Options for Denormal Representation in Logarithmic Arithmetic

Mark G. Arnold 1 Sylvain Collange 2
2 ALF - Amdahl's Law is Forever
Inria Rennes – Bretagne Atlantique , IRISA-D3 - ARCHITECTURE
Abstract : Economical hardware often uses a FiXed-point Number System (FXNS), whose constant absolute precision is acceptable for many signal-processing algorithms. The almost-constant 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 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. This paper proposes three variations of a new number system, respectively called the Denormal LNS (DLNS), Denormal Mitchell LNS (DMLNS) and Denormal Offset Mitchell LNS (DOMLNS), which are all hybrids of the properties of FXNS and SLNS. The inspiration for D(OM)LNS comes from the denormal (aka subnormal) 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. Our first DLNS implementation leverages existing SLNS basic blocks. Synthesis shows the novel circuit primarily consists of traditional SLNS addition and subtraction tables, with additional datapaths that allow the novel arithmetic unit to act on conventional SLNS as well as DLNS and mixed data, for a worst-case area overhead of 26 %. Unlike SLNS, this DLNS implementation is still costly for general (non-constant) multiplication, division and roots. To overcome this difficulty, this paper proposes the other variations called Denormal Mitchell LNS (DMLNS) and Denormal Offset Mitchell LNS (DOMLNS), in which the well-known Mitchell’s method makes the cost of general multiplication, division and roots closer to that of SLNS. Taylor-series computations suggest subnormal values in DMLNS and DOMLNS also behave similarly to those in the IEEE-754 FP standard. Synthesis shows that DMLNS and DOMLNS respectively have average area overheads of 25 % and 17 % compared to an equivalent SLNS 5-operation unit.
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Contributor : Sylvain Collange <>
Submitted on : Tuesday, November 25, 2014 - 2:27:13 PM
Last modification on : Thursday, November 15, 2018 - 11:57:43 AM

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Mark G. Arnold, Sylvain Collange. Options for Denormal Representation in Logarithmic Arithmetic. Journal of VLSI Signal Processing Systems for Signal, Image, and Video Technology, Springer, 2014, 77 (1-2), pp.207-220. ⟨10.1007/s11265-014-0874-3⟩. ⟨hal-01087059⟩



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