Analysis and design of linear finite state machines for signature analysis testing

1034 IEEE TRANSACTIONS ON COMPUTERS, VOL. 40, NO. 9, SEPTEMBER 1991

Analysis and Design of Linear Finite State Machines for Signature Analysis Testing

Maurizio Damiani, Piero Olivo,

Abstruct-This paper presents a theoretical investigation on the aliasing error probability (AEP) in signature analysis testing by means of linear finite state machines (LFSM’s). The analysis is based on the assumption of statistical independence of successive error veetors. The equations of the resulting Markov chain model of the LFSM are solved to determine an exact expression of the AEP as a function of the main LFSM features (such as its number of flip-flops and feedback network) and of the relevant parameters of the testing environment (namely, test length and input e m r probabilities). This expression is used to prove criteria for the synthesis of LFSM’s with minimum asymptotic and transient AEP.

A fundamental lower bound on the AEP is presented. It represents the performance limit of any LFSM with respect to aliasing minimization, and we show that the AEP in machines realizing counters mod 2k - 1 is the closest to such a bound, in particular periodically reaching it. Finally, it is proven that the lower bound on the AEP is independent from any linear reencoding of the inputs, so that no linear interface circuit can improve the maximum performance of a LFSM.

Index Terms- Built-in self-test, design for testability, linear finite state machines, signature analysis, Markov processes.

I. INTRODUCTION

N signature analysis testing, linear finite state machines I ( L F S M ’s) (usually linear-feedback shift-registers [ 11, [2], though recently also linear cellular automata have been pro- posed [3]) are used for the compaction of the data stream coming from a digital circuit under test (CUT). At the end of the test phase, only the state of the machine (called the signature of the CUT) is checked against the expected signature to decide on the CUT correctness [see Fig. (l)].

This testing procedure is prone to aliasing errors, i.e., correct signatures generated by faulty circuits. The occurrence of these errors depends on the entire output sequence of the faulty CUT, and it can be analyzed exactly only through prohibitively long fault simulations. Obviously, this approach can neither provide instruments for understanding the dependencies of aliasing from the main test parameters (such as the test length or the LFSM design), nor informations on the impact of faults not modeled by the simulator on the performance of the signature register. Furthermore, the reliability of such simulation-based measures of aliasing probability is limited by the considered fault models. Such models, although precious

Manuscript received August 1, 1989; revised January 2, 1990. M. Damiani is with the Center for Integrated Systems, Stanford University,

P. Olivo and B. Riccd are with the Dipartimento di Elettronica, Informatica

IEEE Log Number 9101690.

Stanford, CA 94305.

e Sistemistica, Universiti di Bologna, 40136 Bologna, Italy.

and Bruno Riccdi, Member, IEEE

in analyzing several testability issues in a circuit, are in fact inadequate to describe correctly the faulty behavior of real MOS VLSI circuits [4].

The study of the major dependencies of aliasing errors must then rely on probabilistic models of the tested circuits and LFSM’s. The associated performance measure is then the aliasing error probability (AEP). In the past, the CUT was modeled by assuming, admittedly unrealistically, the equal probability of all possible error sequences [SI, [2], [5 ] . A more general assumption of statistical independence of successive error vectors (SI assumption, for short) was first used in [6] to investigate the properties of single-input shift-registers with characteristic polynomial 1 + 2‘. The work on this type of registers was then extended to the parallel-input case in [7].

In [8] it was first recognized that the SI assumption could be used to model the behavior of a linear-feedback shift- register as a Markov chain. It was also first conjectured that, at least in the single-input case, primitive characteristic polynomials could provide a better performance with regards to aliasing, by resulting in a better distribution of the Markov chain eigenvalues. Computational techniques for evaluating the aliasing probability have been analyzed in [9].

In [lo] an analytical approach was presented for the analysis of the Markov chain equations, and a proof was presented of the aforementioned conjecture. Exact and simplified expressions of the aliasing probability, valid for any test length and characteristic polynomial, were also derived.

This work has been then extended to a parallel-input case in [l l] , by considering each output of a CUT as an independent source of random noise. The solution of the Markov chain equations showed in particular that, unlike the single-input case, the aliasing probability in a k-bit LFSM can asymptotically reach values much larger than the “classical” limit 1/2‘

A general theory on the data compaction properties of linear finite-state machines, however, is still missing. For example, the performance limits of such machines with regards to aliasing are not known, and several issues, such as the effects of statistical correlations among inputs or the possible influence of interface circuits between the LFSM and the CUT outputs on aliasing have been the subject of introductory studies [13] but are not yet fully understood.

In this paper, we present a theory of the aliasing probability of sufficient generality to deal with all the problems mentioned above; beyond that, it attempts to contribute to the understanding of the properties of linear (over the Galois field GF(2) [14]) finite structures under random noise.

[SS], [W.

0018-9340/91/0900-1034$01.00 0 1991 IEEE

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I

1035 DAMIANI et al.: LINEAR FINITE STATE MACHINES FOR SIGNATURE ANALYSIS TESTING

Fig. 1. General signature analysis testing setup.

In particular, the SI assumption is used to model the register behavior as a Markov chain [15]. A general, exact solution is derived for the model equations, by means of an original technique based on the use of Hadamard matrices [lo], [ll]. From this expression separate analyses are conducted on the asymptotic and transient behavior of the AEP, respectively, and in particular necessary and sufficient conditions on the LFSM architecture for guaranteeing the minimum asymptotic AEP = 1 / 2 k are presented. It is in particular shown that, similarly to the single-input case [8], [lo], [ l l] , the behavior of LFSM's realizing maximum-period counters (MP-LFSM's) [16], [17] is optimal with regards to aliasing minimization.

A fundamental lower bound on the AEP is derived, which constitutes the performance limit of such machines with regards to aliasing. It is then proven that such a bound: 1) is actually periodically reached by every MP-LFSM; and 2) is independent from any linear reencoding of the CUT outputs, SO

that no linear interface circuit between the CUT and the LFSM can improve the maximum performance of linear signature analysis techniques.

The rest of this paper is organized as follows: the next section contains the formal description of the Markov chain model of a LFSM and the mathematical tools required for the solution of its equations; Section I11 presents the solution technique and a discussion of the main characteristics of the AEP behavior; finally, the fourth section contains a derivation of the lower bound on the AEP and a discussion on its implications in a testing environment. The proofs of the theorems presented in this work have been collected in the Appendix.

11. DEFINITIONS AND NOTATIONS

Let k be the number of available LFSM inputs and register cells. If the CUT has fewer outputs than I C , the remaining free LFSM inputs will be considered connected to dummy CUT outputs, constantly 0. The state of the LFSM and the CUT output can be both represented by k-dimensional binary vectors, denoted by y = ( ~ ~ , . - . , y k ) ~ and a = ( c q , . . . , Qk)* , respectively.

An alternative representation consists in considering each vector as the binary code of an integer i from 0 to 2k - 1. For example, in the case of the state of the LFSM, i = 2("')yl + + 2Oyk. Both notations will be used in this paper, depending on the context. The binary vector corresponding to an integer i will be denoted with y; =

The symbol in will indicate the future state of a given state i in a LFSM in n clock cycles. In particular, i l and

T ( yi, 1 , ' . ' 9 Yi, k ) -

i-1 will indicate the future and past state in one clock cycle, respectively.

A. State Transitions in LFSM's The future state of a LFSM can be computed from the

present state vi and CUT output Q by means of the linear equation [16], [18]

The matrix of binary elements Q = ( q 2 , 3 ) identifies uniquely the function implemented by the linear feedback network. Its nonsingularity (det Q # 0) will be the only constraint imposed to the LFSM structure.

The structure of the matrix Q for linear-feedback shift- registers and linear cellular automata is shown in Fig. 2. Other architectures can be found in the literature on signature analysis [19] and on linear finite state machines [16], [18].

The characteristic polynomial of the matrix Q is said to be irreducible if it cannot be factored. An irreducible polynomial is said to be primitive if it divides x Z k - l + 1, but no other polynomial of the type xq + 1, q < 2k - 1 [16], [17].

The autonomous behavior of a LFSM is obtained by setting CY 5 0 in (1). A LFSM working in its autonomous mode will be referred to as ALFSM. Because of the nonsingularity of Q, there is a unique predecessor i-1 for any state i in an ALFSM; consequently, the state diagram of the ALFSM's considered here consists always of a set of C cycles, of period I , ; c = 1, . . , C. In particular the state 0 always forms a cycle of period 1.

For any k at least an ALFSM can be found whose state diagram is formed by a maximum period cycle, of period 2k - 1, plus the state-0 cycle. Such machines will be referred to as maximum period LFSM's (MP-LFSM's). A well-known theorem states that a linear machine satisfies this MP property if and only if its characteristic polynomial is primitive [16],

The output vector of a faulty CUT at n (i.e., after n test vectors are applied) can be represented as the superposition of the correct-circuit output vector and an error vector; by the linear superposition theorem the state y ( n ) can be represented as the superposition of the state reached by applying the correct output sequence (i.e., the expected signature) with the one reached by considering the register in the initial state 0

~ 7 1 .


I I OJ

I I I I Q t :::a . .

( 4 (4 Fig. 2. (a) Schematic diagram of a k-bit external-exoe shift-register; (b) the corresponding state-transition matrix; (c) schematic diagram of a linear cehlar

automaton; (d) its corresponding state-transition matrix Q.

and applying only the error vectors e ( n ) (a superposed "error signature," identically 0 for a fault-free CUT).

It is thus possible to assume an initial state 0 for the LFSM, and to consider the error vector ~ ( n ) as the effective CUT output. In particular, an aliasing event occurs whenever at the end of the test the is o, namely when the LFSM has returned to the initial state after applying the error sequence alone [5].

vector ~ ( n ) can be computed from [15]

7r(n) = MT(n - 1). (2)

The vector ~ ( 0 ) contains the initial probabilities of each state in the LFSM. In our case, the initial state is 0 with certainty, hence

B. The Markov Chain Model of the LFSM

Hereafter, each possible vector L represented by the corresponding integer e = 2("l)c1 + ... + ~ O Q , will be given a probability p,. From the SI assumption, the vector P = (PO, PI, . + . , &k -1) describes completely the stochastic process at the LFSM inputs.

The input process will be assumed stationary, i.e., p will be assumed constant throughout the test. The validity of the expression of the AEP, however, can be stretched to the more general nonstationary case, in which p = p ( n ) , as long as the SI assumption is maintained.

The actual value of p depends on the fault considered. In particular, po represents the probability of a correct CUT output and consequently 1 - po provides the probability of an error event at the LFSM inputs, a measure of the random detectability of the fault. Note also that this model of the CUT behavior automatically solves the problem of modeling the statistical correlations among different CUT outputs.

The SI assumption allows us to regard the future state of a LFSM as statistically dependent only on the present state and error vector, and therefore to model the register behavior as a Markov chain [15], with states and transitions corresponding to those of the LFSM. Such chains are described completely by their 2k x 2k matrix M of transition probabilities. Their elements M,, (representing the probability of a transition to state i from state j in one clock cycle) are given in our case by the probability of the unique error vector causing that transition in the LFSM.

Let r,(n) indicate the probability of state i at n (i.e., after n clock cycles); the vector ~ ( n ) = (?ro(n), ?r~(n), . . e , r 2 k - 1

( ~ z ) ) ~ will then describe the probability of all states at n. The

T

T(0 ) = (1,0, * . . , o y . (3)

Fig. 3 provides an example of a Markov chain associated to a simple 3-bit LFSM. Further details on Markov chain modeling of signature registers are available in [8].

As discussed in Section 11-A, aliasing events are due to the return to the initial state 0 after n clock cycles when a sequence of n nonidentically 0 error vectors is applied. The probability of state 0 at n is given by r ~ ( n ) , while the probability of the trivial all-0 vectors sequence is given by p:; the aliasing error probability at n (AEP(n)) is then [8]:

Equations (2) and (4) provide a simulation tool the Markov chain and, in particular, for the AEP(n) function. Neverthe- less, such equations do not provide any analytical informations on the AEP behavior or on the synthesis of optimal machines. Furthermore, the simulation complexity is O(n x 2 2 k ) , which may represent a computational problem even for LFSM's of small size (i.e., for lc _> 6 t 8).

The solution procedure that we present in this paper is a generalization of the one presented in [lo]. It relies on the following auxiliary definitions: Definition I: Dual LFSM: The dual of a lc-bit LFSM with

future-state matrix Q is the linear machine defined by the matrix (Q-') . In particular, the dual of an external-EXOR linear feedback shift-register is the internal-ExoR structure obtained by inverting all the feedback taps, except the rightmost one

T

[lo].

1037 DAMIANI et al.: LINEAR FINITE STATE MACHINES FOR SIGNATURE ANALYSIS TESTING

',. ' --/

PO

p2

_ _ _ _ _ _ - - - __.

- - . _. . . . .

'6

( b) Fig. 3. (a) A simple 3-bit LFSM. (b) Its associated Markov chain, assuming

OdY Po, P z , and P6.

Definition 2: Hadamard matrices and vectors: The 2k x Z k Hadamard matrices are defined recursively by [16]

Such matrices are orthogonal and symmetric. As each row of H 2 k contains exactly 2(k-1) 1's and -1's (except for the uppermost row, filled with 2k l's),

- Z k - 1 1 - hi = ( l , O , . * . ,o>?

i = O 2k

The ith column vector hi of the matrix H 2 k will be associated to state i (identified by the binary vector zi = (xi, 1,s. . , xi, k ) T ) in the dual ALFSM. The reason for es- tablishing this correspondence is provided later by Theorem 1 in Section 111.

Definition 3: probability coeficients: The elements $i, a = 0,. . , 2k - 1 of the vector

will be called probability coefficients of the states i in a LFSM. In particular, $0 is the sum of all error probabilities, and therefore is always 1.

The calculation of .rc, can be performed in 0 (k x zk) . Fig. 4 shows the coefficients $i for the Markov chain described in Fig. 3.

It can be easily verified that for i = 0,. . . , 2k - 1: I$,l 5 po + . $i 2 po - (p1 +

+ pp-1 = 1; and . +p2k-1) = 2p0 - 1; in particular

$I; > 0 when po > 0.5.

111. MODELING OF THE ALIASING ERROR PROBABILITY

In this section we present a derivation of exact and simplified expressions for the AEP(n) function. The derivation relies on the following result about the properties of vectors h; with respect to the Markov chain matrix M of a LFSM:

Theorem I : Let i by any state of the dual ALFSM; then

Theorem 1 states a remarkable algebraic property of M : all vectors h, are transformed by M (except for a scale factor) just as the corresponding states are sequenced in the dual ALFSM.

An application of Theorem 1 is given in Fig. 5, referring to the three-input LFSM of Fig. 3.

The following simple result is a first direct consequence of Theorem 1.

Corollary 1: ~ ( n ) = (1/2')ho is an equilibrium probability distribution for the Markov chain.

Proof: It is sufficient to recall that in any LFSM the future state of state 0 is still 0 and that $0 = 1; then from

Q.E.D. Theorem 1 M(1/2')ho = (1/2')ho. If this equilibrium condition is asymptotically reached then,

from (4), AEP(n + CO) = 1/2k, the "classic" limit already derived [l], [5], [20], [21] under more restrictive assumptions. In the next sections, however, it is shown that for most LFSM's this equilibrium condition is not always reachable, and that a much larger asymptotic AEP may result.

A. Exact Modeling of the Aliasing Probability The result of Theorem 1 is here applied to obtain an exact

expression of the aliasing probability. In particular, from (S),

Hence, from (2), (3), and (6), - 2'-1 / n \

(9)

and since, from (5), the uppermost element of each vector hi is 1,

. Z k - 1 1 n \

Equation (11) can be evaluated in O(n x 2k), and therefore is substantially simpler than the matrix model discussed in

1038

0 0.2 0

P =

0 0.1

IEEE TRANSACTIONS ON COMPUTERS, VOL. 40, NO. 9, SEPTEMBER 1991

ho hl hz h3 h4 h5 h6 h7

1 1 1 1 1 1 1 5

0.6 1 -1 -1 -1 -1 0.1

( b) Fig. 4. (a) A possible error probability distribution leading to the Markov chain graph of Fig. 3(a). (b) Computation of the corresponding distribution.

Section 11; its fundamental interest, however, lies in the possi- bility it gives to inspect the behavior of the AEP(n) function and its main dependencies.

Notice that in the nonstationary case, p = p ( n ) ; consequently + = +(n) and A4 = M(n). The derivation of the aliasing probability, however, is still correct if M" is replaced by n:==, M ( v ) in (9) and (10) and if the elements +i , are substituted by $iu(v) in (11).

In Sections 111-B and 111-C, (11) is used in particular to analyze the asymptotic and transient behavior of the aliasing probability, respectively.

B. Asymptotic Behavior of the Aliasing Probability Since the term corresponding to state 0 (and, by Corollary 1,

to equilibrium) is constant,

= AEP,, + AEPn,(n) (12)

where AEP,, and AEPn,(n) describe equilibrium value and nonequlibrium behavior, respectively, of the AEP function.

AEP,, represents also the asymptotic value of AEP(n) if the products in (12) tend to vanish as n -+ 00. A first condition for having

1 lim AEP(n) = - n-w 2 k

can be devised as follows.

state i(') such that If in each cycle c in the dual ALFSM there is at least one

1 < 1, then for any state i in the same

Theorem 3 Present State Future State ~~

OOo (0) 001 (1) 010 (2) 011 (3) 100 (4) 101 (5)

111 (7) 110 (6)

Fig. 5. (a) Autonomous Theorem 1.

Mho = $oh0 Mhi = 47h7

OOo (0) 111 (7)

Mh2 = 4lhl Mh3 = $6h6

001 (1) 110 (6) 010 (2) Mh4 = Q2h2 101 (5 ) Mh5 = $5h5 011 (3) Mh6 = 43h3

(4) Mh7 = $4h4

( b) dual of the LFSM of Fig. 3. @>Result of

cycle as i(") it is possible to write

where Ln/ZcJ represents the minimum number of occurrences of i(') in the dual ALFSM, regardless of the initial state in c. As the right term of (14) vanishes as n ---f 00, also each product in (12) must asymptotically vanish, and consequently (13) is satisfied.

This condition is not easy to check, as it is expressed in terms of the coefficients +i , which depend on the considered fault. It is then obviously desirable to have stronger results related only to the LFSM architecture and independent from any CUT properties. To this regard, the following theorem provides necessary and sufficient conditions on the state transition matrix Q of an ALFSM for obtaining an asymptotic AEP of 1/2k, regardless of the error probability distribution.

Theorem 2: (13) is verified for any nontrivial distribution p if and only if the matrix Q does not have any nontrivial invariant subspaces 1141, [22]. If Q has an invariant subspace of dimension d, then a nontrivial p can be found such that the AEP limit is 1/2d.

Theorem 2 shows implicitly that 1/2k is the lower bound for the asymptotic aliasing probability in a LFSM. No LFSM can then be expected to provide a better asymptotic performance under the SI assumption.

The maximum asymptotic AEP is determined by the minimum dimension of a nontrivial invariant subspace of &. If, for example, Q has an invariant subspace of dimension 1, then the AEP may reach an asymptotic value as high as 1/2.

The result of Theorem 2 can be physically explained as follows.

In the equilibrium condition, all states in the Markov chain have an equal probability 1/2k, as stated by Corollary 1. If the chain graph is connected (i.e., any state is reachable from any other in some steps with a nonzero probability), then sooner or later the Markov chain will reach this equilibrium condition [15]. Suppose there is an invariant subspace S of Q,

DAMIANl et al.: LINEAR FINITE STATE MACHINES FOR SIGNATURE ANALYSIS TESTING 1039

of dimension d < k, and suppose that, for a given fault, the only possible error vectors are in S. By definition of invariant subspace, 1) if y E S, Q y E S; and 2) any linear combination of vectors in S is still in S.

Hence, the future state of the LFSM, Qy @ E will always be in S. Asymptotically, only the 2d states in S will have equal probability; all the other states are unreachable and will always have probability 0. The asymptotic probability of any state in S (in particular, of state 0) will be then 1/2d > 1/2k.

Consider again the example LFSM of Fig. 3. The subspace defined by the base vectors (O1O)T. ( l l l ) T is an invariant subspace of dimension 2 of its state transition matrix. Fig. 6(a) shows the Markov graph resulting from a probability distribution in which only the vectors (OOO)T and (O1O)T (be- longing to the invariant subspace) have nonzero probability. Fig. 6(b) reports the aliasing probability that results from such a distribution, showing in particular the larger asymptotic value 1/2' predicted by Theorem 2. Notice, in particular, that the probability distribution used in this example models any fault detectable only at input 2.

A fundamental result of linear algebra (see, for example, [14] and [22]) states that the matrix Q does not have any nontrivial invariant subspace if and only if its characteristic polynomial is irreducible. Therefore, an asymptotic value 1/2k for the AEP can be guaranteed only by machines with irreducible characteristic polynomial.

C. Markov Chain Analysis and SimpliBed Aliasing Probability In Section 111-B, a link was established between the asymp-

totic AEP and the properties of the state transition matrix Q of an ALFSM, and in particular it was shown that only LFSM's with irreducible feedback polynomial always result in the minimum asymptotic AEP. On the other hand, as the test length is finite, in order to obtain an adequate picture of the aliasing probability behavior it is necessary to analyze also the transient component of the AEP(n) function. To this purpose, in this section an analysis of the Markov chain transients is presented. In particular, the distribution of the eigenvalues of M and its link with the particular ALFSM are derived, and the role played by such eigenvalues in the AEP is shown to be helpful for a simplification of its expression.

It is worth pointing out that from the standpoint of aliasing minimization, short transients of the Markov chain (given by eigenvalues of small magnitude) are desirable [8], [lo], [ll]. In fact, recalling that ~ ( 0 ) = 1 and 7ro(n) decays (possibly) to 1/2k as n grows, it follows from (4) that short transients in the Markov chain may result in a generally lower ~ o ( n ) and therefore a lower AEP for a given n. The understanding of the link between the structure of a LFSM and the Markov chain eigenvalues is therefore important also for the development of optimal synthesis criteria of LFSM's for signature analysis.

The following theorem gives an expression of the eigenvalues of M.

Theorem 3: Each cycle c in the dual ALFSM, of period I,, corresponds to a set of I, eigenvalues of M , to be found by

PO

p2

---..-------. ...............................................

( 4

0.2

0.1

I I

0.1 Y 1 I I I

0 5 10 16 20 25

Number of Patterns (n)

(b) Fig. 6. (a) Markov chain associated to the LFSM of Fig. 3(a) assuming now also p6 = 0. Notice that, differently from Fig. 3@), the chain graph is not connected, and in particular that state 0 appears in a cluster of four states. (b) Aliasing probability in the 3-bit example LFSM, with the probability distribution of Fig. 4(a) (solid line) and the modified probability distribution pg = 0, p2 = 0.3 (dotted line), modeling in particular a fault detectable only at input 2. Notice the larger asymptotic AEP produced by this latter distribution.

solving

= $)i" v=l


where i is any state in c. In particular, such eigenvalues have t all the same magnitude

In previous works [lo], [ll] it has been shown (under more restrictive assumptions on the error probability distribution) that maximum-period linear-feedback shift-registers always present the optimal distribution of eigenvalues with regards to the minimization of the chain transient duration. This property is of immediate extension to the general case of MP-LFSM's. In fact, the eigenvalues corresponding to the maximum period cycle are the solutions of1

2"-1

pk-l = n ?+hi (17) i=l

and have magnitude I ( a)

_( ,_ , . I , . . . ._ . . . . . . . ...... Equation (17) shows, in particular, that all Ic-bit MP-LFSM's give rise to the same distribution of eigenvalues. For any non-MP-LFSM, from (16) and (18) it follows that 0.10

C n+ = r&1

c=l 0.05

and therefore, if there is a cycle c such that r, < TMP, there must exist at least another nontrivial cycle c' such that rCt > TMP. Hence, MP-LFSM's are expected to provide the shortest chain transients and a generally smaller AEP.

0. I I 1

Fig. 7 shows a comparison of the eigenvalue distribution 0 5 10 15

on the complex plane in the case of the example LFSM and a primitive 3-bit LFSM, with the probability distribution of


( b) Fig. 7. (a) Markov chain eigenvalues on the complex plane for a primitive 3-bit LFSM (squares), and the example LFSM (triangles), with the probability distribution p of Fig. 4. @) AEP in the example (solid line) and a MP (dotted line) LFSM's.

Fig. 3. k t US now consider each product appearing in the expres-

sion (11) of the AEP. Since, from (16), for any state i in a cycle c

(20) In the case of MP-LFSM's (22) reduces to

it can be assumed n n $i, 53 T,".

v=l

Such an approximation is exact every 1, clock periods, as long as the first member of (21) is nonnegative. Equation (11) is consequently reduced to

l C AEP(n) 53 - 2 k

Zcr," - p: c= 1

'Notice that, as the product appearing in (17) involves all the elements of the vector .J, except i o , the order in which such elements are accounted is uninfluent, so that the double subscript i, appearing in (15) can actually be replaced by the simple subscript i in (17).

common to all Ic-bit maximum period devices. From the periodic exactness of the approximation (21) it

follows that the terms r," in (22) and (23) are exact estimates of the products in (11) when n is a multiple of I , . In the case of a linear-feedback shift-register, recalling that all periods 1, must divide the basic register period (the one of the cycle passing by state 1 [16]), it can be inferred that (22) is exact with that basic period. In the general case, (22) is exact with the minimum common multiple of all periods in the ALFSM.

DAMIANI et al.: LINEAR FINITE STATE MACHINES FOR SIGNATURE ANALYSIS TESTING 1041

A further property of the simplified AEP is given by the

Theorem 4: For any LFSM and any probability distribution following theorem:

P; PO > 0.5 + c 1

AEP(n) 2 3 1,r; - p t . c=l

The simplified aliasing probability is therefore actually a lower bound on the actual AEP.

Fig. 8 shows a comparison of exact and simplified aliasing probability for different LFSM’s and error probability distribution.

D. Notable Cases The analysis presented so far relied on a very general

assumption on the distribution of the errors at the LFSM inputs. This generality was necessary to deal with the broad variety of behaviors typically presented by faulty circuits.

It is sometimes possible and convenient, however, to make some simplifying assumptions on the input probability distribution. In particular, the q-ary symmetric channel [21] and the independent noise sources [l l] , [20] assumptions have been considered in the literature. This section we show briefly a near derivation of the aliasing probability for these cases, that relies on the approach considered so far.

1) The q-ary Symmeric Channel: A q-ary symmetric channel (q 2 2) is defined as a transmission channel (modeling in our case the CUT output) that introduces errors with the probability distribution p, = (1 - po)/(q - 1); e = 1 , . . , q - 1. We are interested in the case q = 2k. It is immediate to verify that with such a distribution $i = 1 - (1 - po)q/(q - 1); i = 1, + . . , q - 1. Hence, the expression (11) of the AEP reduces to

2k A E P ( n ) = - + I - - I - -

(25) 2: ( a ) ( 2 k - 1

Interestingly (25), substantially simpler and more general than the one shown in [21], is totally independent from the LFSM architecture; under the q-ary symmetric channel assumption, therefore, all k-bit linear machines present the same behavior with regards to aliasing. Furthermore, the asymptotic AEP is always 1 / 2 ~ .

2) Independent Binary Random Sources: This case models the statistical independence of the LFSM input bits. Hence, p can be derived from the k error probabilities &; j = 1 , . . + , k. Under this assumption [13]

k

(26) $Ji = n (1 - 2 4 p j ; j=1

consequently the exact expression of the aliasing probability reduces to

where wj( i ,n ) is the weight (i.e., the count of 1’s) of the bit sequence in the flip-flop j in the dual ALFSM, with initial

I I Exact

Approx.

0 100 200 300 400 500 600


Fig. 8. Exact (solid lines) and simplified (dotted lines) AEP for the external-woa linear-feedback shift-registers defined by the polynomials: 1 + r2 + x5 + r6+ r8 (primitive, curve a); 1 + x1 + r2 + x3 + x6+ r7 + x8 (primitive, curve b); 1 + x2 + x6 + r8 (nonprimitive, curve c); 1 + x 2 + x3 + x6 + x7 + x8 (nonprimitive, curve d). po = 0.98, p40 = 0.007, p4g = 0.006, p56 = 0.007.

state i. The simplified expression is still in the form given by (22), where

k

r, = n (1 - 24j)uj3~; (28) j=1

the exponents uj,, = wj(i,l,)/l, are the frequency of 1’s at the j th bit of the dual ALFSM, during cycle c. Equations (27) and (28) were proved in [ l l ] only for linear-feedback shift- registers. The single-input case can be easily derived from (27) and (28) by assuming that only one of the probabilities q!~j is nonzero. The resulting expression was proven in [lo] for linear-feedback shift-registers.

Iv. FUNDAMENTAL LIMITES OF SIGNATURE ANALYSIS

In this section we present some bounds of the AEP(n) function that implicitly define performance limits of signature analysis techniques by means of linear finite structures, and further illustrate the peculiarities of MP devices with regards to aliasing minimization. The following discussion depends, in particular, on the theorem:

Theorem 5: Let A E P M P ( ~ ) denote the simplified aliasing probability in any k-bit MP-LFSM; then

AEP(n) 2 A E p ~ p ( n ) (29)

for any k-bit LFSM, any test length, and any probability distribution p ; po > 0.5.

From Theorem 5, no linear finite state machine can yield an aliasing probability lower than A E p ~ p ( n ) . Thus, confirming that (as indicated by the analysis of the eigenvalue distribution) MP-LFSM’s should always provide the best performance with regards to aliasing. This is probably true in particular when the test length is a multiple of 2k - 1; in that case, in fact, A E P M P ( ~ ) is the exact estimate of the aliasing probability in such machines.

1042 IEEE TRANSACTIONS ON COMPUTERS, VOL. 40, NO. 9, SEF'TEMBER 1991

The discussion carried so far showed that, in order to ensure the minimum asymptotic and transient AEP for any probability distribution, MP-LFSM's represent the most reliable choice, and that the function AEPMp(n) represents a lower bound of the AEP for such machines as well as the minimum achievable AEP in any LFSM.

The actual computation of AEPMp(n) relies, however, on the calculation of the parameter TMP, which requires the detailed knowledge of the probability distribution p . On the other hand, the only easily available parameter for faulty circuits is usually only an estimate of the fault detectability 1 - P O . The following theorem gives simple bounds to TMP.

Theorem 6:

z ( k - 1 ) 2k (2PO - 5 TMp 5 1 - - (1 - P o ) (30) 2 k - 1

for any probability distribution p ; PO > 0.5. The bounds on TMP in (30) are tight: the lower bound is the

actual TMP in the single-input case [see (28)], while the upper bound is the actual TMP in the q-ary transmission channel model.

By replacing the bounds of Theorem 6 in (23) it is imme- diately possible to obtain bounds for A E p ~ p ( n ) that depend only on the fault detectability 1 - PO:

2k I-+ ;k ( 1 - 7 ;)( 1 - - 2k--1 (1 - P o ) ) n -PE

(31)

Recalling the periodic exactness of AEPMp(n), (31) provides in particular an upper bound to the aliasing probability when n is a multiple of 2k - 1. An interesting consequence of this latter property is that MP devices guarantee (at least periodically) an AEP below 1/2k. This property is not in general verified in non-MP structures.

In a testing environment, the assumption po > 0.5 used to prove Theorems 5 and 6 is not particularly restrictive; physically, it implies that a fault should be detectable by less than half of the test vectors, and in ordinary digital circuits only a few faults have such a large detectability to result in PO < 0.5.

A further limit of signature analysis with linear machines is now considered. After a test length nk = [k/(l - p o ) l , a fault has caused an average number k of error vectors to appear at the LFSM inputs. If po 2 0.5, all coefficients $ ~ i in (11) are nonnegative, so that

k

AEP(nk) 2 1/2k - p;' 2 1/2k - p y . From the inequality PO I e P 0 - l ; 0.5 5 PO 5 1 it follows that

(32)

Equation (33) shows that after a test length nk the aliasing probability is already very close to 1/2k. Hence, it is possible to conclude that given a test length n, any fault with a

random detectability larger than k / n (i.e., po < 1 - k/n) is masked with a probability at least approximately 1/2k by any LFSM, and only faults with a lower detectability have a higher probability of being not masked.

It may be wondered whether, by means of a suitable reencoding of the CUT outputs, it is possible to reduce the aliasing probability introduced by a LFSM. A simpler version of this problem (the effect of a permutation of inputs on aliasing) has been considered, for example, in [23]. Such a reencoding may be realized by means of a linear combinational circuit, acting as interface between the CUT and the LFSM, and described by a simple k x k matrix R. The actual LFSM input vector E' is thus related to the CUT output by the relationship E' = Re. Although in principle any matrix may be used, it is evident that only an interface realizing an 1-to-1 mapping of the CUT output vectors into the LFSM input vectors can preserve the information present at the CUT output, and not reduce the testability of any fault; consequently, the reencoding matrix R should be nonsingular. The problem of the optimal interconnection between the CUT and the LFSM is thus reduced to that of finding the optimal nonsingular reencoding matrix R. This problem is implicitly solved by the following theorem, which also further enlightens the fundamental nature of the bound given by Theorem 5 .

Theorem 7: AEPMp(n) is invariant with respect to any linear nonsingular encoding of the CUT outputs.

Therefore, the maximum performance of a LFSM cannot be improved by any reencoding of its inputs. Theorem 7 states, however, a further remarkable property of MP-LFSM's, namely that their performance is, at first order, unaffected by any linear reencoding of the CUT outputs, thus automatically solving the optimal interconnection problem.

Needless to say, the encoding of the CUT output may instead affect remarkably the performance of a non-MP- LFSM, as it may drastically change even the asymptotic value of the aliasing probability.

V. CONCLUSIONS

In this paper we presented an investigation of the properties of linear finite state machines when used in data compaction for signature analysis. The assumption of independent errors at the LFSM inputs has been used to model the register behavior as a Markov chain. The chain equations have been solved to obtain the exact dependence of the aliasing probability as a function of test length, input error probabilities, and LFSM structure.

The analysis of both the asymptotic and transient behavior of the aliasing probability showed that the most important factor in determining the properties of a LFSM in data compaction is only its characteristic polynomial. In particular, we proved that 1) contrary to previous indications, only machines with an irreducible characteristic polynomial can guarantee the asymptotic 1/2k aliasing probability limit, and 2) amidst such machines, those with a primitive polynomial are also likely to result in the minimum transient AEP, regardless of their particular design (i.e., cellular automata, shift-register, etc.).

DAMIANl et al.: LINEAR FINITE STATE MACHINES FOR SIGNMURE ANALYSIS TESTING

~

1043

Furthermore, we derived a lower bound on the AEP, which represents the performance limit of any linear finite state machine for signature analysis. This bound is in particular reached periodically by every MP-LFSM, and is fundamental in the sense that it cannot be improved by any linear reencoding of the CUT outputs.

Finally, we showed that when an average number k of errors is entered, the aliasing probability in any LFSM has in practice already reached the minimum asymptotic value 1/2k.

APPENDIX The proof of the theorems cited in the text requires some

auxiliary results on Hadamard matrices and on the Hadamard transform 1 ~ , of a probability distribution p.

Lemma a1 [IO]: The ith column h; of H 2 k is the truth table of the linear function of the vector t = ( 2 1 , . . . , Zk)T:

j = 1

expressed in the (1, -1) code (obtained by replacing 0 and 1 with 1 and -1, respectively).

Lemma a2: Given a subspace S, consider a probability distribution p such that p , = 0 V e $! S. Then, $i = 1 for any element i of the subspace S' orthogonal to S.

Proof: For each i E SI

2k-1 2"-1

e=O e=O

On the other hand, p , is assumed to be possibly nonzero only when (z;,ce) = 0. Hence,

2"l

$i = p , = 1. Q.E.D. e=O

Proof of Theorem 1: Let P(e) denote the 2k x 2k permutation matrix describing the behavior of the ALFSM when the input is e, (i.e., P t i = 1 if and only if 1 is the future state of m, when the ALFSM input is e). The matrix M can be then expressed as

2k---I

e=O

The behavior of the matrices P(,) with respect to the vectors hi is explained by the following Lemma.

Lemma 1.1: Let i be any state in the dual ALFSM, and let il denote its future state, then

where [hi,], denotes the eth element of hi,.

Proof: Let j51 denote the predecessor of a state j in the ALFSM, assuming a constant input e. Since Q is nonsingular, from (1) such as predecessor is unique, and its binary vector representation is yjy, = Q-l(yj @ B e ) .

Recalling the definition of P(e) , the element of P(e)hi in position j (indicated by [P(")hiIj) is the element of hi corresponding to the predecesor of j ; in formula

= [hiIj,, .

On the other hand, from Lemma al , [h i ] j y1 is the (1, -1) code, and

(zi,ujyl) = ( z i ~ - l ( v j CB .e>>

= (si1 7 yj CB € e )

= €e) CB (zil > gj) (a71

where zil is the binary vector representation of il, the future state of i in the dual ALFSM. It is then immediate to verify that, from Lemma al , in the (1, -1) code the right term of (a7) is exactly [hi,] ,[h;, l j . Therefore, (as) is verified for each entry j . Q.E.D.

The proof of Theorem 1 now follows directly from the definition of probability coefficients and from the lemma just proved:

2"--1 2 k - - 1

e=O e=O

- - [ g p e [ h i 1 l e ) hi, = $ilhil. Q.E.D. (as>

Proof of Theorem 2: Let S be a nontrivial invariant subspace of Q . With some algebra it can be easily verified that its orthogonal subspace S' is an invariant subspace of (Q-') .

If part-Consider any distribution p ; p , = 0 V e $? S. It follows that for each i E SI: 1) $; = 1 from Lemma a2; and 2) iu E SI, v > 1. Thus, each of the 2"' states i E S' contributes with a term

T

1 " 1 2k

1 " 5 n$i"=- n1=- 2k u = l u = l

to the aliasing probability. The remaining coefficients $;, i $2 S' can be made positive by choosing 1 > po > 0.5, so that each state not in SI contributes to the AEP with a positive term. Hence,

2k-d AEP(n) 2 - -pn (a101 2k

and in particular

Only if part-Suppose now that (all) is satisfied. From (11) and the fact that I$il 5 1 V i, each state in the dual


ALFSM contributes to the AEP with a term that either: 1) is constantly 1/2k; 2) decays to 0; or 3) oscillates periodically between 1/2k and -1/2k. It can be verified that the total contribution of the states in group 3) to the aliasing probability must be 0 (otherwise the AEP would have periodic jumps of finite amplitude and would not converge asymptotically). Only states from groups 1) and 2) are therefore of interest.

Let S denote the set of states l), and consider the minimum subspace S' containing S. It is then to be proved that: a) S' is an invariant subspace, and b) S' has dimension < k.

Part u)-Proof: let i be any state in S. Then i l E S, for otherwise the contribution of state i to the AEP would either be of type 2 or 3. Let then i be any state in S' - S. Then, it must be a linear combination of vectors in S; consequently also its successor i l is a linear combination of vectors in S and in particular is in S'.

Part b)-Proof: let d denote the dimension of SI. Suppose, by contradiction, that d = k. Then, there must be a set S' S of k linearly independent vectors whose probability coefficients are 1. Consider the probability distributions p that can produce this situation. It must be the solution of

2 k - 1

$2 = pe[hz], = 1, i E S' (a121

(a13)

e=O

2 k - 1

$0 = C pe = 1. e=O

For each state i E S', by subtracting the corresponding equation from (a13), it follows that

2"-1

Pe(1- [htj,) = 0. (a141 e=O

The value of the coefficients (1 - [h,],) is either 2 or 0. Therefore, the first member of (a14) is the sum of nonnegative quantities, and the only solution is p, = 0 whenever [h,], = -1, i.e., whenever (z,,~,) = 1. Consequently, the error vectors e, such that p, can be nonzero are solutions of

(z , ,ee) = 0; 2% E S'. (a151

The system (al5) is of k linearly independent equations, and the only solution is the trivial one E , = 0, implying p, # 0 only for e = 0. But in this case p = ( l , O , . . . , O ) T , and therefore AEP(n) 0 which contradicts AEP, > 1/2k; then it must be d < k. Q.E.D.

Proof of Theorem 4: The proof of the following theorems requires a known result concerning arithmetic and geometric means.

Lemma a3: Generalized geometric and arithmetic means

f i x : . 5 (g w,x:)'"; m 2 0 (a211

for any positive real quantities a,; i = 1, . . . , n, with positive real weights w, satisfying the constraint

~ 3 1 .

2 = 1

n

The first and last member of (a21) are usually referred to as generalized geometric and arithmetic mean, respectively.

The quantity r: is the geometric mean of the 1, quantities n

(a231 ( i , n) = $iv ; i E cycle c V=l

with weights wi = 1/lC and therefore, from Lemma a3, . L-1

m=O " C

Consequently,

Proof of Theorem 5: From Theorem 4 it is sufficient to prove that

or, equivalently, that C C

r: 2 rzP = JJ (rc 1 2 -1 (a27)

where it has been assumed that c = 1 corresponds to the trivial state-0 cycle in any LFSM. The first member of (a27) is the generalized arithmetic mean of the quantities r:, with weights w, = ZC/(2' - l), while the right member is the geometric mean of the same quantities. The inequality (a27) then follows from Lemma a3. Q.E.D.

Proof of Theorem 6: The first inequality follows from the convexity of ln(rMp) as a function of the variables $i, on the convex set defined by the inequalities 2p0 - 1 5 $i 5 k 1; i = 1 , . . . , 2k - 1, and the identity $1 + . . . 4- $ 2 k - 1 = 2 PO - 1 [23]. The minimum therefore occurs on the vertices of such a set [23]. On the other hand, these vertices correspond to distributions in which all p',s are zero, except a single one, whose value is therefore 1 - PO. It can be verified that with any of such distributions $i = 1 for exactly 2"' coefficients, and $i = 1 for the remaining ones.

The second inequality follows from observing that TMP is the geometric mean of the coefficients $i, and therefore from Lemma a3

IC

C G c=2 c=2

2 k - 1

Proof of Theorem 7: The 1-to-1 correspondence between the LFSM input vectors and the CUT output vectors can be described by a permutation matrix P such that Pi, = 1 if and

DAMIANl et al.: LINEAR FINITE STATE MACHINES FOR SIGNATURE ANALYSIS TESTING 104s

only if I is the LFSM input corresponding to the CUT output m. After the encoding, the vector p is transformed into p’:

p’ = P p (a291

(a301

and the new coefficients $: are given by

+’ = Hpp’ = H2k P p = H;kp

where H;k = H2hP. The columns of Hkk are therefore a permutation of those of H2k. On the other hand, from Lemma al, (= [H2k], , , ) is the ( 1 , -1) code value of (xi,€(}. But €1 = R-le, (the actual LFSM input vector); therefore,

T (Si, q) = (Si, R-le,) = ((l2-1) Zi, €,,,>

= (xi!, e m ) (a311 T where X ; I = (R-l) q.

row of P’ of the rows of H2k and

Hence, [ h ! ; k : ” ] i , m is the (1, -1) value of The ith is the ith row of H2k, hence H$ is a permutation

+’ = P ’ H p p = PI+. (a34

Equation (a32) shows that +’ is just a permutation of +; since TMP is clearly invariant under such a permutation, the simplified aliasing probability is unchanged. Q.E.D.

131

[91

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R.A. Frohwerk, “Signature analysis: A new digital field services method,” Hewlett-Packurd J., pp. 2-8, May 1977. B. Koenemann, J. Mucha, and G. Ziehoff, “Built-in test for complex digital integrated circuits,” IEEE J. Solid State Circuits, vol. SC-15,

P. D. Hortensius, R. D. McLeod, and H. C. Card, “Cellular automata based logic block observers,” in Proc. Second Technical Workshop: New Directions for IC Testing, Apr. 1987, pp. 1-18. J. A. Abraham and K. Fuchs, “Fault and error models for VLSI,” Proc.

J. E. Smith, “Measures of the effectiveness of fault signature analysis,” IEEE Trans. Comput., vol. C-29, pp. 510-514, 1980. R. David, “Feedback shift register testing,” in Proc. FTCS 1978, June

-, “Signature analysis testing for multioutput circuits,” IEEE Trans. Comput., vol. C-35, pp. 830-837, 1986. T. W. Williams, W. Daehn, M. Gruetzner, and C. W. Starke, “Bounds and analysis of aliasing errors in linear-feedback shift-registers,” IEEE Trans. Comput.-Aided Design, vol. CAD-7, pp. 75-83, Jan. 1988. A. Ivanov and V. K. Aganval, “An iterative technique for calculating aliasing probability in linear-feedback signature registers,” in Proc. FTCS 1988, May 1988, pp. 70-75. M. Damiani, P. Olivo, M. Favalli, and B. Riccb, “An analytical model for aliasing probability in signature analysis testing,” IEEE Trans. Comput.-Aided Design, vol. CAD-8, pp. 1133-1144, Nov. 1989. M. Damiani, P. Olivo, M. Favalli, S. Ercolani, and B. Riccb, “Aliasing errors in signature analysis testing with multiple-input shift-registers,” in Proc. IEEE Euro. Test Conf, Apr. 1989, pp, 346-353. A. Hlawiczka, “Hybrid design of parallel signature analyzers,” in Proc. IEEE Euro. Test Con$, Apr. 1989, pp. 346-360. T. W. Williams and W. Daehn, “Aliasing probability for multiple-input signature analyzers with dependent inputs,” in Proc. IEEE COMPEURO, May 1989, pp. 120-127. G. Birkhoff and S. MacLane, A Survey of Modern Algebra. New York Macmillan, 1977. M. S. Bartlett, An Introduction to Stochastic Processes. Champaign: Matrix. 1976.

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[16] W. W. Peterson and E. J. Weldon, Jr., Error Correcting Codes. Cam- bridge, MA: MIT Press, 1972.

1171 S. W. Golomb, Shifi-Register Sequences. San Francisco, C A Holden- Day, 1967.

[18] T. L. Booth, Sequential Machines and Automata Theory. New York: Wiley, 1967.

I191 L.T. Wang and E.J. McCluskey, “A hybrid design of maximum- length sequence generators,” in Proc. IEEE Int. Test Con$, Sept. 1987,

[20] T. W. Williams and W. Daehn, “Aliasing errors in multiple-input signature analysis registets,” in Proc. IEEE In?. Test Conf, Apr. 1989,

[21] D. K. Pradhan, S. K. Gupta, and M. G. Karpovsky, “Aliasing probability for multiple-input signature analyzer and a new compression technique,” in Proc. IEEE BIST Workshop, Mar. 1989.

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[22] C. C. McDuffee, Vectors and Matrices, Math. Ass. of Amer., 1961. [23] A. W. Roberts and D. E. Varberg, Convex Functions. New York Aca-

demic, 1973.

Mr. Damiani receivec Fellowship in 1988 and

is now also interested and fault simulation.

in

Maurizio Damiani received the Dr.Eng. degree from the University of Bologna, Bologna, Italy, in 1987.

He is currently working towards the Ph.D. degree at Stanford University, Stanford, CA. From 1987 to 1988 he was with the Department of Electronics, University of Bologna, as a Research and Teaching Assistant. His research interests are in the area of logic synthesis and testing, including design for testability and built-in self-test techniques, high- level synthesis, and coding theory.

d the AEI Scholarship and a Rotary International 1989, respectively.

Piem Olivo was born in Bologna, Italy in, 1956. Since 1983 he has been an Assistant Professor of

Applied Electronics at the University of Bologna, where he graduated in 1980 and received the Ph.D. degree in 1987. In 1986-1987 and autumn 1989 he was a visiting scientist at the IBM T. J. Watson Research Center. His scientific interests are in the area of solid state devices, including Si02 physics, electron transport and trapping through thin S i02 structures, oxide breakdown and reliability, MOS measurements techniques, thin oxide properties. He IC testing, with emphasis on design for testability

Bruno %cc6 (M’8.5) was born in Parma, Italy, in 1947. In 1971 he graduated in electrical engineering from the University of Bologna, Italy, and in 1975 received the Ph.D. degree from the University of Cambridge, U.K., where he had been working at the Cavendish Lab.

In 1980, he became Full Professor of Applied Electronics at the University of Padova, Italy. In 1983 he joined the Department of Electronics of the University of Bologna, Italy. In 1981 he was a Visiting Scholar at the University of Stanford and,

later, at the IBM Thomas J. Watson Research Center, Yorktown Heights, NY. His scientific interests concern solid-state devices and integrated circuits. In particular, he has worked on electron transport in polycrystalline silicon, tunneling in heterostructures, silicon dioxide physics, hot electrons effects in MOSFET’s, latch-up in CMOS structures and Monte Carlo simulation. Furthermore, he is also interested in circuit design and testing.

In 1988, Dr. Ricc6 was appointed Associate Editor of Europe for IEEE TRANSACTIONS ON ELECTRON DEVICES.

Analysis and design of linear finite state machines for signature analysis testing

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