XFT: Extending the Digital Application of the Fourier Transform. (arXiv:0911.0952v1 [math.NA])

In recent years there has been a growing interest in the fractional Fourier
transform driven by its great number of applications. The literature in this
field follows two main routes. On the one hand the applications fields where
the ordinary Fourier transform can be applied are being revisited to use this
intermediate time-frequency representation of signals; and on the other hand
fast algorithms for numerical computation of the fractional Fourier transform
are devised. In this paper we derive a Gaussian-like quadrature of the
continuous fractional Fourier transform. This quadrature is given in terms of
the Hermite polynomials and their zeros. By using some asymptotic formulae we
are able to solve the quadrature by a diagonal congruence transformation
equivalent to a chirp-FFT-chirp transformation, yielding a fast discretization
of the fractional Fourier transform and its inverse in closed form. We extend
the range of the fractional Fourier transform by considering arbitrary complex
values inside the unitary circle and not only at the boundary. Interestingly
enough, the congruence transformation evaluated at $z=i$, which gives the
Fourier transform, improves the standard discrete Fourier transform, yielding a
new method to compute a more accurate FFT.

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Original source : http://arxiv.org/abs/0911.0952...