parameterized timing analysis with general delay models and arbitrary variation sources
DESCRIPTION
Parameterized Timing Analysis with General Delay Models and Arbitrary Variation Sources. Khaled R. Heloue and Farid N. Najm University of Toronto {khaled, najm}@eecg.utoronto.ca. Problem. Timing verification is a crucial step More pronounced in current technologies Types of variations - PowerPoint PPT PresentationTRANSCRIPT
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Parameterized Timing Analysis with General Delay Models and Arbitrary
Variation Sources
Khaled R. Heloue and Farid N. Najm
University of Toronto
{khaled, najm}@eecg.utoronto.ca
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Problem
Timing verification is a crucial step More pronounced in current technologies
Types of variations Process variations are random statistical variations Environmental variations are uncertain variations that are
non-statistical
… cause circuit delay variations!
Parameterized Timing Analysis (PTA) Delay is “parameterized” as a function of variations Propagated in the timing graph to determine arrival times Circuit delay becomes parameterized
Useful information: sensitivities, margins, distributions, yield
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Previous Work
Statistical Static Timing Analysis (SSTA) One type of PTA Parameters are random variables with
known distributions Gaussian??
Different delay models Linear, quadratic…
Different correlation models Grid/Quad-tree, Principal Component Analysis (PCA)
Limitations: Can not handle uncertain variables, i.e. non
statistical variables Some have difficulty in handling the Max
operation efficiently In nonlinear/non-Gaussian case
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Previous Work
Multi-Corner Static Timing Analysis (MCSTA) Is another type of PTA Get a conservative bound on maximum
(worst case) corner delay Delay is parameterized using affine (linear) functions
Hyperplanes Parameters can be random variables and/or
uncertain variables
Limitations Linear delay models Does not follow well the spread of the circuit delay
Accuracy guaranteed only at the maximum corner delay Sensitivities are not captured well
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Our Approach
Propose a Parameterized Timing Analysis technique Random parameters with arbitrary distributions Uncertain non-statistical parameters General class of delay models Linear in circuit size (for linear and quadratic models)
Propose two methods to resolve the MAX operation Using guaranteed upper/lower bounds Using an approximation that minimizes the square of the error Both methods preserve the nonlinearities of the delay model
Propose two applications: MCSTA with linear/nonlinear models SSTA with nonlinear models, random & uncertain variables
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General Delay Models
To represent timing quantities, we will use a general class of delay models F
This class of nonlinear functions F has the following three properties:1. F is closed under linear (and/or affine) operations2. All functions in F are bounded3. All functions in F can be maximized and
minimized efficiently
FXfXXfA ApA )(),,( 1
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General Delay Models – Cont’d
Property 1
Property 2
Property 3 Guarantees overall efficiency of approach
FXfcbBaAC
FXfB
FXfA
C
B
A
)(
)(
)(
)(max and )(min
)(
maxmin
maxmin
XfAXfA
AAA
FXfA
AX
AX
A
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),max( 21 DBDAC
Propagation
To propagate arrival times in the timing graph SUM operation MAX operation
SUM can be performed By Property 1 of F
MAX is nonlinear Bound the MAX using functions in F Approximate the MAX using functions in F
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MAX Operation
Let C = max(A,B) be the maximum of A and B and assume that A, B belong to F C does not necessarily belong to F
We want to find
FCCC
CC
CCC
aul
a
ul
,,
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MAX Linear or Nonlinear??
The nonlinearity of the MAX depends on the difference D, between A and B
Note that and that
MAX is linear when Dmin ≥ 0 that is A dominates B C = A Dmax ≤ 0 that is B dominates A C = B
MAX is nonlinear when Dmax ≥ 0 and Dmin ≤ 0
)0,max(
)0,max(
),max(
DB
BAB
BAC
FDmaxmin DDD
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Bounding the MAX
C = B + max(D,0) and Dmax ≥ 0, Dmin ≤ 0
Let Y = max(D,0) Y does not belong to F since MAX is nonlinear
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MAX Upper Bound
Yu is the best ceiling on Y and is exact at the extremes
Since Yu is a linear function of D, then
)( minminmax
max DDDD
DYu
FYu
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MAX Upper Bound – Cont’d
Since C = B + Y, then
Where
SBA
DD
DDB
DD
DA
DD
D
YBC uu
)1()1(minmax
minmax
minmax
min
minmax
max
minmax DDS
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MAX Lower Bound
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MAX Lower Bound – Cont’d
Lower bound on Y
Lower bound on C
otherwise
DD if 0
DD if
minmax
minmax
D
D
Yl
otherwise )1(
DD if
DD if
minmax
minmax
BA
B
A
Cl
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MAX Approximation
Y = max(D,0)
Minimize:
max
min
max
min
2
2
)0,max(D
D
D
D
a
dDDbaD
dDYYE
YbaDYa
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MAX Approximation – Cont’d
Take the partial derivatives with respect to and Set them to zero and solve for the variables
Simple expressions in Dmax and Dmin
a b
3minmax
2min
2max
3minmax
minmax2max
)(
2
)(
)3(
DD
DDb
DD
DDDa
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Summary
Given a general class of nonlinear functions F with certain properties If timing quantities
Then propagation (SUM & MAX) can be done while maintaining the same delay model Bounds LS Approximation
The MAX is “linearized” Coefficients are simple functions of Dmin and Dmax
Independent of whether variables are random or uncertain
Distribution independent
FXfA A )(
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Application 1
Traditional STA Need to check circuit timing at all process corners Exponential number of runs
Multi-corner STA Parameterize delay as a function of
process/environmental parameters Propagate once to get the maximum delay
(also parameterized) Determine the maximum/minimum corner
delays efficiently
Apply our framework to MCSTA with linear/quadratic models
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Linear/quadratic models
Timing quantities are expressed as follows:
Show that all properties hold Linear/quadratic models survive linear
(affine) operations Bounded since -1 ≤ Xi ≤ 1 Maximized efficiently (show in paper how this is done)
11 and
ˆ1
2
1
i
p
i iiiio
p
i iio
X
XaXaa
XaaA
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Results
90nm library and following process parameters: Vtn, Vtp, Ln, Lp
Characterized library to get delay sensitivities
Used ISCAS’85 circuits1. Maximum delay at the maximum/minimum corners are
computed using exhaustive STA2. Maximum/minimum corner delays are determined using
our approach (Bounds and LS-approximation)
Average errors:
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Application 2
SSTA with quadratic delay models random parameters with arbitrary distributions
(Gaussian, uniform, triangular, etc…) uncertain non-random parameters varying in
specified ranges
Delay model:
r
r
Br
p
i iiiio
Ar
p
i iiiio
XbXbXbbB
XaXaXaaA
1
2
1
2
ˆ
ˆ
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The Three properties…
Surviving addition:
Bounded & can be maximized and minimized The maximum and minimum of a quadratic function
depends on whether the vertex is within the range or not (explained in the paper)
rCrr
p
i iiiiiioo
Xba
XbaXbaba
BAC
22
1
2)ˆˆ()()(
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Results
In addition to Xr we use four global variables Xi Truncated Gaussian, Uniform,
and Triangular 10%-20% deviation in
nominal delay
Compared our LS approach to Monte Carlo Metrics: 95%-tile, 99%-tile, σ/μ
Avg error very small < 1%
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CDF Comparison
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Conclusion
Proposed the first Parameterized Timing Analysis technique Random parameters with arbitrary distributions
Gaussian, uniform, triangular, etc… Uncertain non-statistical parameters
Variables in ranges General delay models (some restrictions)
Linear, quadratic, other… Simple and accurate technique
Applied our framework to Multi-corner STA with linear and quadratic models Nonlinear (quadratic) SSTA with arbitrary distributions