On Convergence of Switching Windows Computation in Presence of Crosstalk Noise

1. On Convergence of Switching Windows Computation in Presence of Crosstalk Noise Pinhong Chen*+, Yuji Kukimoto+, Chin-Chi Teng+, Kurt Keutzer* *Dept. of EECS, Univ. of California, Berkeley, CA +Silicon Perspective, A Cadence Company Santa Clara, CA

2. Outline • Introduction • Crosstalk effects • Switching windows computation • Numerical formulation • Fixed point computation • Convergence properties • Discrete models • Conclusion Pinhong Chen, et al.

3. Introduction • Crosstalk effects are important for DSM designs • Static timing analysis needs to consider crosstalk effects: delay variation due to crosstalk noise • Switching windows cannot be computed in one pass • Iterations are required • What are the numerical properties of the iterations? Pinhong Chen, et al.

4. Increasing Coupling Capacitance Ratio in DSM Technologies Wire aspect ratio changes: • Grounded capacitance reduces but coupling capacitance increases! Cc Cc Cs Cs Pinhong Chen, et al.

5. Crosstalk Noise Effects • Crosstalk noise affects the circuit functionality/timing in two ways • Glitch propagation problem • Delay variation Victim Suffering from noise Contributing noise Aggressor Victim Aggressor Pinhong Chen, et al.

6. Victim with noise Crosstalk Noise Inducing Timing Variation Opposite direction switching Vdd/2 t Same direction switching Aggressor Victim Pinhong Chen, et al.

7. Switching Window Definition • What is “switching window” of a net? • A timing interval during which a net could possibly make transitions Earliest arrival time Latest arrival time Rise switching window Pinhong Chen, et al.

8. Importance of Switching Windows • Switching windows help to isolate noise source • No overlap between switching windows => no delay variation Switching window Victim Possible duration of switching Constant Signal Aggressor Pinhong Chen, et al.

9. Chicken-and-Egg Problem • S. S. Sapatnekar, IEPEP, 1999. • Computing the latest arrival time of net a needs to know net b’s latest noisy arrival time • Computing the latest arrival time of net a needs to know net b’s latest noisy arrival time a b Pinhong Chen, et al.

10. Previous Work • H. Zhou, et al. DAC 2001 • Using lattice theory to prove convergence • Showing multiple convergence points • Discrete in nature • Our contributions • Numerical framework and formulation • Numerical fixed point computation • Examining effects of coupling models and overlapping models • Examining properties of convergence Pinhong Chen, et al.

11. Switching Window Overlapping Function Overlapping function Maximum noise 1.0 Fractional noise No noise Delta delay = Maximum delta delay of victim net i due to aggressor j Pinhong Chen, et al.

12. Formulation of Latest Arrival Time Considering Crosstalk Noise Delta delay due to aggressor j Interconnect delay Gate delay Earliest arrival time of net j Latest arrival time of net i Latest arrival time of net k Pinhong Chen, et al.

13. Latest Arrival Time Function Victim Aggressor 2 Aggressor 1 Pinhong Chen, et al.

14. Switching Window Formulation Pinhong Chen, et al.

15. Bounds of Switching Windows • Set to get the upper bound Earliest arrival time considering noise Lower bound (no noise) Latest arrival time considering noise Upper bound (max noise) Pinhong Chen, et al.

16. Convergence of Switching Windows computation • For Nnets, 2N variables are needed • Converged when Fixed point Pinhong Chen, et al.

17. Fixed Point Computation • For any two points in a closed and bounded domain, if there exists a constant such that • The fixed point iteration converges and guarantees a unique convergence point • A sufficient condition for uniqueness, existence, and convergence Pinhong Chen, et al.

18. Multiple Convergence Points • L < 1 is not guaranteed in switching windows calculation • Multiple convergence points, depending on the initial condition b c a Unstable fixed point Pinhong Chen, et al.

19. Tightening Bounds • If the initial condition starts from the maximum switching windows, the fixed point iteration monotonically shrinks the switching windows in the subsequent passes. • Proof by induction • Each pass is still an upper bound Lower bound (no noise) Upper bound (max noise) Pinhong Chen, et al.

20. Growing Lower Bounds • If the initial condition starts from the minimum switching windows, the fixed point iteration monotonically grows the switching windows in the subsequent passes. • Proof by induction • Can obtain the tightest bound when converged Lower bound (no noise) Upper bound (max noise) Pinhong Chen, et al.

21. Proof of Convergence • Starting from the minimum switching windows, the fixed point iteration monotonically grows the switching windows in the subsequent passes. • Switching windows have an upper bound. Lower bound (no noise) Upper bound (max noise) Pinhong Chen, et al.

22. Decreasing Portion in Arrival Time Function a A decreasing portion makes the iteration oscillate. b Aggressor Pinhong Chen, et al.

23. Non-Monotone Property • Reducing a gate delay may increase the total path delay due to noise Victim Aggressor Pinhong Chen, et al.

24. Discrete Overlapping Model Overlapping function Maximum noise 1.0 No noise Delta delay = Step function Maximum delta delay of victim net i due to aggressor j Pinhong Chen, et al.

25. Discrete Overlapping Model (cont’d) • Easier to converge • Compared with continuous models • Complexity , where N is the number of nets, and M is the maximum number of aggressors of any net. • The convergence point is an upper bound of the continuous model • The latest arrival time functions are discontinuous Pinhong Chen, et al.

26. Conclusion • Numerical formulation can easily explain a variety of properties of switching windows convergence • Switching window computation can be well-controlled by careful selection of the underlying models Pinhong Chen, et al.