Logarithmic Number System for Low-power Arithmetic

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Logarithmic Number System for Low-power Arithmetic

• V. Paliouras and T. Stouraitis
• Electrical Computer Engineering Department
• University of Patras
• GREECE

2
The Presentation

• Background and motivation
• Representational capabilities of LNS
• Error behavior
• Equivalence to fixed-point representation
• LNS encoding and signal activity
• LNS architecture and operations
• Conclusions

3
Background

• Several ways to do arithmetic
• Fixed-point arithmetic
• Floating-point arithmetic
• Residue Number System
• Logarithmic Number System
• An old idea use logarithms of operands.
• When are representations equivalent ?
• Reluctance in adoption
• Wasted performance potential

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LNS Definition

• LNS maps a real number X to a triplet
• x has integer and fractional part xI.F
• k integer word length
• l fractional word length

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The Compression Property
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LNS Processing
LNS Data Processing
Converter
Converter
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LNS/FXP Equivalence

• Range
• Precision Different relative error behavior

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An Equivalence Condition

• A (k,l,b)-LNS covers a range at least as long as
  an n-bit fixed-point system with an average
  representational error equal or smaller to that
  of the fixed-point system, when l and k are
  given as

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n b1.5 b1.5 b1.5 b2 b2 b2 b2.5 b2.5 b2.5
n k l neq k l neq k l neq
5 3 2 6 3 3 9 2 3 7
6 4 3 10 3 4 9 2 4 7
7 4 4 10 3 5 9 3 5 12
8 4 5 10 3 6 9 3 6 12
9 4 5 10 4 7 17 3 7 12
10 5 6 20 4 8 17 3 7 12
11 5 7 20 4 9 17 3 8 12
12 5 8 20 4 10 17 4 9 23
13 5 9 20 4 11 17 4 10 23
14 5 10 20 4 12 17 4 11 23
15 5 11 20 4 13 17 4 12 23
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Activity and LNS Encoding
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LNS and Operations

• Z,?, ? linear and x,y,z log images

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LNS and Architecture

• Square root and square of number for free.
• Multiply and divide gt add and subtract.
• Addition and subtraction are more complicated.

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Add/Subtract Architecture

• Techniques for size reduction can be used for
  power dissipation reduction.

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LNS Add/Subtract vs. FXP Multiply
n (bits) Base b LNS LNS LNS n?n Multiplication
n (bits) Base b Addition Subtraction Average n?n Multiplication
8 1.5 0.99 0.99 0.99 2.02
8 2 0.90 0.89 0.89 2.02
8 2.5 0.98 0.98 0.98 2.02
10 1.5 1.14 1.16 1.15 2.64
10 2 1.18 1.18 1.18 2.64
10 2.5 1.09 1.11 1.10 2.64
12 1.5 1.49 1.52 1.50 3.35
12 2 1.58 1.58 1.58 3.35
12 2.5 1.46 1.49 1.48 3.35
14 1.5 2.38 2.46 2.42 4.13
14 2 2.62 2.61 2.62 4.13
14 2.5 2.36 2.36 2.31 4.13
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Conclusions

• Low-power LNS due to impact on
• Activity, by encoding
• Architecture, by operation simplification
• The LNS base is an important design parameter.
• Consider an LNS implementation for
  computationally intensive applications!