Note on the calculation of Fourier series

Cooley and Tukey have recently presented an algorithm for the machine calculation of Fourier series [1]. In this connection mention should be made of the similar method described by Danielson and Lanezos [2]. Although the latter is less elegant and is phrased wholly in terms of real quantities, it yields the same results as the binary form of the Cooley-Tukey algorithm with a comparable number of arithmetical operations. A small-computer program has been written in this laboratory which uses the Danielson-Lanezos method with one minor modification, described below.* In this form cosine and sine series may be evaluated independently of one another, and as with the Cooley-Tukey process, the calculation can be performed by replacing input data with results, without any substantial storage requirement beyond that set by the original number of input coefficients. The procedure used is readily extensible to the computation of [(N/2) + 1] cosine and [(N/2) 1] sine sums, from an original sequence of N real numbers, where N is any power of 2 greater than the fourth. Two executions of this procedure will yield values for a set of N complex Fourier series of N terms each, and can be accomplished by N[log2 N + 3] real multiplications and 7N/2[10g2 N 23/7] real additions. For the same task the