An improved FFT digit-reversal algorithm

Several improvements on the bit-reversal algorithm of B. Gold and C.M. Rader (1969) are presented. The savings in computation are obtained by observing that not all indexes need to be reversed. In particular, a closed-form expression is derived for the largest index that must be digit-reversed (for an arbitrary radix). To limit the number of unnecessary digit-reversals, a closed-form expression (in terms of N and R) is derived for the largest array index that must be reversed. In addition, it is shown that the smallest index that must be reversed is always 1, not 0, as is commonly implemented. A computational analysis is given, comparing the original and modified algorithms. The modifications to the algorithm led to an improved efficiency with almost no increase in the size of the program or its working storage