Complex rectangular transforms

While deriving rectangular transforms Agarwal and cooley have used polynomial factors with real integer coefficients which resulted in real convolution matrices. In this paper, it is shown that the use of polynomial factors with complex integer coefficients yields new algorithms with complex convolution matrices, which require less number of multiplications than rectangular transforms. The paper outlines the derivation of the new algorithms and presents the convolution matrices for N = 4,5,7,8 and 9. The results show that the new approach yields smaller theoretical minimum number of multiplications for N = 4 and 8, and the corresponding algorithms are optimum. In view of the simplest factors used in deriving the algorithms for N = 5,7 and 9, it is believed that the corresponding algorithms are the best among those which achieve the theoretical minimum number of multiplications. The matrices have been verified to satisfy the necessary and sufficient condition derived by Agarwal and Cooley.