Split manageable efficient algorithm for Fourier and Hadamard transforms

In this paper, a general, efficient, manageable split algorithm to compute one-dimensional (1-D) unitary transforms, by using the special partitioning in the frequency domain, is introduced. The partitions determine fast transformations that split the N-point unitary transform into a set of Ni-point transforms i=1: n(N1+...N n=N). Here, we introduce a class of splitting transformations: the so-called paired transforms. Based on these transforms, the decompositions of the Fourier transforms of arbitrary orders are given, and the corresponding algorithms are considered. Comparative estimates revealing the efficiency of the proposed algorithms with respect to the known ones are given. In particular, a proposed method of calculating the 2r-point Fourier transform requires 2^r-1(r-3)+2 multiplications and 2^r-1(r+9)-r2-3r-6 additions. In terms of the paired transforms, the splitting of the 2r-point Hadamard transform is described. As a result, the proposed algorithm for computing this transform uses on the average no more than six operations of additions per sample.