A new approach to solve the sequence-length constraint problem in circular convolution using number theoretic transform

Offers a novel approach to solve the sequence-length constraint problem by proposing a formula to produce generalized modulo a numbers for number theoretic transforms. By selecting a prime M as the modulo number and choosing the least primitive root M as the a in the number theoretic transform, the sequence lengths become exponentially proportional to the word length. The set of generalized modulo numbers includes Mersenne and Fermat numbers. The circular convolution obtained by this method is accurate, i.e., without roundoff error. >

[1]  L. Leibowitz A simplified binary arithmetic for the Fermat number transform , 1976 .

[2]  Miroslav Morháč,et al.  Precise deconvolution using the Fermat number transform , 1986 .

[3]  C. Rader,et al.  On the application of the number theoretic methods of high-speed convolution to two-dimensional filtering , 1975 .

[4]  J. McClellan,et al.  Hardware realization of a Fermat number transform , 1976 .

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

[6]  Zvonko G. Vranesic,et al.  A Many-Valued Algebra for Switching Systems , 1970, IEEE Transactions on Computers.

[7]  E. Vegh,et al.  Fast complex convolution in finite rings , 1976 .

[8]  Andreas Antoniou,et al.  Number theoretic transform based on ternary arithmetic and its application to cyclic convolution , 1983 .

[9]  D. D. Givone,et al.  the allen-givone implementation oriented algebra , 1977 .

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

[11]  Oystein Ore,et al.  Number Theory and Its History , 1949 .

[12]  Graham A. Jullien,et al.  Implementation of Multiplication, Modulo a Prime Number, with Applications to Number Theoretic Transforms , 1980, IEEE Transactions on Computers.

[13]  R. Krishnan,et al.  Implementation of complex number theoretic transforms using quadratic residue number systems , 1986 .

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

[15]  H. Nussbaumer Digital filtering using pseudo fermat number transforms , 1977 .

[16]  J. Brule,et al.  Fast convolution with finite field fast transforms , 1975 .

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

[18]  H. Lu,et al.  MCD-to-m-valued and m-valued-to-MCD converters , 1988, [1988] Proceedings. The Eighteenth International Symposium on Multiple-Valued Logic.

[19]  Henri J. Nussbaumer Digital filtering using complex Mersenne transforms , 1976 .

[20]  P. Chevillat Transform-domain digital filtering with number theoretic transforms and limited word lengths , 1978 .