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LNS Mathematical and Algorithmic Theory

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LNS Mathematical and Algorithmic Theory

Papers on the mathematical basis of the Logarithmic Number System (LNS) and its applied algorithms in theory include:

  1. Behrooz Parhami, 2010, Computer Arithmetic: Algorithms and Hardware Designs, 2010, Oxford University Press, New York, NY, https://web.ece.ucsb.edu/~parhami/text_comp_arit.htm, https://books.google.com.au/books/about/Computer_Arithmetic.html?id=tEo_AQAAIAAJ&redir_esc=y
  2. Molahosseini AS, De Sousa LS, Chang C-H, 2017, Embedded systems design with special arithmetic and number systems, Springer. Book on Amazon: https://www.amazon.com/Embedded-Systems-Design-Special-Arithmetic-ebook/dp/B06XRVG3YF/, https://doi.org/10.1007/978-3-319-49742-6, https://link.springer.com/book/10.1007/978-3-319-49742-6 (A text that contains multiple papers on LNS and RNS.)
  3. B. Parhami, 2020, Computing with logarithmic number system arithmetic: Implementation methods and performance benefits, Computers & Electrical Engineering, vol. 87, p. 106800, 2020. https://www.sciencedirect.com/science/article/abs/pii/S0045790620306534
  4. Arnold, M.G., Bailey, T.A., Cowles, J.R., Winkel, M.D., 1992, Applying features of the IEEE 754 to sign/logarithm arithmetic, IEEE Transactions on Computers 41, 1040–1050 (1992) https://ieeexplore.ieee.org/document/156547
  5. Paliouras, V., Stouraitis, T., 2001, Low-power properties of the Logarithmic Number System, Proceedings of 15th Symposium on Computer Arithmetic (ARITH15), Vail, CO, June 2001, pp. 229–236 (2001) https://ieeexplore.ieee.org/document/930124
  6. Paliouras, V., Stouraitis, T., 2000, Logarithmic number system for low-power arithmetic, In: Soudris, D.J., Pirsch, P., Barke, E. (eds.) PATMOS 2000. LNCS, vol. 1918, pp. 285–294. Springer, Heidelberg (2000), https://link.springer.com/chapter/10.1007/3-540-45373-3_30
  7. T. Stouraitis, 1986, Logarithmic Number System: Theory analysis and design, University of Florida, Ph.D. dissertation, University of Florida ProQuest Dissertations Publishing,  1986. 8704221 https://www.proquest.com/openview/0f48dddc19ec62058062ae1b32ee981d/1, https://openlibrary.org/books/OL25923701M/Logarithmic_number_system_theory_analysis_and_design
  8. F. J. Taylor, 1985, A hybrid floating-point logarithmic number system processor, IEEE Trans. Circuits Syst., vol. CAS-32, pp. 92-95, Jan. 1985. https://ieeexplore.ieee.org/abstract/document/1085588
  9. M. L. Frey and F. J. Taylor, 1985, A table reduction technique for logarithmically architected digital filters, IEEE Trans. Acoust Speech Signal Processing, vol. ASSP-33, pp. 718-719, June 1985. https://ieeexplore.ieee.org/document/1164597
  10. E. E. Swartzlander, D. V. S. Chandra, H. T. Nagle and S. A. Starks, 1983, Sign/logarithm arithmetic for FFT implementation, IEEE Trans. Comput., vol. C-32, pp. 526-534, June 1983. https://ieeexplore.ieee.org/document/1676274
  11. G. L. Sicuranza, 1983, On efficient implementations of 2-D digital filters using logarithmic number systems, IEEE Trans. Acoust. Speech Signal Processing, vol. ASSP-31, pp. 877-885, Aug. 1983. https://ieeexplore.ieee.org/document/1164149 (Algorithms for LNS arithmetic.)
  12. M. L. Frey and F. J. Taylor, 1985, A table reduction technique for logarithmically architected digital filters, IEEE Trans. Acoust. Speech Signal Processing, vol. ASSP-33, pp. 719-719, June 1985. https://ieeexplore.ieee.org/document/1164597 (Reducing lookup table sizes for LNS.)
  13. H. Fu, O. Mencer and W. Luk, 2010, FPGA Designs with Optimized Logarithmic Arithmetic, IEEE Trans. Computers, vol. 59, no. 7, pp. 1000-1006, July 2010. https://ieeexplore.ieee.org/document/5416693 (LNS on FPGAs.)
  14. Chih-Wei Liu; Shih-Hao Ou; Kuo-Chiang Chang; Tzung-Ching Lin; Shin-Kai Chen, 2016, A Low-Error, Cost-Efficient Design Procedure for Evaluating Logarithms to Be Used in a Logarithmic Arithmetic Processor, IEEE Trans. Computers (April 2016) https://ieeexplore.ieee.org/document/7118135 (Algorithms for the initial logarithmic conversion from a floating-point into an LNS representation.)
  15. H. L. Garner, 1965, Number Systems and Arithmetic, in Advances in Computers, Vol. 6, F. L. Alt and M. Rubinoff (eds.), Academic Press, 1965. https://www.sciencedirect.com/science/article/abs/pii/S0065245808604209
  16. N. G. Kingsbury and P. J. W. Rayner, 1971, Digital Filtering Using Logarithmic Arithmetic, Electronics Letters, Vol. 7, pp. 56-58, 1971. https://digital-library.theiet.org/content/journals/10.1049/el_19710039 (Early paper on logarithmic numbers.)
  17. Tso-Bing Juang, Pramod Kumar Meher and Kai-Shiang Jan, 2011, High-Performance Logarithmic Converters Using Novel Two-Region Bit-Level Manipulation Schemes, Proc. of VLSI-DAT (VLSI Symposium on Design, Automation, and Testing), pp. 390-393, April 2011. https://ieeexplore.ieee.org/document/5783555
  18. Tso-Bing Juang, Han-Lung Kuo and Kai-Shiang Jan, 2016, Lower-Error and Area-Efficient Antilogarithmic Converters with Bit-Correction Schemes, Journal of the Chinese Institute of Engineers, Vol. 39, No. 1, pp. 57-63, Jan. 2016. https://www.tandfonline.com/doi/abs/10.1080/02533839.2015.1070692?journalCode=tcie20
  19. Ying Wu, Chuangtao Chen, Weihua Xiao, Xuan Wang, Chenyi Wen, Jie Han, Xunzhao Yin, Weikang Qian, Cheng Zhuo, 2023, A Survey on Approximate Multiplier Designs for Energy Efficiency: From Algorithms to Circuits, ACM Transactions on Design Automation of Electronic Systems, 2023. https://doi.org/10.1145/3610291, https://arxiv.org/abs/2301.12181 (Extensive survey of many approximate multiplication algorithms.)
  20. Patrick Robertson, Emmanuelle Villebrun, Peter Hoeher, et al., 1995, A comparison of optimal and sub-optimal map decoding algorithms operating in the log domain, in IEEE International Conference on Communications, 1995. https://ieeexplore.ieee.org/document/524253
  21. Mark G. Arnold, 2014, LNS References, XLNS Research, http://www.xlnsresearch.com/home.htm (An exhaustive list of LNS research articles up to around 2014.)
  22. N. G. Kingsbury and P .J. W. Rayner, 1971, Digital Filtering Using Logarithmic Arithmetic, Electronics Letters, 7, pp 56-58, 1971, https://www.infona.pl/resource/bwmeta1.element.ieee-art-000004235144
  23. F. Albu; J. Kadlec; N. Coleman; A. Fagan, 2002, The Gauss-Seidel fast affine projection algorithm, IEEE Workshop on Signal Processing Systems, https://ieeexplore.ieee.org/abstract/document/1049694/, PDF: https://www.academia.edu/download/32934948/sips2002.pdf (Simplistic coverage of LNS addition with just exponentiation.)

For more research papers on computational theory for LNS models, see https://www.aussieai.com/research/logarithmic#theory.

 

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