Parameterized Timing Analysis with General Delay Models and Arbitrary Variation Sources

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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_at_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 withknown
  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.
  nonstatistical variables
• Some have difficulty in handling the
  Maxoperation 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/oruncertain 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
• F is closed under linear (and/or affine)
  operations
• All functions in F are bounded
• All functions in F can be maximized
  andminimized efficiently

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General Delay Models Contd

• Property 1
• Property 2
• Property 3
• Guarantees overall efficiency of approach

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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

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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

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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

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MAX Upper Bound Contd

• Since C B Y, then
• Where

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MAX Lower Bound
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MAX Lower Bound Contd

• Lower bound on Y
• Lower bound on C

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MAX Approximation

• Y max(D,0)
• Minimize

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MAX Approximation Contd

• Take the partial derivatives with respect to
  and
• Set them to zero and solve for the variables
• Simple expressions in Dmax and Dmin

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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

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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 cornerdelays
  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)

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Results

• 90nm library and following process parameters
• Vtn, Vtp, Ln, Lp
• Characterized library to get delay sensitivities
• Used ISCAS85 circuits
• Maximum delay at the maximum/minimum corners are
  computed using exhaustive STA
• 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
  inspecified ranges
• Delay model

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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)

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Results

• In addition to Xr we use four global variables Xi
• Truncated Gaussian, Uniform, and Triangular
• 10-20 deviation innominal delay
• Compared our LS approach to Monte Carlo
• Metrics 95-tile, 99-tile, s/µ
• Avg error very small lt 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