A Residue Arithmetic Implementation of the FFT

Abstract Recently, a new method of performing complex arithmetic in a residue arithmetic number system was reported. This paper extends these techniques into several application areas including the design of an FFT system. In addition, a new single-modulus complex residue number system (RNS) and sign management system is developed for use as a basic computational engine. Various RNS and conventional design options are developed and compared in the context of speed and complexity. The result of this analysis suggests that the new single-modulus complex RNS may be significantly superior to the alternative FFT design choices.

[1]  Charles M. Rader,et al.  Number theory in digital signal processing , 1979 .

[2]  W. Kenneth Jenkins,et al.  The use of residue number systems in the design of finite impulse response digital filters , 1977 .

[3]  Graham A. Jullien,et al.  Hardware implementation of convolution using number theoretic transforms , 1979, ICASSP.

[4]  Dilip K. Banerji,et al.  On Translation Algorithms in Residue Number Systems , 1972, IEEE Transactions on Computers.

[5]  Fred J. Taylor,et al.  An Autoscale Residue Multiplier , 1982, IEEE Transactions on Computers.

[6]  Fred J. Taylor,et al.  An efficient residue-to-decimal converter , 1981 .

[7]  D. Agrawal,et al.  Modulo (2n+1) arithmetic logic , 1978 .

[8]  Fred J. Taylor,et al.  A Radix-4 FFT Using Complex RNS Arithmetic , 1985, IEEE Transactions on Computers.

[9]  Graham A. Jullien,et al.  Residue Number Scaling and Other Operations Using ROM Arrays , 1978, IEEE Transactions on Computers.

[10]  Fred J. Taylor,et al.  A VLSI Residue Arithmetic Multiplier , 1982, IEEE Transactions on Computers.

[11]  Fred J. Taylor,et al.  A more efficient residue arithmetic implementation of the FFT , 1985, 1985 IEEE 7th Symposium on Computer Arithmetic (ARITH).

[12]  F. J. Taylor,et al.  Residue Arithmetic A Tutorial with Examples , 1984, Computer.

[13]  Graham A. Jullien,et al.  Implementation of FFT Structures Using the Residue Number System , 1979, IEEE Transactions on Computers.

[14]  I. Herstein,et al.  Topics in algebra , 1964 .

[15]  Allen M. Peterson,et al.  Fast fourier transform processors using Gaussian residue arithmetic , 1985, J. Parallel Distributed Comput..

[16]  H. Garner.  The residue number system , 1959, IRE-AIEE-ACM '59 (Western).

[17]  F. Taylor.  A single modulus complex ALU for signal processing , 1985, IEEE Trans. Acoust. Speech Signal Process..

[18]  Richard I. Tanaka,et al.  Residue arithmetic and its applications to computer technology , 1967 .

[19]  A. S. Ramnarayan.  Practical realisation of mod p, p prime multiplier , 1980 .

[20]  F J Taylor,et al.  Comparison of DFT algorithms using a residue architecture , 1981 .

[21]  David M. Burton.  Elementary number theory , 1980 .