Statistical Timing Analysis Statistical Timing Analysis g g y y Considering Spatially and Considering Spatially and Temporally Correlated Dynamic Temporally Correlated Dynamic Temporally Correlated Dynamic Temporally Correlated Dynamic Power Supply Noise Power Supply Noise Takashi Enami Shinyu Ninomiya Masanori Hashimoto Takashi Enami Shinyu Ninomiya Masanori Hashimoto Dept. Information Systems Engineering, Osaka Univ., Japan { {enami.takashi, ninomiya.shinyu, hasimoto}@ist.osaka-u.ac.jp i t k hi i i hi h i t }@i t k j
Agenda • Background and objective • Statistical modeling of power supply noise • SSTA with statistical model of power supply noise SSTA with statistical model of power supply noise • Experimental results • Conclusion C l i ISPD 2008 2008/04/16 2
Background Voltage drop at power wire. • Power supply noise is becoming more influential on timing due to – increasing current consumption, – decreasing power supply voltage. Demand for timing analysis considering power supply noise. ISPD 2008 2008/04/16 3
Objective • Conventional analysis method – Dynamic analysis • Analysis with test patterns. => Combination space of test patterns is tremendous => Combination space of test patterns is tremendous. – Static analysis • Analysis with constant voltage drop. y g p => Irrelevant voltage drop leads to optimistic or excessively pessimistic estimation. Exact worst-case delay cannot be obtained practically. Propose a statistical timing analysis method that gives reasonable worst-case timing. ISPD 2008 2008/04/16 4
Difficulties of noise aware timing analysis • Cell position (spatial) affect gate delay. affect gate delay • Switching timing (temporal) A B temporal difference spatial difference Voltage e A: delay(t 1 ) > delay(t 2 ) A d l t t 1 : delay(A) > d l (t ) > d l (A) > (t ) A delay(B) B: delay(t 1 ) < delay(t 2 ) t 2 : delay(A) < B delay(B) delay(B) t 1 t 2 Time Position of critical paths and spatial and temporal variation p p must be considered simultaneously. ISPD 2008 2008/04/16 5
Proposed approach (Overview) including spatial and temporal variation Estimate the of power supply noise worst-case delay. Voltage bility Statistical Probab model model V SSTA Circuit Time structure Delay Power supply noise Delay distribution ISPD 2008 2008/04/16 6
Agenda • Background and objective • Statistical modeling of power supply noise • SSTA with statistical model of power supply noise SSTA with statistical model of power supply noise • Experimental results • Conclusion C l i ISPD 2008 2008/04/16 7
Modeling flow Power supply noise Spatial and temporal division. Power variables 1 1. Gaussianization: if necessary Gaussianization: if necessary improve Gaussianity of the variables. 2. Orthogonalization: g transform the correlated variables into the uncorrelated variables. Statistical model Statistical model ISPD 2008 2008/04/16 8
Spatial and temporal division remove spatial and temporal continuity for variable assignment for variable assignment 1 Cycle oltage Vo V x,y,3 V V V x,y,1 V x,y,2 V x,y,3 V V V x,y,1 V Time ex) divide a clock cycle into 3 time spans. ) y p spatial division temporal division choose a representative value choose a representative value use average voltage in each span use average voltage in each span. for each partition. ISPD 2008 2008/04/16 9
Assigning variables and obtaining distribution 1 Cycle V x,y,1 ty Probabilit Voltage different clock cycle different clock cycle V P Voltage => different sample V x,y,2 V x,y,3 V x y 3 V x,y,1 V x,y,2 V x,y,3 V x,y,1 V x y 1 V x y 2 V x y 3 V x y 1 Probability y Time P Data set of Voltage power variables calculate average, V x,y,3 standard deviation and standard deviation and Probability y correlation coefficient P Voltage can model spatial and temporal variation. ISPD 2008 2008/04/16 10
Correlation of power variables 1 cycle ltage tage Volt oltage Vo Time spatial correlation Vo Time temporal correlation Time Voltage drop tends to be similar. g p Voltage drop lasts awhile. g p spatial correlation spatial correlation temporal correlation temporal correlation Correlation of power variables is strong. ISPD 2008 2008/04/16 11
Correlation’s effect on delay Correlation between variables affects delay distribution. V 1 ,V 2 : ex) SUM operation no correlation obability V 1 ,V 2 : positive correlation positive correlation Pro Delay V 1 V 2 Delay calculation considering correlation i i is inevitable. i bl Computationally expensive. ISPD 2008 2008/04/16 12
Orthogonalization • Orthogonalization by principal component analysis (PCA) – Delay calculation considering correlation is facilitated [1]. – Compact statistical model is derived when strong correlation exists. – Compatibility with SSTA developed for manufacturing variability [1] variability [1]. • z i : original variable • μ i : average of z i • σ i : standard deviation of z i compac • λ j t : j th largest eigenvalue • e ij : j th eigenvector corresponding to z i • pc j p j : j th principal component (PC) j t p c pa co po e t ( C) [1] H Chang et al [1] H. Chang, et. al., Statistical Timing Analysis “Statistical Timing Analysis Under Spatial Correlations,” IEEE TCAD , • k, k ’ : number of PCs Vol. 24, No. 9, Sep. 2005. ISPD 2008 2008/04/16 13
Gaussianization PCA assumes Gaussian distribution. � Some variables might be different from Gaussian. S i bl i ht b diff t f G i improve Gaussianity of the variables before PCA. improve Gaussianity of the variables before PCA Box-Cox Λ : Every variable has transformation transformation the optimum value individually. h i l i di id ll original distribution transformed distribution Box-Cox Box Cox transformation ISPD 2008 2008/04/16 14
Agenda • Background and objective • Statistical modeling of power supply noise • SSTA with statistical model of power supply noise SSTA with statistical model of power supply noise • Experimental results • Conclusion C l i ISPD 2008 2008/04/16 15
Gate delay calculation canonical gate delay model receiver side receiver side considering power supply noise. id i l i driver side σ V** : standard deviation of V** : sensitivity of V** to d* e (V**),j : j th eigenvector corresponding to V** How are these parameters( μ r , A r,j ) decided? ISPD 2008 2008/04/16 16
Parameter calculation • Parameters of canonical delay model must consider not only cell position but also switching timing, – Parameters are set based on arrival time. 1 cycle μ r , a r,j If switching timing crosses over If switching timing crosses over the boundary of two time spans. Time => calculate weighted-average g g of two spans. arrival time arrival time ISPD 2008 2008/04/16 17
Agenda • Background and objective • Statistical modeling of power supply noise • SSTA with statistical model of power supply noise SSTA with statistical model of power supply noise • Experimental results – Accuracy of proposed SSTA Accuracy of proposed SSTA – SSTA with reduced model – SSTA considering power supply noise and SSTA id i l i d manufacturing variability • Conclusion Conclusion ISPD 2008 2008/04/16 18
Experimental conditions • noise generators, – two circuits, • FPU[2] (90nm process, 39k gates), • tiny64 processor[2] (90nm process 20k gates) tiny64 processor[2] (90nm process, 20k gates), – size: 1mm x 1mm, – input vector: random, 2000 clock cycles, • circuits for timing analysis, – ISCAS85 (5 circuits), – 64-bit multiplier, – ALU, – H-tree (evaluation of clock jitter), H tree (evaluation of clock jitter) FPUの電源網 [2] OPENCORES.ORG, http://www.opencores.org/. ISPD 2008 2008/04/16 19
Procedure of accuracy evaluation compare proposed SSTA and Monte Carlo STA (MC) using the same noise information given to PCA using the same noise information given to PCA. delay distribution (MC) (MC) power MC supply noise comparison modeling statistical model statistical model proposed SSTA proposed SSTA delay distribution (proposed SSTA) (p p ) • SSTA includes errors that originate from – discretization, – PCA for incomplete Gaussian distribution, – SSTA operation. ISPD 2008 2008/04/16 20
Accuracy of proposed SSTA |SSTA – MC| |SSTA MC| • Proposed SSTA estimates circuit # cells MC the delay accurately. avg (%) sd (%) c432 232 0.478 4.21 • Estimation errors E ti ti c1355 329 1.27 29.4 without Box-Cox transformation c1908 387 0.472 24.1 (avg: 0.465%, sd: 14.4%), (avg: 0 465% sd: 14 4%) c6288 c6288 3382 3382 0.370 0.370 9.15 9.15 c7552 2070 0.172 13.8 => non-Gaussianity was multiplier 41629 0.0969 7.23 not significant but ALU ALU 14655 14655 0.216 0.216 2.87 2.87 Box-Cox transformation H-tree 7 0.576 10.8 improved results slightly. average - 0.456 12.7 noise generator: FPU spatial division number: 10x10 temporal division number: 10 ISPD 2008 2008/04/16 21
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