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LNS Subtraction Using Novel Contransformation and/or Interpolation. Panagiotis Vouzis 1 , Sylvain Collange 2 , and Mark Arnold 1 {vouzis@lehigh.edu, sylvain.collange@ens-lyon.fr, maab@lehigh.edu} 1 Computer Engineering, Lehigh University, Bethlehem, USA 2 É cole Normale Sup é rieure de Lyon

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lns subtraction using novel contransformation and or interpolation

LNS Subtraction Using Novel Contransformation and/or Interpolation

Panagiotis Vouzis1, Sylvain Collange2, and Mark Arnold1

{vouzis@lehigh.edu, sylvain.collange@ens-lyon.fr, maab@lehigh.edu}

1Computer Engineering,

Lehigh University,

Bethlehem, USA

2École Normale Supérieure de Lyon

46 Allée d’Italie

6934 Lyon Cedex 07, France

ASAP 2007

outline
Outline
  • Logarithmic Number System
  • Addition/Subtraction Optimization
  • Cotransformations
  • Synthesis, Simulation and Error Behavior
  • Conclusions
outline1
Outline
  • Logarithmic Number System
  • Addition/Subtraction Optimization
  • Cotransformations
  • Synthesis, Simulation and Error Behavior
  • Conclusions
use of lns on embedded control

Input (16 bits)

16-bit

µP

Core

Matrix

Coprocessor

+

LNS

Output (16 bits)

Write

Read

CS

Data or Status

Use of LNS on Embedded Control

FPGA

Plant

Set point

u(k)

_

use of lns on embedded control1
Use of LNS on Embedded Control
  • Drug Delivery
  • Robotics
  • Microfluidic control
the logarithmic number system lns
The Logarithmic Number System (LNS)
  • x is a two’s complement representation with
    • k integer bits
    • l fractional bits

l

k

I. Koren, Computer Arithmetic Algorithms, MA: Brookside Court Publishers,

1998. Chapter 10: Logarithmic Number System

addition subtraction with z 0
Addition/Subtraction with z>0

x

(|x-y|,zs)

f(|x-y|),zs)

Preprocessing

y

t

+

min(x,y)

addition subtraction with z 01
Addition/Subtraction with z<0

10% – 20% Memory Savings

x

(-|x-y|,zs)

f(-|x-y|),zs)

Preprocessing

y

t

+

max(x,y)

addition subtraction
Addition/Subtraction

x

(z,zs)

f(z,zs)

Preprocessing

y

t

+

w

outline2
Outline
  • Logarithmic Number System
  • Addition/Subtraction Optimization
  • Cotransformations
  • Synthesis, Simulation and Error Behavior
  • Conclusions
addition subtraction optimization1
Addition/Subtraction Optimization
  • Tabulate only for z ≤ 0

E. E. Swartzlander and A. G. Alexopoulos, “The Sign/Logarithm Number System,” IEEE Transactions on Computers, No. 24, vol. 12, pp. 1238–1242, Dec. 1975.

addition subtraction optimization2
Addition/Subtraction Optimization
  • Tabulate only for z ≤ 0
  • Tabulate only for sb(z)>2–l, db(z)<–2–l

T. Stouraitis. Logarithmic Number System Theory, Analysis, and Design. PhD thesis, Univ. of Florida, Gainesville, Florida, 1986.

addition subtraction optimization3
Addition/Subtraction Optimization
  • Tabulate only for z ≤ 0
  • Tabulate only for sb(z)>2–l, db(z)<–2–l
  • Interpolate—Multipartite Tables

D. M. Lewis, “An Architecture for Addition and Subtraction of Long Word Length Numbers in the Logarithmic Number System,” IEEE Transactions on Computers, vol. 39, no. 11, pp. 1325–1336, 1990.

F. de Dinechin, and A. Tisserand, “Some Improvements on Multipartite Table Methods,” In Proceedings of the 15thSymposium on Computer Arithmetic, pp. 128–135, Vail, Colorado, 11–13 June 2001.

addition subtraction optimization4
Addition/Subtraction Optimization
  • Tabulate only for z ≤ 0
  • Tabulate only for sb(z)>2–l, db(z)<–2–l
  • Interpolate—Multipartite Tables
  • Cotransformation: db(z)=f(sb(t))

M. G. Arnold, “An Improved Cotransformation for Logarithmic Subtraction,” In Proceedings of the International Symposium on Circuits and Systems, pp. 752–755, Scottsdale, AZ, 26–29 May 2002.

outline3
Outline
  • Logarithmic Number System
  • Addition/Subtraction Optimization
  • Cotransformations
  • Synthesis, Simulation and Error Behavior
  • Conclusions
no cotransformation
No Cotransformation

x

z < 0

-|x-y|

max

+

s/d

y

Advantages

z < 0 means narrower s(z) width

Fastest logic

Disadvantages

size of d(z) > size of s(z)

Large d(z) table

coleman s cotransformation
Coleman’s Cotransformation

x

z < 0

-|x-y|

max

Coleman

t1

t2

+

s/d

y

Advantages

z < 0 means s(z) width slightly narrower

size of d(z)  size of s(z)

Disadvantage

still need d(z)

arnold s cotransformation
Arnold’s Cotransformation

x

z > 0

+|x-y|

min

Arnold

t1

t2

+

s

y

Advantage

Can eliminate d(z): low cost fast s(z) table

Disadvantage

z > 0 means slightly wider s

vouzis cotransformation
Vouzis’ Cotransformation

x

z < 0

-|x-y|

max

Vouzis

t1

t2

t3

+

s

y

Advantages

Can eliminate d(z): low cost fast s(z) table

z < 0 means s(z) width slightly narrower

Easiest HDL coding/fix bug in 2002 “improved” Arnold

Disadvantage

Need extra cotransformation table

novel cotransformation
Novel Cotransformation

x

z < 0

+|x-y|

min/max

Novel

t1

t2

+

y

s

Advantages

Can eliminate d(z): low cost fast s(z) table

z < 0 means s(z) width slightly narrower

No extra cotransformation table

Disadvantage

Tiny extra mux logic

coleman s cotransformation1
Coleman’s Cotransformation

z1

z2

J. N. Coleman, “Simplification of Table Structure in Logarithmic Arithmetic,” IEE Electronic Letters, vol 31, no. 22, pp. 1905-1906, 26 Oct. 1996.

coleman s cotransformation2
Coleman’s Cotransformation

z1

db(z1)

x

+

(-|x-y|)

sb(z)

Mux

1 0

Preprocessing

db(z2)

(-|x-y|)

y

t

+

zs

max(x,y)

arnold s cotransformation1
Arnold’s Cotransformation

z1

z2

M. G. Arnold, T. A. Bailey, J. R. Cowles, and M. D. Winkel, “Arithmetic Cotransformations in the Real and Complex

Logarithmic Number Systems,” IEEE Transactions on Computers, vol. 47, pp. 777-786, July 1998.

improved cotransformation
Improved Cotransformation

zh (k+(l –j) bits)

zl (j bits)

100 . . . 0 (h=2j –l, i.e., smallest zh  0)

M. G. Arnold, “An Improved Cotransformation for Logarithmic Subtraction,” In Proceedings of the International Symposium on Circuits and Systems, pp. 752–755, Scottsdale, AZ, 26–29 May 2002.

vouzis cotransformation1
Vouzis’ Cotransformation

P. Vouzis, S. Collange, and M. Arnold, “Cotransformation providesArea and Accuracy Improvements in an HDL library

for LNS Subtraction,” Accepted for The 10th EuroMicro Conference on Digital Systems and Design, Lübeck, Germany,

27–31 August, 2007.

graphics example
Graphics Example

Corrected by using a

2-value LUT

Caused by absence of

new special case

k=5, l=8, j=5

novel cotransformation combination
Novel Cotransformation Combination
  • Addition Precondition: z < 0
  • Subtraction Precondition: z > 0

Special cases needed for z1= 0 and z2= 0.

Special cases can be eliminated by storing appropriate values in the LUTs.

eliminating special cases
Eliminating Special Cases

l

  • Let
    • Case A: z1 = 0, z2 > 0
      • It is proven that t = min(x,y)+db(z2)
    • Case A: z1 > 0, z2 = 0
      • It is proven than t = min(x,y)+db(z1)

zH (k+n bits)

zL (l−n bits)

k

n

outline4
Outline
  • Logarithmic Number System
  • Addition/Subtraction Optimization
  • Cotransformations
  • Synthesis, Simulation and Error Behavior
  • Conclusions
guard bit simulation
Guard-bit Simulation

Minimal cotransformation guard bits and next-nearest probabilities.

Effect of (g) interpolator- and (h) cotransformation-guard bits on error and rounding for l=12.

l

optimized interpolation cotransformation hybrid
Optimized Interpolation/Cotransformation Hybrid

l

k

11…110

zH

zL

Assume power-of-two partitioning

n

Partial Interpolation

(k+l-2m)2n

Cotransformation

2∙2m

Full Interpolation

(k+l-n)2n

error behavior
Error Behavior

sb(z) by 2nd-ord. multip.

db(z) by vouzis’ cotran.

sb(z) by 1st-ord. interp.

db(z) by novel cotran.

sb(z) by 2nd-ord. multip.

db(z) by 2nd-ord. multip.

sb(z) by multip.

db(z) by multip.

sb(z) by multip.

db(z) by cotran.

outline5
Outline
  • Logarithmic Number System
  • Addition/Subtraction Optimization
  • Cotransformations
  • Synthesis, Simulation and Error Behavior
  • Conclusions
conclusions
Conclusions
  • LNS is useful for application-specific systems
  • Overview of existing techniques for LNS addition/subtraction
  • Overview of existing cotransformation techniques
  • New cotransformation proposed
  • New cotransformation studied in terms of area, latency, and error behavior
  • Cotransformation improves error behavior and area, with a slight cost for latency
acknowledgements
Acknowledgements
  • Nicolas Frantzen and Jesus Garcia for their contributions
  • The ASAP organizing committee for the shared best-paper award
thank you for your attention

Thank you for your attention

Questions?

ASAP 2007