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| [[File:Gaussian logarithm.svg|thumb|300px|right|The <math>s_b(z)</math> and <math>d_b(z)</math> functions.]]
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| A '''logarithmic number system''' ('''LNS''') is an arithmetic system used for representing real numbers in computer and [[digital hardware]], especially for [[digital signal processing]].
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| ==Theory==
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| In LNS, a number, <math>X</math>, is represented by the [[logarithm]], <math>x</math>, of its [[absolute value]] as follows:
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| :<math>X\rightarrow\{s,x=\log_b(|X|)\},</math>
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| where <math>s</math> is a bit denoting the sign of <math>X</math> (<math>s=0</math> if <math>X>0</math> and <math>s=1</math> if <math>X<0</math>).
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| The number <math>x</math> is represented by a binary word which usually is in the [[two's complement]] format. LNS can be considered as a [[floating-point]] number with the [[significand]] being always equal to 1. This formulation simplifies the operations of multiplication, division, powers and roots, since they are reduced down to addition, subtraction, multiplication and division, respectively. | |
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| On the other hand, the operations of addition and subtraction are more complicated and they are calculated by the formula:
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| :<math>\log_b(|X|+|Y|)=x+s_b(z)</math>
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| :<math> \log_b(||X|-|Y||)=x+d_b(z),</math>
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| where <math>z=y-x</math> is the difference between the logarithms of the operands, the "sum" function is <math>s_b(z)=\log_b(1+b^z)</math>, and the "difference" function is <math>d_b(z)=\log_b(|1-b^z|)</math>. These functions <math>s_b(z)</math> and <math>d_b(z)</math>, depicted in the figures to the right, 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 LNS primarily for increasing the precision of floating-point math operations.
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| == History ==
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| Logarithmic number systems have been independently invented and published at least three times, as an alternative to [[Fixed-point arithmetic|fixed-point]] and [[floating-point]] number systems.<ref>{{cite journal | author = S. C. Lee and A. D. Edgar | title = Addendum to "The Focus Number System" | journal = IEEE Transactions on Computers | volume = C-28 | issue = 9 | date = Sept. 1979 | pages = 693 | url = http://ieeexplore.ieee.org/xpls/abs_all.jsp?arnumber=1675442 | doi = 10.1109/TC.1979.1675442 }}</ref>
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| Kingsbury and Rayner introduced "logarithmic arithmetic" for [[digital signal processing]] in 1971.<ref>{{cite journal | author = N. G. Kingsburg and P. J. W. Rayner | title = Digital filtering using logarithmic arithmetic | journal = Electronics Letters | volume = 7 | date = Jan. 28, 1971 | pages = 55 | url = http://scitation.aip.org/getabs/servlet/GetabsServlet?prog=normal&id=ELLEAK000007000002000056000001 }}</ref>
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| A similar LNS was described in 1975 by Swartzlander and Alexopoulos; rather than use two's complement notation for the logarithms, they offset them (scale the numbers being represented) to avoid negative logs.<ref>{{cite journal |author=E. E. Swartzlander and A. G. Alexopoulos |title=The Sign/Logarithm Number System |journal=IEEE Transactions on Computers |volume=C-24 |issue=12 |pages=1238–1242 |date=December 1975 | url = http://ieeexplore.ieee.org/xpls/abs_all.jsp?arnumber=1672765 |doi=10.1109/T-C.1975.224172 }}</ref>
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| Lee and Edgar described a similar system, which they called the "focus" number system, in 1977.<ref>{{cite journal | author = S. C. Lee and A. D. Edgar | title = The focus number system | journal = IEEE Transactions on Computers | volume = C-26 | issue = 11 | date = Nov. 1977 | pages = 1167–1170 | url = http://ieeexplore.ieee.org/xpls/abs_all.jsp?arnumber=1674770 | doi = 10.1109/TC.1977.1674770}}</ref>
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| The mathematical foundations for addition and subtraction in an LNS trace back to [[Carl Friedrich Gauss]] and Z. Leonelli.<ref>{{cite web |url=http://www.1911encyclopedia.org/Logarithm#Addition_and_Subtraction.2C_or_Gaussian_Logarithms |title=Logarithm: Addition and Subtraction, or Gaussian Logarithms |publisher=[[Encyclopædia Britannica Eleventh Edition]]}}</ref><ref>{{cite book | title = Carl Friedrich Gauss | author = G. Waldo Dunnington | publisher = MAA | year = 2002 | isbn = 0-88385-547-X | url = http://books.google.com/?id=_dce9Jaq7iQC&pg=PA112&dq=Leonelli+gaussian#PPA112,M1 }}</ref>
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| == Applications ==
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| LNS has been used in the [[Gravity Pipe]] (GRAPE) special-purpose supercomputer<ref>{{cite book |author=J Makino and M. Taiji |title=Scientific Simulations with Special Purpose Computers: The GRAPE Systems |publisher=Wiley |year=1998 |isbn=978-0-471-96946-4 | url = http://books.google.com/?id=JHJQAAAAMAAJ&dq=intitle%3A%22GRAPE+Systems%22+logarithmic&q=logarithmic#search }}</ref> that won the [[Gordon Bell Prize]] in 1999.
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| A substantial effort to explore the applicability of LNS 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).<ref>{{cite journal |author=J. N. Coleman, C. I. Softley, J. Kadlec, R. Matousek, M. Tichy, Z. Pohl, A. Hermanek, and N. F. Benschop |title=The European Logarithmic Microprocessor |journal=IEEE Transactions on Computers |volume=57 |issue=4 |pages=532–546 |date=April 2008 |doi=10.1109/TC.2007.70791}}</ref> A fabricated prototype of the processor, which has a 32-bit cotransformation-based LNS arithmetic logic unit (ALU), demonstrated LNS as a "more accurate alternative to floating-point," with improved speed. Further improvement of the LNS design based on the ELM architecture has again shown its capability to offer significantly better in speed and more accurate than the floating-point.<ref>{{cite journal |author=R.C Ismail and J.N Coleman|title=ROM-less LNS |journal=2011 20th IEEE Symposium on Computer Arithmetic (ARITH) |pages=43–51 |date=July 2011 |doi=10.1109/ARITH.2011.15}}</ref>
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| LNS is sometimes used in [[FPGA]]-based applications where most arithmetic operations are multiplication or division.<ref>{{cite journal|doi=10.1109/FPT.2006.270342|title=Comparing Floating-point and Logarithmic Number Representations for Reconfigurable Acceleration|author=Haohuan Fu, Oskar Mencer, Wayne Luk| url=http://ieeexplore.ieee.org/xpl/freeabs_all.jsp?arnumber=4042464|journal=IEEE Conference on Field Programmable Technology |month=December|pages=337 |year=2006 |date=June 2010}}</ref>
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| == References ==
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| {{reflist}}
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| ==External links==
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| *[http://www.xlnsresearch.com A site that lists LNS papers]
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| *[http://www.ens-lyon.fr/LIP/Arenaire/Ware/FPLibrary/ A VHDL Library for LNS hardware generation]
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| [[Category:Computer arithmetic]]
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| [[Category:Digital signal processing]]
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| [[Category:Logarithms]]
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