Asymptotic Probability Extraction for Non-Normal Distributions of Circuit Performance By: Sedigheh Hashemi 201C-Spring2009

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Asymptotic Probability Extraction for Non-Normal
Distributions of Circuit PerformanceBy
Sedigheh Hashemi201C-Spring2009
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Asymptotic Probability Extraction for Non-Normal
Distributions of Circuit Performance

• X. Li, P. Gopalakrishnan and L. Pileggi, CMU
• J. Le, Extreme DA

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Overview

• Introduction
• Asymptotic Probability EXtraction (APEX)
• Implementation of APEX
• Numerical examples
• Conclusion

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IC Technology Scaling
Feature Size Scale Down
Year Leff (nm) W L Tox Vth H ?
1997 250 25.0 32.0 8.0 10.0 25.0 22.2
1999 180 26.2 33.3 8.0 10.0 30.0 24.0
2002 130 28.0 34.6 9.8 10.0 30.0 27.3
2005 100 30.0 40.0 12.0 11.4 33.8 31.7
2006 70 33.3 47.1 16.0 13.3 35.7 33.3
Process Variations (3s / Nominal) Nassif 01
Process variation is becoming relatively larger!
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Statistical Problems in IC

• Statistical methods have been proposed to address
  various statistical problems
• We focus on analysis problem in this work

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Modeling Process Variations

• Assumption
• Process variations ?xi satisfy Normal
  distributions N(0,si)
• Principle component analysis (PCA)
• ?xi can be decomposed into independent ?yi
  N(0,1)

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Response Surface Model
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Response Surface Model

• A low noise amplifier example designed in IBM
  0.25 µm process

Normal Distribution ?yi
Performance Linear Quadratic
F0 1.76 0.14
S11 6.40 1.32
S12 3.44 0.61
S21 2.94 0.34
S22 5.56 3.47
NF 2.38 0.23
IIP3 4.49 0.91
Power 3.79 0.70
Nonlinear Transform
Non-Normal Distribution p
Regression Modeling Error for LNA
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Moment Matching

• Key idea
• Conceptually consider PDF as the impulse response
  of an LTI system

Nonlinear Transform
Unknown PDF
Normal Distribution
Match Moments
LTI System
Impulse Excitation
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Moment Matching

• Impulse response
• Moments

• Match the first 2M moments

Impulse Excitation
Impulse Response
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Connection to Probability Theory

• F(?) is called characteristic function in
  probability theory
• We actually match the first 2M terms of Taylor
  expansion at ? 0

System Theory
Probability Theory
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Connection to Probability Theory

• Proposition 1
• Proposition 2
• Typical characteristic functions are "low-pass
  filters"
• A low-pass system is determined by its behavior
  in low-freq band (? 0)
• Taylor expansion is accurate around expansion
  point (? 0)
• Moment matching is efficient in approximating
  low-pass systems Celik 02

Characteristic Function for Typical Random
Distributions
Celik 02 IC Interconnect Analysis, Kluwer
Academic Publishers, 2002
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The Classical Moment Problem
T. Stieltjes 1894
RSM
Moment Matching
Probability Extraction
pdf(p)
pdf(p)
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APEX Asymptotic Probability Extraction

• Classical moment problem
• Existence uniqueness of the solution
• Find complete bases to expand PDF function space

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Direct Moment Evaluation

• If ?y1, ?y2,... are independent standard Normal
  distribution N(0,1)
• Require computing symbolic expression for pk(Y)

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Binomial Moment Evaluation

• Key idea
• Recursively compute high order moments
• Derived from eigenvalue decomposition
  statistical independence theory

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Step 1 Model Diagonalization
?zi are independent N(0,1) since eigenvectors U
are orthogonal !
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Step 2 Moment Evaluation

• NOT compute symbolic expression for pk(Y)
• Achieve more than 106x speedup compared with
  direct evaluation

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Overview

• Introduction
• Asymptotic Probability EXtraction (APEX)
• Implementation of APEX
• PDF/CDF shifting
• Reverse PDF/CDF evaluation
• Numerical examples
• Conclusion

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PDF/CDF Shifting

• PDF/CDF shifting is required in two cases
• Over-shifting results in large approximation
  error
• The challenging problem is to accurately
  determine ?

?
?
pdf(p)
pdf(p)
p
0
0
p
Mean µ
Mean µ
Case 1 Not Causal
Case 2 Large Delay
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PDF/CDF Shifting

• Exact ? doesn't exist since pdf(p) is unbounded
• Define a bound ? such that the probability P(p
  µ-?) is sufficiently small
• Propose a generalized Chebyshev inequality to
  estimate ? using central moments

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Reverse PDF/CDF Evaluation

• Final value theorem of Laplace transform
• Moment matching is accurate for estimating upper
  bound
• Use flipped pdf(-p) for estimating lower bound

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Overview

• Introduction
• Asymptotic Probability EXtraction (APEX)
• Implementation of APEX
• Numerical examples
• Conclusion

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ISCAS'89 S27
Longest Path in ISCAS'89 S27

• ST 0.13 µm process
• 6 principal random factors
• MOSFET variations
• No intra-die variation
• No interconnect variation

• Linear delay modeling error
• 4.48
• Quadratic delay modeling error
• 1.10 (4x smaller)

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ISCAS'89 S27

• Binomial moment evaluation achieves more than
  106x speedup

Moment Order Direct Direct Binomial
Moment Order of Terms Time (Sec.) Time (Sec.)
1 28 1.00 ? 10-2 0.01
3 924 3.02 ? 100 0.01
5 8008 2.33 ? 102 0.01
6 18564 1.57 ? 103 0.01
7 38760 8.43 ? 103 0.02
8 74613 3.73 ? 104 0.02
15 0.04
20 0.07
Moment Evaluation
Computation Time for Moment Evaluation
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ISCAS'89 S27
Delay

• Numerical oscillation for low order approximation
• Increasing approx. order provides better accuracy
• Typical approx. order is 7 10

Cumulative Distribution Function for Delay
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ISCAS'89 S27

• APEX is the most accurate approach
• APEX achieves more than 200x speedup compared
  with MC 104 runs
• APEX 0.18 seconds
• MC 104 runs 43.44 seconds

Linear Legendre APEX
1 Point 1.43 0.87 0.04
10 Point 4.63 0.02 0.01
25 Point 5.76 0.12 0.03
50 Point 6.24 0.05 0.02
75 Point 5.77 0.03 0.02
90 Point 4.53 0.16 0.03
99 Point 0.18 0.78 0.09
Comparison on Estimation Error
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Low Noise Amplifier

• IBM 0.25 µm process
• 8 principal random factors
• MOSFET RCL variations
• No mismatches

Performance Linear Quadratic
F0 1.76 0.14
S11 6.40 1.32
S12 3.44 0.61
S21 2.94 0.34
S22 5.56 3.47
NF 2.38 0.23
IIP3 4.49 0.91
Power 3.79 0.70
Regression Modeling Error for LNA
Circuit Schematic for LNA
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Low Noise Amplifier

• APEX is the most accurate approach
• APEX achieves more than 200x speedup compared
  with MC 104 runs
• APEX 1.29 seconds
• MC 104 runs 334.37 seconds

Performance Corner Corner Linear Linear Legendre Legendre APEX APEX
Performance 1 99 1 99 1 99 1 99
F0 15.8 20.1 1.11 1.10 0.20 0.55 0.06 0.05
S11 45.4 51.5 5.78 1.40 2.94 3.28 0.09 0.08
S12 38.9 44.6 3.88 1.16 0.39 0.27 0.14 0.28
S21 60.3 51.6 2.91 4.69 0.37 0.01 0.17 0.19
S22 23.1 36.0 1.01 5.61 1.11 0.84 0.07 0.19
NF 51.9 72.8 3.70 3.52 0.34 0.37 0.06 0.12
IIP3 54.6 59.7 5.02 5.93 0.29 0.43 0.33 0.26
Power 16.6 42.5 0.01 1.24 0.92 0.93 0.09 0.02
Comparison on Estimation Error
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Operational Amplifier

• IBM 0.25 µm process
• 49 principal random factors
• MOSFET variations from design kit
• Include mismatches

Performance Linear Quadratic
Gain 3.92 1.57
Offset 21.80 7.49
UGF 1.14 0.45
GM 0.96 0.52
PM 1.11 0.41
SR (P) 0.82 0.66
SR (N) 1.27 0.44
SW (P) 0.38 0.16
SW (N) 0.36 0.12
Power 1.00 0.64
Circuit Schematic for OpAmp
Regression Modeling Error for OpAmp
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Operational Amplifier

• APEX achieve more than 100x speedup compared with
  MC 104 runs

Performance Linear Linear Legendre Legendre APEX APEX
Performance 1 99 1 99 1 99
Gain 22.7 10.4 22.0 81.7 1.45 0.32
Offset 11.5 74.7 222 159 0.58 3.20
UGF 3.78 4.30 0.39 0.33 0.03 0.18
GM 2.72 2.46 0.37 0.20 0.08 0.04
PM 4.41 3.79 0.40 0.52 0.13 0.02
SR (P) 0.81 0.97 0.35 0.34 0.11 0.07
SR (N) 3.83 4.31 0.24 0.27 0.13 0.24
SW (P) 0.13 0.03 0.37 0.37 0.16 0.06
SW (N) 0.06 0.03 0.34 0.43 0.09 0.01
Power 0.69 0.65 0.35 0.41 0.11 0.00
Comparison on Estimation Error
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Application of APEX

• APEX can be incorporated into statistical
  analysis/synthesis tools
• E.g. robust analog design Li 04

Optimization Engine
Unsized Topology
Optimized Circuit Size
Simulation Engine
APEX
Li 04 Robust analog/RF circuit design with
projection-based posynomial modeling, IEEE ICCAD,
2004
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Conclusion

• APEX applies moment matching for PDF/CDF
  extraction
• Propose a binomial moment evaluation for
  computing high order moments
• Moments are efficiently matched to a pole/residue
  formulation
• Solve several implementation issues of APEX
• PDF/CDF shifting using generalized Chebyshev
  inequality
• Reverse PDF/CDF Evaluation
• APEX can be incorporated into statistical
  analysis/synthesis tools
• Statistical timing analysis
• Yield optimization