Split manageable efficient algorithm for Fourier and Hadamard transforms

• Published in: IEEE transactions on signal processing Vol. 48; no. 1; pp. 172 - 183
• Main Authors: Grigoryan, A.M., Agaian, S.S.
• Format: Journal Article
• Language: English
• Published: New York, NY IEEE 01-01-2000; Institute of Electrical and Electronics Engineers; The Institute of Electrical and Electronics Engineers, Inc. (IEEE)
• Subjects: Algorithm design and analysis; Algorithms; Applied sciences; Computing time; Convolution; Data compression; Discrete Fourier transforms; Exact sciences and technology; Fast Fourier transforms; Fourier transforms; Frequency domain analysis; Frequency domains; Image processing; Information, signal and communications theory; Mathematical methods; Partitioning; Partitioning algorithms; Signal processing; Splitting; Studies; Telecommunications and information theory; Transformations; Transforms; Signal processing; Hadamard transformation; Frequency domain method; Fourier transformation; Fast algorithm
• ISSN: 1053-587X; 1941-0476
• DOI: 10.1109/78.815487

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 N/sub i/-point transforms i=1: n(N/sub 1/+...N/sub 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 2/sup r/-point Fourier transform requires 2/sup r-1/(r-3)+2 multiplications and 2/sup r-1/(r+9)-r/sup 2/-3r-6 additions. In terms of the paired transforms, the splitting of the 2/sup r/-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.