots,N-1$

## 3 Group theoretic interpretation and generalization

This procedure may be interpreted in terms of group theory as follows: The basis functions $\chi_{k}(j)e^{-2\pi\nu_{k}ij/f}$ are the characters   of the irreducible representations of the group $\mathbb{Z}_{M}$. The orthogonality relation $\langle\chi_{m},\chi_{n}\rangle=\delta_{mn}$ which was used to obtain the transform as a projection is the orthogonality relation for characters.

Once we view the discrete Fourier transform in this fashion, it is not hard to generalize it by replacing $\mathbb{Z}_{M}$ with any discrete group $G$. As it turns out, there are two versions of the generalization which turn out to coincide exactly in the case of commutative groups (such as $\mathbb{Z}_{M}$).

First generalization: Let $f\colon G\to\mathbb{C}$ be conjugation  invariant, which means that, for every $y\in G$, it is the case that $f(yxy^{-1})=f(x)$. Then one has

 $f(x)=\sum_{k}\frac{1}{d_{k}}\langle f,\chi_{k}\rangle\chi_{k}(x)$
 $\langle\mathbf{u},\mathbf{v}\rangle=\sum_{g\in G}\mathbf{u}(g)\bar{\mathbf{v}}% (g)$

The justification for this comes from the fact that the set of irreducible characters spans the space of conjugation invariant functions and the orthogonality of these characters.

Second generalization: If $f\colon G\to\mathbb{C}$ is any function on a group (not necessarily conjugation invariant), then we have the transform

 $f(x)=\sum_{k}\sum_{i,j=1}^{d_{k}}\langle f,\rho^{k}_{mn}\rangle\rho^{k}_{mn}(x)$

where $k$ and $d_{k}$ are as before and $\rho^{k}_{mn}$ are the matrix elements of the representation $k$. The justification for this comes from the fact that the set of matrix elements of representations spans the space of functions on the group and the orthogonality relation for matrix elements.

References: Paul Bourke, http://astronomy.swin.edu.au/ pbourke/analysis/dft/”Discrete Fourier Transform”

 Title discrete Fourier transform Canonical name DiscreteFourierTransform Date of creation 2013-03-22 12:39:45 Last modified on 2013-03-22 12:39:45 Owner stitch (17269) Last modified by stitch (17269) Numerical id 25 Author stitch (17269) Entry type Definition Classification msc 65T50 Synonym DFT Related topic DiscreteSineTransform Related topic DiscreteCosineTransform Related topic FourierTransform Related topic LaplaceTransform Related topic TwoDimensionalFourierTransforms Related topic FourierStieltjesTransform Defines inverse discrete Fourier transform Defines IDFT