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

ISPD 2002Pinhong Chen, et al.2 Outline Introduction Introduction Crosstalk effects Crosstalk effects Switching windows computation Switching windows computation Numerical formulation Numerical formulation Fixed point computation Fixed point computation Convergence properties Convergence properties Discrete models Discrete models Conclusion Conclusion

ISPD 2002Pinhong Chen, et al.3 Introduction Crosstalk effects are important for DSM designs Crosstalk effects are important for DSM designs Static timing analysis needs to consider crosstalk effects: delay variation due to crosstalk noise 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?

ISPD 2002Pinhong Chen, et al.4 Increasing Coupling Capacitance Ratio in DSM Technologies Cs Cc Cs Wire aspect ratio changes: Grounded capacitance reduces but coupling capacitance increases! Grounded capacitance reduces but coupling capacitance increases!

ISPD 2002Pinhong Chen, et al.5 Crosstalk Noise Effects Crosstalk noise affects the circuit functionality/timing in two ways Crosstalk noise affects the circuit functionality/timing in two ways –Glitch propagation problem –Delay variation Aggressor Victim Aggressor Suffering from noise Contributing noise

ISPD 2002Pinhong Chen, et al.6 Crosstalk Noise Inducing Timing Variation Victim with noise Vdd/2 Opposite direction switching Same direction switching Aggressor Victim t

ISPD 2002Pinhong Chen, et al.7 Switching Window Definition What is “switching window” of a net? What is “switching window” of a net? –A timing interval during which a net could possibly make transitions Rise switching window Latest arrival time Earliest arrival time

ISPD 2002Pinhong Chen, et al.8 Importance of Switching Windows Switching windows help to isolate noise source Switching windows help to isolate noise source –No overlap between switching windows => no delay variation Switching window Victim Constant Signal Possible duration of switching Aggressor

ISPD 2002Pinhong Chen, et al.9 Chicken-and-Egg Problem S. S. Sapatnekar, IEPEP, S. S. Sapatnekar, IEPEP, 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

ISPD 2002Pinhong Chen, et al.10 Previous Work H. Zhou, et al. DAC 2001 H. Zhou, et al. DAC 2001 –Using lattice theory to prove convergence –Showing multiple convergence points –Discrete in nature Our contributions Our contributions –Numerical framework and formulation –Numerical fixed point computation –Examining effects of coupling models and overlapping models –Examining properties of convergence

ISPD 2002Pinhong Chen, et al.11 Switching Window Overlapping Function 1.0 Delta delay = Maximum delta delay of victim net i due to aggressor j Overlapping function No noise Fractional noise Maximum noise

ISPD 2002Pinhong Chen, et al.12 Formulation of Latest Arrival Time Considering Crosstalk Noise Latest arrival time of net i Interconnect delay Gate delay Latest arrival time of net k Earliest arrival time of net j Delta delay due to aggressor j

ISPD 2002Pinhong Chen, et al.13 Latest Arrival Time Function Victim Aggressor 1 Aggressor 2

ISPD 2002Pinhong Chen, et al.14 Switching Window Formulation

ISPD 2002Pinhong Chen, et al.15 Bounds of Switching Windows Earliest arrival time considering noise Latest arrival time considering noise Lower bound (no noise) Upper bound (max noise) Set to get the upper bound Set to get the upper bound

ISPD 2002Pinhong Chen, et al.16 Convergence of Switching Windows computation For N nets, 2N variables are needed For N nets, 2N variables are needed Converged when Converged when Fixed point

ISPD 2002Pinhong Chen, et al.17 Fixed Point Computation For any two points in a closed and bounded domain, if there exists a constant such that 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

ISPD 2002Pinhong Chen, et al.18 Multiple Convergence Points L < 1 is not guaranteed in switching windows calculation L < 1 is not guaranteed in switching windows calculation Multiple convergence points, depending on the initial condition Multiple convergence points, depending on the initial condition a b c Unstable fixed point

ISPD 2002Pinhong 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. 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)

ISPD 2002Pinhong 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. 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)

ISPD 2002Pinhong 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. 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. Switching windows have an upper bound. Lower bound (no noise) Upper bound (max noise)

ISPD 2002Pinhong Chen, et al.22 Decreasing Portion in Arrival Time Function Aggressor a b A decreasing portion makes the iteration oscillate.

ISPD 2002Pinhong Chen, et al.23 Non-Monotone Property Reducing a gate delay may increase the total path delay due to noise Reducing a gate delay may increase the total path delay due to noise Aggressor Victim

ISPD 2002Pinhong Chen, et al.24 Discrete Overlapping Model 1.0 Delta delay = Maximum delta delay of victim net i due to aggressor j Overlapping function No noise Maximum noise Step function

ISPD 2002Pinhong Chen, et al.25 Discrete Overlapping Model (cont’d) Easier to converge 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 convergence point is an upper bound of the continuous model The latest arrival time functions are discontinuous The latest arrival time functions are discontinuous

ISPD 2002Pinhong Chen, et al.26 Conclusion Numerical formulation can easily explain a variety of properties of switching windows convergence 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 Switching window computation can be well-controlled by careful selection of the underlying models