Built-in Mathematica Symbol

Fourier

Fourier[list]
finds the discrete Fourier transform of a list of complex numbers.

MORE INFORMATION

The discrete Fourier transform v_s of a list u_r of length n is by default defined to be . »

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 . »

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 can be used on SparseArray objects.

EXAMPLES

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Basic Examples (2)

Find a discrete Fourier transform:

Find a power spectrum:

Find a discrete Fourier transform:

In[1]:=

Out[1]=

Find a power spectrum:

In[1]:=

Out[1]=

Scope (4)

Fourier always gives approximate numerical results:

Use Chop to remove negligible imaginary parts:

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)

No normalization:

Normalization by

:

Normalization by

:

Data from a Sinc function with noise:

Ordinary spectrum without normalization:

Partial spectrum:

Applications (9)

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:

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:

Here is some periodic data with some noise:

Find the maximum mode in the spectrum:

Find a high-resolution spectrum between modes where the maximum was found:

Determine the period from the frequencies:

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:

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 matrix multiplication:

The conjugate transpose of the matrix is equivalent to InverseFourier:

Possible Issues (2)

If b is not relatively prime to n, the transform may not be invertible:

Lengths that are powers of 2 or factorizable into a product of small primes will be faster: