1. 1 Oct 24-26, 2006 ITC'06 Fault Coverage Estimation for Non-Random Functional Input Sequences Soumitra Bose Intel Corporation, Design Technology, Folsom, CA 95630 Vishwani D. Agrawal Auburn University, Dept. of ECE, Auburn, AL 36849

2. 2 Oct 24-26, 2006 ITC'06 Outline Problem statement Background Motivating examples New algorithms Results Conclusion

3. 3 Oct 24-26, 2006 ITC'06 Problem Statement Fault simulation of non-scan circuits is needed for Finding compact high fault coverage vectors for testing Finding test points for DFT Expensive for Large non-scan or partial-scan circuits Long functional (non-random) verification vector sequences Motivation Exact fault simulation is too expensive Statistical fault simulator is a useful tool, but needs accuracy –In coverage estimation –In identifying faults not detectable by vector sequence

4. 4 Oct 24-26, 2006 ITC'06 Background Approximate fault simulation –Per-vector analysis Critical path tracing (CPT), Abramovici et al., IEEE D&T 1984. Necessary conditions, Akers et al., ITC 1990. –Post-simulation analysis, Stafan, Jain and Agrawal, IEEE-D&T 1985. Dominator analysis in ATPG, Kirkland and Mercer, ITC 1987. Fault detection at fanout stem depends on signal states in this part and the observability of the dominator. Fanout stem Dominator

5. 5 Oct 24-26, 2006 ITC'06 Stafan: A Tutorial Example 11001 00110 00000 C0=0.4 C1=0.6 C0=1.0 C1=0.0 C0=0.6, C1=0.4 S=0.6 C0=0.4 C1=0.6 S=1.0 C0=0.4, C1=0.6 (controllabilities) S=0.4 (sensitization count) sa1 sa0 sa1 Detected faults Incorrectly detected faults OB0=1.0 OB1=1.0 OB0=0.0 OB1=1.0 OB0=1.0, OB1=0.0 (observabilities) OB0=1.0 OB1=1.0 OB0=1.0, OB1=0.0 pd: Prob(sa0 detected) = C1 × OB1, Prob(sa1 detected) = C0 × OB0 Threshold detection by N vectors: PD(N) = 1 – (1 – pd) N ≥ 0.5

6. 6 Oct 24-26, 2006 ITC'06 Motivation: A Three-Vector Example [000] [011] [110] [000] [111] [001] [110] sa1 (1) b0 = 1 (2) b1 = 1/3 (3) b0 = 1/3 * 2/3 = 2/9 Probability of detection (PD) over entire sequence depends on: (1) Number of vectors (N) (2) Per vector detection probability (pd) PD = 1 – (1 – 2 / 9) 3 = 0.53 ≥ 0.5  Fault is detected (correct decision) PD = 1 – ( 1 – pd ) N

7. 7 Oct 24-26, 2006 ITC'06 Motivation: Example of Correlated Vectors Same example with first vector repeated 5 times at the end: [00000000] [01100000] [11011111] [00000000] [11111111] [00100000] [11011111] sa1 (1) b0 = 1 (2) b1 = 1/8 (3) b0 = 1/8 * 2/8 = 1/32 PD = 1 – (1 – 1/32) 8 = 0.22 ≤ 0.5  Fault not detected (incorrect decision).

8. 8 Oct 24-26, 2006 ITC'06 Outline of New Algorithm Upper Bounding: find faults that cannot be detected 1)Graph dominators: necessary conditions for detection 2)Opposite parity reconvergence: blocked propagation 3)Equal parity reconvergence: multiple path propagation Effective length of sequence: estimate “new” content 1)Track input combinations for each gate 2)Collect Stafan-like statistics only for new vectors Detection Prob, PD = 1 – (1 – pd) N eff

9. 9 Oct 24-26, 2006 ITC'06 Algorithm: Fanout Reconvergence (I) 11 00 11 00 11 Same vector applied twice: sa0 G2 G3 G5 sa1 Negative errors: Stafan finds G1 output to be unobservable. Structural analysis: Stem G1 has two same parity paths {G2 and G3}. Signal monitoring: Only pattern 0XX0 can make stem G1 unobservable Since 0XX0 did not occur  G1 is observable. G1 Portion of c17: Fanout G1 reconverges with same inversion parity at G5. Detected faults found not detected due to stafan error

10. 10 Oct 24-26, 2006 ITC'06 Algorithm: Fanout Reconvergence (II) G0 G2 G4 10 11 10 01 11 10 sa0 00 Positive error: When observable fanouts of I2 are treated as independent. Structural analysis: Fanouts paths {G0, and G1-G2} have different parities. Signal monitoring: Propagation through G0: I1=1 and (I3=0 or I4=0) is false Propagation through G1-G2: {I3=1 and I4=1} and I1=0 is false  I2 is unobservable I1 I2 I3 I4 G1 Portion of c17: Fanout I2 reconverges with different inversion parities at G4. Undetected fault found detected due to stafan error

11. 11 Oct 24-26, 2006 ITC'06 Algorithm: Effective Length Estimation Rework previous example of repeated vectors: [00000000] [01100000] [11011111] [00000000] [11111111] [00100000] [11011111] sa1 (1) b0 = 1 (2) b1 = 1/3 (3) b0 = 1/3 × 2/3 = 2/9 N eff = 3, PD = 1 – (1 – 2/9) = 0.53 ≥ 0.5  Correctly estimated as detected. N eff

12. 12 Oct 24-26, 2006 ITC'06 100 Random vectors Results for Two Vector Sequences Simulation of combinational benchmark circuits. Short sequence: 100 random vectors. Long sequence: 2000+ vectors. Constructed from short sequence. Both sequences have same coverage. Simulation of combinational benchmark circuits. Short sequence: 100 random vectors. Long sequence: 2000+ vectors. Constructed from short sequence. Both sequences have same coverage. 50 Vectors repeated 1-100 times

13. 13 Oct 24-26, 2006 ITC'06 Setting Threshold for Fault Detection Consider a fault as detected if PD(N) = 1 – [1 – pd(N eff )] N eff ≥ p Upper bound coverage, p = 0.99; result is very close to analytical upper bound coverage (VTS’06). Lower bound coverage, p = 0.01. For average value of coverage: p = 0.50 (traditionally used value) p = 0.63 (an improved threshold proposed in this paper)

14. 14 Oct 24-26, 2006 ITC'06 Results: Benchmark Circuits Circuit Fault coverage (%) CPU time (%) over logic simulation Exact Fault Sim. Old Stafan New Stafan for both seq. Short seq. Long seq. N eff Cov. c43293.294.998.38992.933.3 c88091.190.091.36390.120.0 c135587.918.737.25088.422.2 c190869.875.781.78272.930.8 c267077.363.572.27477.336.8 c354070.876.384.410069.530.8 c531591.776.693.710092.234.9 c628899.792.998.84699.716.4 c755284.574.587.38986.252.2 Coverage estimation error 5 – 69%

15. 15 Oct 24-26, 2006 ITC'06 New Fault Coverage Estimate for c2670 110100100010000 0 20 40 60 80 Number of vectors Fault coverage (%) Statistical lower bound, PD ≥ 0.01 Statistical upper bound, PD ≥ 0.99 or VTS’06 paper Exact fault simulation Stafan New estimate, PD ≥ 0.63

16. 16 Oct 24-26, 2006 ITC'06 Coverage Estimation for a Large Design Individual sequence coverages by fault simulation Upper bound (VTS’06) New estimate Fault simulation 05101520253040 Functional vector sequences Fault coverage (%)

17. 17 Oct 24-26, 2006 ITC'06 Conclusion Estimation errors in Stafan are due to reconverging fanouts: –Positive error: Undetected faults estimated as detected. –Negative error: Detected faults estimated as undetected. Upper Bounding improvements: –Positive errors reduced by dominator analysis. –Negative errors reduced by reconvergence analysis. Structural analysis of dominators and signal monitoring finds the effective length (N eff ) of non-random functional input sequences. Improved analysis is useful for Test development – vector selection from functional sequences. DFT – test point selection (VTS’06 paper).