# Documentation

### This is machine translation

Translated by
Mouse over text to see original. Click the button below to return to the English verison of the page.

# Fourier Analysis and Filtering

Fourier transforms, convolution, digital filtering

Transforms and filters are tools for processing and analyzing discrete data, and are commonly used in signal processing applications and computational mathematics. When data is represented as a function of time or space, the Fourier transform decomposes the data into frequency components. The `fft` function uses a fast Fourier transform algorithm that reduces its computational cost compared to other direct implementations. For a more detailed introduction to Fourier analysis, see Fourier Transforms. The `conv` and `filter` functions are also useful tools for modifying the amplitude or phase of input data using a transfer function.

## Functions

 `fft` Fast Fourier transform `fft2` 2-D fast Fourier transform `fftn` N-D fast Fourier transform `fftshift` Shift zero-frequency component to center of spectrum `fftw` Interface to FFTW library run-time algorithm tuning control `ifft` Inverse fast Fourier transform `ifft2` 2-D inverse fast Fourier transform `ifftn` N-D inverse fast Fourier transform `ifftshift` Inverse FFT shift `nextpow2` Exponent of next higher power of 2
 `conv` Convolution and polynomial multiplication `conv2` 2-D convolution `convn` N-D convolution `deconv` Deconvolution and polynomial division
 `filter` 1-D digital filter `filter2` 2-D digital filter `ss2tf` Convert state-space representation to transfer function

## Topics

Fourier Transforms

This topic defines the discrete Fourier transform and its implementations, and introduces an example of basic Fourier analysis for signal processing applications.

Basic Spectral Analysis

This topic introduces frequency and power spectrum analysis of two time-domain signals.

Polynomial Interpolation Using FFT

This example shows how to use the fast Fourier transform to estimate coefficients of a polynomial interpolant.

Analyze 2-D Optics with the Fourier Transform

This topic defines the two-dimensional Fourier transform, and uses the `fft2` function to transform a 2-D optical mask into frequency space.

Smooth Data with Convolution

This example uses convolution to smooth noisy, two-dimensional data.

Filter Data

This topic defines the `filter` function in MATLAB®, and presents two examples of filters that modify input data.