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37
Enhancing Path Delay Fault Coverage by Weighted Pseudorandom Test Generation
0ystein Gjermundnes, Einar J. Aas
Abstract - The implementation of a system for analyzing circuits with respect to their path-delay fault testability is presented. It includes a path-delay fault simulator, and an ATPG for path-delay faults combined into a test tool. The test tool is used to evaluate the performance of several different test vector generators. The test generators exploit weighted pseudo-random stimuli generation, based on arithmetic BIST and SIC patterns. The main goal is to find efficient heuristics that improves path-delay fault detection efficiency in terms of test time. We show that weighted ABIST stimuli are productive for detecting the K-longest path-delay faults for most circuits. On the average, we obtained fault coverage of 92.6% for the 20.000 longest paths on iscas’85 circuits.
Index Terms - Built-in testing, Fault diagnosis, Automatic testing.
I. Introduction
Defect oriented testing is gaining attention, and Path Delay Fault (PDF) testing is one of the more challenging problems to study [9]. The test method toolbox has expanded significantly over the last decade. Various trade-offs on test methodology, test quality (measured by various fault coverage metrics), design-for-test development costs, silicon overhead, and cost of Automatic Test Equipment, including test application time, are performed.
For PDF testing, deterministic test pattern pairs, or Built-In Self-Test (BIST) generated patterns may be exploited. We have chosen to explore the possible usage of BIST methods. This paper describes the implementation of a system for analyzing circuits with respect to their path-delay fault testability. The system includes a path-delay fault simulator, and an Automatic Test Pattern Generator (ATPG) for path-delay faults, combined into a test tool. The test tool is used to evaluate the performance of different test vector generators that may be used in various BIST
Manuscript received February 4, 2008.
The work was done while Gjermundnes was affiliated with NTNU.
0ystein Gjermundnes with the ARM Norway, PBox N-2182, NO-7412
Trondheim Norway, e-mail: oystein. sjermundnes@arm. com
Einar J. Aas with the Norwegian University of Science and Technology -
NTNU, NO-7491 Trondheim, Norway, e-mail: ejaas@iet.ntnu.no
arrangements. The test generators exploit weighted pseudorandom stimuli generation, based on arithmetic BIST principles. We show that this is a viable BIST method for detecting the K-longest path-delay faults with satisfactory PDF coverage for many circuits, but not for all circuits. We employ the tool on iscas’85 circuits. Our focus is on the methodology, not on specific stimuli generators. We envision the use of compact software programs, like published [8], to be loaded into the system under test. An in-depth presentation of this test project is found in [5].
II. Path Delay Fault Simulation Model
The path-delay fault model was proposed by Smith [9]. A definition of the path-delay fault model from [1] is:
The delay defect in the circuit is assumed to cause the cumulative delay of a combinational path to exceed some specified duration. The combinational path begins at a primary input or a clocked flip-flop, contains a connected chain of gates, and ends at a primary output or a clocked flip-flop. The specified time duration can be the duration of the clock period (or phase), or the vector period. The propagation delay is the time that a signal event (transition) takes to traverse the path. Both switching delays of devices and transport delays of interconnects on the path contribute to the propagation delay.
There are two path-delay faults associated with each physical path in the circuit: slow-to-rise, and slow-to-fall. Fig. 1 shows one path. The path-delay fault model has the ability to detect distributed defects caused by statistical process variations. A test for a path-delay fault will also detect any spot defects along the path. The number of paths,
and thus path-delay faults, may be exponential in the number of gates in the circuit.
The selection of proper simulation algebra (alphabet and logic rules) is crucial for any logic/fault simulator. Our simulator PDFSim uses the 6-valued algebra developed by [9]. Several features to obtain an efficient simulator are presented in [5], see also [4]. Of course, a two-pattern test vector is needed for delay fault testing. We adopt SIC (Single Input Change) vectors, because it was shown in [11] that such vectors are more effective than Multiple Input Change vectors for robust and non-robust testing.
III. Automatic Test Pattern Generation
It is intractable to test all path-delay faults in a circuit. There are nearly 1020 paths in one of the iscas’85 circuits! one accepted strategy is to test a subset of all possible path delay faults. The longest testable paths are of particular importance for high quality delay testing. An algorithm for extracting the K-longest testable path-delay faults (K-LT-PDF) in a circuit has been developed, and integrated with the fault simulator. The test generators employed will be evaluated against the fault lists containing K-LT-PDF.
The earliest attempts at creating an ATPG that could extract the K-LT-PDF were very inefficient. ATPGs normally employed two separate phases. Usually, a lot of paths are untestable, and a structural path extractor would find and pass a lot of untestable paths to the test generator. Fortunately, by combining the structural path extractor and the test generator, it is possible to prune the search space significantly by sorting out untestable sets of paths at an early stage. This approach was originally used by Qiu and Walker [12]. We have introduced several improvements in terms of efficiency, including recursive learning [6], and FAN-like [3] justifications. Recursive learning is a method for extracting all logical dependencies between signals in a circuit, and to perform precise implications for a given set of value assignments.
IV. BIST-based Stimuli Generators
A. Basis Vectors
First, we wanted to investigate whether ABIST generators of a simple kind, namely accumulator based stimuli generators, would provide sufficient basis for pseudo-random patterns. In particular, the generator described in [8] was investigated:
Ai = Ai-1 + C (mod 2n), A0=I, i=1,2,3, ..., V (1)
By carefully selecting the parameters C and I, one may exhaustively cover every subinterval of size r within the first 2r test vectors. This generator may be implemented as a compact software program in a micro controller. It will generate uniformly distributed values. Let us call these patterns UDB (Uniformly Distributed Basis) patterns.
But are these generated values of adequate statistical quality? We compared the generator against a Mersenne
Twister (MT) generator [7]. This generator is considered as an excellent benchmark for uniformly generated pseudorandom numbers. But it is much more complex to implement in SW or HW. The simple ABIST generator given in (1) was not as efficient. But by combining three generators of type (1), and proper weighting, we developed a better basis, called GAU (U -for uniform). This generator yields considerably shorter test application times than a Mersenne-based generator will.
The rationale behind the use of weighted test patterns is as follows: consider Fig. 1. For a path to be sensitized from input to output, proper controlling values must be applied to the inputs not included in the path. We are looking for ABIST patterns that exhibit statistical properties inductive to fault detection. It is known that proper weighting of input values, i.e. non-uniform distribution of ones and zeros, might enhance the efficiency of fault detection.
Thus, we devised various schemes for weighting the random patterns. These schemes employed the basic generator, with added features for weighing. Transitions on input pins were generated by so-called Single Input Change (SIC) vectors. From a basis vector, we toggle one bit at a time to obtain two-pattern test vectors. For an N-input circuit, 2N vectors are generated this way.
First, we define the GA1 generator: use of the GAU generator, and SIC vectors. This yields a uniform generator, which we will compare potential weighting heuristics against.
B. GA2: stuck-at test set weights
Weights are based on a deterministic test set (obtained from a commercial ATPG) for stuck-at faults. For each input pin, we counted the relative number of ones and zeros, and used these numbers as weights. Don’t cares were counted in both the one and the zero set. Basis vectors are generated from (1), with r=16, and three sets of (C, I) values.
The rationale is that these patterns have contributed to controlling values on the inputs for efficient stuck-at fault detection, and may be promising as candidates for path delay fault testing as well.
C. GA3: counting based weights
Weights are generated based on fault coverage measurements. The circuit is first fed from a pseudorandom generator of type GA1. Two counters (S0Ctr, S1Ctr) are associated with each input. These counters store the number of path-delay faults detected when the input has a stable value (S0 or S1). When a predetermined number of basis patterns (10M) has been applied, the weighting factors can be computed for every input according to:
p0 = S0Ctr/(S1Ctr + S0Ctr), (2)
p1 = S1Ctr/(S1Ctr + S0Ctr). (3)
Subsequently, we rerun the fault simulator with these weights. This yields the generator GA3. This heuristic is
inspired from the fact that patterns with more weight on the HIGH value are productive for AND/NAND gate testing.
Notice that the counting is not activated before 100 basis patterns have been applied. This will leave out the easy-to-detect faults. These faults will be detected anyway.
D. GA4 and GA5
Two less successful schemes were GA4 and GA5. GA4: similar to GA2, but weights were computed with “reseeding”. One output pin at a time was considered when recording fault detection of a test vector. The weight set was recomputed once for every output pin.
GA5 is similar to GA4, but the sequence of seeds was optimized somehow.
E. GA6: weights based on deterministic tests
Similar to GA2, except that weights are generated based on a deterministic test set for path-delay faults. First, a test set for the 20.000 longest paths of non-robust faults was generated. Then, for each pin, we computed the ratio of ones (zeros) that occurred in the complete test set. Don’t cares were counted twice, both as 0 and 1. These values were used as weights throughout the experiment, similar to GA3 above.
V. Experiments
Armed with the tools and generators described above, several experimental runs were set up.
A. Benchmark circuit properties
Circuits from the iscas’85 benchmark suite were engaged in the experiments presented below. Some information about each circuit is provided in this section.
The number of inputs (I), outputs (O), gates (G), logical levels (L) and physical paths (P) for each circuit is shown in Table 1 (the two last columns will be discussed in Section 5.2.1). The number of paths is much larger than the stuck-at fault set. Notice in particular the huge number of paths for benchmark c6288 (a 16x16 bit array multiplier).
The circuits c432 and c499 are omitted from most of the experiments because they contain xOR-gates, which are not currently supported by the ATPG. Another circuit that is omitted from most experiments is c6288. The large number of paths in this circuit causes problems for both the simulator and the ATPG. C17 is discarded for its simplicity. The rest of the benchmarks are used in all experiments.
TABLE 1
Benchmark properties
Circuit I /O G/L P UB PF
c880 60/26 469/25 8642 16652 16652
c1355 41/32 619/25 4173216 1110076 20000
c1908 33/25 938/41 729057 355197 20000
c2670 233/140 1566/33 679960 1306884 20000
c3540 50/22 1741/48 28676671 12330969 20000
c5315 178/123 2608/50 1341305 353300 20000
c6288 32/32 2480/125 10**20 - -
c7552 207/108 3827/44 726494 282752 20000
B. Experimental results
This section presents some statistical properties of the sequences generated by the different test generators described in Section 4. This information can be used as an aid in the interpretation of the results.
EX1: Find the K-longest testable paths
The objective of this experiment was to find the longest non-robust testable paths of each benchmark circuit, which was done by using the ATPG tool described in Section 3. Provided unlimited time and memory, the tool would list all testable faults in each circuit. Unfortunately, some of the circuits contain a huge number of testable path-delay faults, and this would cause the size of the data structure inside the ATPG tool to blow up. In order to keep the whole path store inside computer memory, the size of the path store was set to a maximum of 1M. The ATPG was asked to find the 20.000 longest non-robust testable paths in each of the benchmarks. The number of such paths found (PF) for the different circuits are listed in the last column of Table 1, together with an upper bound (UB) [2] of all non-robust path delay faults. Since all circuits except c880 contain more than 20.000 testable paths, the ATPG had no problem finding 20.000 testable paths. It is reassuring to notice that the upper bound of c880 from [2] coincides with the number of paths we found.
EX2: Determine no. of paths detected by unweighted pseudo-random stimuli
In this experiment test vectors were applied, and the number of detected path-delay faults and their length were logged. The test vectors were generated with an unweighted Mersenne Twister pseudo-random generator (GT1). The purpose was to obtain information about the number of paths of different lengths detected by a standard pseudorandom generator. Typical results are presented in Figure 2.
Fig 2. No. of detected paths vs. path length for various no. of test vectors
The longer paths are not as easily detected as the shorter paths. This is to be expected; long paths need more constraints than shorter paths.
In fact, we found that for all circuits, a randomly selected physical path is longer than the average length of the paths detected within 4M test vectors. The average physical path was from 9.8% to 72 % longer. Clearly, UDB patterns are not effective for detecting the longest PDFs.
EX3: Comparison of GA1 - GA5
In this experiment the performances of GA1 - GA5 were evaluated. The experiment exploited the fault simulator described in Section 2. 10M test patterns were simulated for each circuit and generator. Each simulation run was repeated 10 times with different seeds in order to cover statistical variations. Table 2 presents the average number of detected faults over 10 trials after 10M applied test vectors.
TABLE 2
Detected faults after 10M applied test vectors
Circuit GA1 GA2 GA3 GA4 GA5
c880 8714 16194 16550 16470 16473
c1355 1050139 1085021 1110297 1110264 1110258
c1908 269846 283665 349613 349579 349568
c2670 51739 85948 107711 102734 104141
c3540 588541 996001 1062718 1050384 1050579
c5315 173526 309498 339396 339122 339157
c7552 146754 185983 185687 185264 185383
Sum 2289259 2962310 3171972 3153817 3155559
The best result, i.e. the highest number of detected faults, is shown in bold in Table 2 for each circuit. The stimuli generator with the poorest performance is the unweighted pseudo-random generator GA1. This generator detected the fewest number of non-robust path delay faults in all tests. Generator GA2, which is a weighted pseudo-random generator with weights based on a deterministic test set for stuck-at faults, is somewhat better than GA1. The best generators are GA3, GA4, GA5 and GA6.
The performances of GA3, GA4 and GA5 do not differ by much, but the results point in favor of GA3, which detects most path-delay faults for all but one benchmark. GA3 is a weighted pseudo-random generator with weights based on the counting scheme described in section 4.
We performed the same experiments with the MT as the basic pseudorandom generator. To summarize, the results were in general only slightly better than for the ABIST generator. For example, the equivalent of GA3 exhibited a total improvement (summed over all circuits) of 0.12% more detected path delay faults.
EX4: Weighted pseudo-random patterns to find the K-longest testable path-delay faults
The purpose of this experiment was to find out if proper weighting of pseudorandom stimuli, based on K=20.000 deterministic test patterns for path-delay faults, would yield more efficient path delay tests than using uniformly distributed patterns. The experiments were conducted as follows:
First, the K=20.000 longest testable path-delay faults were extracted for each circuit as described in EX1. For each detected path, the path number was stored in a file together with the corresponding path length and test vector. Weights for GA6 and GT6 were then extracted based on each test set as described in section 4. Notice that generators labeled GTi refers to the use of Mersenne Twister random numbers, but with same heuristics as the corresponding GAi ( i= 1-6).
prior to each simulation run a fault list with the 20.000 longest testable path delay faults was uploaded to the simulator. 10M single-input-change test patterns were then applied to each circuit for each generator. Each simulation run was repeated 10 times with different seeds in order to cover statistical variations. six different generators were used: GA1, GA3, GA6, GT1, GT3 and GT6.
The three generators GA1, GA3 and GA6 are using the exact same underlying accumulator based pseudo-random generator. GA3 and GA6 are weighted pseudo-random generators, and will be compared against GA1 (uniform
weights). The three generators GT1, GT3 and GT6 are using the exact same underlying MT pseudo-random generator. GT3 and GT6 are weighted pseudo-random
generators, and will be compared against GT1 (uniform weights).
Two measures were recorded:
Fault coverage in relation to the size (K) of the fault list. (K=20000 for all circuits except c880 which contains only 16652 non-robust testable paths).
Test time speedup, defined as the ratio:
Rimp(methx) =NTP(uniform)/NTP(methx), (4)
where NTP represents the number of test patterns. The name of the stimuli generator is used as argument (methx).
TABLE 3
Fault coverage, FC, of best method after 10M applied test
VECTORS
Circuit FC(GAx)
c880 99.3% (GA3)
c1355 100% (GA3)
c1908 97.6% (GA3)
c2670 67.9% (GA6)
c3540 86.9% (GA6)
c5315 96.7% (GA6)
c7552 99.8% (GA3)
Average 92.6%
During simulation the fault coverage, FC, was sampled from time to time until 10M test patterns had been applied. Figure 3 shows two typical curves of fault detection vs. no. of test vectors applied. The lowest curves represent unweighted stimuli, while the other curves are given for four different weighting schemes. The improvements are notable for GA3 and GA5.
Table 3 shows the fault coverage after 10M test vectors. The numbers in the second column represent fault coverage achieved with the best generator of GA3 and GA6.
We observed that the GT methods are slightly better than the GA methods. Furthermore, 5 out of 7 circuits attain 97.6% fault coverage, or more. Two circuits exhibit inferior fault coverage, and need more test patterns or other methods of path-delay fault detection.
As mentioned, similar experiments were carried out with the MT as the basic pseudorandom generator, in order to
check possible improvement when using a more authoritative pseudorandom generator. These generators are called GT1-GT6. The MT resulted only in slight improvements. The average fault coverage increased from 92.6% to 93.6%.
The standard deviation of the sample fault coverage after 10M applied test patterns over the 10 trials was also computed. It varied from 0% to 1.5%. Thus, the seed value does not influence the outcome much.
C. Test time speedup
One important goal in testing is the ability to obtain a desired test quality for less cost. In our case, test time, i.e. no. of test vectors to be applied for a given test quality, should be kept at a minimum. In order to measure the speedup of a weighted generator over that of a uniformly distributed pseudo-random generator, one can compare the number of test vectors needed in order to achieve the same fault coverage. The target coverage in our case was set to the fault coverage attained with the unweighted generator after application of 10M stimuli. The improvement factors of the best-weighted generator over uniformly distributed stimuli, defined in (4), are listed in Table 4.
TABLE 4
Time speedup of best method over uniformly distributed stimuli
Circuit Rimp(GAx) Rimp(GTx)
c880 11.9 15.1
c1355 1.5 2.7
c1908 8.0 10.8
c2670 10.7 14.3
c3540 7.1 9.1
c5315 4.7 7.0
c7552 1.0 1.0
We observe time speedups from nothing to a factor 11.9 (GA) or 15.1 (GT)! However, it is unfortunately not possible to devise an a priori metric that may predict speedup. But given the potential of substantial savings in
test time, and thus savings of test cost, it can be recommended to experiment with GA3 and GA6 for a newly designed circuit, and use this method whenever beneficial.
6. Conclusion
A system for analyzing circuits with respect to their path-delay fault testability has been presented. It includes a path-delay fault simulator, and an ATPG for path-delay faults combined into a test tool. This tool was used to evaluate the performance of different test vector generators for various BIST arrangements. The test generators exploit weighted pseudo-random stimuli generation, based on arithmetic BIST principles. We did find useful heuristics that improve path-delay fault efficiency in terms of test time. We showed that weighted ABIST stimuli are productive for detecting the K-longest path-delay faults for many circuits. On the average, we obtained fault coverage of 92.6% for the 20.000 longest paths on a subset of iscas’85 circuits. We observed time speedups from nothing to a factor 12 with the accumulator based stimuli generator, making it well worth the effort of experimenting with such methods for potential high quality path-delay fault testing. However, it should be noted that our methods do not always give significant improvements, and are not generally applicable.
Future work would involve using the simulator and the ATPG to create better generators based upon knowledge about the structure of the circuit. We might also investigate the influence the number of longest paths will have on the test quality obtainable.
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