The Denormal Logarithmic Number System

Abstract

Abstract-Economical hardware often uses a FiXed-point Number System (FXNS), whose constant absolute precision is acceptable for many signal-processing algorithms. The almostconstant relative precision of the more expensive Floating-Point (FP) number system simplifies design, for example, by eliminating worries about FXNS overflow because the range of FP is much larger than FXNS for the same wordsize; however, primitive FP introduces another problem: underflow. The conventional Signed Logarithmic Number System (SLNS) offers similar range and precision as FP with much better performance (in terms of power, speed and area) for multiplication, division, powers and roots. Moderate-precision addition in SLNS uses table lookup with properties similar to FP (including underflow). This paper proposes a new number system, called the Denormal LNS (DLNS), which is a hybrid of the properties of FXNS and SLNS. The inspiration for DLNS comes from the denormal numbers found in IEEE-754 (that provide better, gradual underflow) and the µ-law often used for speech encoding; the novel DLNS circuit here allows arithmetic to be performed directly on such encoded data. The proposed approach allows customizing the range in which gradual underflow occurs. A wide gradual underflow range acts like FXNS; a narrow one acts like SLNS. Simulation of an FFT application illustrates a moderate gradual underflow decreasing bit-switching activity 15% compared to underflowfree SLNS, at the cost of increasing application error by 30%. DLNS reduces switching activity 5% to 20% more than an abruptly-underflowing SLNS with one-half the error. Synthesis shows the novel circuit primarily consists of traditional SLNS addition and subtraction tables, with additional datapaths that allow the novel ALU to act on conventional SLNS as well as DLNS and mixed data, for a worst-case area overhead of 26%.

Options for Denormal Representation in Logarithmic Arithmetic

“…Furthermore, while [2] only applies to denormals patterned after IEEE-754, the novel approach in this paper suggests a range of denormal representations (from one similar to IEEE-754 to a fully-denormal one similar to the µ-law for speech encoding [30]). This paper is an extended version of [47].…”

RISC Conversions for LNS Arithmetic in Embedded Systems

“…In a very interesting paper [10], it is proposed how numbers close to zero can be represented in the denormal LNS method (DLNS) using either fixed-point or LNS representations, guaranteeing constant absolute or constant relative precisions, respectively. Up to now, LNS have not been standardized.…”