Fast Gray Code Kernel Algorithm for the Sliding Conjugate Symmetric Sequency-Ordered Complex Hadamard Transform

Abstract

A fast algorithm based on the gray code kernel (GCK) for computing the conjugate symmetric sequency-ordered complex Hadamard transform (CS-SCHT) in a sliding window is presented. The proposed algorithm computes the current projection value from the previously computed ones. In order to obtain the peculiar computation order of the projection values, we construct the CS-SCHT matrix tree and also introduce the $\alpha $ -related concept. The properties of the elements of the CS-SCHT matrix are also given for deriving the GCK sliding CS-SCHT algorithm. The proposed algorithm only needs N/2+log $_{2}N-2$ (or log $_{2}N-1$ ) multiplications with $j$ and $4N-2$ (or $2N-1$ ) real additions for complex (or real) input data, which is more efficient than the block-based CS-SCHT and other existing sliding complex transform algorithms, such as the radix-4 sliding CS-SCHT algorithm, sliding FFT algorithm, and sliding DFT algorithm. A comparison of the proposed algorithm with other sliding transforms in terms of computation time is also presented to validate the theoretical results.