# Documentation

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# dftmtx

Discrete Fourier transform matrix

## Syntax

```A = dftmtx(n) ```

## Description

A discrete Fourier transform matrix is a complex matrix of values around the unit circle whose matrix product with a vector computes the discrete Fourier transform of the vector.

`A = dftmtx(n)` returns the `n`-by-`n` complex matrix, `A`, that, when multiplied into a length-`n` column vector, `x`, computes the discrete Fourier transform of `x`. In other words, `y = A*x` is the same as `y = fft(x)`.

The inverse discrete Fourier transform matrix is

```Ai = conj(dftmtx(n))/n ```

## Examples

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In practice, it is more efficient to compute the discrete Fourier transform with the FFT than with the DFT matrix. The FFT also uses less memory. The two procedures give the same result.

```x = 1:256; y1 = fft(x); n = length(x); y2 = x*dftmtx(n); norm(y1-y2)```
```ans = 8.0399e-12 ```

## Algorithms

`dftmtx` takes the FFT of the identity matrix to generate the transform matrix.