Plenary Lecture

Plenary Lecture

Logarithmic Number Systems

Professor Mark Arnold
Lehigh University
Bethlehem, PA

Abstract: The Logarithmic Number System (LNS) represents real numbers using a finite precision logarithm. Like any finite representation, the number of bits chosen determines the resolution of the system and therefore the application performance. LNS offers better performance and lower cost for "easy" real operations such as multiplication, division, roots and powers compared to fixed- and floating-point number systems where such operations are thought to be hard. The problems with LNS are that addition and especially subtraction are increasingly expensive when performed with extreme accuracy, because these operations involve table lookup and possibly interpolation. Also, conversion to and from conventional representations can be similarly expensive. Another inconvenience is the fact the logarithm of zero is undefined. This talk will consider how certain special-purpose applications have overcome these problems to exploit LNS advantages, giving hardware that is faster, cheaper and consumes less power than those based on traditional arithmetic.
Examples of special-purpose systems that have adopted LNS successfully include neural networks, multimedia encoders/decoders, control systems, speech recognition and N-body simulators. In each of these applications, designers have reformulated the algorithm to avoid certain LNS weaknesses. LNS works for such applications because they have a large share of "easy" operations and they tolerate lower-precision results. Traditionally, LNS sum and difference calculation have carried out with enough accuracy to be faithful to the number of bits of precision required for the application, however this can be relaxed in some cases. To minimize the cost of the LNS, simulative studies determine the minimum number of bits for in the LNS representation for the application to operate successfully.
This talk will explore many LNS techniques. These include interpolation methods, cotransformation of difficult subtractions into easier additions, elimination of subtractions through redundancy, bit-serial arithmetic and ROM-less approximations. Also, this talk will consider recent implementations that generalize LNS, such as for complex values.

Brief Biography of the Speaker:
Mark G. Arnold received the BS and MS from the University of Wyoming (USA), and the PhD from the University of Manchester (UK) Institute of Science and Technology (UMIST). From 1982 to 2000, he was on the faculty of the University of Wyoming. From 2000 to 2002, he was a lecturer at UMIST. In 2002, he joined the faculty of Lehigh University (USA). In 1976, he co-developed SCELBAL, the first open-source floating-point high-level language for personal computers. In 1997, he received the best paper award from Open Verilog International for describing the Verilog Implicit To One-hot (VITO) tool he co-developed. In 2007, he received the best paper award from the Application-specific Systems, Architectures and Processors (ASAP) conference for a paper describing novel cotransformations for Logarithmic Number Systems (LNS). He is the author of over one hundred technical papers (the majority on LNS) and the book Verilog Digital Computer Design. His research interests include computer arithmetic, hardware description languages, microrobotics and embedded, control, multimedia and application-specific systems.

WSEAS Unifying the Science