The Fourier Transform of a signal is ordinarily considered simply to be a representation of a periodic time-signal using an alternative basis space. A signal of period T comprises frequency components in the ordered set { (0/T), (1/T), (2/T) … }. If the signal is a discrete time signal, this set is finite. For a continuous time signal, this set is infinite. In either case, the signal is described fully by an ordered set of coefficients.

The response of a linear dynamic system to such a periodic signal will contain frequency components in the same set and this response is also described fully by an ordered set of coefficients. A given set of coefficients might represent either a signal or a frequency response. This trivial observation is reminiscent of Clifford's early comment (to the effect) that every number represents both a quantity and a transformation.

The dual role of a set of Fourier coefficients motivates us to consider what is the reference periodic signal - defined by the criterion that when this signal is transformed by a dynamic system represented by an arbitrary set of Fourier coefficients, the result is a signal having the same Fourier coefficients. This reference periodic signal is found to be the periodic unit impulse of arbitrary phase relative to a fixed time marker.

With the frequency-response interpretation in mind, a useful geometric view of the Fast Fourier Transform begins to develop. We consider the continuous signal, x(t), for which x(t+T)=x(t). This signal is sampled at 2N equally spaced instants per cycle and we can exercise some control over the phasing of the sampling process such that the first of the 2N samples is taken at t0 < (T/2N). Arranging the sampled data in a vector (having 2N real components) produces yN(t0). Through Fourier transformation this data can be cast in frequency-domain form as another vector, YN(t0), having 2N real components.

This paper expands and explores this algebraic representation and interprets the Fast Fourier Transform in these terms. Its primary aim is to elucidate the operation of the FFT.

Seamus.Garvey@nottingham.ac.uk