1. Cellular Automata as BIST pattern generators
Presented by Jeffrey Dwoskin
4/19/2002
Advisor: Dr. Michael Bushnell

2. Jeffrey Dwoskin - Cellular Automata as BIST pattern generators
References
P Pal Chaudhuri et al. ‘Additive Cellular Automata Theory and Applications’, IEEE Computer Society Press, California, USA, 1997
N Ganguly, B K Sikdar, P Pal Chaudhuri,  Design of An on-chip Test Pattern Generator without Prohibited Set (PPS), 15th International Conference on VLSI Design, 2002, Bangalore, India.
M Bushnell, V Agrawal, ‘Essentials of Electronic Testing for Digital Memory & Mixed Signal VLSI Circuits’, Kluwer Academic Publishers, Boston, MA, USA, 2000
Jeffrey Dwoskin - Cellular Automata as BIST pattern generators
4/19/2002

3. Current Pattern Generators - LFSR
Linear Feedback Shift Register
Chain of flip flops with feedback taps
High auto-correlation
Non-local feedback
Jeffrey Dwoskin - Cellular Automata as BIST pattern generators
4/19/2002

4. Cellular Automata as Pattern Generators
Each CA cell is a flip flop with its input based only on its local neighbors
Regular/local design allows compact layout
Better pseudo-random patterns without auto-correlation
Jeffrey Dwoskin - Cellular Automata as BIST pattern generators
4/19/2002

5. Cellular Automata Cells
Each CA cell consists of a flip flop and an XOR gate to determine its next state
The XOR inputs can potentially be from itself, its left neighbor and/or its right neighbor
Which of these inputs is present determines the type of CA cell
Xc-1(t)
Xc+1(t)
Left
Neighbor
Right
Xc(t)
Xc(t+1)
Jeffrey Dwoskin - Cellular Automata as BIST pattern generators
4/19/2002

6. Characterization of CA Cells
We characterize a CA cell by its truth table
The binary number created becomes the rule
= = 15010
Xc-1(t) Xc(t) Xc+1(t) 27
111 26
110 25
101 24
100 23
011 22
010 21
001 20
000
Rule #
Xc-1(t) Xc+1(t) 1 90
150
Jeffrey Dwoskin - Cellular Automata as BIST pattern generators
4/19/2002

7. Characterization of CA
We can also characterize a CA cell by which neighbors it connects to.
Xc-1(t) Xc+1(t) would be 101 since it connects to the left and right neighbors but not itself.
We can do the same for an entire CA using a matrix
Jeffrey Dwoskin - Cellular Automata as BIST pattern generators
4/19/2002

8. Matrix Characterization of CA
Such a matrix is called a characteristic matrix or CA matrix
The CA matrix (T) is defined by:
1, if the next state of the ith cell depends on the present state of the jth cell 0, otherwise
T[i,j] =
Jeffrey Dwoskin - Cellular Automata as BIST pattern generators
4/19/2002

9. Jeffrey Dwoskin - Cellular Automata as BIST pattern generators
Example CA Matrix
4 Cell CA 1
T =
Jeffrey Dwoskin - Cellular Automata as BIST pattern generators
4/19/2002

10. State Transitions Using CA Matrix
If the current state of the CA is ft(x), ft+1(x) = T·ft(x):
Addition operator is XOR
ft(x) T
ft+1(x) 1 1 1 =
0(1) 1(1) 0(0) 1(0) = 1
Jeffrey Dwoskin - Cellular Automata as BIST pattern generators
4/19/2002

11. Characteristic Polynomial
We can find the characteristic polynomial of a CA by constructing the matrix T + xI and computing its determinant: T
T+xI 1
1+x 1 x
P(x) = det(T + xI) = 1 +x3 +x4
Jeffrey Dwoskin - Cellular Automata as BIST pattern generators
4/19/2002

12. Jeffrey Dwoskin - Cellular Automata as BIST pattern generators
Group CA
If the CA characterized by T forms a cyclic group, then: Tm = I (identity matrix)
Where m is the order/length of the cycle
Such a CA where this holds is called a Group CA
We also find that for a Group CA: det T = 1
Jeffrey Dwoskin - Cellular Automata as BIST pattern generators
4/19/2002

13. Maximum Length Group CA
A Group CA can be classified as maximum-length by the presence of a cycle of length 2n-1 with all non-zero states
Additionally, the characteristic polynomial will be primitive – i.e. the polynomial has no factors
Jeffrey Dwoskin - Cellular Automata as BIST pattern generators
4/19/2002

14. Non-Maximum Length Group CA
Multiple cycles
Non-primitive characteristic polynomial
If the order (m) of the group CA is non-prime, then the lengths of the cycles are the factors of m
Jeffrey Dwoskin - Cellular Automata as BIST pattern generators
4/19/2002

15. Design of An On-Chip Test Pattern Generator without Prohibited Set (PPS)
N Ganguly, B K Sikdar, P Pal Chaudhuri
15th International Conference on VLSI Design, 2002, Bangalore, India

16. Jeffrey Dwoskin - Cellular Automata as BIST pattern generators
Introduction
Problem: Some circuits to which we want to add BIST hardware, have a set of prohibited patterns (vectors) that we must avoid while testing
May place circuit in an undesirable state or damage the circuit
Any solution should maintain the randomness qualities of the test patterns to maintain high fault efficiency for the CUT (circuit under test)
Jeffrey Dwoskin - Cellular Automata as BIST pattern generators
4/19/2002

17. Proposed Design of the TPG
We will use an n-cell non-maximum length group CA
State space divided into multiple cycles
The prohibited patterns will be made to fall in the smaller length cycles while one of the bigger cycles will be used to generate the test patterns
Jeffrey Dwoskin - Cellular Automata as BIST pattern generators
4/19/2002

18. Jeffrey Dwoskin - Cellular Automata as BIST pattern generators
Example – CUT with 7 PI’s
We will use a 7-cell group CA shown (T)
The CA has cycles of length 1, 7, 15 & 105
Out of the given PPS, the length 7 cycle contains 3 patterns and the length 15 cycles contains 5 more
Only 2 of the prohibited patterns fall in the length 105 cycle, and are only separated by 10 time steps
PPS =
T =
7x7
Jeffrey Dwoskin - Cellular Automata as BIST pattern generators
4/19/2002

19. Jeffrey Dwoskin - Cellular Automata as BIST pattern generators
Terminology
Target Cycle (TC) – The cycle of largest length generated by the CA
Redundant Cycle (RC) – The cycles other than the TC.
They are redundant in the sense that they are not used for TPG
Jeffrey Dwoskin - Cellular Automata as BIST pattern generators
4/19/2002

20. Jeffrey Dwoskin - Cellular Automata as BIST pattern generators
Example (cont’d)
In our example, the cycles of length 1,7, and 15 are Redundant cycles and the cycle of length 105 is the Target cycle
Since the 2 prohibited patterns in the TC are separated by 10 time steps, we start at the 11th time step and clock for 94 clock cycles for TPG
Jeffrey Dwoskin - Cellular Automata as BIST pattern generators
4/19/2002

21. Design Constraints for an n-PI CUT
C1: The TPG is synthesized out on an n-cell non-maximal length group CA having a number of cycles. One cycle (the TC) can be used for generation of pseudo-random test patterns
C2: Most of the patterns of the PPS lie in the redundant cycles
C3: The remaining members of the PPS, if any, should get clustered in the TC within a distance of Dmax so that most of the patterns of TC can be used for testing the CUT in a single run
Jeffrey Dwoskin - Cellular Automata as BIST pattern generators
4/19/2002

22. Satisfying Constraint C1
An n-cell CA based TPG for a given CUT with n-PI, should have a TC with length greater than or equal to:
3(2n-1)/4 for n < 16
(2n-1)/2 for n ≥ 16
Synthesis Algorithm
Input: n, length of TC
Output: T matrix of CA, resulting cycle structure
Generates a set SCA of CA satisfying the input constraints for C1
Now we must find the subset of SCA that satisfy the constraints C2 and C3
Jeffrey Dwoskin - Cellular Automata as BIST pattern generators
4/19/2002

23. Jeffrey Dwoskin - Cellular Automata as BIST pattern generators
Synthesis Algorithm
Input: n, length of TC
Step 1: Generate the numbers a & b such that:
a & b are mutually prime
a + b = n
(2a – 1)(2b – 1) is close to length of TC
Step 2: Generate T matrices Ta & Tb corresponding to maximal length CA of size a and b respectively
Step 3: Place Ta & Tb in block diagonal form to derive Tnxn corresponding to the desired CA
Output: T matrix of the non-maximal length group CA, the resulting cycle structure
Note: These CA will all have an all 0 cycle, 2 RCs, and a TC
Jeffrey Dwoskin - Cellular Automata as BIST pattern generators
4/19/2002

24. The Heuristic Solution
The problem we have defined is too hard to solve outright
However, we can easily verify whether each solution satisfies the necessary conditions
The length of the PPS for all practical purposes is very small (assumed to be at most 25)
Additionally, the subset of SCA with a valid 3-neighborhood CA is small
Jeffrey Dwoskin - Cellular Automata as BIST pattern generators
4/19/2002

25. Acceptability Criteria
An approximate solution (member of SCA with a valid 3-neighborhood) is acceptable only if:
At least 75% of the PPS fall in the RCs
The TC generating the test pattern sequence is long enough (C1)
The value of Dmax (maximum distance lost in the TC to avoid generation of any prohibited pattern) is at most 10% of the cycle length
Jeffrey Dwoskin - Cellular Automata as BIST pattern generators
4/19/2002

26. Verification Algorithm
Input: A candidate CA from SCA produced by the synthesis algorithm
Step 1: Find the vector basis of the RCs
Every vector in a cycle can be uniquely written as a linear combination of the basis vectors of the cycle
Step 2: Estimate the number of prohibited patterns that fall in the RCs
For each vector in the PPS, determine if it can be generated by a linear combination of the basis vectors of either RC
Jeffrey Dwoskin - Cellular Automata as BIST pattern generators
4/19/2002

27. Verification Algorithm
Step 3: Compute Dmax for all patterns in the PPS that fall in the TC
Let PPSTC represent the subset of PPS that is contained in the TC
For each pattern in PPSTC, load the CA with the pattern and then run for Di time steps to cover all of the patterns in PPSTC
Compute Dmax as: Dmax = min(Di)
If the results meet the acceptability criteria, then the CA is accepted. Otherwise, we reject the candidate CA and try the next CA from SCA
Jeffrey Dwoskin - Cellular Automata as BIST pattern generators
4/19/2002

28. Jeffrey Dwoskin - Cellular Automata as BIST pattern generators
Complete Algorithm
Input: Prohibited Pattern Set for the n-PI CUT
Randomly generate a non-maximal length group CA (a member of SCA) that satisfies constraint C1
Identify the TC and RCs
Verify that the TC and RCs meet the acceptability criteria (C2 & C3)
If it does, select the CA as the TPG, otherwise iterate for the next CA
Find the seed value for the TC and the length of the test pattern that avoids the prohibited patterns in the TC
Evaluate the fault coverage of the CUT with this test pattern
Output: CA based TPG, seed value, and test results (fault coverage, # of test patterns, etc) for the CUT
Jeffrey Dwoskin - Cellular Automata as BIST pattern generators
4/19/2002

29. Jeffrey Dwoskin - Cellular Automata as BIST pattern generators
Experimental Results
Real life data about PPS for a CUT is proprietary in nature and not usually available
Used randomly generated PPS of 25 patterns
The success rate is expected to improve substantially with real life PPS data which are expected to have correlation rather than being random
Jeffrey Dwoskin - Cellular Automata as BIST pattern generators
4/19/2002

30. Success Rate of TPG Design
# Cells
|PPS|
TC length
RC lengths
(%) PPS in RCs
Dmax
Avg # Iter 9
465
15, 31 75 48 25 14 15
14329
7, 2047 80
1223 20
8191
1, 8191 95
106 23 16
57337
7, 8191 65
21259 50
32767
1, 32767 97
259 17
65535
1, 65535 94
1000 18
131072 1, 98
336 13 24
1, (223-1) 84
18121 26
1, (225-1) 78
42342 32 *
(215-1), (217-1) 89
33571 33
(216-1), (217-1)
17498 21 35
(217-1), (218-1)
7853 12 36
(217-1), (219-1)
14322 41
(220-1), (221-1) 82
31132 43
(221-1), (222-1) 93
20211
* Indicates that the cycle length ≈ 2n – 2n/2
Jeffrey Dwoskin - Cellular Automata as BIST pattern generators
4/19/2002

31. Comparison of Test Results
Jeffrey Dwoskin - Cellular Automata as BIST pattern generators
4/19/2002

32. Design of CA TPG for pairs of test vectors
Proposed work by Michael Bushnell & Jeffrey Dwoskin

33. Jeffrey Dwoskin - Cellular Automata as BIST pattern generators
Problem Definition
Use a CA to generate test pattern pairs for delay fault/capacitive coupling faults
Number of vector pairs should be ~ 500
Try to fit as many pairs in the TPG as possible. The rest will have to be stored in a ROM for a second test epoch
Jeffrey Dwoskin - Cellular Automata as BIST pattern generators
4/19/2002

34. Jeffrey Dwoskin - Cellular Automata as BIST pattern generators
Proposed Method
For each bit position of each vector pair, produce a signature that represents compatible CA
Represents whether the XORing of the selected neighbors in the first vector will produce the second vector
Jeffrey Dwoskin - Cellular Automata as BIST pattern generators
4/19/2002

35. Jeffrey Dwoskin - Cellular Automata as BIST pattern generators
Signature
Neighbor
Patterns:
Vector pair 1:
For 2nd bit position, given neighbors in first vector
of 101, we get:
Pattern:
Result:
The necessary 2nd bit position is a 1, so patterns 0, 1, 3, 4, and 6 are a match
If we use XNOR instead of XOR, then the results are inverted, so the other patterns, 2, 5, and 7 match with XNOR
We come up with the following signature for the 2nd bit position for this pair:
XOR
XNOR
= DA2516
Jeffrey Dwoskin - Cellular Automata as BIST pattern generators
4/19/2002

36. Jeffrey Dwoskin - Cellular Automata as BIST pattern generators
Combining Signatures
We repeat this process to find a signature for each bit position of each pair of vectors.
We can combine the signatures for 2 pairs by using a bitwise AND on their signatures.
The result is the CA cells for each bit position that will function for both vector pairs
We can continue to add additional vector pairs as long as each bit position has at least 1 matching pattern among all of the pairs.
We can also consider adding an additional 16 bits to each vector to represent patterns using AND, NAND, OR, and NOR instead of XOR/XNOR.
This may not still be considered a CA, however it may allow for more pairs to be combined
Jeffrey Dwoskin - Cellular Automata as BIST pattern generators
4/19/2002

37. Jeffrey Dwoskin - Cellular Automata as BIST pattern generators
Test Generation
After the signatures are combined, we will have some number of necessary CA to produce all of the desired pairs
If this number is small (2-3), we can use these CA to generate tests
If it is too large, we may have to move some difficult pairs to a ROM
Jeffrey Dwoskin - Cellular Automata as BIST pattern generators
4/19/2002

38. Jeffrey Dwoskin - Cellular Automata as BIST pattern generators
What’s Next
Write C program to generate and combine signatures
Find method for combining larger sets of pairs efficiently
Determine whether these pattern generators will provide enough fault coverage for normal SA-faults or whether we need to add an additional CA or LFSR for these tests
Jeffrey Dwoskin - Cellular Automata as BIST pattern generators
4/19/2002

39. Thank you Questions?