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

2

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

3

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

4

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

5

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

6

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

7

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

8

),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

9

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

,,

10

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

11

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

12

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

13

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

14

MAX Lower Bound

15

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

16

MAX Approximation

Y = max(D,0)

Minimize:

max

min

max

min

2

2

)0,max(D

D

D

D

a

dDDbaD

dDYYE

YbaDYa

17

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

18

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 )(

19

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

20

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

21

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:

22

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

23

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()()(

24

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%

25

CDF Comparison

26

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