A connection between bit reversal and matrix transposition: hardware and software consequences

A simple connection between bit reversal and matrix transposition is explained in the context of two-dimensional fast Fourier transform (FFT). This connection is then shown to be useful for hardware and software implementations of two-dimensional FFTs based on the row-column algorithm: the bit-reversal operations involved in the row and column one-dimensional transforms of length M can merge with the matrix transposition in between, to form a single bit reversal of length N=M², which is more efficient than considering these operations separately. Furthermore, this connection is also the base of new bit-reversal algorithms that are explained. The first one can be implemented in two different versions, depending on the availability of an auxiliary memory of size √N. One of these versions turns out to be as fast as a recently proposed algorithm by D.M. Evans (1987), while the second is twice as fast as the usual one. The second algorithm is a divide-and-conquer approach to the problem that allows easy parallelization