Fourier Analysis

The Wolfram Language provides broad coverage of both numeric and symbolic Fourier analysis, supporting all standard forms of Fourier transforms on data, functions, and sequences, in any number of dimensions, and with uniform coverage of multiple conventions.

Fourier Transforms

FourierTransform, InverseFourierTransform complex Fourier transforms (FT, IFT)

FourierSinTransform  ▪  InverseFourierSinTransform

FourierCosTransform  ▪  InverseFourierCosTransform

Fourier Series

FourierSeries truncated complex Fourier series to any order

FourierSinSeries  ▪  FourierCosSeries  ▪  FourierTrigSeries

FourierCoefficient n coefficient in a Fourier series

FourierSinCoefficient  ▪  FourierCosCoefficient

Fourier Sequence Transforms

FourierSequenceTransform discrete-time Fourier transform (DTFT)

InverseFourierSequenceTransform inverse transform (IDTFT)

Discrete Fourier Transforms

Fourier, InverseFourier discrete transforms of lists of data in any dimension (DFT)

FourierDST, FourierDCT discrete sine, cosine transforms (DST, DCT, types I-IV)

Fourier Analysis in Signal and Image Processing

PeriodogramArray squared magnitude of the discrete Fourier transform

SpectrogramArray short-time discrete Fourier transform

Periodogram plot the power spectrum

Spectrogram plot the short-time Fourier transform

ImagePeriodogram  ▪  CepstrumArray  ▪  CepstrogramArray  ▪  Cepstrogram

Fourier Analysis in Control Systems

TransferFunctionModel frequency response of a system

BodePlot plot of the frequency response for a system

NyquistPlot  ▪  NicholsPlot  ▪  SingularValuePlot  ▪  ...

Fourier Analysis in Probability & Statistics

CharacteristicFunction Fourier transform of the PDF for a random variable

PowerSpectralDensity Fourier transform of the CovarianceFunction for a random process

FourierParameters select between Fourier analysis conventions

Fourier Convolutions

Convolve  ▪  DiscreteConvolve  ▪  ListConvolve  ▪  ImageConvolve

Related Transforms

LaplaceTransform  ▪  ZTransform  ▪  HankelTransform

Related Functions

DiracDelta  ▪  HeavisideTheta  ▪  DiscreteDelta  ▪  KroneckerDelta

SquareWave  ▪  TriangleWave  ▪  SawtoothWave

Wavelet Analysis »

DiscreteWaveletTransform  ▪  ContinuousWaveletTransform  ▪  ...