Abstract
Several Residue number systems which lead to certain advantages in Signal Processing applications have been described in literature. These are based on concepts of Quadratic Residues, Polynomial Residue Number systems, Modulus replication, logarithmic number systems and those using specialized moduli. These are considered in detail. Applications of these concepts and techniques for achieving fault tolerance are described in later Chapters.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Similar content being viewed by others
References
W.K. Jenkins, J.V. Krogmeier, The design of dual-mode complex signal processors based on Quadratic modular number codes. IEEE Trans. Circuits Syst. 34, 354–364 (1987)
G.A. Jullien, R. Krishnan, W.C. Miller, Complex digital signal processing over finite rings. IEEE Trans. Circuits Syst. 34, 365–377 (1987)
M.A. Soderstrand, G.D. Poe, Application of quadratic-like complex residue number systems to ultrasonics, in IEEE International Conference on ASSP, vol. 2, pp. 28.A5.1–28.A5.4 (1984)
R. Krishnan, G.A. Jullien, W.C. Miller, Complex digital signal processing using quadratic residue number systems. IEEE Trans. ASSP 34, 166–177 (1986)
V. Paliouras, T. Stouraitis, Novel high radix residue number system architectures. IEEE Trans. Circuits Syst. II 47, 1059–1073 (2000)
V. Paliouras, T. Stouraitis, Novel high-radix residue number system multipliers and adders, in Proceedings of ISCAS, pp. 451–454 (1999)
I. Kouretas, V. Paliouras, A low-complexity high-radix RNS multiplier. IEEE Trans. Circuits Syst. Regul. Pap. 56, 2449–2462 (2009)
I. Kouretas, V. Paliouras, High radix redundant circuits for RNS moduli r n-1, rn and rn + 1, in Proceedings of IEEE ISCAS, vol. V, pp. 229–232 (2003)
I. Kouretas, V. Paliouras, High-radix r n-1 modulo multipliers and adders, in Proceedings of 9th IEEE International Conference on Electronics, Circuits and Systems, vol. II, pp. 561–564 (2002)
M. Abdallah, A. Skavantzos, On multi-moduli residue number systems with moduli of the form r a, r b-1 and r c+1. IEEE Trans. Circuits Syst. 52, 1253–1266 (2005)
A. Skavantzos, F.J. Taylor, On the polynomial residue number system. IEEE Trans. Signal Process. 39, 376–382 (1991)
A. Skavantzos, T. Stouraitis, Polynomial residue complex signal processing. IEEE Trans. Circuits Syst. 40, 342–344 (1993)
M.C. Yang, J.L. Wu, A new interpretation of “Polynomial Residue Number System”. IEEE Trans. Signal Process. 42, 2190–2191 (1994)
V. Paliouras, A. Skavantzos, T. Stouraitis, Multi-voltage low power convolvers using the Polynomial Residue Number System, in Proceedings 12th ACM Great Lakes Symposium on VLSI, pp. 7–11 (2002)
M. Abdallah, A. Skavantzos, The multipolynomial Channel Polynomial Residue Arithmetic System. IEEE Trans. Circuits Syst. II Analog Digit. Signal Process. 46, 165–171 (1999)
V. Paliouras, A. Skavantzos, Novel forward and inverse PRNS converters of reduced computational complexity, in 36th Asilomar Conference on Signals, Systems and Computers, pp. 1603–1607 (2002)
H.C. Shyu, T.K. Truong, I.S. Reed, A complex integer multiplier using the quadratic-polynomial residue number system with numbers of form 22n+1. IEEE Trans. Comput. C-36, 1255–1258 (1987)
A. Skavantzos, N. Mitash, Implementation issues of 2-dimensional polynomial multipliers for signal processing using residue arithmetic, in IEE Proceedings-E, vol. 140, pp. 45–53 (1993)
A. Skavantzos, N. Mitash, Computing large polynomial products using modular arithmetic. IEEE Trans. Circuits Syst. II Analog Digit. Signal Process. 39, 252–254 (1992)
B. Singh, M.U. Siddiqui, Multivariate polynomial products over modular rings using residue arithmetic. IEEE Trans. Signal Process. 43, 1310–1312 (1995)
P.E. Beckmann, B.R. Musicus, Fast fault-tolerant digital convolution using a polynomial Residue Number System. IEEE Trans. Signal Process. 41, 2300–2313 (1993)
M.G. Parker, M. Benaissa, GF(pm) multiplication using Polynomial Residue umber Systems. IEEE Trans. Circuits Syst. II Analog Digit. Signal Process. 42, 718–721 (1995)
J. Chu, M. Benaissa, Polynomial residue number system GF(2m) multiplier using trinomials, in 17th European Signal Processing Conference (EUSIPCO 2009), Glasgow, Scotland, pp. 958–962 (2009)
J. Chu, M. Benaissa, GF(2m) Multiplier using Polynomial Residue Number System, in IEEE Asia Pacific Conference on Circuits and Systems, pp. 1514–1517 (2008)
J. Chu, M. Benaissa, A novel architecture of implementing error detecting AES using PRNS, in 14th Euromicro Conference on Digital System Design, pp. 667–673 (2011)
N.M. Wigley, G.A. Jullien, D. Reaume, Large dynamic range computations over small finite rings. IEEE Trans. Comput. 43, 78–86 (1994)
L. Imbert, G.A. Jullien, Fault tolerant computation of large inner products. Electron. Lett. 37, 551–552 (2001)
L. Imbert, V. Dimitrov, G.A. Jullien, Fault-tolerant computations over replicated finite rings. IEEE Trans. Circuits Syst. I Fundam. Theory Appl. 50, 858–864 (2003)
D.E. Knuth, The Art of Computer Programming. Seminumerical algorithms, vol. 2, 3rd edn. (Addison-Wesley, Boston, 1997)
L. Imbert, G. A. Jullien. Efficient fault-tolerant arithmetic using a symmetrical modulus replication RNS. in 2001 IEEE Workshop on Signal Processing Systems, Design and Implementation, SIPS’01, pp. 93–100 (2001)
L. Imbert, G.A. Jullien, V. Dimitrov, A. Garg, Fault tolerant complex FIR filter architectures using a redundant MRRNS, in Conference Records of The 35th Asilomar Conference on Signals, Systems, and Computers, vol. 2, pp. 1222–1226 (2001)
N.M. Wigley, G.A. Jullien, On modulus replication for residue arithmetic computations of complex inner products. IEEE Trans. Comput. 39(8), 1065–1076 (1990)
P. Chan, G.A. Jullien, L. Imbert, V. Dimitrov, G.H. McGibney, Fault-tolerant computations within complex FIR filters, in 2004 IEEE Workshop on Signal Processing Systems, Design and Implementation, SIPS’04, pp. 316–320 (2004)
I. Steiner, P. Chan, L. Imbert, G.A. Jullien, V. Dimitrov, G.H. McGibney, A fault-tolerant modulus replication complex FIR filter, in Proceedings 16th IEEE International Conference on Application-Specific Systems, Architecture Processors, ASAP’05, pp. 387–392 (2005)
C. Radhakrishnan, W.K. Jenkins, Z. Raza, R.M. Nickel, Fault tolerant Fermat Number Transform domain adaptive filters based on modulus replication RNS architectures, in Proceedings of 24th Asilomar Conference on Signals, Systems and Computers, pp. 1365–1369 (2009)
A.P. Preethy, D. Radhakrishnan, RNS-based logarithmic adder, in IEE Proceedings—Computers and Digital Techniques, vol. 147, pp. 283–287 (2000)
M.L. Gardner, L. Yu, J.W. Muthumbi, O.B. Mbowe, D. Radhakrishnan, A.P. Preethy, ROM efficient logarithmic addition in RNS, in Proceedings of 7th International Symposium on Consumer Electronics, ISCE-2003, Sydney (December 2003)
M.G. Arnold, The residue Logarithmic Number system: theory and implementation, in 17th IEEE Symposium on Computer Arithmetic, Cape Code, pp. 196–205 (2005)
Further Reading
J.H. Cozzens, L.A. Fenkelstein, Computing the discrete Fourier transform using residue number systems in a ring of algebraic integers. IEEE Trans. Inf. Theory 31, 580–588 (1985)
H.K. Garg, F.V.C. Mendis, On fault-tolerant Polynomial Residue Number systems, in Conference Record of the 31st Asilomar Conference on Signals, Systems and Computers, pp. 206–209 (1997)
G.A. Jullien, W. Luo, N.M. Wigley, High throughput VLSI DSP using replicated finite rings, J. VLSI Signal Process. 14(2), 207–220 (1996)
J.B. Martens, M.C. Vanwormhoudt, Convolutions of long integer sequences by means of number theoretic transforms over residue class polynomial rings. IEEE Trans. Acoust. Speech Signal Process. 31, 1125–1134 (1983)
J.D. Mellott, J.C. Smith, F.J. Taylor, The Gauss machine: a Galois enhanced Quadratic residue Number system Systolic array, in Proceedings of 11th Symposium on Computer Arithmetic, pp. 156–162 (1993)
M. Shahkarami, G.A. Jullien, R. Muscedere, B. Li, W.C. Miller, General purpose FIR filter arrays using optimized redundancy over direct product polynomial rings, in 32nd Asilomar Conference on Signals, Systems & Computers, vol. 2, pp. 1209–1213 (1998)
N. Wigley, G.A. Jullien, W.C. Miller, The modular replication RNS (MRRNS): a comparative study, in Proceedings of 24th Asilomar Conference on Signals, Systems and Computers, pp. 836–840 (1990)
N. Wigley, G.A. Jullien, D. Reaume, W.C. Miller, Small moduli replications in the MRRNS, in Proceedings of the 10th IEEE Symposium on Computer Arithmetic, Grenoble, France, June 26–28, pp. 92–99 (1991)
G.S. Zelniker, F.J. Taylor, Prime blocklength discrete Fourier transforms using the Polynomial Residue Number System, in 24th Asilomar Conference on Signals, Systems and Computers, pp. 314–318 (1990)
G.S. Zelniker, F.J. Taylor, On the reduction in multiplicative complexity achieved by the Polynomial Residue Number System. IEEE Trans. Signal Process. 40, 2318–2320 (1992)
Author information
Authors and Affiliations
Rights and permissions
Copyright information
© 2016 Springer International Publishing Switzerland
About this chapter
Cite this chapter
Ananda Mohan, P.V. (2016). Specialized Residue Number Systems. In: Residue Number Systems. Birkhäuser, Cham. https://doi.org/10.1007/978-3-319-41385-3_8
Download citation
DOI: https://doi.org/10.1007/978-3-319-41385-3_8
Published:
Publisher Name: Birkhäuser, Cham
Print ISBN: 978-3-319-41383-9
Online ISBN: 978-3-319-41385-3
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)