A logarithmic number system (LNS) is an arithmetic system used for representing real numbers in computer and digital hardware, especially for digital signal processing.
A number,
, is represented in an LNS by two components: the logarithm (
) of its absolute value (as a binary word usually in two's complement), and its sign bit (
):
![{\displaystyle X\rightarrow {\begin{cases}x=\log _{b}{\big |}X{\big |}\,,\\s={\begin{cases}0{\text{ if }}X>0\,,\\1{\text{ if }}X<0\,.\end{cases}}\end{cases}}}](//wikimedia.org/api/rest_v1/media/math/render/svg/4e0a85b3541a7b9112ae2e0d312fff35fe6170ed)
An LNS can be considered as a floating-point number with the significand being always equal to 1 and a non-integer exponent. This formulation simplifies the operations of multiplication, division, powers and roots, since they are reduced down to addition, subtraction, multiplication, and division, respectively.
On the other hand, the operations of addition and subtraction are more complicated and they are calculated by the formulae:
![{\displaystyle \log _{b}(|X|+|Y|)=x+s_{b}(y-x)\,,}](//wikimedia.org/api/rest_v1/media/math/render/svg/d0179f7a66798c0d147bdebe5197c907c7d1adda)
![{\displaystyle \log _{b}{\bigg |}|X|-|Y|{\bigg |}=x+d_{b}(y-x)\,,}](//wikimedia.org/api/rest_v1/media/math/render/svg/71215ec530ac6b5654ba9305072e1040b8e0f5ac)
where the "sum" function is defined by
, and the "difference" function by
. These functions
and
are also known as Gaussian logarithms.
The simplification of multiplication, division, roots, and powers is counterbalanced by the cost of evaluating these functions for addition and subtraction. This added cost of evaluation may not be critical when using an LNS primarily for increasing the precision of floating-point math operations.
Logarithmic number systems have been independently invented and published at least three times as an alternative to fixed-point and floating-point number systems.[1]
Nicholas Kingsbury and Peter Rayner introduced "logarithmic arithmetic" for digital signal processing (DSP) in 1971.[2]
A similar LNS named "signed logarithmic number system" (SLNS) was described in 1975 by Earl Swartzlander and Aristides Alexopoulos; rather than use two's complement notation for the logarithms, they offset them (scale the numbers being represented) to avoid negative logs.[3]
Samuel Lee and Albert Edgar described a similar system, which they called the "Focus" number system, in 1977.[4][1][5][6]
The mathematical foundations for addition and subtraction in an LNS trace back to Zecchini Leonelli and Carl Friedrich Gauss in the early 1800s.[7][8][9][10][11]
In the late 1800s, the Spanish engineer Leonardo Torres Quevedo conceived a series of analogue calculating mechanical machines[12][13] and developed one that could solve algebraic equations with eight terms, finding the roots, including the complex ones. One part of this machine called an "endless spindle" allowed the mechanical expression of the relation
,[14] with the aim of extracting the logarithm of a sum as a sum of logarithms.
A LNS has been used in the Gravity Pipe (GRAPE-5) special-purpose supercomputer[15] that won the Gordon Bell Prize in 1999.
A substantial effort to explore the applicability of LNSs as a viable alternative to floating point for general-purpose processing of single-precision real numbers is described in the context of the European Logarithmic Microprocessor (ELM).[16][17] A fabricated prototype of the processor, which has a 32-bit cotransformation-based LNS arithmetic logic unit (ALU), demonstrated LNSs as a "more accurate alternative to floating-point", with improved speed. Further improvement of the LNS design based on the ELM architecture has shown its capability to offer significantly higher speed and accuracy than floating-point as well.[18]
LNSs are sometimes used in FPGA-based applications where most arithmetic operations are multiplication or division.[19]
Leonelli, Zecchini (1803) [1802]. Supplément logarithmique. Théorie des logarithmes additionels et diductifs (in French). Bordeaux: Brossier. (NB. 1802/1803 is the year XI. in the French Republican Calendar.) Leonhardi, Gottfried Wilhelm (1806). LEONELLIs logarithmische Supplemente, als ein Beitrag, Mängel der gewöhnlichen Logarithmentafeln zu ersetzen. Aus dem Französischen nebst einigen Zusätzen von GOTTFRIED WILHELM LEONHARDI, Souslieutenant beim kurfürstlichen sächsischen Feldartilleriecorps (in German). Dresden: Walther'sche Hofbuchhandlung. (NB. An expanded translation of Zecchini Leonelli's Supplément logarithmique. Théorie des logarithmes additionels et diductifs.) Gauß, Johann Carl Friedrich (1808-02-12). "LEONELLI, Logarithmische Supplemente". Allgemeine Literaturzeitung (in German) (45). Halle-Leipzig: 353–356. Mehmke, Rudolf [in German] (1908). "I23". Encyclopédie des sciences mathematiques pures et appliquées. Paris, France: Gauthier-Villars. p. 351. Coleman, John Nicholas; Softley, Christopher I.; Kadlec, Jiri; Matousek, Rudolf; Licko, Miroslav; Pohl, Zdenek; Hermanek, Antonin (2002-08-07) [2001-11-04]. "The European Logarithmic Microprocessor – a QR RLS application". Conference Record of Thirty-Fifth Asilomar Conference on Signals, Systems and Computers (Cat.No.01CH37256). Vol. 1. Monterey, CA, USA: IEEE. pp. 155–159. doi:10.1109/ACSSC.2001.986897. ISBN 0-7803-7147-X. ISSN 1058-6393. Fu, Haohuan; Mencer, Oskar; Luk, Wayne (2007-01-02) [2006-12-13]. "Comparing floating-point and logarithmic number representations for reconfigurable acceleration". 2006 IEEE International Conference on Field Programmable Technology. IEEE. pp. 337–340. doi:10.1109/FPT.2006.270342. ISBN 978-0-7803-9728-6.
- Muller, Jean-Michel; Scherbyna, Alexandre; Tisserand, Arnaud (February 1998). "Semi-Logarithmic Number Systems" (PDF). IEEE Transactions on Computers. 47 (2): 145–151. doi:10.1109/12.663760. ISSN 0018-9340. Archived (PDF) from the original on 2018-07-13. Retrieved 2018-07-11. Previously published in: Muller, Jean-Michel; Scherbyna, Alexandre; Tisserand, Arnaud (July 1995). "Semi-Logarithmic Number Systems". Proceedings of the 12th IEEE Symposium on Computer Arithmetic (ARITH 12). Bath, UK.
- Kahrs, Mark; Brandenburg, Karlheinz, eds. (2002) [1998]. Applications of Digital Signal Processing to Audio and Acoustics (PDF). Kluwer Academic Publishing. ISBN 0-7923-8130-0. Archived (PDF) from the original on 2018-07-07. Retrieved 2018-07-07. (NB. Describes a 13-bit LNS used in Yamaha music synthesizers during the 1980s.)
- Kremer, Hermann (2002-08-29). "Gauss'sche Additionslogarithmen feiern 200. Geburtstag". de.sci.mathematik (in German). Archived from the original on 2018-07-07. Retrieved 2018-07-07.
- Zehendner, Eberhard (Summer 2008). "Rechnerarithmetik: Logarithmische Zahlensysteme" (PDF) (Lecture script) (in German). Friedrich-Schiller-Universität Jena. Archived (PDF) from the original on 2018-07-09. Retrieved 2018-07-09.
- Hayes, Brian (September–October 2009). "The Higher Arithmetic". American Scientist. 97 (5): 364–368. doi:10.1511/2009.80.364. Archived from the original on 2018-07-09. Retrieved 2018-07-09. . Also reprinted in: Hayes, Brian (2017). "Chapter 8: Higher Arithmetic". Foolproof, and Other Mathematical Meditations (1 ed.). The MIT Press. pp. 113–126. ISBN 978-0-26203686-3. ISBN 0-26203686-X.
- Amir Sabbagh, Molahosseini; de Sousa, Leonel Seabra; Chip-Hong Chang, eds. (2017-03-21). Embedded Systems Design with Special Arithmetic and Number Systems (1 ed.). Springer International Publishing AG. doi:10.1007/978-3-319-49742-6. ISBN 978-3-319-49741-9. LCCN 2017934074. (389 pages)