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# Fourier

 Fourier[list]finds the discrete Fourier transform of a list of complex numbers.
• The discrete Fourier transform vs of a list ur 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.
• 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.
Find a discrete Fourier transform:
Find a power spectrum:
Find a discrete Fourier transform:
 Out[1]=

Find a power spectrum:
 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:
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:
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:
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: