1 On Convergence of On Convergence of Switching Windows Switching Windows Computation in Computation in Presence of Crosstalk Presence of Crosstalk Noise Noise Pinhong Chen* Pinhong Chen* + , Yuji , Yuji Kukimoto Kukimoto + , Chin-Chi Teng , Chin-Chi Teng + , , Kurt Keutzer* Kurt Keutzer* *Dept. of EECS, Univ. of California, *Dept. of EECS, Univ. of California, Berkeley, CA Berkeley, CA + Silicon Perspective, A Cadence Company Silicon Perspective, A Cadence Company Santa Clara, CA Santa Clara, CA
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1 On Convergence of Switching Windows Computation in Presence of Crosstalk Noise Pinhong Chen* +, Yuji Kukimoto +, Chin-Chi Teng +, Kurt Keutzer* *Dept.
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On Convergence of Switching On Convergence of Switching Windows Computation in Windows Computation in Presence of Crosstalk NoisePresence of Crosstalk Noise
Pinhong Chen*Pinhong Chen*++, Yuji Kukimoto, Yuji Kukimoto++, , Chin-Chi TengChin-Chi Teng++, Kurt Keutzer*, Kurt Keutzer*
*Dept. of EECS, Univ. of California, *Dept. of EECS, Univ. of California,
Berkeley, CABerkeley, CA
++Silicon Perspective, A Cadence CompanySilicon Perspective, A Cadence Company
Santa Clara, CASanta Clara, CA
ISPD 2002 Pinhong Chen, et al. 2
OutlineOutline IntroductionIntroduction Crosstalk effectsCrosstalk effects Switching windows computationSwitching windows computation Numerical formulationNumerical formulation Fixed point computationFixed point computation Convergence propertiesConvergence properties Discrete modelsDiscrete models ConclusionConclusion
ISPD 2002 Pinhong Chen, et al. 3
IntroductionIntroduction
Crosstalk effects are important for DSM Crosstalk effects are important for DSM designs designs
Static timing analysis needs to consider Static timing analysis needs to consider crosstalk effects: delay variation due to crosstalk effects: delay variation due to crosstalk noisecrosstalk noise– Switching windows cannot be computed in Switching windows cannot be computed in
one passone pass– Iterations are requiredIterations are required– What are the numerical properties of the What are the numerical properties of the
iterations?iterations?
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Increasing Coupling Capacitance Increasing Coupling Capacitance Ratio in DSM Technologies Ratio in DSM Technologies
Cs
Cc Cc
Cs
Wire aspect ratio changes:Wire aspect ratio changes:Grounded capacitance reduces but coupling
capacitance increases!
Grounded capacitance reduces but coupling
capacitance increases!
μm.130μm.50
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Crosstalk Noise EffectsCrosstalk Noise Effects
Crosstalk noise affects the circuit Crosstalk noise affects the circuit functionality/timing in two waysfunctionality/timing in two ways– Glitch propagation problemGlitch propagation problem
What is “switching window” of a net?What is “switching window” of a net?– A timing interval during which a net could A timing interval during which a net could
possibly make transitionspossibly make transitions
Rise switching windowLatest arrival time
Earliest arrival time
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Importance of Switching WindowsImportance of Switching Windows
Switching windows help to isolate Switching windows help to isolate noise sourcenoise source– No overlap between switching windows No overlap between switching windows
=> no delay variation => no delay variation
Switching window
Victim
Constant Signal
Possible duration of switching
Aggressor
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Chicken-and-Egg ProblemChicken-and-Egg Problem S. S. Sapatnekar, IEPEP, 1999.S. S. Sapatnekar, IEPEP, 1999. Computing the latest arrival time of net Computing the latest arrival time of net a a needs needs
to know net to know net bb’s latest noisy arrival time’s latest noisy arrival time Computing the latest arrival time of net Computing the latest arrival time of net aa needs needs
to know net to know net bb’s latest noisy arrival time’s latest noisy arrival timea
b
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Previous WorkPrevious Work H. Zhou, H. Zhou, et al. et al. DAC 2001DAC 2001
– Using lattice theory to prove convergenceUsing lattice theory to prove convergence– Showing multiple convergence pointsShowing multiple convergence points– Discrete in natureDiscrete in nature
Our contributionsOur contributions– Numerical framework and formulationNumerical framework and formulation– Numerical fixed point computationNumerical fixed point computation– Examining effects of coupling models and Examining effects of coupling models and
overlapping modelsoverlapping models– Examining properties of convergenceExamining properties of convergence
Bounds of Switching WindowsBounds of Switching Windows
Earliest arrival timeconsidering noise
Latest arrival timeconsidering noise
Lower bound (no noise)
Upper bound (max noise)
i
iAjikk
Dkiiji xdtx )(max maxmaxmax
iiAj
ikkDk
iiji ydty )(min minmaxmin
1ijSet to get the upper boundSet to get the upper bound
ISPD 2002 Pinhong Chen, et al. 16
Convergence of Switching Convergence of Switching Windows computationWindows computationFor For NN nets, nets, 2N 2N variables are neededvariables are needed
Converged whenConverged when),,,( 00 NN yyxx x
)( )0()1( xfx
)( )1()2( xfx
*)(* xfx
Fixed point
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Fixed Point ComputationFixed Point Computation For any two points in a closed and For any two points in a closed and
bounded domain, if there exists a constant bounded domain, if there exists a constant such that such that
– The fixed point iteration converges and The fixed point iteration converges and guarantees a unique convergence pointguarantees a unique convergence point
– A sufficient condition for uniqueness, A sufficient condition for uniqueness, existence, and convergence existence, and convergence
10 and )()( 2121 LL xxxfxf
21,xxL
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Multiple Convergence PointsMultiple Convergence PointsL < L < 1 1 is not guaranteed in switching is not guaranteed in switching
windows calculationwindows calculationMultiple convergence points, Multiple convergence points,
depending on the initial conditiondepending on the initial condition
x
)(xf
a
b
cUnstable fixed point
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Tightening BoundsTightening Bounds If the initial condition starts from the maximum If the initial condition starts from the maximum
switching windows, the fixed point iteration switching windows, the fixed point iteration monotonically shrinks the switching windows monotonically shrinks the switching windows in the subsequent passes.in the subsequent passes.– Proof by inductionProof by induction– Each pass is still an upper boundEach pass is still an upper bound
Lower bound (no noise)
Upper bound (max noise)
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Growing Lower BoundsGrowing Lower Bounds If the initial condition starts from the minimum If the initial condition starts from the minimum
switching windows, the fixed point iteration switching windows, the fixed point iteration monotonically grows the switching windows in monotonically grows the switching windows in the subsequent passes.the subsequent passes.– Proof by inductionProof by induction– Can obtain the tightest bound when convergedCan obtain the tightest bound when converged
Lower bound (no noise)
Upper bound (max noise)
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Proof of ConvergenceProof of Convergence Starting from the minimum switching Starting from the minimum switching
windows, the fixed point iteration windows, the fixed point iteration monotonically grows the switching monotonically grows the switching windows in the subsequent passes.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)
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Decreasing Portion in Arrival Decreasing Portion in Arrival Time Function Time Function
i
iAjikk
Dkiijjijjij xdtxxyxxf )(max))()(()( max
x
Aggressor
a
b
A decreasing portion makes the iteration oscillate.
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Non-Monotone PropertyNon-Monotone PropertyReducing a gate delay may increase Reducing a gate delay may increase
the total path delay due to noisethe total path delay due to noise
i
iAjikk
Dkiijjiiji xdtyxx )(max)( max
ijiijj xyxy )(
jy
ix
Aggressor
Victim
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Discrete Overlapping ModelDiscrete Overlapping Model
max)( ijjiij tyx
ij
1.0
ji yx
Delta delay =
Maximum delta delay of victim net i due to aggressor j
ix
jy
Overlapping function
No noise
Maximum noise
Step function
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Discrete Overlapping Model Discrete Overlapping Model (cont’d)(cont’d) Easier to convergeEasier to converge
– Compared with continuous modelsCompared with continuous models– Complexity , where Complexity , where NN is the number of is the number of
nets, and nets, and MM is the maximum number of is the maximum number of aggressors of any net.aggressors of any net.
The convergence point is an upper bound The convergence point is an upper bound of the continuous modelof the continuous model
The latest arrival time functions are The latest arrival time functions are discontinuousdiscontinuous
)( 22MNO
ISPD 2002 Pinhong Chen, et al. 26
ConclusionConclusion
Numerical formulation can easily Numerical formulation can easily explain a variety of properties of explain a variety of properties of switching windows convergenceswitching windows convergence
Switching window computation can Switching window computation can be well-controlled by careful be well-controlled by careful selection of the underlying modelsselection of the underlying models