An Algorithm for the Machine Calculation of Complex Fourier Series

Abstract

An efficient method for the calculation of the interactions of a 2m factorial experiment was introduced by Yates and is widely known by his name. The generalization to 3m was given by Box et al. [1]. Good [2] generalized these methods and gave elegant algorithms for which one class of applications is the calculation of Fourier series. In their full generality, Good's methods are applicable to certain problems in which one must multiply an JV-vector by an JV X N matrix which can be factored into m sparse matrices, where m is proportional to log JV. This results in a procedure requiring a number of operations proportional to JV log JV rather than JV2. These methods are applied here to the calculation of complex Fourier series. They are useful in situations where the number of data points is, or can be chosen to be, a highly composite number. The algorithm is here derived and presented in a rather different form. Attention is given to the choice of JV. It is also shown how special advantage can be obtained in the use of a binary computer with JV = 2m and how the entire calculation can be performed within the array of JV data storage locations used for the given Fourier coefficients.Consider the problem of calculating the complex Fourier series A straightforward calculation using ( 1 ) would require JV2 operations where "operation" means, as it will throughout this note, a complex multiplication followed by a complex addition.The algorithm described here iterates on the array of given complex Fourier amplitudes and yields the result in less than 2JV Iog2 JV operations without requiring more data storage than is required for the given array A. To derive the algorithm, suppose JV is a composite, i.e., JV = rvr2. Then let the indices in (1)

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