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 th 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
Spectrogram — plot the short-time Fourier transform
ShortTimeFourier ▪ InverseShortTimeFourier ▪ InverseSpectrogram ▪ ShortTimeFourierData ▪ SpectrogramArray
Periodogram — plot the power spectrum
PeriodogramArray ▪ ImagePeriodogram
Cepstrogram — plot the power cepstra computed on partitions of data
CepstrumArray ▪ CepstrogramArray
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