No normalization:

Normalization by :

Normalization by :

Data from a Sinc function with noise:

Ordinary spectrum without normalization:

Partial spectrum:

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 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:

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:

Define a fractional Fourier transform using different choices of FourierParameters: