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Path Delay Fault Classification Based on ENF Analysis

Path Delay Fault Classification Based on ENF Analysis. Matrosova A., Nikolaeva E. Tomsk State University, Russia. Abstract.

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Path Delay Fault Classification Based on ENF Analysis

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  1. Path Delay Fault Classification Based on ENF Analysis Matrosova A., Nikolaeva E. Tomsk State University, Russia

  2. Abstract Path delay fault classification based on using ENF and an addition of its product is suggested. It allows clarifying a nature of single and multiple PDF manifestation and getting test patterns. This approach is also used for investigation of combinational circuit testability properties concerning PDFs. Key words: path delay faults, equivalent normal form (ENF).

  3. Introduction In recent years with development of nanometer technologies, delay testing has become very important problem. The objective of delay testing is to detect timing defects degrading the performance of a circuit. Among the proposed delay fault models, the path delay fault model (PDF model) is considered the most accurate and has received wide attention.

  4. Introduction In order to observe delay defects, it is necessary to generate and propagate transitions in the circuit. This requires application of a pair of vectors v1, v2. The first vector v1 stabilizes all signals in the circuit. It means the signal value on any circuit pole coincides with the function value corresponding to this pole and vector v1. The function depends on the circuit input variables. The second vector v2 causes the desired transition. Take into account that delays of falling (1/0) transition and rising (0/1) transition along of the same path from a primary input to a primary output in a circuit may be different. We need a pair of vectors v1, v2 for each transition of a path.

  5. Introduction Single and multiple PDFs are distinguished. In this paper we first consider single PDFs. In accordance with the conditions of fault manifestation single PDFs are divided into robust, non robust and functional ones. Robust fault manifestation does not depend on delays of other paths of a circuit. Non robust fault shows itself only if all other paths of a circuit are free fault. Functional fault may manifest itself only if the certain other paths are sensitized together with the path considered. When several paths are fault we say about multiple PDF.

  6. Introduction Several attempts were made to formalize these informal notions for gates of circuit, for vector pairs (in the case of robust PDF) and so on. Here we try to formalize above mentioned notions for vector pairs and ENF of arbitrary combinational circuit using addition of ENF product. In this way we hope to clarify a nature of PDF manifestation in order to facilitate getting a vector pair for different types of PDFs.

  7. Equivalent normal form (ENF) Consider equivalent normal form (ENF) that represents a function implementing with a circuit and all circuit paths. Each ENF literal is supplied with index sequence enumerating gates of the path. It should be noted that literal with the same index sequence may appear in different ENF products. Literals of the same product of ENF have different index sequences. ENF of the circuit (Fig.1) is as follows (1). Fig.1. The combinational circuit.

  8. Equivalent normal form (ENF) This sum may have products with both the same literals and inverse literals. Product having the same literals or/and inverse literals will be called non ordinary product. In the example considered ____ is non ordinary product. Otherwise a product is ordinary one. Further we will consider ENF and S together. Except ENF let consider sum S of products derived from ENF with removing index sequences of literals:

  9. Robust and non robust PDFs. Notions A notion of orthogonal pair of products can be extended to non ordinary products. If one product of the pair has literal xiand another product has literal __ then these products are orthogonal otherwise they are not orthogonal. If a non ordinary product has literals __,__ then this product is empty. Let __ be non empty (possibly non ordinary) product from a sum S of products, xi be literal of __ and α – the path corresponding to this literal in ENF. Eliminate in __ all repetitive literals. As a result we have got ordinary product K comprising xi. Now change in K a literal xi for __ and obtain product __. Product __is called an addition of K relative to literal xi. For example, ___________, xi = e, ________.

  10. Robust and non robust PDFs. Notions First formalize notions of robust PDFs separately for falling and rising sequences. Let Kα be set of products from S so that each product is not orthogonal to K and comprises literal xi marked with the index sequence representing the path α in the corresponding ENF. Kα presents additional possibilities of sensitizing the path α. In product sum S for ________, and _____ we have: __________.

  11. Robust and non robust PDFs. Notions Choose a Boolean vector v of input variables of a circuit. Subset of Kα in which any product takes the 1 value on vector v denote as Kα(v). If v = 01101, then Kα(v) is empty (in our example). Let M be derived from the sum S of products by eliminating __ and Kα(v). Mand Kα(v) are originated byv. Consider falling sequence corresponding to α. Let v1 be vector that stabilizes all signals in the circuit ensuring the 1 value of the function and v2 – provides the desired (falling) transition. Denote as u minimal cube covering v1, v2 and as k(u) – product corresponding to u.

  12. Robust and non robust PDFs. Dfinitins Definition1. PDF of falling sequence manifests itself as robust one under the following conditions. 1. Product _____ takes the 1 value on a vector v1. 2. Sum M of products takes the 0 value on a vector v1. 3. Product __ takes the 1 value on a vector v2. 4. Sum S of products takes the 0 value on a vector v2. 5. Product k(u) is orthogonal to each product from M. 6. There exists a product from a set _________ in which literal xi appears only once.

  13. Robust and non robust PDFs. Dfinitins Now consider rising sequence corresponding to α. Let v1 be a vector that stabilizes all signals in the circuit ensuring the 0 value of the function and v2 is a vector that provides the desired (rising) transition. Definition2. PDF of rising sequence manifests itself as robust one under the following conditions. 1. Product _ takes the 1 value on a vector v1. 2. Sum S of products takes the 0 value on a vector v1 3. Product K takes the 1 value on a vector v2. 4. Sum M of products takes the 0 value on a vector v2. 5. Product k(u) is orthogonal to each product from M. 6. There exists a product from set ___________in which literal xi appears only once. The conditions of robust PDF manifestation of rising sequence are derived from the conditions of robust PDF manifestation of falling sequence (for the same path α) through changing v1 for v2 and conversely.

  14. Robust and non robust PDFs. Dfinitins Now regard conditions of non robust PDF manifestation for α. Definition3. PDF of falling sequence manifests itself as non robust one under the following conditions. 1. Product K(__) takes the 1 value on a vector v1. 2. Sum M of products takes the 0 value on a vector v1. 3. Product __takes the 1 value on a vector v2. 4. Sum S of products takes the 0 value on a vector v2. 5. Product k(u) is not orthogonal to some products from M. 6. There exists a product from a set __________in which literal xi appears only once. The condition 5 means that some products from M may take the 1 value when changing v1 for v2. If other paths in a circuit are not free fault then masking PDF of falling sequence ofαis possible. To exclude masking we have to suppose that all other paths are free fault. The conditions of non robust PDF manifestation of rising sequence are derived from the conditions of non robust

  15. Robust and non robust PDFs. Dfinitins The conditions of non robust PDF manifestation of rising sequence are derived from the conditions of non robust PDF manifestation of falling sequence through changing v1 for v2 and conversely and eliminating the condition 6. Definition4. PDF of rising sequence manifests itself as non robust one under the following conditions. 1. Product __ takes the 1 value on a vector v1. 2. Sum S of products takes the 0 value on a vector v1. 3. Product K takes the 1 value on a vector v2. 4. Sum M of products takes the 0 value on a vector v2. 5. Product k(u) is not orthogonal to some products from M. Take into consideration that the condition 5 in the definitions 1–4 divides path delay faults into robust and non robust ones.

  16. Robust and non robust PDFs. Theorems Call ordinary product K expansible relative to literal xi if new product obtained with elimination of literal xi be implicant of the function. (In our case a function is represented with sum S of products). Then xi is expanding literal. Otherwise K is non expansible relative to literal xi and xi is not expanding literal. Notice that prime implicant is non expansible relative to each literal. Theorem1. To detect either robust or non robust manifestation of PDF originated with non empty product __(K) and literal xi it is necessary that K be non expansible with respect to literal xi. Let K be non expansible with respect to xi and P be obtained from S with elimination of product __ and a set of products Kα.

  17. Robust and non robust PDFs. Theorems Theorem2. To detect either robust or non robust PDF (originated with non empty product __(K) and not expanding literal xi) it is necessary that the cube corresponding to K be not completely covered with the cubes corresponding to the products of P. Let Q be set of Boolean vectors representing minterms of K not covered with P. Find among a set {__,Kα} products that don’t contain repeating literalsxi. Join them into a set __. Let __ be result of intersection of Q and cubes corresponding to __. Theorem3. To detect PDF of both robust sequences and non robust falling sequence it is necessary that a set __ be not empty. Theorem4. To detect PDF of non robust rising sequence it is necessary that a set Q be not empty

  18. Robust and non robust PDFs. Resume Taking into consideration Theorems 1-3 and definitions 1-4 we conclude the following. 1. Test patterns of a pair v1, v2 that detects PDF of robust falling sequence may be applied for detecting PDF of robust rising sequence and inversely. 2. Test patterns of a pair v1, v2 that detects PDF of non robust falling sequence may be applied for detecting PDF of non robust rising sequence. 3. Test patterns of a pair v1, v2 detecting PDF of non robust rising sequence not always detects non robust falling sequence.

  19. Robust and non robust PDFs. Vector pair generation Represent __ as sum Dof all prime implicants. Also represent all minterms of __ product on which S takes the 0 value as sum __ of all prime implicants. Let k be product from D and _ be product from __. Let __,__ be derived from __,__ removing literals __,__, correspondingly. Consider all pairs ___,___ originated by __,__. Theorem5. If a pair __,__ consists of not orthogonal products then the pair originates vectors v1, v2 that detect PDF of robust falling and rising sequences. Vector v1(v2) turns into 1 product ___ and vector v2 (v1) turns into 1 product ___ for falling (rising) sequence. If __,__ are not empty, then checking pairs __,__ we may find vectorsv1, v2 for detecting PDF of both robust sequences. In this paper we discuss only possibilities of getting pairv1, v2 without consideration of calculation problems.

  20. Robust and non robust PDFs. Vector pair generation

  21. Robust and non robust PDFs. Examples Fig.1. The combinational circuit.

  22. Robust and non robust PDFs. Examples Consider next example. We want to find vector pair for path ____ using product _______,_______. The product is non expansible with respect to d. Product ______, _________, __ contains repeated literal d, but product ___ from Kα contains the only literal, _______________________. Representing Q as sum of products we obtain the only product __ (Fig.2). Then we derive __ and represent it as sum of products, _________. ____________________. Consider _______ and _______,_______,_______. According to the Theorem 5 there exists vector pair detecting PDF of robust sequences. Vector v1 = 10000 turns into 1 expression ___________ and vector v2 = 10010 turns into 1 expression ___________ for falling sequence of the path ____. M = P, u = 100-0, ___________, k(u) is orthogonal to M. It means the fault considered manifests itself as robust one.

  23. Robust and non robust PDFs. Examples The pair v2, v1 detects robust PDF of rising sequence of the path _____. Fig. 3. Sensitizing the falling robust sequence of the path _____ . Notice that in the paper S. Devadas, K. Keitzer “Synthesis of Robust Delay-Fault-Testable Circuits: Theory”. IEEE Transactions on Computer-Aided Design, vol.11, NO 1. January 1992. p.87-101. when formalizing conditions of robust path delay manifestation authors don’t consider the condition 6. Keeping this condition allowed to find vector pair for the path _____from product ________.The authors suppose that this product must be ignored.

  24. Functional PDFs. Notions Consider product __ comprising literal xi corresponding to path α. LetK be derived from __ with elimination of repeated literals. Product K is expansible relative to xi. (In the expression S the product _________ is expansible relative to d). If we want to derive from K new product which is not function implicant (S represents function), then we must exclude certain subset of literals together with xi. Let Xj be minimal subset of literals that originates the product __ from (___) K so that ___ is not expansible relative to xi. Let δ be subset of paths corresponding to Xjin __ and K. Each path from δ is free fault. Next change in K each literal from {xi, Xj} for inverse literal and obtain product __. __ is called an addition of K relative to literals {xi, Xj}.

  25. Functional PDFs. Notions Let Kα be subset of S in which every product contains all literals of a set {xi, Xj} with the corresponding index sequences in ENF and every product is not orthogonal to K (the same literals have the same index sequence). Subset Kα represents additional possibilities of sensitizing the path α together with a set δ of paths. Subset of Kα in which any product takes the 1 value on a vector v denote asKα(v). Subset M is derived from S by eliminating __ andKα(v).

  26. Functional PDFs. Notions As each path from δ is free fault then functional falling sequence is not detectable. Then consider rising sequence corresponding to α and a set δ. Let v1 be a vector that stabilizes all signals in a circuit ensuring the 0 value of the function and v2 be a vector that provides the desired (rising) transition. Definition5. PDF of rising sequence manifests itself as functional one under the following conditions. 1. Product __ takes the 1 value on a vector v1. 2. Sum S of products takes the 0 value on a vector v1. 3. Product K(___) takes the 1 value on a vector v2. 4. Sum M of products takes the 0 value on a vector v2. 5. Product k(u) may be not orthogonal to some products from M.

  27. Functional PDFs. Example Consider a circuit of Fig.4. We have the following ENF: _________________. ____________, choose _____, K is non expansible per a and b. Let ____, ______, _____, k=D=ab,_________, v1 = 00, v2 = 11. PDF of rising sequence of the path a36 manifests itself as functional one together with the path b36. Fig4. Manifestation of functional PDF for the path __

  28. Functional PDFs. Definitions Let K(__) be expansible relative to literal xi and non expansible relative to literal xj. Elimination of xj originates the product ___ from K so that it becomes non expansible relative to xi. Let γ be free fault path corresponding to xj inK (___). Change in K each literal from {xi, xj} for inverse literals and obtain product __ that is an addition to K. Let Kα be subset of S in which every product contains all literals of a set {xi, xj} with the corresponding index sequences in ENF and every product is not orthogonal to K (the same literals have the same index sequence). A set Kα represents additional possibilities of sensitizing the path α together with the paths γ. Subset of Kα in which any product takes the 1 value on vector v denote asKα(v). Let M be derived in the regular way. PDF of rising sequence of α appears as functional one under the conditions suggested in the definition5.Deriving D, __ is similar to above mentioned way for functional PDFs.

  29. Functional PDFs. Example Illustrate functional PDF of rising sequence for the circuit of Fig 5. Extract from S product _________ and consider literal ____. The product is expansible relative to d, xi = d. There is no other literal for which the product is expansible. Let xj be equal to e. The product is non expansible relative to xj. Kα is empty, _____________________________, _____________, v2 = 11011. ________, __________v1 = 01000. Fig5. Manifestation of functional PDF for the path ____

  30. Functional PDFs. Resume Take into account that we considered the conditions of functional PDF manifestation only for expanding literal xiunder suggestion that the number of additional sensitizing paths is minimal. It is possible to increase the number of additional sensitizing paths using the approach based on the definition 5. It is also possible to apply this approach for non expanding literal xi. It may be useful, for example, when PDF corresponding to xi does not manifest itself either as robust or non robust one.

  31. Multiple PDFs. Notions Consider multiple fault originated by __, a set {Xj} of its literals and the corresponding set δ of paths. Let product K be derived from __ with elimination of repeated literals. Next change in K each literal from {Xj} for inverse literals and obtain product __. __ is called an addition of K relative to literals of a set {Xj}. Let Kα be subset of S in which every product contains all literals of a set {Xj} correlating to a set δ of the sequences and each product is non orthogonal to K. Notice that {Xj} doesn’t contain repeated literals. Subset Kα represents additional possibilities of sensitizing a set δ of paths. Subset of Kα in which any product takes the 1 value on a vector v denote as _____. Let M be derived in the regular way.

  32. Multiple PDFs. Defnitions Definition 6. Multiple PDF of falling sequences manifests itself under the following conditions. 1. Product K(__) takes the 1 value on a vector v1. 2. Sum M of products takes the 0 value on a vector v1. 3. Product __ takes the 1 value on a vector v2. 4. Sum S of products takes the 0 value on a vector v2. 5. Product k(u) may be not orthogonal to some products from M. 6. There exists a product from a set _________ in which each literal from {Xj} appears only once.

  33. Multiple PDFs. Defnitions Definition 7. Multiple PDF of rising sequence manifests itself under the following conditions. 1. Product __ takes the 1 value on a vector v1. 2. Sum S of products takes the 0 value on a vector v1. 3. Product ______ takes the 1 value on a vector v2. 4. Sum M of products takes the 0 value on a vector v2. 5. Product k(u) may be not orthogonal to some products from M. Notice that multiple PDFs may be originated by not only one product but also intersection of several products and several paths corresponding to the same input variables. It is out of our consideration here.

  34. Multiple PDFs. Example Illustrate multiple PDF manifestation. The fault is caused with one product and its literals. Consider the circuit of Fig.6. _________, multiple fault is represented with two paths: {Xj} = {e59, b59}. _______________________________, _____________. _________. __________. Consider falling sequence choosing v1 = 11011, v2 = 00010. On Fig.6 we see multiple PDF manifestation of falling sequences. If changing v1 for v2 the conditions for rising sequence for the same fault are fulfilled. Fig.6. Sensitizing the multiple falling sequences originated by paths e59, b59.

  35. Conclusion Path delay fault classification based on using ENF and a addition of product is suggested. It allows getting the nature of single and multiple PDF manifestation and finding vector pairs that detect PDFs. This approach is useful for investigation of testability properties of circuits concerning PDFs. In partly we found out that all single PDFs in a circuit obtained by covering Shared ROBDD with CLBs manifest themselves as robust ones. Test patterns from pairs are contained among test patterns for single stuck-at faults at the CLB poles of a circuit.

  36. Thanks for your attention!

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