Wavelet Toolbox

nstdfft

Nonstandard 1-D fast Fourier transform.

Syntax

[XHAT,OMEGA] = nstdfft(X,LOWB,UPPB)

Description

nstdfft is a general mathematical utility.

[XHAT,OMEGA] = nstdfft(X,LOWB,UPPB) returns a nonstandard FFT of signal X sampled on a power-of-2 regular grid (not necessarily integers) on the interval [LOWB,UPPB].

Output arguments are XHAT, the shifted FFT of X computed on the interval OMEGA given by OMEGA = [-n:2:n-2] / (2*(UPPB - LOWB)), where n is the length of X. Outputs are vectors of length n.

Length of X must be a power of two.

Algorithm

Given observations between two bounds l and u: x₁, x₂, . . . , x_N, which are regularly sampled from a continuous signal f:

x_k = f(l + (k - 1)) for k = 1 to N where = (u - 1) / N

nstdfft computes approximations of the continuous Fourier transform

coefficients:

for

:

:

using the standard discrete fast fourier transform fft.

For a given frequency :

can be rewritten as:

using t = sN + l.

The integral term can be approximated by the finite sum:

Since

:

:

then

: 1 :

,

which are the usual frequencies of the discrete Fourier transform.

It turns out that

can be approximated by:

where

which can be computed using standard fft and a normalization. The function instdfft inverts this transform in three steps: normalization, use of ifft, and translation in time.

The length of X must be a power of two.

See Also
fft, fftshift, instdfft

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