Fast Reverse Jacket Transform and its Inverse Transform with Application to Fast Fourier Transform

The recently introduced Reverse Jacket matrix (RJM) has a generalized form of the Hadamard matrix. Thus the RJM is closely related to the matrix for a fast Fourier transform (FFT). It also has a very interesting structure, i.e., its inverse can be easily obtained and has the reversal form of the original matrix. By decomposition of the RJM a fast Reverse Jacket transform (FRJT) is obtained, which is remarkably efficient than the center-weighted Hadamard transform (CWHT). In this paper, we present an inverse fast Reverse Jacket transform (IFRJT) and express the inverse in explicit form. The advantage of the IFRJT is to get the inverse of a given matrix much easier and faster than using the inverse CWHT. The FRJT and the IFRJT can be generalized in terms of the Kronecker product of the Hadamard matrix. We illustrate how simple the IFRJT can be achieved, and provide an example in order to show the relation between the FRJT and FFT