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Ones Count
The number of ones in the CUT output response is counted. In this method the number of ones is the signature. It requires a simple counter to accomplish the goal. Figure 40.10 shows the test structure of ones count for a single output CUT. For multiple output ones, a counter for each output or one output at a time with the same input sequence can be used. Input test sequence can be permuted without changing the count.
Fig. 40.10 Ones count compression circuit structure
For Nbit test length with r ones the masking probability is shown as follows:
Number of masking sequences
2^{N }possible output sequences with only one fault free.
The masking probabilities:
It has low masking probability for very small and very large r. It always detects odd number of errors and it may detect even number of errors.
Transition Count
It is very similar to ones count technique. In this method the number of transitions in the CUT response, zero to one and/or one to zero is counted. Figure 40.11 shows a test structure of transition counting. It has simple hardware DFF with EXOR to detect a transition and counter to count number of transitions. It has less aliasing probability than ones counting. Test sequences cannot be permuted. Permutation of input sequences will change the number of transitions. On the other hand, one can reorder the test sequence to maximize or minimize the transitions, hence, minimize the aliasing probability.
Fig. 40.11 Transition count compression circuit structure
For Nbit test length with r transitions the masking probability is shown as follows: For the test length of N, there are N1 transitions.
Number of masking sequences
Hence, is the number of sequences that has r transitions.
Since the first output can be either one or zero, therefore, the total number must be multiplied by
2. Therefore total number of sequences with same transition counts : . Again, only one of them is faultfree.
Masking probabilities:
Syndrome Testing
Syndrome is defined as the probability of ones of the CUT output response. The syndrome is 1/8 for a 3input AND gate and 7/8 for a 3input OR gate if the inputs has equal probability of ones and zeros. Figure 40.12 shows a BIST circuit structure for the syndrome count. It is very similar to ones count and transition count. The difference is that the final count is divided by the number of patterns being applied. The most distinguished feature of syndrome testing is that the syndrome is independent of the implementation. It is solely determined by its function of the circuit.
Fig. 40.12 Syndrome testing circuit structure
The originally design of syndrome test applies exhaustive patterns. Hence, the syndrome is S = K /2^{n} , where n is the number of inputs and K is the number of minterms. A circuit is syndrome testable if all single stuckat faults are syndrome detectable. The interesting part of syndrome testing is that any function can be designed as being syndrome testable.
LFSR Structure
Fig. 40.13 Two types of LFSR
One of the most important properties of LFSRs is their recurrence relationship. The recurrence relation guarantees that the states of a LFSR are repeated in a certain order. For a given sequence of numbers a_{0}, a_{1}, a_{2},…………a_{n},…….. We can define a generating function:
G(x) = a_{0} + a_{1}x + a_{2}x^{2} + …………+ a_{m}x^{m} + ……
{α_{m} } = {α_{0},α_{1},α_{2}}
where α_{i}or 0 depending on the out put stage and time t_{i}.
The initial states are a_{n}, a_{n+1},…….,a_{2}, a_{1}. The recurrent relation defining {a_{m}}is
where c_{i}=0, means output is not fed back
=1,otherwise
G(x) has been expressed in terms of the initial state and the feedback coefficients. The denominator of the polynomial G(x), f (x) = is called the characteristic polynomial of the LFSR.
LFSR for Response Compaction: Signature Analysis
For an output sequence of length N, there is a total of 2^{N}1 faulty sequence. Let the input sequence is represented as P(x) as P(x)=Q(X)G(x)+R(x). G(x) is the characteristic polynomial; Q(x) is the quotient; and R(x) is the remainder or signature. For those aliasing faulty sequence, the remainder R(x) will be the same as the faultfree one. Since, P(x) is of order N and G(x) is of order r, hence Q(x) has an order of Nr. Hence, there are 2Nr possible Q(x) or P(x). One of them is faultfree. Therefore, the aliasing probability is shown as follows:
for large N. Masking probabilities is independent of input sequence. Figure 40.14 illustrates a modular LFSR as a response compactor.
Fig. 40.14 Modular LFSR as a response compactor
MultipleInput Signature Register (MISR)
Fig. 40.15 Multiple input signature register
Figure 40.15 illustrates a mstage MISR. After test cycle i, the test responses are stable on CUT outputs, but the shifting clock has not yet been applied.
Ri(x)= (m1)th polynomial representing the test responses after test cycle i.
Si(x)=polynomial representing the state of the MISR after test cycle i.
G (x) is thecharacteristicpolynomial
Assume initial state of MISR is 0. So,
This is the signature left in MISR after n patterns are applied. Let us consider a nbit response compactor with mbit error polynomial. Then the error polynomial is of (m+n2) degree that gives (2^{m+n1}1) nonzero values. G(x) has 2^{n1}1 nonzero multiples that result m polynomials of degree <=m+n2.
Probability of masking
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