An efficient recursive algorithm and an explicit formula for calculating update vectors of running walsh-hadamard transform

Walsh-Hadamard transform (WHT) has many applications in digital signal processing including bioinformatics. While there are efficient algorithms for implementing this transform such as fast WHT (FWHT), performing WHT continuously on a sliding window over a long sequence is time consuming. As it is not reasonable to compute a separate WHT upon arrival of every new sample, another implementation named running WHT (RWHT) has been introduced in (G. Deng and A. Ling, 1996) which needs to have some update vectors pre-calculated. In this paper we report on an efficient recursive algorithm to find these vectors. Also we propose an easy-to-compute explicit formula for computing coefficients of these update vectors in a computation time independent of the vector size. A proof of the formula and a comparison between the proposed recursive algorithm and an algorithm based on the formula are also given.