On-line Testing for Feedback Bridging Faults
4.3 Efficient construction of F N -detector for bridging faultsfaults
4.3.3 OBDD based procedure for generation of exhaustive set of F D-transitions for feedback bridging faultsF D-transitions for feedback bridging faults
As already discussed, the bridging between two lines is called feedback bridging if there exists at least one path between these two lines. We refer the two lines involved in the feedback
D−FF
D−FF v1
v2 Primary
input v3
1
v2
Clock e1
v1+
v2+
NSF block
v
e2 s−a−0 at e1
0
=0
=0
=1
1
0
1 1/0
1/0 1/0
0
(a) NSF with fault, partitioned into cones, whene2 dominatese1.
D−FF
D−FF v1
v2 Primary
input v3
v1
v2
Clock
e2
v1+
v2+ s−a−0 at e
NSF block
e1
2 0
(b) NSF with fault, partitioned into cones, whene1 dominatese2.
Figure 4.8: NSF with faults (e2 dominates e1 and e1 dominates e2)
0 1
0 1
1
0
0
1
1 0 v1
v2 v2
v3
(a) Normal OBDD for cone of NSF output v+1.
v1
v2
1 0
0
1
0 1
(b) Fault OBDD for cone of NSF outputv+1 (s-a-0 ate1).
v1
v2 v2
v3
0 1
0 1
0 0
1
0 1
1
satisfy−all−1
(c) XORed OBDD for the normal and faulty OBDDs.
v1
v2 0
0 1
1
0
1 satisfy−all−0
(d) Dominating OBDD for linee2.
Figure 4.9: OBDDs for non-feedback bridging fault whene2 dominates e1
bridging fault as the back line(b) and the f ront line(f), where b is closer to the primary inputs and f is closer to the primary outputs. In other words, there is path from b to f.
The procedures of generation of test patterns for feedback bridging faults are discussed in two parts. The first part discusses the procedure wheref becomes the dominating line and b becomes the dominated line and the second part deals with the reverse of that.
Part 1: Generation of exhaustive set of F D-transitions for feedback bridging fault– the f ront line dominates the back line
The procedure for generation of the exhaustive set of F D-transitions for feedback bridging faults (where f dominates b) involves all the steps required for the non-feedback bridging faults (Subsection 4.3.2) namely, partitioning the normal NSF using cones of influence for each of the outputs, introduction of fault by inserting s-a-0 fault at the dominated line, partitioning the faulty NSF and finally generating the F D-transitions using OBDD based operations.
However, additional steps are required in this case of feedback bridging fault to determine if the F D-transitions cause oscillations. To elaborate, a test pattern that drives 0 to thedominating line (heref) and detects s-a-0 fault at the dominatedline (hereb) may not qualify to become anF D-transition, if the fault effect at line b propagates through line f and makes it 1. It is easy to observe that for sensitizing the fault, line f is driven to 0 and effect of s-a-0 fault at line b is propagated to the NSF output, however, if the fault effect makesf = 1, then there is oscillation.
The following steps based on OBDD are performed to check if the test patterns generated using the scheme for non-feedback bridging faults (Subsection 4.3.2) cause oscillations for the given feedback bridging fault where f dominates b.
Let T P be the set of test patterns obtained from the intersection of IPs−a−0,dominated
andIP0,dominating. T P is obtained using the steps discussed in Subsection 4.3.2, for the case whene1, the dominated line is b and e2, the dominating line isf.
1. Generate a cone for the dominating line (i.e., f) in the faulty NSF (s-a-0 at b). Let
“dominating faulty OBDD” be the OBDD representation for the Boolean expression corresponding to the dominating line f under the fault.
Apply the “satisfy-all-1” operation on the “dominating faulty OBDD”, which generates all the input patterns for which the value at dominating line f becomes 1 under the fault at dominated line b. The set of such input patterns be IPf=1,s−a−0at dominated. 2. If there is a test pattentpin the intersection ofT P andIPf=1,s−a−0at dominated, thentp
needs to be eliminated from the setT P. It may be noted that test patterntp, implies (i) 0 in thedominating line f under normal condition, (ii) different values in the NSF outputv+munder normal condition compared to s-a-0 fault at thedominatedlineb, (iii) however, under fault, thedominating linef is pulled to 1. So,tp results in oscillation under the fault.
The remaining set of test patterns inT P are mapped toF D-transitions.
A test pattern which remains in T P after the check implies (i) 0 in the dominating line f under normal condition as well as faulty condition, (ii) different values in the NSF outputvm+ under normal condition compared to s-a-0 fault at the dominatedline b. So such a test pattern can detect the fault without oscillation.
input (v3) Primary
v1
v2
v1+
D−ff
D−ff clock
b
v2+
NSF block
(s−a−0) e1
e2 f 1
0
0 0
1/0
0
1/0 1
0 0
1/0
1/0
feedback bridging fault(b,f) cone for line "f"
Figure 4.10: NSF with fault, partitioned into cones, whenf dominatesb.
Now we explain the above procedure with the help of a simple sequential circuit with feedback bridging fault F1 between lines e1 and e2, as shown in Figure 4.10. Let e1 be the dominated line (b) and e2 be the dominating line (f). The partitioning of the NSF into cones on its outputs and on the dominating line are also shown in the figure.
The OBDDs for the cone of NSF output v1+ under normal condition (expression v1v20 + v01v2 + v10v3) and s-a-0 condition at e1 (expression v01v3 + v1v20v3) are shown in Figure 4.11(a) and Figure 4.11(b), respectively. The OBDD obtained from XORing normal
v1
v2 v2
v3
1 0
0 1
0
1 1
0 0
1
(a) Normal OBDD for cone of NSF outputv1+.
v1
v2
v3
0 1
1 0
0
1
0 1
(b) Faulty OBDD for cone of NSF output v1+(s-a-0 at line b).
v1
v2 v2
v3
0 1
0 1
0
1 0
1
0 1
Satisfy_all_1
(c) XORed OBDD for the normal and faulty OBDDs.
0
1 0
1
1 0
1 0
v1
v2
v3
Satisfy_all_0
(d) Dominating OBDD for the line f under normal condition.
v1
v3 0
1
0
1 0 1 Satisfy_all_1
(e) Dominating faulty OBDD for the linef (under s−a−0at b).
Figure 4.11: OBDDs for feedback bridging fault when f dominates b
and faulty OBDDs is shown in Figure 4.11(c). The “satisfy-all-1” operation on XORed OBDD generates the set IPs−a−0,dominated as {v1 = 1, v2 = 0, v3 = 0;v1 = 0, v2 = 1, v3 = 0}. Here, the dominating line is e2 (f) and vdomminating =v01v2 +v01v3, which is represented by the dominating OBDD shown in Figure 4.11(d). The “satisfy-all-0” operation is applied on this dominating OBDD which generates the setIP0,dominating ={v1 = 1, v2 = 0, v3 = 0;v1 = 1, v2 = 0, v3 = 1;v1 = 1, v2 = 1, v3 = 0;v1 = 1, v2 = 1, v3 = 1;v1 = 0, v2 = 0, v3 = 0}. The intersection of IPs−a−0,dominated and IP0,dominating gives the test pattern set T P as {v1 = 1, v2 = 0, v3 = 0}.
Now, we check if the test pattern {v1 = 1, v2 = 0, v3 = 0} causes oscillation under fault. The “dominating faulty OBDD” for the line e2(f) (expression is v10v3) is shown in Figure 4.11(e). Then “satisfy-all-1” operation is applied on this OBDD to generate the set IPf=1,s−a−0at dominated as {v1 = 0, v2 = 0, v3 = 1;v1 = 0, v2 = 1, v3 = 1}. Now the intersection of T P and IPf=1,s−a−0at dominated is φ, which implies that test pattern {v1 = 1, v2 = 0, v3 = 0} can detect the feedback bridging fault F1, when f dominates b, as it does not cause oscillation. The fault (s-a-0) manifestation at v+1 for the test pattern {v1 = 1, v2 = 0, v3 = 0} is 0, thereby mapping to F D1-transition as h10,0,0di.
As discussed before, to generate the exhaustive set ofF D1-transitions, now we need to repeat this procedure when the roles off ront line andback lineare reversed i.e., generation of exhaustive set of F D-transitions for feedback bridging fault, when back line dominates the f ront line.
Part 2: Generation of exhaustive set of F D-transitions for feedback bridging fault– the back line dominates the f ront line
In the circuit considered in Part 1 (Figure 4.10), if the roles of f ront line and back lineare reversed we obtain the circuit given in Figure 4.12. In this case, asback line(e1) dominates the f ront line (e2), we drive 0 to the e1 and test for s-a-0 fault at e2. Using the OBDD based operations to generate the test patterns (discussed in Subsection 4.3.2), we obtainT P as{v1 = 0, v2 = 0, v3 = 1}. Now, it needs to be verified if the test pattern causes oscillation under fault i.e., under fault, the dominating line b is pulled to 1. It may be noted that this is not possible because fault effect cannot be propagated from the dominated line to the dominating line. The reason is obvious, as there is no path from the dominated line to the dominating line. So F D1-transition corresponding to this test pattern is h00,1,0di, as NSF output v1+ for this test pattern under fault is 0. So the exhaustive set of F D1-transitions is {h00,1,0di,h10,0,0di}.
input (v3) Primary
v1
v2
v1+
D−ff
D−ff clock
v2+
NSF block
(s−a−0) f cone for line "b"
1
0
0
e1 0 b
1
1 0 1
0
0 1
1/0
1/0 e2