Stratified sampling for fault coverage of vlsi systems
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Stratified Sampling for Fault Coverage of VLSI Systems. Vishwani D. Agrawal Agere Systems, Murray Hill, NJ 07974 [email protected] September 26, 2001 Collaborators: Pradip Thaker, Acorn Networks, and Mona Zaghloul, GWU. VLSI System Design.

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Stratified Sampling for Fault Coverage of VLSI Systems

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Stratified Sampling for Fault Coverage of VLSI Systems

Vishwani D. Agrawal

Agere Systems, Murray Hill, NJ 07974

[email protected]

September 26, 2001

Collaborators: Pradip Thaker, Acorn Networks, and Mona Zaghloul, GWU

Agrawal: Stratified Sampling

VLSI System Design

Register-transfer level (RTL)

design and verification



fault coverage


Logic synthesis

Test generation

Timing and physical design

Design and test data

for manufacturing

Agrawal: Stratified Sampling


  • Accurately estimate the gate-level fault coverage for a VLSI system at the RT-level

  • Advantages:

    • Improve test

    • Improve design

    • Avoid expensive design changes

  • Previous approaches do not accurately represent gate-level fault coverage (function errors, mutation, statement faults, branch faults, etc.)

Agrawal: Stratified Sampling


  • Model faults as representative sample of the targeted (gate-level stuck-at) faults.

  • Treat the coverage in an RTL module as a statistical sampling estimate.

  • For a multi-module VLSI system, combine module coverages according to the stratified sampling technique.

Agrawal: Stratified Sampling

Outline of Talk

  • Introduction to fault sampling.

  • RTL fault model and application to modules.

  • Coverage in a multi-module system:

    • Need for stratified sampling

    • Stratum weights

    • Experimental results

  • Conclusion

  • References

Agrawal: Stratified Sampling

Fault Sampling

  • A randomly selected subset (sample) of faults is simulated.

  • Measured coverage in the sample is used to estimate fault coverage in the entire circuit.

  • Advantage: Saving in computing resources (CPU time and memory.)

  • Disadvantage: Limited data on undetected faults.

Agrawal: Stratified Sampling

Random Sampling Model





All faults with

a fixed but





Np = total number of faults

(population size)

C = fault coverage (unknown)

Ns = sample size

Ns << Np

c = sample coverage

(a random variable)

Agrawal: Stratified Sampling

Probability Density of Sample Coverage, c

(x--C )2

-- ------------

1 2s2

p (x ) = Prob(x < c < x +dx ) = -------------- e

s (2 p)1/2

C (1 - C)

Variance, s 2 = ------------






p (x )

Mean = C



C +3s

C -3s



Sample coverage

Agrawal: Stratified Sampling

Sampling Error Bounds

C (1 - C )

| x - C | = 3 [ -------------- ]1/2


Millot, 1923

Solving the quadratic equation for C, we get the 3-sigma

(99.8% confidence) estimate (Agrawal-Kato, 1990):


C3s = x ------- [1 + 0.44 Nsx (1 - x )]1/2


Where Ns is sample size and x is the measured fault

coverage in the sample.

Example: A circuit with 39,096 faults has an actual

fault coverage of 87.1%. The measured coverage in

a random sample of 1,000 faults is 88.7%. The above

formula gives an estimate of 88.7% 3%. CPU time for sample simulation was about 10% of that for all faults.

Agrawal: Stratified Sampling

An RTL Fault Model(ITC-2000)

  • Language operators are assumed to be fault-free

  • Variables (map onto signal lines) contain faults

    • stuck-at-0

    • stuck-at-1

  • Only one fault is applied at a time (single fault assumption)

  • Agrawal: Stratified Sampling

    RTL Fault Injection

    • Not affected by faults:

      • Synthetic operators + - * >= <= == !=

      • Boolean operators & | ^ ~

      • Logical operators && || !

      • Sequential elements (flip-flops & latches)

    • Faults introduced in signal variables (stems and fan-outs)

    • Separate faults for bits of data words

    Agrawal: Stratified Sampling

    Fault Modeling for Boolean Operators

    Agrawal: Stratified Sampling

    Stem and Fan-out Fault Modeling

    • RTL fan-out faults: if(X) then Z=Y; else Z=!Y;

    • Unique RTL fault is placed on each fan-out of each bit of a variable

    • Unique RTL fault on each stem

    Agrawal: Stratified Sampling

    More RTL Faults

    Agrawal: Stratified Sampling

    Observations and Assumption: RTL Faults

    • RTL faults may have detection probability distribution similar to that of collapsed gate-level faults

    • Statistically, an RTL fault-list approximates a random sample from the gate-level fault-list

    • Number of RTL faults vs. gate-level faults depends on

      • Level of RTL description

      • Synthesis procedure used to convert RTL to gate level

    Agrawal: Stratified Sampling

    RTL Fault Simulation

    • Analogous to gate-level approach

    • Faults injected in RTL code of the design description by a C++ parser; a simulatable logic buffer element inserted at fault site

    • Fault report contains statistics on detected and undetected RTL faults

    • Cadence’s Verifault-XL used as RTL fault simulator

    Agrawal: Stratified Sampling

    Estimation Error for Module Fault Coverage

    • RTL fault coverage assumed to be an estimate of the collapsed gate-fault coverage within statistical bound [Agrawal and Kato, D&T, 1990]:

    a = 3.00 for confidence probability of 99.8%

    c = ratio of detected to total number of RTL faults

    M = number of gate faults

    N = number of RTL faults, k = 1 - N/M

    Agrawal: Stratified Sampling

    DSP Interface Module(3,168 Gates)

    Agrawal: Stratified Sampling

    RTL Faults and VLSI System Coverage

    • Experimental results demonstrate RTL fault coverage of a module to be a good statistical estimate of the gate-level fault coverage

    • A VLSI system consists of many interconnected modules

    • Overall RTL fault-list of a VLSI system does not constitute a representative sample of the gate-level fault-list

    Agrawal: Stratified Sampling

    Error at System Level




    150 faults

    90% cov.



    100 faults

    91% cov.

    RTL Coverage = (0.91 x 100 + 0.39 x 100) / 200 = 65%

    Gate Coverage = (0.90 x 150 + 0.40 x 400) / 550 = 54%

    • A correct estimation of gate-level fault coverage from RTL coverage:


    400 faults

    40% cov.


    100 faults

    39% cov.

    91 x (150 / 550) + 39 x (400 / 550) = 53%

    Agrawal: Stratified Sampling

    Application of Stratified Sampling

    • Fault population of a VLSI system divided into strata according to RTL module boundaries

    • RTL faults in each module are considered a sample of corresponding gate-level faults

    • The stratified RTL coverage is an estimate of the gate-level coverage:

    Wm = stratum weight of mth module = Gm/G

    cm = RTL fault coverage of mth module

    Gm = number of gate-level faults in mth module

    G = number of all gate-level faults in the system

    M = number of RTL modules in the system




    Agrawal: Stratified Sampling

    Application of Stratified Sampling

    C + t s

    • Range of coverage,

    s2= --------cm(1 - cm)





    rm- 1


    rm = number of RTL faults in mth module

    t = value from tables of normal distribution

    The technique requires knowledge of stratum weights and not absolute values of Gm and G

    Agrawal: Stratified Sampling

    Stratum Weight Extraction Techniques

    • Logic synthesis based weight extraction

      Wm = Gm/G

    • Floor-planning based weight extraction

      Wm = Am/A

    • Entropy-measure based weight extraction

    Agrawal: Stratified Sampling

    Experimental Procedure

    • Technology-dependent weight extraction

      • Several unique gate-level netlists obtained by logic synthesis from the same RTL code

      • Each synthesis run performed using a different set of constraints, e.g., area optimization (netlist 1), speed optimization (netlist 2), or combined area and speed optimizations (netlists 3 and 4)

      • Strata weights calculated using gate-level fault lists of various synthesized netlists

    • Technology-independent weight extraction

      • Stratum weights calculated using area distribution among modules

    • Each set of stratum weights used to calculate RTL fault coverage and error bounds

    • Impact of estimation error investigated

    Agrawal: Stratified Sampling

    Experimental Data: Weight Distributions

    Agrawal: Stratified Sampling

    Experimental Data: RTL Fault Coverage

    Agrawal: Stratified Sampling

    Experimental Data: Error Bounds

    Agrawal: Stratified Sampling

    Timing Controller ASIC (17,126 Gates)

    Agrawal: Stratified Sampling

    A DSP ASIC(104,881 Gates)

    Agrawal: Stratified Sampling


    • Main ideas of RTL fault modeling

      • A small or high-level RTL module contributes few RTL faults, but large statistical tolerance gives a correct coverage estimate

      • Stratified sampling accounts for varying module sizes and for different RTL details that may be used

      • Stratum weights appear to be insensitive to specific details of synthesis

    • Advantages of the proposed RTL fault model

      • High-level test generation and evaluation

      • Early identification of hard-to-test RTL architectures

      • Potential for significantly reducing run-time penalty of the gate-level fault simulation

    Agrawal: Stratified Sampling


    • V. D. Agrawal, “Sampling Techniques for Determining Fault Coverage in LSI Circuits,” J. Digital Systems, vol. V, no. 3, pp. 189-202, 1981.

    • V. D. Agrawal and H. Kato, “Fault Sampling Revisited,” IEEE Design & Test of Computers, vol. 7, no. 4, pp. 32-35, Aug. 1990.

    • P. A. Thaker, M. E. Zaghloul, and M. B. Amin, “Study of Correlation of Testability Aspects of RTL Description and Resulting Structural Implementation,” Proc. 12th Int. Conf. VLSI Design, Jan. 1999, pp. 256-259.

    • P. A. Thaker, V. D. Agrawal, and M. E. Zaghloul, “Validation Vector Grade (VVG): A New Coverage Metric for Validation and Test,” Proc. 17th IEEE VLSI Test Symp., Apr. 1999, pp. 182-188.

    • P. A. Thaker, Register-Transfer Level Fault Modeling and Evaluation Techniques, PhD Thesis, George Washington University, Washington, D.C., May 2000.

    • P. A. Thaker, V. D. Agrawal, and M. E. Zaghloul, “Register-Transfer Level Fault Modeling and Test Evaluation Techniques for VLSI Circuits,” Proc. Int. Test Conf., Oct. 2000, pp. 940-949.

    • This presentation is available from the website

    Agrawal: Stratified Sampling

    Thank you

    Agrawal: Stratified Sampling

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