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Arithmetic Co-Transformations in the Real and Complex Logarithmic Number Systems
July 1998 (vol. 47 no. 7)
pp. 777-786

Abstract—The real logarithmic number system, which represents a value with a sign bit and a quantized logarithm, can be generalized to create the complex logarithmic number system, which replaces the sign bit with a quantized angle in a log/polar coordinate system. Although multiplication and related operations are easy in both real and complex systems, addition and subtraction are hard, especially when interpolation is used to implement the system. Both real and complex logarithmic arithmetic benefit from the use of co-transformation, which converts an addition or subtraction from a region where interpolation is expensive to a region where it is easier. Two co-transformations that accomplish this goal are introduced. The first is an approximation based on real analysis of the subtraction logarithm. The second is based on simple algebra that applies for both real and complex values and that works for both addition and subtraction.

[1] M.G. Arnold, T.A. Bailey, J.R. Cowles, and J.J. Cupal, "Redundant Logarithmic Arithmetic," IEEE Trans. Computers, vol. 39, no. 8, pp. 1,077-1,086, Aug. 1990.
[2] M. Arnold, T. Bailey, J. Cowles, and J. Cupal, "Initializing RAM-Based Logarithmic Processors," J. VLSI Signal Processing, vol. 4, pp. 243-252, 1992.
[3] M. Arnold, T. Bailey, and J. Cowles, "Comments on 'An Architecture for Addition and Subtraction of Long Word Length Numbers in the Logarithmic Number System'," IEEE Trans. Computers, vol. 41, no. 6, pp. 786-788, June 1992.
[4] M.G. Arnold et al., "Applying Features of IEEE 754 to Sign/Logarithm Arithmetic," IEEE Trans. Computers, Aug. 1992, pp. 1,040-1,050.
[5] M.G. Arnold, "Method and Apparatus for Fast Logarithmic Addition and Subtraction," U. S. Patent 5,337,266, 9 Aug. 1994.
[6] M.G. Arnold, T.A. Bailey, J.J. Cupal, and M.D. Winkel, "On the Cost Effectiveness of Logarithmic Arithmetic for Back-Propagation Training on SIMD Processors," Proc. 1997 Int'l Conf. Neural Networks,Houston, vol. 2, pp. 933-936,9-12 June 1997.
[7] J.L. Barlow and E.H. Bareiss, "On Roundoff Distribution in Floating Point and Logarithmic Arithmetic," Computing, vol. 34, pp. 325-364, 1985.
[8] J.N. Coleman, "Simplification of Table Structure in Logarithmic Arithmetic," Electronic Letters, vol. 31, pp. 1,905-1,906, Oct. 1995.
[9] T.C. Chen, "Automatic Computation of Exponentials, Logarithms, Ratios and Square Roots," IBM J. Research&Development, pp. 380-388, July 1972.
[10] "FastMath: Software Faster Than a Coprocessor," C User's J., vol. 9, no. 7, p. 12, July 1991.
[11] S. Gundelfinger, "Zur Berechnung der Gausschen Logarithmen für Kliene Werthe von B resp. zugehörige Werthe von A," Journal für die reine und angewandte Mathematik, vol. 124, pp. 87-92, 1902.
[12] W.N. Holmes, "Composite Arithmetic: Proposal for a New Standard," Computer, vol. 30, no. 3, pp. 65-73, Mar. 1997.
[13] T. Kurokawa and T. Mizukoshi, "A Fast and Simple Method for Curve Drawing—A New Approach Using Logarithmic Number Systems," J. Information Processing, vol. 14, pp. 144-152, 1991.
[14] D.M. Lewis, "An Architecture for Addition and Subtraction of Long Word Length Numbers in the Logarithmic Number System," IEEE Trans. Computers, pp. 1,325-1,336, 1990.
[15] D.M. Lewis, "Interleaved Memory Function Interpolators with Application to an Accurate LNS Arithmetic Unit," IEEE Trans. Computers, Aug. 1994, pp. 974-982.
[16] D.M. Lewis, "114 MFLOPS Logarithmic Number System Arithmetic Unit for DSP Applications," Int'l Solid-State Circuits Conf., pp. 1,547-1,553,San Francisco, Feb. 1995.
[17] J.D. Marasa and D.W. Matula, "A Simulative Study of Correlated Error in Various Finite-Precision Arithmetics," IEEE Trans. Computers, vol. 22, no. 6, pp. 587-597, June 1973.
[18] R. Mehmke, "Additionslogarithmen für Complexe Grössen," Zeitschrift für Mathematik und Physik, vol. 40, pp. 15-30, 1895.
[19] J.M. Muller, "Une Mèthodologie du Calcul Hardware des FonctionsÈlèmentaires," Mathematical Modeling and Numerical Analysis, vol. 20, pp. 667-695, Sept. 1985.
[20] J.M. Muller, A. Tisserand, and A. Scherbyna, "Semi-Logarithmic Number System," Proc. 12th Symp. Computer Arithmetic, pp. 201-207,Bath, England,19-21 July 1995.
[21] R.E. Morley, T.J. Sullivan, and G.L. Engel, "VLSI Based Design of a Battery-Operated Hearing Aid," Proc. Southcon/90, pp. 55-59,Orlando, Fla.,20-22 Mar. 1990.
[22] Nell, "Ueber die Interpolationsrechnungen bei grosseren Logarithmentafeln," Zeitschrift fur Vermessungswesen, vol. 20, pp. 442-446, 1891.
[23] I. Orginos, V. Paliouras, and T. Stouraitis, "Novel Algorithm for Multi-Operand Logarithmic Number System Addition and Subtraction Using Polynomial Approximation," Proc. Int'l Symp. Circuits and Systems, pp. 1,992-1,995,Seattle, Apr. 30- May5 1995.
[24] V. Paliouras and T. Stouraitis, "Novel Algorithm for Accurate Logarithmic Number System Subtraction," Proc. Int'l Symp. Circuits and Systems, pp. 268-271,Atlanta, May12-15 1996.
[25] L. Pickett Private communication, Nov.17 1996.
[26] L. Pickett, "Method and Apparatus for Exponential/Logarithmic Computation," U. S. Patent 5,197,024, Mar.23 1993.
[27] T. Stouraitis, "Logarithmic Number System Theory, Analysis, and Design," PhD dissertation, Univ. of Florida, Gainesville, 1986.
[28] S. Shanks, "New Soft Co-Processor for Fujitsu's High-Performance Embedded Controllers," Embedded Control Design, vol. 1, p. 10, Fall 1994.
[29] E.E. Swartzlander and A.G. Alexopoulos, "The Sign/Logarithm Number System," IEEE Trans. Computers, vol. 24, no. 12, pp. 1,238-1,242, Dec 1975.
[30] E.E. Swartzlander et al., "Arithmetic for Ultrahigh Speed Tomography," IEEE Trans. Computers, vol. 29, pp. 341-353, 1980.
[31] E.E. Swartzlander et al., "Sign/Logarithm Arithmetic for FFT Implementation," IEEE Trans. Computers, vol. 32, pp. 526-534, 1983.
[32] F.J. Taylor, R. Gill, J. Joseph, and J. Radke, "A 20 Bit Logarithmic Number System Processor," IEEE Trans. Computers, vol. 37, pp. 190-199, 1988.
[33] P.R. Turner, "Complex SLI Arithmetic: Representation, Algorithms and Analysis," Proc. 11th Symp. Computer Arithmetic, pp. 18-25,Windsor, Ontario, Canada, June 29- July4 1993.

Index Terms:
Arithmetic co-transforamtions, logarithmic number systems, complex logarithms.
Citation:
Mark G. Arnold, Thomas A. Bailey, John R. Cowles, Mark D. Winkel, "Arithmetic Co-Transformations in the Real and Complex Logarithmic Number Systems," IEEE Transactions on Computers, vol. 47, no. 7, pp. 777-786, July 1998, doi:10.1109/12.709377
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