Important: The FFT and iFFT functions have been deprecated.  For FFT, use the superseding functions DiscreteTransforms[FourierTransform] and SignalProcessing[FFT] instead.  For iFFT, use the superseding functions DiscreteTransforms[InverseFourierTransform] and SignalProcessing[InverseFFT] instead. These new functions can compute the fast Fourier transform for sequences of arbitrary length (not restricted to a power of 2).

The FFT(m,x,y) and iFFT(m,x,y) commands compute in place the fast Fourier transform and the inverse fast Fourier transform of a complex sequence of length $2^m$.

The first argument m should be a non-negative integer and the second and third arguments x and y should be arrays of floats indexed from 1 to $2^m$ .  The array x contains the real part of the complex sequence on input and contains the real part of the fast Fourier transform on output.  The array y contains the imaginary part of the complex sequence on input and contains the imaginary part of the fast Fourier transform on output. Both procedures return 2^m, the number of points in the complex sequence.

These procedures may be invoked with evalhf, which uses the hardware floating-point number system.

$x≔\mathrm{array}\left(\left[7.\,5.\,6.\,9.\right]\right)\:$

$y≔\mathrm{array}\left(\left[0\,0\,0\,0\right]\right)\:$

$\mathrm{FFT}\left(2\,x\,y\right)$

$4$

$\mathrm{print}\left(x\right)$

$\left[\begin{array}{cccc}27.& 1.& \mathrm{-1.}& 1.\end{array}\right]$

$\mathrm{print}\left(y\right)$

$\left[\begin{array}{cccc}0& 4.& 0.& \mathrm{-4.}\end{array}\right]$

$\mathrm{iFFT}\left(2\,x\,y\right)$

$4$

$\mathrm{print}\left(x\right)$

$\left[\begin{array}{cccc}7.000000000& 5.000000000& 6.000000000& 9.000000000\end{array}\right]$

$\mathrm{print}\left(y\right)$

$\left[\begin{array}{cccc}0.& 0.& 0.& 0.\end{array}\right]$

Oppenheim, Allan, V., and Schafer, Ronald W. Digital Signal Processing. New Jersey: Prentice-Hall, 1975.  See Fig. 6.5, p. 332.