Numerical Calculations in Clifford Algebra

21Fast Fourier Transform

Standard fast Fourier transform (FFT) algorithms support at best complex signals (functions of time) or waveforms (functions of space). In Chapter 23, it is of interest to use the FFT to calculate derivatives of an electromagnetic field in order to simulate wave propagation using the finite difference time domain (FDTD) method. For that application, the signal is a Clifford bivector in four dimensions (both space and time), making it necessary to construct a Fourier transform which operates at least on bivectors, and preferably on all other Clifford entities as well.

21.1 Theory

In general, the spectrum is calculated from a signal by the Fourier transform as follows:

(21.1)

When the signal is bandlimited and periodic, the same spectral values are obtained¹ if the original signal is replaced by a new function:

(21.2)

consisting of samples covering one ...