1. Algorithm for Fast Statistical Timing Analysis Jakob Salzmann, Frank Sill, Dirk Timmermann SOC 2007, Nov ‘07, Tampere, Finland Institute of Applied Microelectronics and Computer Engineering University of Rostock

2. 2 Outline Motivation Static Timing Analysis Statistical Timing Analysis Simulation Results Conclusion & Outlook

3. 3 Motivation(1) Progressive transistor scaling leads to higher impact of parameter variations Physical on-chip variations due to Imprecise fabrication process Gate oxide thickness Transistor width, length Doping Environment Ambient temperature Cooling Time Electro migration Mechanical stress Thermal stress

4. 4 Motivation(2) Parameter variations lead to unpredictable timing behavior Chips compete against each other Before market entry, knowledge about maximum speed in the worst case Step forward: Information on speed distribution of a chip production set Most chips are faster than worst case speed! I1 I2 Y I1 I2 Y Time I1 I2 Y Time Delay Delay ? ?

5. 5 Static Timing Analysis Classic Approach: Worst case analysis Estimate margins of all parameters Find parameter set which results in worst case delay Simulate gate delay with worst case inputs Add delays of each data path to get resulting delay of the circuit No realistic representation of timing behavior Overstated circuit delay I1 I2 Y Time Worst case delay

6. 6 Statistical Timing Analysis(1) σ – standard deviation μ – mean value PDF: Innovative Approach Estimate margins and Gaussian distribution of all possible parameters Monte-Carlo simulations to get delay probabilities Estimate Probability Density Function (PDF) of gate delay Calculate PDF of overall delay More realistic representation of timing behavior Prediction how many circuits match estimated delay μ σ

7. 7 Our new approach Only one simulation set per gate No underestimation of standard deviation Simple extension to gates with more than 2 inputs Low calculation effort Multi input switching (MIS) Former approach [Aga04] Lot of simulation sets per gate Imprecise calculation of standard deviation #Simulation sets ~ #input² High calculation effort Single input switching (SIS) Simple mathematical approach µ Y = µ 1 + µ 2, σ Y ² = σ 1 ² + σ 2 ² Correlations between gates Not within scope of this presentation Statistical Timing Analysis(2)

8. 8 Statistical Timing Analysis(3) Multi input switching – Simulation Theses: Resulting mean value depends on Gate PDF Input PDF Order and time differences of inputs µ Y increases in case of proximate inputs with high standard deviations µ G increases by proximate inputs Resulting standard deviation depends on dominating input Difference between Input Mean Values [ps]

9. 9 Statistical Timing Analysis(4) Approach to calculate proximate effect and dominating input Separate behavior of gate into impact of Inputs and Gate itself “Resulting Input PDF“ by convolution of all Input-PDFs Addition of “Resulting Input PDF“ and ”Gate PDF” by Single Input Switching algorithm II σ Y ² = σ I ² + σ G ² µ Y = µ I + µ G

10. 10 Statistical Timing Analysis(4) Approach for Resulting „Input PDF“ by convolution of all Input-PDFs Integration of all „Input PDF“ to obtain their Cumulative Density Function (CDF) Approximation of all „Input CDF“ by a set of linear equations Multiply the edges of the „Input CDFs“ approximations to get a “Resulting Input CDF“ Mean value by intersection with probability 0.5 Standard deviation by root-mean-square deviation of the points of the “Resulting Input CDF” from mean value

11. 11 Simulation Results(1) Algorithm must not underestimate gate delay! Calculated mean value ≥ Simulated mean value Calculated standard deviation ≥ Simulated standard deviation Three example casesHSpice Case1: σ A = 5ps, σ B = 10ps Case2:σ A = 20ps,σ B = 40ps Case3:σ A = 40ps, σ B = 80ps Approximation NAND2 - Gate

12. 12 Simulation Results(2) Tree structure – worst case of switching behavior I1 I2 I3... I31 I32 Y Different input switching cases 0 → 1 Case1Case 2 Case 3Simulation time HSpice simulation μ 1052 ps975 ps1001 ps 20 min σ 32 ps35 ps38 ps Presented algorithm μ 1082 ps1039 ps 75 ms σ 33 ps42 ps Former Approach (Agarwal) μ 1044 ps969 ps989 ps 1.4 s σ 27 ps Static worst case timing analysis μ 1295 ps 60ms σ ---

13. 13 Conclusion & Outlook Goal: Developing algorithm for calculating statistical timing behavior of a Multi Input Gate Only one simulation set per gate No underestimation of gate delay Simple extension to gates with more than 2 inputs Low calculation effort Automatic tool for calculating statistical timing behavior of larger (and real) circuits

14. 14 Questions? Thank you for your attention! [AGA04]A. Agarwal, F. Dartu, and D. Blaauw; Statistical Gate Delay Model Considering Multiple Input Switching, 41st Design Automation Conference, USA, 2004 References