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

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.

[1]  J. Cooley,et al.  New algorithms for digital convolution , 1977 .

[2]  S. Winograd.  On computing the Discrete Fourier Transform. , 1976, Proceedings of the National Academy of Sciences of the United States of America.

[3]  V. Umapathi Reddy,et al.  Combination of complex rectangular transforms and f.n.t. to implement fast convolution , 1979 .

[4]  C. Rader.  Discrete Fourier transforms when the number of data samples is prime , 1968 .

[5]  Trieu-Kien Truong,et al.  The use of finite fields to compute convolutions , 1975, IEEE Trans. Inf. Theory.

[6]  C. Burrus,et al.  Number theoretic transforms to implement fast digital convolution , 1975 .

[7]  Trieu-Kien Truong,et al.  Complex integer convolutions over a direct sum of Galois fields , 1975, IEEE Trans. Inf. Theory.

[8]  V. Umapathi Reddy,et al.  Convolution algorithms for small-word-length digital-filtering applications , 1979 .

[9]  C. Burrus,et al.  Fast one-dimensional digital convolution by multidimensional techniques , 1974 .

[10]  H. Nussbaumer,et al.  Fast polynomial transform algorithms for digital convolution , 1980 .

[11]  V. Umapathi Reddy,et al.  Complex rectangular transforms , 1979, ICASSP.

[12]  Elwyn R. Berlekamp,et al.  Algebraic coding theory , 1984, McGraw-Hill series in systems science.

[13]  Charles M. Rader,et al.  Discrete Convolutions via Mersenne Transrorms , 1972, IEEE Transactions on Computers.

[14]  C. Burrus,et al.  Fast Convolution using fermat number transforms with applications to digital filtering , 1974 .

[15]  Henri J. Nussbaumer.  Complex convolutions via Fermat number transforms , 1976 .

[16]  J. Cooley,et al.  New algorithms for digital convolution , 1977 .

[17]  Henri J. Nussbaumer,et al.  Computation of Convolutions and Discrete Fourier Transforms by Polynomial Transforms , 1978, IBM J. Res. Dev..