1 on convergence of switching windows computation in presence of crosstalk noise pinhong chen* +,...

1

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


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?


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


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

– Delay variationDelay variationAggressor

Victim

Victim

Aggressor

Suffering from noiseContributing noise


Crosstalk Noise Inducing Timing Crosstalk Noise Inducing Timing VariationVariation

Victim with noise

Vdd/2Vdd/2 Opposite direction switching

Same direction switching

Aggressor

Victim

t


Switching Window DefinitionSwitching Window Definition

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


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


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


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


Switching Window Overlapping Switching Window Overlapping FunctionFunction

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

Fractional noise

Maximum noise

ixjy

ix

jy


Formulation of Latest Arrival Time Formulation of Latest Arrival Time Considering Crosstalk NoiseConsidering Crosstalk Noise

kx ixik id

jy

ijt

i

iAjikk

Dkiijjiiji xdtyxx )(max)( max

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


Latest Arrival Time FunctionLatest Arrival Time Function

i

iAjikk

Dkiijjij xdtyxxf )(max)()( max

x

Victim

Aggressor 1

Aggressor 2


Switching Window FormulationSwitching Window Formulation

i

iAjikk


iiAj

ikkDk

iijijiji ydtyxy )(min)( max

iy ix


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


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


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


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


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




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




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)



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.


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


ijiijj xyxy )(

jy

ix

Aggressor

Victim


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


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


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

1 on convergence of switching windows computation in presence of crosstalk noise pinhong chen* +,...

Documents