An Investigation of Cooley-Tukey Decompositions for the FFT

The goals of the research discussed in this report are to determine the impact of di erent Cooley-Tukey decompositions on the performance of computer programs that compute the FFT, evaluate di erent methods for nding the most e cient decompositions, and determine the characteristics of the most e cient decompositions. Experiments are conducted using three di erent FFT programs and three dynamic programming (DP) methods for searching for e cient decompositions. The results show that even for FFTs of sizes under 211, the runtime for the average decomposition may be up to three times the runtime for the optimal decomposition. The results also show that a basic implementation of DP performs as well as an exhaustive search at nding fast decompositions for FFTs of sizes up to 210, and that the simple DP performs as well as two more sophisticated versions of DP for FFT sizes up to 220. Furthermore, the results show that for an out-of-place implementation of the FFT, right-expanded decompositions are optimal because they require memory storage only for the input and output data arrays, whereas other decompositions require additional temporary storage. Moreover, the results show that for an in-place implementation of the FFT, balanced decompositions are optimal if the algorithm is iterative, and right-expanded decompositions are optimal if the algorithm is recursive.