Selection of the flip-flops for partial enhanced scan techniques Текст научной статьи по специальности «Математика»

2012

ВЕСТНИК ТОМСКОГО ГОСУДАРСТВЕННОГО УНИВЕРСИТЕТА Управление, вычислительная техника и информатика

№ 2(19)

ПРОЕКТИРОВАНИЕ И ДИАГНОСТИКА ВЫЧИСЛИТЕЛЬНЫХ СИСТЕМ

УДК 004.312

A.Yu. Matrosova, A.V. Melnikov, R.V. Mukhamedov, S.A. Ostanin, V. Singh

SELECTION OF THE FLIP-FLOPS FOR PARTIAL ENHANCED SCAN TECHNIQUES1

Structural scan based delay testing is used to detect delay faults. Because of the architectural limitations not each test pair vb v2 can be applied by scan delay testing. That declines test coverage. Partial enhanced scan approach based on selection of flip-flops was suggested to permit using arbitrary test pairs vb v2. The problem of selection of flip-flops may be solved with applying estimations of controllability and observability of the state variables corresponding to the flipflops. Calculation of controllability and observability estimations is based on 2-length combinational equivalent analyses and PDF testing.

Keywords: Path delay fault (PDF); robust PDF; equivalent normal form (ENF).

Introduction

Delay testing has become very important problem in nanometer technologies. Structural scan based delay testing is used for detecting the circuit delays. Because of the architectural limitations not each pair vb v2 can be applied by a scan delay test. Enhanced scan techniques were developed to remove the restrictions on vector pairs. Unfortunately these techniques have rarely been used in practice because of the near doubling of the flip-flop area. Most promising are partial enhanced scan approaches that are based on selection of the proper flip-flops for including them in enhanced scan chains [1]. In the paper [2] it is suggested to include in scan chains flip-flops with low estimations of the 0 values of the corresponding state variables (low controllability) but facilities of propagation of changing signal values from the input to the output (observability) are not considered. In the paper [3] were suggested deriving estimations of both controllability and observability. We suggest to appreciate flip-flop controllability as sum of 01 transition probability and 10 transition probability for the corresponding state variable. The flip-flop observability is appreciated as a probability of robust PDF manifestation for the paths of the same state variable. The algorithms of controllability and observability calculation are developed.

In Section 1 the problem of calculation 01(10) transition probability for the state variable is discussed. In Section 2 the method of calculation of robust PDF manifestation probability for the paths corresponding to a state variable is suggested.

1 Работа выполнена при финансовой поддержке Российского фонда фундаментальных исследований (грант № 11-08-92694-ИНД).

x 1

Xn

y i

1. Calculation of a probability of 01(10) transitions of state variables

Let we have a synchronous circuit (Fig. 1) in which are input variables,

y1,.,yp are state variables, z1,.,zm are output variables, and d1,...,dp are flip flops.

Random input sequence of a sequential circuit is described with the probability distribution p(x1),.,p(xn). Here p(x;), /=1...n, is the probability of 1 value of input variable x;. Let assume that the probability distribution p(y1),... ,p(yP) of state variables is also known. Here p(x;), /=1...n, is the probability of 1 value of input variable xt.

We want to calculate the probability 01(10) transition for a state variable y. For that consider the 2-length combinational equivalent of Fig. 2.

Having the structural description of a combinational equivalent copy we may represent the copy functions corresponding to the combinational circuit C2 by the shared BDD which roots correspond

d1

yp

Fig. 1. Synchronous circuit

to the state variables

yi

yp. These functions depend on the input variables

x12,...,x„2 and the state variables y1 2,...,yp2 (Fig. 2).

Select root correlating with the variable y f . All paths connecting this root with the 1-terminal node of the shared BDD represent the corresponding function y,2 (x12,.,xn2, y12,.,yp2) as a disjoint sum of products (DSoP) D, that is as a sum in which products are orthogonal each other. All paths connecting this root with 0-terminal node represent an inversion of this function as DSoP D , in the same way.

Two products are orthogonal if some variable appears in one product without inversion and in another - with inversion.

Xi

y i

yp

C 1

zi Xi

i zm x: Xn

1 V1 yi

Vp1 yp2

C 2

->zi

Vi

Vp

-> 2'

yi

-> 21

yP

Fig. 2. 2-length combinational equivalent

2

2

2

X

n

m

2

1.1. 01 transition probability

Extract a product K from the Dt that either contains literal y2 or variable y,2 is absent in K. In the last case we add literal y2 to product K. This product encloses in general case both the input and the state variables.

Notice that the state variables correspond to the transition functions that

depend on the input variables x11,.,xn1 and the state variables y11,.,yp1 and all these variables are statistically independent on the input variables x12,.,xn2 [2]. It means that we may substitute instead of the variables x12,.,xn2 of the product K their probabilities and do that with each product from the D,, selected above mentioned way. We have got the following sum: ^bj kj

Here bj is result of multiplications of the probabilities p(xi),...,p(x„) of the proper input variables from a set {x12,.,xn2} and kj consists of some state variables (possibly with inversions) from a set {y2,...,^2}. The different functions corresponding to the state variables of the product kj (they are represented with the C1 copy) in general case depend on the same variables and these functions are statistically dependent.

Represent corresponding to the product kj conjunction of the state variable functions with the BDD-graph.

We may calculate a probability of this conjunction using the given probability distribution for the input and the state variables.

Multiply the result by bj. Execute this with each product K from the Dt. Summing results we calculate the 01 transition probabilities.

yi

yi

1.2. Example of calculating 01 transition probability

Consider an example to illustrate the above mentioned results. The transition functions of a copy of the combination equivalent are represented with the following shared

BDD (Fig. 3). For simplicity we don’t use upper indexes of variables.

Let probabilities of the 1 value of the internal and the state variables be the same and equal to 1/2: p(xi) = pfo) = p(yi) = p(y) = 1/2.

Select the state variable y2 (Fig. 2). Extract the corresponding DSoP from the shared BDD (Fig. 3). Here we take into consideration

• . 2' 2 2—22 upper indexes: yl = xl x2 v xl y2 .

All products do not contain y2. Consequently we need add the literal y 12 to each

product. As a result we have got DSoP D*:

2 2—2 —2—2 2 xl x2 y1 v x1 y1 y2 .

Substitute probabilities instead of the variables x12, x22 and obtain the following result: 1/4 y2 v 1/2 y2 y22.

Extract the functions y J, , corresponding to the y2,

y22 from the shared BDD (Fig. 2 and 3):

—2 —1—1 1—1

yi = x1 y2v x1 x2;

p( yi2) = 1/4+1/4 = 1/2;

2—1 111

J2 = x1 V x1 y ^2 .

The conjunction of these functions is represented with the expression: (xj1 y-5 v xj) a (xj1 v xjy1 y\).

The BDD of this expression is represented on Fig. 4.

Extract DSoP, representing the conjunction yj2y^:

xjy2 v xj1x2yj1 y2 = 1/4 +1/16 = 516.

Fig. 3. Shared BDD

Fig. 4. BDD representation of the conjunction y2 y\

We could calculate p( y1 y2) directly from the BDD of Fig. 4.

Substitute into the expression marked with (*) the obtained probabilities and calculate the probability of 01 transition: p (01) = 1/4 * 1/2 + 1/2*5/16 = 9/32.

1.3. 10 transition probability

To calculate 10 transition probabilities we need obtain DSoP of D; from shared BDD selecting root that correlates with the variable yf . Then we choose the products from D; with the literal yi and the products that does not contain the variable yi adding to the last products the literal yi. For each chosen product we execute the steps of the point A.

2. A probability calculation of manifestation of robust PDFs for state variable

Consider the problem of probability calculation of manifestation of robust PDFs for state variable. First we consider robust PDF test pair properties [3]. For that we will examine the equivalent normal form (ENF) of a combinational equivalent copy.

2.1. Robust PDF test pair properties

An equivalent normal form represents the function implementing with the circuit and all the circuit paths. Each ENF literal is supplied with the index sequence enumerating the gates of the path. It should be noted that a literal with the same index sequence may appear in different ENF products. The ENF of the circuit (Fig. 5) is as follows (1).

d

Fig. 5. The combinational circuit a1459 b59e59 v b59c23459 d3459e59 v

Va14689 b789 c234689 V a14689 c234689 d789 V a14689 b789 d34689 V a14689 d34689 d789

If literals have the same index sequences but their variables are opposite in sign then we call them as opposite literals. The same variables xt,xt being opposite in sign we call opposite variables. Opposite literals are absent in an ENF but opposite variables may present. We will spread all operations on products of a SoP (Sum of Products) on ENF products. We call an ENF product empty if it contains opposite variables. Notice that an ENF empty product consists of different literals.

Examine non-empty products of ENF. Any such product turns into SoP product after elimination of index sequences from literals followed with elimination of repeated variables. Sum of obtained products is SoP representing the function f. Non empty ENF product call implicant of the function f.

We consider single robust PDF of the path a in the circuit for the appropriate transitions along the path [3] as the temporary ap (b^)-fault of the ENF literal xia. This fault

a

c

lasts during time ro, ro>T. Here t is the time passing between adjacent synchronizing signals of the circuit.

The literal xia is changed for the constant 1 (b^-fault) in ENF products when the corresponding PDF activates the 1 value instead of the expected 0 value on the relevant circuit output and a vector v2.This PDF corresponds to falling transition.

The literal xia is changed for the constant 0 (ap-fault) in ENF products when the corresponding PDF activates the 0 value instead of the expected 1 value on the relevant circuit output and a vector v2. This PDF corresponds to rising transition.

A test pattern v2 in the PDF test pair vi, v2 is a test pattern for b^(a^)-fault of ENF.

Let K be ENF product (in partly K may be empty ENF product). K is expansible with the literal xia if elimination of this literal gives rises to product K that is implicant of the function f Otherwise K is not expansible with the xia.

Elimination of the literal from a product modifies the given ENF (K changes for K ). If K is non-empty product and K is non implicant of the functionf thenf changes along with ENF. It means that the 1 value area of the function f increases.

K * is the result of glue of products K and K . Here K is obtained from K with changing the literal xia for opposite literal. We will call K as addition of K .

We suppose that the variable xi in the literal xia doesn’t have an inversion. First consider b^-fault of the literal xia.

Let K be a set of ENF products so that each product does not contain xia.

Divide the rest ENF products into two sets: one of them Krxi consists of products so that each of them has repeating variable xi (the same variables with the same sign of inversion and the different index sequences). Products of Krxi don’t change the function f when b^-fault takes place and consequently not generate test pattern vb for b^-fault.

Another set Ksxi consists of the products (empty and non-empty) without repeating variable xi.

Let K be non-empty product from Ksxi. If K is non-expansible product by literal xia then changing that literal for the constant 1 in K alters the function f. That may be detected with a test pattern vb which turns into 1 the product K and turns into 0 fault free ENF. This test pattern is at the same time test pattern v2 from a test pair v1, v2 for the corresponding PDF. Notice that vb turns into one the product K possibly together with other products derived from Ksxi by changing the literal xia for the constant 1.

If K is expansible product by literal xia then changing that literal for the constant 1 does not alter the function f It means that there is no test pattern vb for b^-fault originated by the product K.

bp-faw/?

Consider b^-fault of the literal xia and test pattern vb that satisfies above mentioned conditions. Variable xi take the 1 value on v2.

Denote as w the minimal cube covering v1, v2 and as k(w) - the product representing w.

Theorem 1. To derive robust PDF test pair v1, v2 corresponding to b^-fault we need the following.

1. v2 is a test pattern for b^-fault;

2. Variable xi in v1, v2 takes the opposite values;

3. k(w) is orthogonal to each product from K;

4. Test pattern v1 turns into 1 product K from Ksxi that generates test pattern vb .

Corollary 1. The function f takes the 0 value on v2 and takes the 1 value on vi. Corollary 2. Empty product K from Ksxi doesn’t originate robust PDF test pair. Corollary 3. v1 is a test pattern for a^-fault.

As k(w) is orthogonal to all products of K then v1 is orthogonal to all products of K but v1 turns into 1the functionf It means that v1 is a test pattern for a^-fault.

Opfaw/i

Now consider ap-fault of the literal xia. All products from K remain in the fault ENF.

The rest products of the fault free ENF are divided into two sets: one of them Kexi consists of the empty products and another Knexi consists of the non-empty products. The products of Kexi do not change the functionf when a^-fault takes place and consequently not generate test pattern va .

Consider the set Knexi. All its products disappear when a^-fault takes place. If it changes the function f then there exists a test pattern va that detects a^-fault. The test pattern turns into 1 some products from Knexi and turns into 0 the rest products of the fault free ENF. The test pattern va also turns into 1 the variable x;.

Theorem 2. To derive robust PDF test pair v1, v2 corresponding to a^-fault we need the following.

1. v2 is a test pattern for a^-fault;

2. Variable xi in v1, v2 takes the opposite values;

3. k(w) is orthogonal to each product from K;

4. There exists product K from Knexi that does not contain repeated variable xi so that values of the variables of the cube representing this product and values of the variables of test patterns v1, v2 (except variable xi) coincide.

Corollary 1. The function f takes the 1 value on v2 and takes the 0 value on v1.

Corollary 2. vi is a test pattern for bp-fault.

As k(w) is orthogonal to all products of K then v1 is orthogonal to all products of K and by the construction v1 turns into 1 the addition of the product K. It means that v1 is a test pattern for b^-fault.

2.2. A probability calculation of robust PDF manifestation for the given path

Represent as the DSoP all robust test pairs for the given path originated by one product of ENF. For that we have to find product K from Knexi that does not contain repeated variable xi and obtain the SoP D from the set K fixing the variables of the product K except xi. All roots of the equation D = 0 are represented either as ROBDD or Free BDD. Each path from the root till the1-terminal node arises to the product, corresponding to 2n-1-r robust test pairs consisting of neighboring Boolean vectors. Here r is a rank of the product and n is the number of ENF variables.

(Notice that the variable xi is absent in the product). Having got the BDD for each K from Knexi that does not contain

repeated variable x and corresponds to the same path a we pig. g. bdd representation may execute disjunction operation on these BDDs and ob- of the product d

tain BDD representing all robust test pairs. Using the last BDD we may calculate a probability of the robust PDF manifestation for the path a substituting instead of the variables their probabilities from the probability distribution.

Illustrate that procedure by an example. Let the variable c be a state variable. First consider the literal c234689. Take into consideration that for the product abc , k(w) = a b , and the corresponding D, D = d v d d . The BDD represents the only product d.

Consider forth product of the ENF that contains the literal c234689. In this case K = ac d , k(w) = a d , D = b v 1 = 1.The equation D = 0 has no roots.

Consequently a probability of robust test pair manifestation for the path a corresponding to the literal c234689 is represented by the formula abd and is equal to 1/2*1/2*1/2 = 1/8.

Here we suppose 1 value probability of each variable is equal to 1/2.

2.3. A probability calculation of robust PDF manifestation for state variable

When we consider a state variable we have to regard all paths (literals) connected with this variable. Derive for each path a probability of robust PDF manifestation in the above mentioned way. Then we have to summarize these probabilities.

Notice that a test pair for one path does not change the signal values of other path corresponding to the same variable as all products representing another paths are contained in a set K formed for the path considered. It means that sensitizations of different paths of the same variable are statistically non-compatible events.

Let c be state variable. We additionally have to find a probability of robust PDF manifestation for the path corresponding to the literal c2 . Then K = b c de,

k(w) = bde, D = a .

BDD originates the only root: product a. Consequently a probability of robust PDFs manifestation for the path corresponding to the literal c is represented by the formula abde. It is equal to 1/2*1/2*1/2*1/2 = 1/16. Then a probability of manifestation of robust PDFs for the state variable c is as follows: 1/8+1/16 = 3/16.

We have got the following experimental results.

Controllability Pc and observability Po have been calculated for each state variable for some benchmarks. Flip-flops corresponding to state variables with low controllability or/and observability can be selected for including in enhanced scan chains. The additional investigations are necessary for choosing the threshold values for probability and observability We may only say that flip-flops with zero observability must not be included in enhanced scan chains.

The algorithm of estimation calculation of flip-flop observability is based on ENF analysis and using BDDs. ENF is very complicate formula for real circuits. It is possible to use OR/AND tree to present all paths of a circuit [6]. These trees are used for finding the estimations for benchmarks of the Table 1.

We may use for description of the ENF of a circuit the system OR-AND trees. The complexity of the system is linear function of the number of the circuit gates [6]. To accelerate the calculation of flip-flop observability it is possible to joint application of the system OR-AND trees and the corresponding system SSBDDs [7]. We hope that these techniques will allow calculating flip-flop observability for more complicate circuits.

Experimental results on Iscas89 benchmarks

Circuit Flip-flops State Variable Pc Po

s27 3 G11 0.172 0.125

G10 0.457 0.063

G13 0.344 0.250

s344 15 ACVG3VD1 0.331 0.219

ACVG2VD1 0.260 0.219

AM3 0.125 0.875

ACVG4VD1 0.180 0.219

CNTVG2VD 0.188 0.375

AM2 0.281 0.875

CNTVG3VD 0.188 0.500

AM1 0.281 0.875

AM0 0.281 0.875

MRVG3VD 0.500 0.125

MRVG4VD 0.641 0.125

MRVG1VD 0.406 0.125

MRVG2VD 0.406 0.125

CNTVG1VD 0.594 0.375

ACVG1VD1 0.280 0.219

s444 21 G58 0.209 0.141

G112 0.335 0.000

G49 0.063 0.063

G111 0.224 0.000

G45 0.094 0.188

G41 0.125 0.125

G113 0.130 0.000

G162BF 0.129 0.453

G80 0.133 0.343

G70 0.091 0.186

G101 0.375 0.500

G66 0.113 0.275

G110 0.233 0.000

G62 0.119 0.230

G109 0.099 0.000

G84 0.120 0.382

G92 0.108 0.362

G155 0.375 0.500

G88 0.115 0.402

G114 0.026 0.000

G37 0.578 0.000

Conclusion

The special estimations of flip-flop controllability and observability are developed. They may be used for including flip-flops into partial enhanced scan chains. Possibilities of their application to more complicate circuits are noted.

REFERENCES

1. Xu G., Singh A. D. Achieving high transition delay fault coverage with partial DTSFF enhanced scan chains // Proceedings of International Test Conference. 2007. P. 1-9.

2. Xu G., Singh A. D. Flip-flop selection to maximize TDF coverage with partial enhanced scan // Proc. ATS2007. 2007. P. 335-340.

3. Wang S. Low overhead partial enhanced scan technique for compact and high fault coverage transition delay test patterns / S. Wang, W. Wei // Proc. ETS2008. 2008. P. 125 -130.

4. Matrosova А. Random simulation of logical circuits //Automation and Remote Control. 1995. No. 1. P. 156-164.

5. Matrosova A., Lipsky V., Melnikov A., Singh V. Path delay faults and ENF // Proc. EWDT Symposium, Russia, St. Petersburg, September, 2010. P. 164-167.

6. Matrosova A., Andreeva A., Melnikov A., Nikolaeva E. Multiple stuck-at fault and path delay fault testable circuits // Proc. EWDT Symposium. 2008. P. 356-364.

7. Ubar R. Multi-valued simulation of digital circuits with structurally synthesized binary decision diagrams // OPA (Overseas Publishers Assotiation) N.V. Gordon and Breach Publishers, Multiple Valued Logic. 2001. V. 4. P. 141-157.

Matrosova Anzhela Yu., Melnikov Alexey V.,

Mukhamedov Ruslan V., Ostanin Sergey A.

Tomsk State University

Singh Virendra. Indian Institute of Technology Bombay

E-mail: [email protected]; [email protected]; [email protected];

[email protected]; [email protected] Поступила в редакцию 12 января 2012 г.

Матросова А.Ю., Мельников А.В., Мухамедов Р.В., Останин С.А. (Томский государственный университет), Сингх В. (Индийский институт технологий, Бомбей). Выбор триггеров в частичные цепи для методов сканирования схем с памятью.

Ключевые слова: неисправность задержки пути, робастно тестируемый путь, эквивалентная нормальная форма (ЭНФ).

При тестировании неисправностей задержек путей схем с памятью используется метод сканирования путей. Существующие архитектурные решения, обеспечивающие реализацию метода сканирования, не позволяют подавать на тестируемую схему любую пару тестовых наборов. Одним из выходов в этой ситуации является использование частичных цепей сканирования с дублирующими триггерами. В эти цепи включаются лишь некоторые триггеры схемы с памятью. В данной работе предлагается включать в частичные цепи сканирования триггеры с низкими оценками управляемости и наблюдаемости. Разработан алгоритм вычисления управляемости переменной состояния, основанный на анализе комбинационного эквивалента длины два. Предложен алгоритм вычисления наблюдаемости переменной состояния, основанный на анализе эквивалентной нормальной формы (ЭНФ). Проведены испытания обоих алгоритмов на контрольных примерах ISCAS’86.