- See Also
-
Related Guides
- Summation Transforms
- Fourier Analysis
- Signal Processing
- GPU Computing
- Data Transforms and Smoothing
- Signal Transforms
- Signal Filtering & Filter Design
- Signal Visualization & Analysis
- Scientific Data Analysis
- GPU Computing with NVIDIA
- GPU Computing with Apple
- Integral Transforms
- Computer Vision
- Audio Analysis
- Image Filtering & Neighborhood Processing
- Numerical Data
- Audio Representation
- GPU Programming
- Tech Notes
-
- See Also
-
Related Guides
- Summation Transforms
- Fourier Analysis
- Signal Processing
- GPU Computing
- Data Transforms and Smoothing
- Signal Transforms
- Signal Filtering & Filter Design
- Signal Visualization & Analysis
- Scientific Data Analysis
- GPU Computing with NVIDIA
- GPU Computing with Apple
- Integral Transforms
- Computer Vision
- Audio Analysis
- Image Filtering & Neighborhood Processing
- Numerical Data
- Audio Representation
- GPU Programming
- Tech Notes
Fourier
Details and Options
- The discrete Fourier transform vs of a list ur of length n is by default defined to be ure2πi(r-1)(s-1)/n. »
- Note that the zero frequency term appears at position 1 in the resulting list.
- Other definitions are used in some scientific and technical fields.
- Different choices of definitions can be specified using the option FourierParameters .
- With the setting FourierParameters->{a,b}, the discrete Fourier transform computed by Fourier is ure2π ib(r-1)(s-1)/n. »
- Some common choices for {a,b} are {0,1} (default), {-1,1} (data analysis), {1,-1} (signal processing).
- The setting effectively corresponds to conjugating both input and output lists.
- To ensure a unique inverse discrete Fourier transform, b must be relatively prime to n. »
- The list of data supplied to Fourier need not have a length equal to a power of two.
- The list given in Fourier [list] can be nested to represent an array of data in any number of dimensions.
- The array of data must be rectangular.
- If the elements of list are exact numbers, Fourier begins by applying N to them.
- Fourier [list,{p1,p2,…}] is typically equivalent to Extract [Fourier [list],{p1,p2,…}]. Cases with just a few positions p are computed using an algorithm that takes less time and memory but is more subject to numerical error, particularly when the length of list is long.
- Fourier can be used on SparseArray objects.
Examples
open all close allBasic Examples (2)
Find a discrete Fourier transform:
Find a power spectrum:
Scope (3)
x is a list of real values:
Compute the Fourier transform with machine arithmetic:
Compute using 24-digit precision arithmetic:
Compute a 2D Fourier transform:
x is a rank 3 tensor with nonzero diagonal:
Compute the 3D Fourier transform:
Options (2)
FourierParameters (2)
No normalization:
Normalization by :
Normalization by :
Data from a Sinc function with noise:
Ordinary spectrum without normalization:
Partial spectrum:
Applications (10)
Computing Spectra (6)
Fourier spectrum of "white noise":
Show the logarithmic spectrum, including the first (DC) component:
The spectrum of a "pulse" is completely flat:
Power spectrum of the Thue–Morse nested sequence [more info]:
Power spectrum of the Fibonacci nested sequence [more info]:
2D power spectrum of a nested pattern:
Plot the nested pattern:
Find the logarithmic power spectrum:
Find the Fourier transform of the rule 30 cellular automaton pattern:
Logarithmic power spectrum:
Filtering Data (1)
Compute discrete cyclic convolutions to smooth a discontinuous function with a Gaussian:
Compute the cyclic convolution:
Show the original and smoothed function:
The convolution is consistent with ListConvolve :
Frequency Identification (1)
Here is some periodic data with some noise:
Find the maximum modes in the spectrum:
Get the position corresponding to positive frequencies and show the position on a plot of the spectrum:
Find a high-resolution spectrum between modes where the maximum was found:
Determine the period from the frequencies:
Computing Eigenvectors (1)
m is a circulant differentiation matrix:
Because , the eigenvalues of m are:
The eigenvectors are the columns of the DFT matrix, so Fourier diagonalizes m:
This allows very efficient computation of MatrixExp [m,r] for a particular vector:
Show the approximate evolution of the heat equation on the unit interval:
Fractional Fourier Transform (1)
Define a fractional Fourier transform using different choices of FourierParameters :
Properties & Relations (6)
InverseFourier inverts Fourier :
For real inputs, all elements after the first come in complex conjugate pairs:
The power spectrum is symmetric:
Cyclic convolution corresponds to multiplication of Fourier transforms:
is given by :
Fourier is equivalent to multiplication with FourierMatrix :
The conjugate transpose of the matrix is equivalent to InverseFourier :
Possible Issues (4)
If is not relatively prime to , the transform may not be invertible:
Lengths that are powers of 2 or factorizable into a product of small primes will be faster:
Fourier uses an efficient algorithm when only a small number of coefficients is needed:
The fast and efficient implementation may result in significant numerical error:
See Also
InverseFourier FourierMatrix FourierDCT FourierDST FourierTransform FourierSinTransform FindRepeat FindTransientRepeat Fit FindPeaks PeriodogramArray ShortTimeFourier CepstrumArray Periodogram ImagePeriodogram Spectrogram Cepstrogram
Function Repository: SymbolicFourier NumberTheoreticTransform FourierShift FastFourierGraph TrigFit WelchSpectralEstimate
Related Guides
-
▪
- Summation Transforms ▪
- Fourier Analysis ▪
- Signal Processing ▪
- GPU Computing ▪
- Data Transforms and Smoothing ▪
- Signal Transforms ▪
- Signal Filtering & Filter Design ▪
- Signal Visualization & Analysis ▪
- Scientific Data Analysis ▪
- GPU Computing with NVIDIA ▪
- GPU Computing with Apple ▪
- Integral Transforms ▪
- Computer Vision ▪
- Audio Analysis ▪
- Image Filtering & Neighborhood Processing ▪
- Numerical Data ▪
- Audio Representation ▪
- GPU Programming
Related Links
History
Introduced in 1988 (1.0) | Updated in 1996 (3.0) ▪ 1999 (4.0) ▪ 2000 (4.1) ▪ 2002 (4.2) ▪ 2003 (5.0) ▪ 2012 (9.0)
Text
Wolfram Research (1988), Fourier, Wolfram Language function, https://reference.wolfram.com/language/ref/Fourier.html (updated 2012).
CMS
Wolfram Language. 1988. "Fourier." Wolfram Language & System Documentation Center. Wolfram Research. Last Modified 2012. https://reference.wolfram.com/language/ref/Fourier.html.
APA
Wolfram Language. (1988). Fourier. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/Fourier.html
BibTeX
@misc{reference.wolfram_2025_fourier, author="Wolfram Research", title="{Fourier}", year="2012", howpublished="\url{https://reference.wolfram.com/language/ref/Fourier.html}", note=[Accessed: 16-November-2025]}
BibLaTeX
@online{reference.wolfram_2025_fourier, organization={Wolfram Research}, title={Fourier}, year={2012}, url={https://reference.wolfram.com/language/ref/Fourier.html}, note=[Accessed: 16-November-2025]}