dftmtx (Signal Processing Toolbox)

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.

```
y = A*x
```

computes the discrete Fourier transform of x.

The inverse discrete Fourier transform matrix is

```
Ai = conj(dftmtx(n))/n
```

Examples

In practice, the discrete Fourier transform is computed more efficiently and uses less memory with an FFT algorithm

```
x = 1:256;
y1 = fft(x);
```

than by using the Fourier transform matrix.

n = length(x);
y2 = x*dftmtx(n);
norm(y1-y2)
ans = 
    1.8297e-009

Algorithm

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

See Also

convmtx, fft

dfilt.statespace digitrevorder