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
Power spectrum of the Fibonacci nested sequence
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
for a particular vector:
Show the approximate evolution of the heat equation
on the unit interval:
of a list 
of length n is by default defined to be 
. »
, the discrete Fourier transform computed by Fourier is 
. »
are 
(default), 
(data analysis), 
(signal processing). 
effectively corresponds to conjugating both input and output lists. 
b
:
:
is given by 
:
EXAMPLES
Basic Examples 
Scope 