Efficient energy gradient calculations for binary and polyphase sequences

Methods for optimization of phase coded waveforms often entail a nearest-neighbor search used in reducing autocorrelation or crosscorrelation sidelobes. Computing the energy gradient - the change in sidelobe energy that results from single-element modifications to the sequence - is a computationally expensive component of the nearest-neighbor search. A recent paper showed that the autocorrelation sidelobe energy gradient for binary sequences could be computed with O(N logN) operations in initialization and O(N) operations in iteration, substantially faster than previous methods which required O(N²) operations both initially and in iteration. In this paper the same approach is extended to additional sequence optimizations - polyphase sequences, biphase cross-correlations, and biphase sequence filtering. It is shown that similarly efficient equations are available for those gradient calculations.