Unscrambling for fast DFT algorithms

Several methods are reviewed for removing the unscrambler in the prime factor algorithms (PFA) and the types of unscrambler necessary for the Cooley-Tukey FFT (fast Fourier transform) are discussed. It is shown that a radix-4, radix-8, radix-16, or any radix-2^m FFT can be written to give the output in the same bit-reversed order as the radix-2 FFT. This applies to programs which mix radix-8, radix-4, and radix-2 stages to have the high efficiency of radix-8 and radix-4 and the variety of lengths of radix-2. In a more general form, the method will allow a radix-16 to give its output in the same order as as radix-4 FFT. The method can be used with radix-2^m Harley, cosine, sine, number-theoretic, and special real-data transforms. This result has practical importance because it allows a single software or hardware bit-reverse counter to unscramble the more efficient radix-4, radix-8, and mixed-radix FFTs