## Fast Fourier Transform functions

slowFFT | Basic in-place FFT.
FFT | Alternate name for slowFFT
slowIFFT | Basic inverse in-place FFT int slowFFT
IFFT | Alternate name for slowIFFT
power_spectrum | Power spectrum using the fastFFT function.
power_spectrum_slow | Power spectrum using the slowFFT function
fastFFT | Fast FFT An optimised implementation by Tony Robinson to be used in preference to slowFFT

> <para> These are the low level functions where the actual FFT is performed. Both slow and fast implementations are available for historical reasons. They have identical functionality. At this time, vectors of complex numbers are handled as pairs of vectors of real and imaginary numbers. </para>

> <formalpara> <title>What is a Fourier Transform ?</title>

> <para> The Fourier transform of a signal gives us a frequency-domain representation of a time-domain signal. In discrete time, the Fourier Transform is called a Discrete Fourier Transform (DFT) and is given by:

> where \(y = (y_0,y_1,... y_{n-1})\) is the DFT (of order \(n\) ) of the signal \(x = (x_0,x_1,... x_{n-1})\), where \(\omega_{n}^{0},\omega_{n}^{1},... \omega_{n}^{n-1}\) are the n complex nth roots of 1. </para>

> <para> The Fast Fourier Transform (FFT) is a very efficient implementation of a Discrete Fourier Transform. See, for example "Algorithms" by Thomas H. Cormen, Charles E. Leiserson and Ronald L. Rivest (pub. MIT Press), or any signal processing textbook. </para>

> </formalpara>

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This page is part of the Edinburgh Speech Tools Library documentation
Copyright University of Edinburgh 1997
Contact: speech_tools@cstr.ed.ac.uk