Performing Fourier transforms on extremely long data streams

In many computer applications, it is necessary to perform discrete Fourier transforms on data sets that are too long to fit into computer memory. Standard ‘‘fast Fourier transform’’ (FFT) algorithms are then not applicable. This article shows how such data sequences may still be Fourier transformed relatively quickly by applying a ‘‘partial FFT’’ process that involves performing several discrete FFTs on smaller subsets formed from the original data sequence, and then mixing and reordering the series of data points produced. The result has the same frequency resolution as, and indeed produces identical results to, the results that would be produced by performing a Fourier transform on the full data set. There is no limit to the length of the data sets that can be transformed by this method. The procedure is illustrated by Fourier transforming a data set of 4×10⁶ points, but it is possible to Fourier transform much longer data sets.