ESE535: Electronic Design Automation. Delay PDFs? (2a) Today. Central Problem. Oxide Thickness. Line Edge Roughness

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1 ESE535: Electronic Design Automation Delay PDFs? (2a) Day 23: April 10, 2013 Statistical Static Timing Analysis Penn ESE535 Spring DeHon 1 Penn ESE535 Spring DeHon 2 Today Sources of Variation Limits of Worst Case Optimization for Parametric Yield Statistical Analysis Behavioral (C, MATLAB, ) Arch. Select Schedule RTL FSM assign Gate Netlist Placement Routing Layout Masks Two-level Multilevel opt. Covering Retiming Penn ESE535 Spring DeHon 3 Mean Number of Dopant Atoms Central Problem As our devices approach the atomic scale, we must deal with statistical effects governing the placement and behavior of individual atoms and electrons Technology Node (nm) Transistor critical dimensions Atomic discreteness Subwavelength litho Etch/polish rates Focus Number of dopants Dopant Placement Penn ESE535 Spring DeHon 4 Oxide Thickness Line Edge Roughness 1.2µm and 2.4µm lines [Asenov et al. TRED 2002] Penn ESE535 Spring DeHon 5 From: Penn ESE535 Spring DeHon 6 1

2 Light Phase Shift Masking What is wavelength of visible light? Source Penn ESE535 Spring DeHon 7 Penn ESE535 Spring DeHon 8 Line Edges (PSM) Intel 65nm SRAM (PSM) Source: Penn ESE535 Spring DeHon 9 Source: Penn ESE535 Spring DeHon 10 Statistical Dopant Placement V th 65nm Penn ESE535 Spring DeHon 11 [Bernstein et al, IBM JRD 2006] Penn ESE535 Spring DeHon [Bernstein et al, IBM JRD 2006] 12 2

3 Gaussian Distribution ITRS 2005 Variation (3σ) From: Penn ESE535 Spring DeHon 13 Penn ESE535 Spring DeHon 14 Example: V th Many physical effects impact V th Doping, dimensions, roughness Behavior highly dependent on V th I DS = µ nc OX 2 W I DS = I S e L V GS nkt / q W L [( ) 2 ] V GS V T 1 e V DS kt / q 1+ λv DS ( ) Penn ESE535 Spring DeHon 15 Impact Performance V th I ds Delay (R on * C load ) Penn ESE535 Spring DeHon 16 Impact of V th Variation Variation in Current FPGAs Penn ESE535 Spring DeHon 17 [Gojman et al., FPGA2013] Penn ESE535 Spring DeHon 18 3

4 Reduce Vdd (Cyclone IV 60nm LP) [Gojman et al., FPGA2013] Penn ESE535 Spring DeHon 19 Source: Noel Menezes, Intel ISPD2007 Penn ESE535 Spring DeHon 20 Old Way Characterize gates by corner cases Fast, nominal, slow Add up corners to estimate range Source: Noel Menezes, Intel ISPD2007 Penn ESE535 Spring DeHon 21 Preclass: Slow corner: 1.1 Nominal: 1.0 Fast corner: 0.9 Penn ESE535 Spring DeHon 22 Corners Misleading Parameteric Yield Probability Distribution Sell Premium Delay Sell nominal Sell cheap Discard [Orshansky+Keutzer DAC 2002] Penn ESE535 Spring DeHon 24 Penn ESE535 Spring DeHon 23 4

5 Phenomena 1: Path Averaging Sequential Paths T path = t 0 +t 1 +t 2 +t 3 + t (d-1) T i iid random variables Mean τ Variance σ Tpath Mean d τ Variance = d σ Penn ESE535 Spring DeHon 25 T path = t 0 +t 1 +t 2 +t 3 + t (d-1) T path Mean d τ Variance = d σ 3 sigma delay on path: d τ + 3 d σ Worst case per component would be: d (τ+3 σ) Overestimate d vs. d Penn ESE535 Spring DeHon 26 SSTA vs. Corner Models Phenomena 2: Parallel Paths STA with corners predicts 225ps SSTA predicts 162ps at 3σ SSTA reduces pessimism by 28% [Slide composed by Nikil Mehta] Penn ESE535 Spring DeHon Source: IBM, TRCAD Cycle time limited by slowest path T cycle = max(t p0,t p1,t p2, T p(n-1) ) P(T cycle <T 0 ) = P(T p0 <T 0 ) P(T p1 <T 0 ) = [P(T p <T 0 )] n 0.5 = [P(T p <T 50 )] n P(T p <T 50 ) = (0.5) (1/n) Penn ESE535 Spring DeHon 28 System Delay P(T p <T 50 ) = (0.5) (1/n) N= N= Probability Distribution Delay Gaussian Distribution Penn ESE535 Spring DeHon 29 From: Penn ESE535 Spring DeHon 30 5

6 System Delay P(T p <T 50 ) = (0.5) (1/n) N= N= For 50% yield want 6 to 7 σ Probability Distribution Delay System Delay T 50 =T mean +7σ path Penn ESE535 Spring DeHon 31 Penn ESE535 Spring DeHon 32 System Delay Corners Misleading Phenomena 2 Phenomena 1 Penn ESE535 Spring DeHon 33 [Orshansky+Keutzer DAC 2002] Penn ESE535 Spring DeHon 34 But does worst-case mislead? STA with worst-case says these are equivalent: Source: Noel Menezes, Intel ISPD2007 Penn ESE535 Spring DeHon 35 Penn ESE535 Spring DeHon 36 6

7 But does worst-case mislead? STA Worst case delay for this? But does worst-case mislead? STA Worst case delay for this? Penn ESE535 Spring DeHon 37 Penn ESE535 Spring DeHon 38 Does Worst-Case Mislead? Delay of off-critical path may matter Best case delay of critical path? Wost-case delay of noncritical path? What do we need to do? Ideal: Compute PDF for delay at each gate Compute delay of a gate as a PDF from: PDF of inputs PDF of gate delay Penn ESE535 Spring DeHon 39 Penn ESE535 Spring DeHon 40 AND rules Delay Calculation Day 22 What do we need to do? Ideal: compute PDF for delay at each gate Compute delay of a gate as a PDF from: PDF of inputs PDF of gate delay Need to compute for distributions SUM MAX (maybe MIN) Penn ESE535 Spring DeHon 41 Penn ESE535 Spring DeHon 42 7

8 Consider entry: MAX(u 1,u 2 )+d For Example MAX Compute MAX of two input distributions. Penn ESE535 Spring DeHon 43 Penn ESE535 Spring DeHon 44 SUM Add that distribution to gate distribution. Continuous Can roughly carry through PDF calculations with Guassian distributions rather than discrete PDFs Penn ESE535 Spring DeHon 45 Penn ESE535 Spring DeHon 46 Dealing with PDFs MAX of Two Identical Gaussians Simple model assume all PDFs are Gaussian Model with mean, σ Imperfect Not all phenomena are Gaussian Sum of Gaussians is Gaussian Max of Gaussians is not a Gaussian Penn ESE535 Spring DeHon 47 Max of Gaussians is not a Gaussian Can try to approximate as Guassian Given two identical Gaussians A and B with µ and σ Plug into equations E[MAX(A,B)] = µ + σ/(π) 1/2 VAR[MAX(A,B)] = σ 2 σ/π [Source: Nikil Mehta] Penn ESE535 Spring DeHon 48 8

9 Probability of Path Being Critical More Technicalities Correlation Physical on die In path (reconvergent fanout) Makes result conservative Gives upper bound Can compute lower [Source: Intel DAC 2005] Penn ESE535 Spring DeHon 49 Graphics from: Noel Menezes (top) and Nikil Mehta (bottom) Penn ESE535 Spring DeHon 50 3 Max of Gaussians with Correlation MAX of Two Identical Gaussians Given two identical Gaussians A and B with µ and σ Plug into equations E[MAX(A,B)] = µ + σ/(π) 1/2 VAR[MAX(A,B)] = σ 2 σ/π [Source: Nikil Mehta] Max of identical Gaussians Extreme of correlated: is just the input Gaussian [Blaauw et al. TRCAD v27n4p589] Penn ESE535 Spring DeHon 52 Penn ESE535 Spring DeHon 51 SSTA vs. Monte Carlo Verification Time Using SSTA in FPGA CAD Le Hei FPGA2007 SSTA Synthesis, Place, Route Kia FPGA2007 Route with SSTA [Slide composed by Nikil Mehta] Source: IBM, TRCAD 2006 Penn ESE535 Spring DeHon 53 Penn ESE535 Spring DeHon 54 9

10 Impact of SSTA in High-Level Synthesis Scheduling and provisioning Summary Nanoscale fabrication is a statistical process Delays are PDFs Assuming each device is worst-case delay is too pessimistic Wrong prediction about timing Leads optimization in wrong direction Reformulate timing analysis as statistical calculation Estimate the PDF of circuit delays Use this to drive optimizations ALU/MUL σ=5% t nominal [Jung&Kim ICCAD2007] Penn ESE535 Spring DeHon 56 Penn ESE535 Spring DeHon 55 Big Ideas: Coping with uncertainty Statistical Reasoning and Calculation Admin Reading for Monday on blackboard Milestone Mondays Penn ESE535 Spring DeHon 57 Penn ESE535 Spring DeHon 58 10