The structure of bilinear cyclic convolution algorithms is explored over finite fields. The algorithms derived are valid for any length not divisible by the field characteristic and are based upon the small length polynomial multiplication algorithms. The multiplicative complexity of these algorithms is small and depends on the field of constants. The linear transformation matrices A, B (premultiplication), and C (postmultiplication) defining the algorithm have block structures which are related to one another. The rows of A and B and the columns of C are maximal length recurrent sequences. Because of the highly regular structure of A, B , and C , the algorithms can be very easily designed even for large lengths. The application of these algorithms to the decoding of Reed-Solomon codes is also examined.

A new structured design method for convolutions over finite fields, Part I

Agarwal and Cooley have presented "rectangular transforms" for cyclic convolution, which are derived with the aid of the Chinese Remainder Theorem and factorization of the polynomial (zN- 1) into irreducible polynomials. The set of polynomials used by Agarwal and Cooley has coefficients from the field of real integers. The authors [3] have considered the set of polynomials with complex integer coefficients and derived new algorithms, referred to as "complex rectangular transforms," In this correspondence we present the detailed algorithms and show that, in many cases, they require fewer arithmetic operations than rectangular transforms for convolution of complex sequences.

Complex rectangular transforms for digital convolution

The Chinese remainder theorem plays a central role in the design of fast algorithms for computing cyclic and acyclic convolution of data sequences. A mathematical analysis of the theorem and related aspects form the topic of this investigation. The focus is almost exclusively on cyclic convolution algorithms. The number system that is studied is the field of rational numbers. Several properties related to the mathematical structure of the algorithms are derived.Note: This work is being reported in two parts. In Part I, we analyze the structure of the cyclic convolution algorithms over the rational number system. In Part II, we describe complex cyclotomic polynomials and the structure of the resulting cyclic convolution algorithms over the complex rational number system.

Analysis of the Chinese remainder theorem and cyclotomic polynomials-based algorithms for cyclic convolution - Part I: Rational number system

Rectangular transforms (RT's) were proposed for efficient computation of cyclic convolution. The algorithms were derived with the aid of the Chinese remainder theorem (CRT). In this letter, we present a simpler procedure to derive the RT algorithms by posing the problem as a set of over-determined linear systems of equations. Any standard subroutine may be used to solve such systems of equations.

http://ieeexplore.ieee.org/iel5/31/23538/01085292.pdf

On deriving rectangular transform matrices for cyclic convolution

This paper investigates the factorization properties of cyclotomic polynomials over the field of complex rational numbers. Based on this factorization and the Chinese remainder theorem, we analyze the mathematical structure of the associated algorithms for computing the cyclic convolution of data sequences. >

Complex cyclotomic polynomials and the related algorithms for cyclic convolution

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.

Complex rectangular transforms

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Complex rectangular transforms