Cooley - Tukey FFT on the Connection MachineS

We describe an implementation of the Cooley Tukey complex-to-complex FFT on the Connection Machine. The implementation is designed to make eeective use of the communications bandwidth of the architecture, its memory bandwidth, and storage with precomputed twiddle factors. The peak data motion rate that is achieved for the interprocessor communication stages is in excess of 7 Gbytes/s for a Connection Machine system CM-200 with 2048 oating-point processors. The peak rate of FFT computations local to a processor is 12.9 GGops/s in 32-bit precision, and 10.7 GGops/s in 64-bit precision. The same FFT routine is used to perform both one-and multi-dimensional FFT without any explicit data rearrangement. The peak performance for a one-dimensional FFT on data distributed over all processors is 5.4 GGops/s in 32-bit precision and 3.2 GGops/s in 64-bit precision. The peak performance for square, two-dimensional transforms, is 3.1 GGops/s in 32-bit precision, and for cubic, three dimensional transforms, the peak is 2.0 GGops/s in 64-bit precision. Certain oblong shapes yield better performance. The number of twiddle factors stored in each processor is P 2N + log 2 N for an FFT on P complex points uniformly distributed among N processors. To achieve this level of storage eeciency we show that a decimation-in-time FFT is required for normal order input, and a decimation-in-frequency FFT is required for bit-reversed input order.