ISOMORPHIC TRANSFORMATION AND ITS APPLICATION TO THE MODULO (2^n+1) CHANNEL FOR RNS BASED FIR FILTER DESIGN
Abstract
In this paper, the implementation of a Finite Impulse Response (FIR) filter in the Residue Number System (RNS), is presented, in which a modulo multiplier based on the isomorphism technique is used to perform multiplication in the (2n+1) channel. An RNS modular multiplication in the Galois Field GF(2n+1) is presented in detail in this paper. The multiplication is based on the isomorphic mapping technique adapted to the residue arithmetic. The isomorphic encoder and decoder look-up tables in the GF(28+1) are given. An architecture for FIR filter design based on distributed arithmetic for multiplication and accumulation in mentioned (2n+1) channel is also presented. This architecture is discussed in details and compared with with architecture based on isomorphing technique.
References
Chen, G., Bai, G., & Chen, H. 2007. A New Systolic Architecture for Modular Division. IEEE Transactions on Computers, 56(2), pp. 282-286. doi:10.1109/tc.2007.20
Chen, R., Fan, J., & Liao, C. 2014. Reconfigurable Galois Field multiplier. In 2014 International Symposium on Biometrics and Security Technologies (ISBAST). Institute of Electrical and Electronics Engineers (IEEE)., pp. 112-115. doi:10.1109/isbast.2014.7013104
Jullien, 1980. Implementation of Multiplication, Modulo a Prime Number, with Applications to Number Theoretic Transforms. IEEE Transactions on Computers, C-29(10), pp. 899-905. doi:10.1109/tc.1980.1675473
Kitsos, P., Theodoridis, G., & Koufopavlou, O. 2003. An efficient reconfigurable multiplier architecture for Galois field GF(2m). Microelectronics Journal, 34(10), pp. 975-980. doi:10.1016/s0026-2692(03)00172-1
NagaJyothi, G., & SriDevi, S. 2017. Distributed arithmetic architectures for FIR filters-A comparative review. In 2017 International Conference on Wireless Communications, Signal Processing and Networking (WiSPNET). Institute of Electrical and Electronics Engineers (IEEE)., pp. 2684-2690. doi:10.1109/wispnet.2017.8300250
Nannarelli, A., Cardarilli, G. C., & Re, M. 2003. Power-delay tradeoffs in residue number system. In Proceedings of the 2003 International Symposium on Circuits and Systems, 2003. ISCAS '03..Institute of Electrical and Electronics Engineers (IEEE)., pp. 413-416. doi:10.1109/iscas.2003.1206300
Padmavathy, R., & Bhagvati, C. 2012. Discrete logarithm problem using index calculus method. Mathematical and Computer Modelling, 55(1-2), pp. 161-169. doi:10.1016/j.mcm.2011.02.022
Pradhan, 1978. A Theory of Galois Switching Functions. IEEE Transactions on Computers, C-27(3), pp. 239-248. doi:10.1109/tc.1978.1675077
Qi, H., Kim, Y., & Choi, M. 2012. A high speed low power modulo 2n + 1 multiplier design using carbon-nanotube technology. In 2012 IEEE 55th International Midwest Symposium on Circuits and Systems (MWSCAS). Institute of Electrical and Electronics Engineers (IEEE)., pp. 406-409. doi:10.1109/mwscas.2012.6292043
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