Download FPGA Power Reduction Using Configurable Dual-Vdd

Non-Gaussian Statistical Timing
Analysis Using Second Order
Polynomial Fitting
Lerong Cheng1, Jinjun Xiong2,
and Lei He1
1EE
Department, UCLA
*2IBM
Research Center
Address comments to [email protected]
*Dr.
Xiong's work was finished while he was with UCLA. This work was partially sponsored by NSF
and Actel.
Outline
 Background and motivation
 Second order polynomial fitting for max operation
 Quadratic SSTA
 Experiment results
 Conclusions and future work
Background and Motivation
 Gaussian variation sources



Linear delay model, tightness probability [C.V DAC’04]
Quadratic delay model, tightness probability [L.Z DAC’05]
Quadratic delay model, moment matching [Y.Z DAC’05]
 Non-Gaussian variation sources



Non-linear delay model, tightness probability [C.V DAC’05]
Linear delay model, ICA and moment matching [J.S DAC’06]
Non-linear delay model, Fourier Series [Cheng DAC’06]
 Need fast and accurate SSTA for Non-linear Delay model with
Non-Gaussian variation sources
Outline
 Background and motivation
 Second order polynomial fitting for max operation
 Quadratic SSTA
 Experiment results
 Conclusions and future work
Linear Fitting: Tightness Probability
 Previously, max operation is approximated by tightness
probability [Chandu DAC04]
where


Tightness probability approximation is a linear fitting
Tightness probability is hard to obtain when A and B are nonGaussian random variables
 Max operation is a non-linear operation

Linear fitting is not accurate enough
 Need a more accurate and efficient non-linear approximation
Non-linear fitting of Max: Second Order
Polynomial Fitting
 Using second order polynomial instead of linear function to
approximate the max operation
where
arbitrary distribution
and V is a random variable with any
How to obtain these
Fitting Coefficients?
 Fitting coefficients are computed by matching the mean of the
max operation while minimizing the square error between h
and max(V,0) within the 3σrange
where
Mean of the Max Operation
 When V is a non-Gaussian random variable, it is hard to
compute E[max(V,0)]
 Two step solution

We approximate the non-Gaussian random variable V as a
quadratic function of a standard Gaussian random variable W by
matching the first 3 moments [Zhang’ISPD05]


c2, c1, and c0 can be computed by close form formulas
Use E[max(g(W), 0)] to approximate E[max(V,0)]
Compute Fitting Coefficients
 Recall the constraint that matching the mean of the max operation
we have
 The constrained optimization equation can be written as the
unconstrained optimization equation:
where
 Expanding the integral, the square error can be represented as
quadratic form of
can be computed easily by letting partial derivative of SE to be 0
Comparison between Second Order Fitting
and Linear Fitting
1
max(V,0)
Linear Fit
Second-Order Fit
4
3
0.8
2
0.6
1
0.4
0
0.2
-1
-2
-2.5
-1.5
-0.5
0.5
1.5
 Assume V~N(0.7, 1)
2.5
3.5
0
-2.5
-1.5
-0.5
0.5
1.5
2.5
3.5
Outline
 Background and motivation
 Second order polynomial fitting for max operation
 Quadratic SSTA
 Experiment results
 Conclusions and future work
Quadratic Delay Model
 Delay is quadratic function of variation sources



Xi’s are independent random variables with arbitrary distribution
Xi’s are with zero mean and unit standard deviation
R is local random variation, which is modeled as a Gaussian
random variable
Atomic Operations for SSTA
 Two atomic operations for block based SSTA, max and add

Given

Compute
Max Operation Flow
Compute mean,
variance, and
skewness of Dp=D1-D2
Obtain the fitting
coefficients Θ for
max(Dp,0)
Reconstruct
Dm=max(D1,D2) to
quadratic delay form
Moments of Dp
 Quadratic form of Dp
where
 First three moments of DP

Because Dp is in quadratic form of variation sources Xi’s, the
moments of Dp can be computed by close form formulas
 With the first three moments, it is easy to obtain the fitting
coefficients Θ for max(Dp,0)=h(Dp,Θ)
Reconstruct Dm to Quadratic Form
 Fitting result of Dm

Dm is a 4th order polynomial of Xi’s
 Moment Matching [Zhan DAC2005]

Joint moments between Dm and Xi’s can be computed by close
form formulas
Add Operation
 Just add the correspondent parameters to get the parameters
of Ds
Computational Complexity
 The computational complexity of one step max operation is
O(n3), where n is the number of variation sources
 The computational complexity of one step add operation is
O(n2)
 The complexity measured as the total number of max and add
operations of the SSTA is linear with respect to the circuit size
Semi-Quadratic SSTA
 Effect of the crossing terms in the quadratic model is weak
[Zhan DAC2005], ignoring crossing terms will not affects the
accuracy too much
 Simplified delay model without crossing terms (semi-quadratic
delay model)
Where
 The SSTA flow for the semi-quadratric delay model is similar
to that of the quadratic delay model, but much simpler

The computational complexity of both max and add operation for
semi-quadratic SSTA is O(n)
Outline
 Background and motivation
 Second order polynomial fitting for max operation
 Quadratic SSTA
 Experiment results
 Conclusions
Experiment Setting
 ISCAS89 benchmark set
 65nm technology
 Two types of variation sources, both with skew-normal distribution


Leff
Vth
 Three types of variation



Inter-die variation
Intra-die spatial variation (grid based model)
Intra-die random variation
 Three comparison cases



Linear SSTA [Chandu DAC2004]
Nonlinear SSTA using Fourier Series [Cheng DAC2007]
100,000 sample Monte-Carlo simulation
PDF Comparison for s15850
Error and Run Time Comparison
 For quadratic SSTA, the error of mean, standard deviation, and
skewness is within 1%, 1%, and 5%, respectively.
 For Semi-Quadratic SSTA, the error of mean, standard deviation,
and skewness is within 1%, 2%, and 25% error.

Semi-quadratic SSTA ignores crossing terms which affects skewness
 Semi-quadratic SSTA results similar error as Fourier SSTA, but 20X
faster
 Semi-quadratic SSTA is more accurate than linear SSTA with similar
run time
Outline
 Background and motivation
 Second order polynomial fitting for max operation
 Quadratic SSTA
 Experiment results
 Conclusions
Conclusion
 A new second order polynomial fitting of the max operation is
proposed
 All the SSTA operations are based on close form formulas
 The quadratic SSTA predicts the error of mean, standard
deviation, and skewnss withing 1%, 1%, and 5% error,
respectively
 The semi-quadric SSTA has similar accuracy as the SSTA with
Fourier Series, but 20X faster