# Fourier Transform Functions

The manual describes the following implementations of the fast Fourier transform functions available in Intel® MKL:

- Fast Fourier transform (FFT) functions for single-processor or shared-memory systems (see FFT Functions)

- Cluster FFT functions for distributed-memory architectures (available with Intel® MKL for the Linux* and Windows* OSs only).

The general form of the discrete Fourier transform is

for `k1 = 0,±1,±2, ... ,` where σ is an arbitrary real-valued scale factor and `δ = ±1`. The forward transform is defined by `σ = 1` and `δ = -1`. In most common situations, the domain of the forward transform, that is, the set where the input (periodic) sequence `{wj1, j2, ..., jd}` belongs, can be either the set of complex-valued sequences, real-valued sequences, and complex-valued conjugate even sequences.

Both groups of FFT functions above present a uniform and easy-to-use applications programmer interface providing fast computation of discrete Fourier transform via the fast Fourier transform algorithm.

The FFT functions support arbitrary lengths. See the Intel MKL User's Guide for the specific radices supported.

### Note

The FFT functions assume the Cartesian representation of complex data (that is, the real and imaginary parts define a complex number). The Intel MKL Vector Mathematical Functions provide an efficient tool for conversion to and from the polar representation (see Example “Conversion from Cartesian to polar representation of complex data” and Example “Conversion from polar to Cartesian representation of complex data”).

- FFT Functions
- Cluster FFT Functions