A new reverse jacket transform and its fast algorithm

This paper presents the reverse jacket transform [RJT] and a simple decomposition of its matrix, which is used to develop a fast algorithm for the RJT. The matrix decomposition is of the form of the matrix products of Hadamard matrices and successively lower order coefficient matrices. This decomposition very clearly leads to a block circular sparse matrix factorization of the reverse jacket [RJ]_N matrix. The main property of [RJ]_N is that the inverse matrices of its elements can be obtained very easily and have a special structure. [RJ]_N is derived using the weighted Hadamard transform corresponding to the Hadamard matrix [H]_N and a basic symmetric matrix Λ. Each element of [RJ]_N is a generalized for polygonal subsampling and canonical Smith form. In this paper we represent in particular the systematical block-wise sparse matrix of extending-method for [RJ]_N