REVISED FAST FOURIER TRANSFORM

Abstract

The problem of realisation of the Discrete Fourier Transform in on-line is analysed because of non-efficient consuming a time for a newrecalculation of spectrum samples if one discrete-time signal sample or even some small portion of samples in period are replaced by newsample or by new samples, respectively. Using Fast Fourier Transform (FFT) procedure it is assumed that some signal samples in the respective period available for processing digitally are updated by a sensor in real time. It is urgent for every new sample that emerges to have a new spectrum. The ordinary recalculation of spectrum samples even with highly efficient Cooley-Tukey FFT algorithm is not suitable due to speedy varying in time real process to be observed. The idea is that FFT procedure should not be recalculated with every new sample, it is needed just to modify it when the new sample emerges and replaces the old one. We retrieve the recursive formulas for FFT algorithms that refer to the spectrum samples modification. In a case of appearing one new sample, the recursive algorithm calculates a new spectrum samples by simple addition of a residual between an old and new samples, multiplied on respective row of Fourier ‘code’ matrix, to a vector of old spectrum samples. An example of 8-point FFT is presented.