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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

http://cm.bell-labs.com/cm/cs/who/va

September 26, 2001

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

Agrawal: Stratified Sampling

Register-transfer level (RTL)

design and verification

90-100%

stuck-at

fault coverage

required

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

- 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

- 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

Detected

fault

Undetected

fault

All faults with

a fixed but

unknown

coverage

Random

picking

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

(x--C )2

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

1 2s2

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

s (2 p)1/2

C (1 - C)

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

Ns

Sampling

error

s

s

p (x )

Mean = C

x

1.0

C +3s

C -3s

x

C

Sample coverage

Agrawal: Stratified Sampling

C (1 - C )

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

Ns

Millot, 1923

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

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

4.5

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

Ns

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

- Language operators are assumed to be fault-free
- Variables (map onto signal lines) contain faults
- stuck-at-0
- stuck-at-1

Agrawal: Stratified Sampling

- 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

Agrawal: Stratified Sampling

- 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

Agrawal: Stratified Sampling

- 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

- 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

- 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

Agrawal: Stratified Sampling

- 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

Gate-

level

M1

150 faults

90% cov.

RTL

M1

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:

M2

400 faults

40% cov.

M2

100 faults

39% cov.

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

Agrawal: 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

M

C=SWmcm

m=1

Agrawal: Stratified Sampling

C + t s

- Range of coverage,

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

M

Wm

S

where,

rm- 1

m=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

- Logic synthesis based weight extraction
Wm = Gm/G

- Floor-planning based weight extraction
Wm = Am/A

- Entropy-measure based weight extraction

Agrawal: Stratified Sampling

- 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

Agrawal: Stratified Sampling

Agrawal: Stratified Sampling

Agrawal: Stratified Sampling

Agrawal: Stratified Sampling

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 http://cm.bell-labs.com/cm/cs/who/va

Agrawal: Stratified Sampling

Thank you

Agrawal: Stratified Sampling