Evaluation in built-in self-test

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E v a lu a tio n in B u ilt-I n S elf-T est

Shujian Zhang

M.Sc.. University of V ictoria. 1993

D issertation Subm itted in P artial Fulfillment of the Requirements for th e Degree of

DOCTOR OF PH ILO SO PH Y in th e Departm ent of C om puter Science

We accept this dissertation as conforming to the required standard

Dr. D.M. Kliller. Co-'Super\isor (D epartm ent of Com puter Science)

Dr. .LC. MuzIoT Co-Supervisor (D epartm ent of Com puter Science)

Dr. M. Fellows. D epartm ental Member (D epartm ent of C om puter Science)

^G * F r M c L e a n (^ u ts id e ^ ^ m b e r(D e p a rtm e n t of Mechanical Engineering)

Dr. I\"anov. External Exam iner (D epartm ent of Electrical and Com puter Engineering, University of British Columbia)

© Shujian Zhang. 1998 University of V ictoria

.A.11 rights reserved. This dissertation may not be reproduced in whole or in part, by photocopy or other means, without the permission of th e author.

11

Co-Supervisors: Dr. D.M. M iller and Dr. .J.C. Muzio

A b stra ct

This dissertation addresses two m ajor issues associated w ith a built-in self-test environm ent: (1) how to m easure w hether a given test vector generator is suitable for testing faults with sequential behavior, and (2) how to m easure th e safety of self-checking circuits.

Measuring the two-vector transition capability for a given test vector gener­ ato r is a key to the selection of the generators for stim ulating sequential faults. T he dissertation studies general properties for the transitions and presents a novel, comprehensive analysis for th e linear feedback shift registers and the linear hybrid cellular autom ata. .\s a result, the analysis solves th e open problem as to "how to properly separate the inputs when the LHC.A.-based generator is used for detecting delay faults".

In general, a self-checking circuit has additional hardw are redundancy th an the original circuit and as a result, the self-checking circuit may have a higher failure ra te than the original one. T he dissertation proposes a féûl-safe evaluation to predict th e probability of the circuit not being in the fail-state. C om pared w ith existing evaluation m ethods, the fail-safe evaluation is more practical because it estim ates th e safety of the circuit, which is decreasing cis tim e goes on. instead of giving a constant probability measure.

Various other results about improving fault coverage for tran sitio n delay faults and testing in macro-based com binational circuits are derived as well.

Il l

Examiners:

Dr. D.M. Miller. Co-Supervisor (D epartm ent of Com puter Science)

Dr. J.C . Muzio, Co-Supervisor (D epartm ent of C om puter Science)

Dr. M. Fellows. D epartm ental Member (D epartm ent of Com puter Science)

' t ÿ ÿ ^ --- :--- -

;---McLean. Outside Member (D epartm ent of Mechanical Engineering)

Dr. .\. l\"anov. External Examiner (D epartm ent of Electrical and C om puter Engineering. University of British Colum bia)

IV

C o n te n ts

A b stract ii

C on ten ts iv

L ist o f T ables v ii

L ist o f F igu res v iii

A ck n ow led gem en ts ix

1 In trod u ction 1

1.1 M o tiv a tio n ... 1

1.2 O utline of th e D is s e r ta tio n ... 7

2 Linear F in ite S ta te M achines 9 2.1 I n tr o d u c tio n ... 9

2.2 Linear F inite S tate M ach in es... 11

2.2.1 Linear Feedback Shift R e g iste r... 11

2.2.2 Linear Hybrid Cellular .-V utom ata... 13

2.2.3 R e m a r k s ... 14

2.3 General Properties for T ra n s itio n s ... 16

C O N T E N T S V

3 T ran sition s o f L H C A and L F SR 28

3.1 Transitions of L H C A ...28

3.2 Transitions of L F S R ( I ) ... 30

3.3 Transitions of LFSR (II) ...32

3.4 S u m m a r y ... 34

4 T ran sition F aults in C om b in ation al C ircu its 38 4.1 In tr o d u c tio n ...38

4.2 Transition (G ate Delay) Fault M o d e l ... 39

4.3 Experim ental R e s u l t s ...41

4.4 Discussion and C o n c lu s io n ... 44

5 T ran sition F aults in L ook -u p T able B ased F P G A C ircu its 46 5.1 In tr o d u c tio n ...46

•5.2 Transition Fault M o d e l ...48

5.3 Technology M a p p i n g ... 50

5.3.1 Problem F o r m a liz a tio n ... 51

5.3.2 4-R L M P/3-R L M P and :3-SAT...52

5.3.3 R e m a r k s ... 56

5.3.4 Technology M apping E x p e rim e n ts... 56

5.4 Experim ental Sim ulation Results ...57

5.5 Detection R e q u i r e m e n t s ... 60

5.6 S u m m a r y ...62

6 P erform ance E v alu ation o f Self-C heck ing C ircu its 63 6.1 In tr o d u c tio n ... 64

6.2 Totally Self-Checking G o a l ...65

6.2.1 P r e l i m i n a r i e s ... 65

6.2.2 D e fin itio n s... 66

6.2.3 General Designs for TSC C i r c u i t s ... 67

6.3 Existing Evaluations of Self-Checking C i r c u i t s ... 71

6.3.1 R elated W o r k ... 71

C O N T E N T S vi

7 F ail-Safe E valu ation for S elf-C h eck in g C ircu its 75

7.1 Fail-Safe E v a lu a tio n ... 75

7.1.1 Generic Markov M o d e l ...76

7.1.2 Simplified Markov M o d e l... 78

7.1.3 Generalizations of P aram eters and M o d e l...S3 7.2 E xperim ental Results and C o m p a riso n s ... 85

7.2.1 R e s u lts ... 85 7.2.2 R e m a r k s ...86 7.3 Sum m ary ... 88 8 C on clu sion s 90 8.1 Sum m ary ...90 8.1.1 M ajor C o n tr ib u tio n s ...90 8.1.2 R e m a r k s ... 91 8.2 F u rth er W o r k ... 91 B ib liograp h y 93 A p p en d ix 101

vi l

L ist o f T ab les

2.1 Test vectors produced by th e LFSR(I), LFSR (II). and L H C A ...15 2.2 An example for the num ber of the tran sitio n s... IS 2.3 An example of selecting cells and partners... 25

3.1 N um ber of selections for Ar-cell substate with 2^* transitions for odd n. 36 3.2 N um ber of selections for A;-cell substate with 2^^ transitions for even n. 37

4.1 A sum m ary of transition fault sim ulation results for 100 different random connections... 43

5.1 An example of transition faults for a 3-input LL’T ...50 5.2 Numbers of LL’Ts and num bers of faults for ISC.\S85 circuits... 57 5.3 A summ ary of the fault sim ulation results for 100 randomly chosen

connections... 59

6.1 Detection capability for the two-rail checker in Figure 6.2...69 6.2 T ruth table for the BCD-to-excess-3 code conversion...71

VIll

L ist o f F ig u res

2.1 (a) LFSR(I) ajid (b) LFSR(II) w ith characteristic polynomial r ’ + x ’+ l . 12

2.2 .A. 5-cell LHCA with characteristic polynomial x"* H- x" + 1...13

2.3 .An exam ple for the m aximal m atching and p artn er s e t... 22

2.4 .An algorithm for determ ining a p artner set p s(w ) for an LHC.A. . . . 26

2.5 .An algorithm for determ ining a partner set p s(w ) for an LFSR(II). . 27

3.1 .A simplified structure of LHCA... 28

3.2 -A simplified structure of L F SR (I)...30

3.3 -A simplified structure of L F S R (Il)... 33

4.1 Simplified exam ple for detecting transition fa u lt...41

5.1 .An exam ple of the construction for 4-RLM P from 3-S.AT... 54

5.2 .An exam ple of the construction for 3-RLM P from 3-S.AT... 55

5.3 -An exam ple of the detection requirem ent for th e transition fault. . . . 60

6.1 .A totally self-checking circu it...68

6.2 .A TSC two-rail code checker... 68

6.3 -A BCD-to-excess-3 circuit... 70

6.4 .A design for a self-checking BCD-to-excess-3 circu it... 72

7.1 .A generic Markov model for th e self-checking circuit in Figure 6.4. . . 76

7.2 -A simplified Markov model for the self-checking circuit in Figure 6.4. 78 7.3 .A simplified Markov model for the self-checking circuit in general. . . 83 7.4 Safety and reliability for the circuit in Figure 6.4 w ith Ac = Ax = 10“ ^. 87

IX

A ck n o w le d g e m e n ts

I would like to express my appreciation to my supervisors. Dr. D.M. Miller and Dr. .J.C. Muzio. for their encouragem ent, invaluable guidance, and constructive discussions throughout my graduate studies a t Victoria.

I would like to thank my com m ittee an d my external exam iner. Dr. .\. Ivanov, for th eir com m ents and suggestions.

T hanks to all my friends both at V ictoria and at O ttaw a for th eir invaluable friendship and profound concern for me.

I am greatly indebted to my parents for th eir unending support, understanding, encouragem ent, and immense love. 1 would like to express my g ra titu d e to my sisters. Meiling. Meiqi. Meijue. and M eiyu. for their consistent encouragem ent and support, especially for taking care of our parents when I am studying abroad.

Finally. 1 would like to thank my lovely wife. Lan. for her support and encour­ agem ent.

The work was supported in part by a P o stg rad u ate Scholarship from th e N atural Sciences and Engineering Research Council of Canada and Scholarships from the L niversitv of Victoria.

C h a p te r 1

I n tr o d u c tio n

1.1

M o tiv a tio n

During its lifetime, a digital system is tested and diagnosed on numerous occasions. For the system to perform its intended mission with high availability, both testing and diagnosis m ust be performed quickly and effectively[.A.KS93].

The prim ary objective of testing digital circuits at the chip, board, or system level is to detect the presence of hardware failures induced by faults in the manufac­ turing processes or by operating stress or wear-out m echanism s. T here is nothing fundam entally difficult in the functional testing of digital circuits. .\11 th a t is nec­ essary is to apply some input test vectors and observe the resulting digital output signals. T he problem , basically, is the volume of d a ta and resultant tim e to test. For circuits w ith a small num ber of logic gates, fully-exhaustive functional testing may be possible. .A.s circuit size increases, techniques to ease this problem become increasingly necessary [Hur98].

B uilt-in self-test (BIST), i.e.. self-test im plem ented in the hardw are itself, is a general approach to test a digital system [ABF90]. .A widely accepted approach to BIST is to use a pseudo-random vector generator and a d a ta com pactor. The

C H A P T E R 1. I N T R O D U C T I O N 2

generator produces th e test vectors applied to a circuit under test and th e com pactor reduces the response to these vectors to a single value (e.^.. 16 or 32 bits) known as a signature.

In general. LFSMs (Linear Finite S tate Machines) [Sto73]. such as an LFSR (Linear Feedback Shift Register) [BMS87] and an LH C.\ (Linear Hybrid Cellular .Automata) [HMP'^SQ. HMC89. SSMM90. ZBM92] are used as BIST test vector generators. Note th a t it is not practical to apply all 2" — I vectors to an n-input circuit with n over 25 when using an n-cell generator because it takes too much tim e. Therefore, minim izing the num ber of test vectors applied to the circuit for detecting all possible faults is one of the m ajor concerns.

On the other hand, an im portant problem one faces during th e design of a BIST circuit is not only to detect stuck-at faults but also to detect delay faults, norm ally due to random variation in process param eters th a t often cause propagation delays to exceed their lim its. T hat is. we need to consider the effectiveness of th e test vector generator for detecting faults with sequential behavior in the circuit under test.

.A. general approach for measuring the quality of a generator is based on fault sim ulation to see how m any faults considered can be detected when a test vector sequence, produced by the generator, is applied to the circuit. Sim ulation is a useful approach for com paring the perform ance of different BIST generators for different fault models. However, w ithout analyzing the fault models considered and th e properties of the generators, it is not easy to draw any general conclusion.

In a BIST environm ent, a checker, which compares two values to see w hether they are the same, is required. For exam ple, the signature generated via th e test should be checked to see if it is correct. Self-checking techniques should be applied to such a checker, so th a t any fault in the checker can be detected as well. In general, self-checking circuits are tested during norm al operation. Therefore, evaluating th e

C H A P T E R I. I N T R O D U C T I O N 3

quality of th e self-checking circuit should be under the normzd op eratio n of the circuit.

A lthough a considerable am ount of work has been done regarding perform ance evaluation and improvement in BIST, there still exist gaps between m odels (or m ethodologies), used in the evaluation and th e perform ance im provem ent, and the realities o f testing in circuits. Consequently, researchers are continuously working in this area in order to expose new approaches which are more reliable in practice. This dissertation addresses the following issues about the evaluation and th e perform ance im provem ent in BIST.

(a) Q u an titativ e Measures of Test Vector Generators

In a BIST environm ent, choosing a test vector generator for stim u latin g defects in a circuit under test is a complex issue. Since it is not easy to a b stra c t fault m odels from physical defects, researchers use stuck-at faults as a general fault m odel to m easure the effectiveness of the stim ulation source used. However, th ere are serious lim itations in predicting defect levels based on stu ck -at fault coverage [PXK'"'94]. Faults with sequential behavior, such as delay faults, are an increasingly frequent problem with m odern circuits and have been subject to considerable study.

We place em phasis on m easuring the effectiveness of a generator for stim u­ lating delay faults and are particularly interested in atte m p tin g to develop a sound theoretical approach to com pare generators which are su itab le for test­ ing sequential type faults (such as delay faults). Such faults require a pair of vectors for stim ulation, the first to set-up th e fault, and th e second to propagate th e fault to a circuit o u tp u t.

A com prehensive description of th e development of delay testin g is given in [Sav92] and an approach to th e selection of a proper test vector

genera-C H A P T E R 1. I N T R O D U genera-C T I O N 4

tor to apply to a circuit under test is proposed in [Sav95. Sav97]. It is noted th a t th e two-vector testing capabilities for th e generators are playing a key role in testing the delay faults.

-A. m ethod for assessing the two-vector testing capabilities of linear test vector generator circuits is given in [FM91]. where tra n sitio n coverage is introduced as a m etric. In [ZBM92. Zha93]. an analysis of tran sitio n coverage is presented, which derives the exact number of distinct tran sitio n s for a linear feedback shift register (LFSR). a linear hybrid cellular a u to m a ta (LHC.A). and two modified generators. XLFSR and .XLHC.A. for a num ber of specific substate vectors. For a 2n-cell LHC.A, an approach is given in [XVCC94] to select an n-cell su b state vector such th at the corresponding vector sequence has transitions.

O ur goal is to concentrate on any Ar-cell su b sta te vector and evaluate the num ber of distinct A:-cell substate vectors which produce th e m axim um num ber 2"*. 1 < A: < [n /2 J. of distinct transitions for any n-cell LHC.A and LFSR with m axim um length cycles. VVe derive a general theoretical answer to the question as to why the LHC.A are b e tte r th a n th e LFSR as BIST generators for sequential faults.

We develop an approach to calculate th e nu m b er of transitions for a given su b state vector for an LFSR or LHC.A. T he approach is efficient because it directly exam ines the feedback for each cell in th e substate vector instead of evaluating the rank of the corresponding su b m atrix . In such a way. we can derive the num bers of distinct Ar-cell su b state vectors th a t have 2"^' transitions, which gives an indication as to how good th e test vector generator is for detecting sequential faults. .As a result, th e analysis solves th e open problem of “how to properly separate the inputs w hen an LHC.A-based generator is used for detecting delay faults” , proposed in [Sav95. Sav97].

C H A P T E R 1. I N T R O D U C T I O N -5

(b ) Im proving Fault Coverage for Transition Delay Faults

For a given n-input circuit under test, we need to apply a test sequence pro­ duced by an n-cell test vector generator for detecting sequential faults. In general, we have n! different ways to assign connections between the circuit input and the generator output. .\n interesting open question is proposed in [ZBM92, Zha93] as to whether a proper connection can be identified based on specific knowledge of the circuit topology or th e functionality of the circuit on knowing the sequential faults considered and th e transition property for a

BIST vector generator used, so that the fault coverage is maximized.

Considerable work toward improving stuck-at fault coverage has been pre­ sented recently {e.g., [Avr94. HK93. KT94. LGB94. MMR94. TM94. HRT'''9.5. MV95]). However, not much attention is being paid to the im provem ent of fault coverage for sequential faults. One of the m ajo r reasons is th at th e la tter problem is much harder than the former. For exam ple, if all transition delay faults are detected, then all stuck-at faults are covered automatically.

To m eet the needs of current application requirem ents, however, we a tte m p t to derive the best possible result using limited com puter resources and specific knowledge about the problem. VVe present our results based on random con­ nections between the circuits and test vector generators, which look promising.

(c) Testing in Macro-Based Combinational Circuits

M acro-based com binational circuits considered are those constructed by look­ up tab le (LUT) based field-programmable gate arrays (FPG.-V). It is m entioned in [Tri92j th a t FPG.-\s can be fully tested after m anufacture, so users' designs do not require test program generation, autom atic test pattern generation, and design for testability. In fact, the possibility still exists that the circuit may not work properly, resulting from component defects, e.g.. component

wear-C H A P T E R L I N T R O D U wear-C T I O N 6

out. Therefore, in [PR94], the problem of testing delay faults in m acro-based com binational circuits is considered an d two delay fault models are proposed.

M acro-based circuits are obtained as a result of technology" m apping. It is instructive to apply logic synthesis techniques to the whole design in which th e testing issues are included. Considering th e key issues, such as circuit representation, technology m apping, and testing techniques used, we intend to analyze the problems deeply and exam ine a general testing strateg y for m acro-based circuits.

(d ) Safety of Self-Checking Circuits

Self-checking circuits can detect the presence of transient and perm anent faults because they are designed with additional sub-circuits th a t are used to m onitor w hether th e circuits work correctly during norm al operation. However, self­ checking circuits may not necessarily be safe because some faults in th e circuits may not necessarily be detected. Therefore, we need a proper evaluation for such circuits.

.A. prim ary difficulty with self-checking circuits is th a t, while both a circuit and a checker may be totally self-checking, the resulting com posite circuit is not necessarily totally self-checking. T hat is. it is possible for there to be erroneous outputs from the circuit which are not caught by the checker. This is caused by the presence of one or m ore faults in the checker which have not been exposed by an appropriate stim ulus from the circuit. The problem is th a t one of the two prim ary assum ptions for totally self-checking circuits ( i.e.. faults occur one at a tim e, and the tim e interval between occurrences of any two faults is long enough for all input codewords to be applied to the circuit) is often not m et during normal operation of th e circuit.

C H A P T E R 1. IN T R O D U C T I O N T

years. However, comparatively little has been done on the analytical evalua­ tion of th eir performance. The existing evaluations are bcised on determining w hether a given circuit satisfies the totally self-checking goal or calculating how much of th e goal has been achieved by th e given circuit [LM84. FMMS4. FM87. LF93].

C onsider th e evaluation of the safety of a self-checking circuit with combina­ tional logic. Since the circuit is tested under norm al operation, as tim e goes on. it m ay be in a different state from a perfect s ta te in which any erroneous o u tp u t can be detected. It could be in an unstable s ta te in which an erroneous o u tp u t may be detected or may not. a safe-state when th e erroneous output has been caught, or a fail-state because the erroneous o u tp u t is undetected. Consequently, we propose a fail-safe e\"aluation. using a Markov model to de­ scribe th e s ta te transitions and predict th e probability of the circuit not being in the fail-state.

1.2

O u tlin e o f th e D iss e r ta tio n

The chapters of this dissertation cover general properties for transitions for linear finite s ta te m achines (Chapter 2), analysis of the tra n sitio n property for linear feed­ back shift register and linear hybrid cellular au to m ata (C h ap ter 3). transition faults in com binational circuits (Chapter 4). transition faults in look-up table based FPG.A. circuits (C h ap ter 5), the safety of self-checking circuits (C h ap ter 6 and Chapter 7). and general conclusions and further work (C hapter 8).

C h ap ter 2 provides knowledge of linear finite s ta te m achines and their general properties for transitions.

C h ap ter 3 presents a method of evaluating the effectiveness of the LHC.A. and LFSR éis test vector generators for stim ulating faults requiring a pair of vectors.

C H A P T E R 1. IN T R O D U C T I O N 8

C hapter 4 reviews the transition fault model and detection requirem ents. Then, it provides em pirical comparisons to show th a t th e analysis of the transition property presented in C hapter 3 is a reasonable m etric of the effectiveness of th e test vector generator in testing delay faults.

C h ap ter 5 examines th e testing of look-up table based FPG.'X circuits, and pro­ poses the transition fault models and detection requirem ents for such circuits. It also discusses the technology m apping problem and proves the XP com pleteness of the K-RLM P (Restricted K-LUT M inimization Problem ) problem for A = 3 and 4.

C hapter 6 examines performance evaluation of self-checking circuits, reviews the existing m ethods, and discusses potential problems with them .

C hapter 7 develops a new approach to evaluate the safety of self-checking circuits. Com pared with existing evaluation m ethods, the proposed approach is m ore prac­ tical because it estim ates the safety of the circuits, which is decreasing as tim e goes on. instead of giving a constant probability measure.

C hapter 8 summarizes the m ajor contributions of the dissertation and raises several related problems as further work.

C h a p te r 2

L inear F in ite S ta te M a c h in e s

This chapter presents a combinatorial m ethod of evaluating the effectiveness of linear hybrid cellular a u to m a ta (LHCA) and linear feedback shift registers (LFSR) as test vector generators for stim ulating faults requiring a pair of vectors. We provide a theoretical cinalysis and empirical comparisons to see why the LHC.A. are better than the LFSRs as the generators for sequential-type faults in a built-in self-test (BIST) environm ent. Based on the concept of a p artn er set. the m ethod derives the num ber of distinct Ar-cell substate vectors which have 2^*^. 1 < A: < [n /2 J. transition capability for an n-cell LHC.A and an n-cell LFSR w ith m axim um length cycles.

2.1

In tr o d u c tio n

In this chapter, we place emphasis on measuring th e effectiveness of a generator for stim ulating faults with sequential behavior, e.g., delay faults, in a com binational circuit, since those faults are an increasingly serious problem w ith m odern circuits and have been subject to considerable study. Lsually linear finite sta te machines (LFSM), e.g., linear feedback shift registers (LFSR) [BMS87] or linear hybrid cel­ lular autom ata (LHC.A) [HMP'*'S9. HMCS9, SSMM90, ZBM92]. are used as BIST

C H A P T E R 2. L I N E A R FINITE S T A T E M A C H I N E S 10

generators. Such a generator is sequenced through a num ber of states w ith each sta te serving as a test vector. The coverage of com binational faults, such as stuck- at faults, depends entirely on the inclusion of th e ap p ro p riate vectors w ithin th e sequence generated. It is well known th a t the perform ance of LFSM in this con­ text is determ ined only by the characteristic polynom ial of th e LFSM. and actu al im plem entation (w hether it is as an LFSR. an LHCA or some others) is largely irrelevant.

On the o th er hand, the question of generators for stim u latin g sequential faults is much more interesting. VVe are particularly in terested in attem p tin g to develop a sound theoretical m etric for comparing generators which are suitable for testin g faults w ith sequential behavior, e.g.. delay faults, which require a pair of vectors for stim ulation, the first to set-up the fault, and the second to propagate the fault to a circuit o u tp u t. T he effectiveness of a particular test vector generator in stim ulating sequential faults is dependent upon its state transition sequence, i.e.. the num ber of distinct pairs of vectors produced by the generator.

.A. m ethod of assessing the two-vector testing capabilities of linear test vector generator circuits is given in [FM9I]. where tran sitio n coverage is introduced as a m etric. In [ZBM92], an analysis of transition coverage is presented, which derives the exact num ber of distinct transitions for the LFSR. LHC.A. and two modified generators. X LFSR and XLHC.A. for a num ber of specific su b state vectors. For a 2n-cell LHC.A. an approach is given in [XVCC94] to select an n-cell substate vector such th a t the corresponding vector sequence has 2^" transitions.

VVe concentrate on any fc-cell substate vector and evaluate th e number of distinct Ar-cell su b state vectors which produce the m axim um num ber 2“^'. 1 < t < [n /2 J . distinct transitions for any n-cell LHC.A and LFSR w ith m axim um length cycles and give a general theoretical answer to the question as to why LHC.A are b e tte r th an the LFSRs as BIST generators for sequential faults.

C H A P T E R 2. L I N E A R FINITE S T A T E M AC H IN ES 11

T he rest of th e chapter is organized as follows. Section 2.2 reviews LFSR and LHC.A.. Section 2.3 introduces general definitions for transitions and th e ir properties. In Section 2.4. we present algorithms of determ ining th e partner sets.

In th e next chapter, we give details of derivation for the num ber of different fc-cell su b state vectors which have 2^* transition capability for the LHC.A and LFSR and claim th a t LHC.A have a much higher transition space th an LFSR. and sum m arize our m ajo r results.

2 .2

L in ea r F in ite S ta te M a ch in es

Linear finite sta te machines (LFSM) th a t are of interest are called autonom ous linear m achines [Sto73]. i.e.. they have no inputs. In the rem ainder of this ch ap ter. LFSM refers to an autonom ous linear machine. In general, th e next sta te function of an LFSM is represented by a state transition m atrix T. For an n-cell s ta te vector s. th e next sta te vector s'*" is given by s'*" = s T. where all operations are carried out over G F{2). the Galois Field o f order 2 [Sto73].

In this section, we briefly review practical linear finite state machines, ciz.. linear feedback shift registers and linear hybrid one-dimensional cellular a u to m a ta . Since they are specific instances of LFSM. they have all the general properties of LFSM. The only difference is th at they have different state transition m atrices which are described as follows.

2 .2 .1

L in ea r F eedb ack S h ift R e g iste r

A linear feedback shift register (LFSR) [BMS87] is an LFSM. Each sta te is uniquely- determ ined from the previous state. T here are two configurations for an LFSR: a T ype I LFSR which has the exclusive-0R gates between the cells, and a Type II LFSR which has exclusive-OR gates on the feedback path . VVe assum e th a t shifting

C H A P T E R 2. L I N E A R F I N I T E S T A T E M A C H IN E S 12

and num bering of the cells are from left to right. For convenience, the Type 1 and T ype II LFSRs are sim ply nam ed LFSR(I) and LFSR (II). respectively.

F ig u r e 2.1 (a) LFSR(I) and (b) LFSR(II) with characteristic polynom ial x ^ + j " - f l .

(a)

(b)

S4

•S4

E x a m p le 2.1 Figure 2.1 shows the two types o f LFSRs derived from the same poly­ nomial x'’ + j " + 1. IVe can w^rite a set o f state transition equations fo r any LFSR. For example, the equations fo r the LFSR(I) of Figure 2.1(a) are:

4

4

4

st

where is the present state o f the cell i and s f is the next .state with all operations being carried out over 6-'F(2). Hence

= {Si.S-2.S3,S4.S5) ■ Tl.

C H A P T E R 2. L I N E A R FINITE S T A T E M A C H IN E S 13 Ti = \ 0 I 0 0 0 0 0 I 0 0 0 0 0 1 0 0 0 0 0 1 I 0 I 0 0

Similarly, the state transition matrix fo r the L F S R (II) o f Figure 2.1(b) is given by

Til = 0 I 0 0 0 0 0 I 0 0 1 0 0 I 0 0 0 0 0 I I 0 0 0 0

2 .2 .2

L in ear H y b rid C ellu la r A u to m a ta

The linear hybrid cellular autom ata (LHCA) considered are LFSM each composed of a one-dimensional array of cells, which only com m unicate with th e ir im m ediate neighbors [SSMM90]. Figure 2.2 is an example of an LHC.A.. The LHC.A is said to have null boundary conditions since the two end connections are fixed at 0.

F ig u r e 2.2 A 5-cell LHC.A with characteristic polynom ial x"* -H x* -h I.

OH

Si 52 53 54 55

D e f in itio n 2.1 L etsi. \ < i < n, be the present state o f cell i o f the LHC.A identified by the n-tuple ( c i.c j c„). Its next state, s f . is given by

C H A P T E R 2. L I N E A R F I N I T E S T A T E M A C H IN E S 14

where jq = ân+i = 0. = 0 fo r a Rule 90 cell and c, = I fo r a Rule 150 cell.

VVe only consider LHCA com posed of Rule 90 and R ule 150 cells since it has been shown in [SSMM90] th a t this is a necessary condition for th e LHCA to have m axim um length cycle. These have been term ed hybrid since th e cells are not all th e same. Throughout this chapter we are concerned with one-dim ensional linear hybrid rule 90/150 cellular a u to m a ta and we refer to th em sim ply as LHC.A. for brevity. .A. general sta te tra n sitio n m atrix for th e LHC.A is given by

T = ( Cl 1 1 Co 0 0 0 1 \ 1 C n —I 1 Cn (2.1)

For exam ple, for the LHC.A shown in Figure 2.2. the state tran sitio n m atrix is

T = 1 1 0 0 0 1 1 1 0 0 0 1 1 1 0 0 0 1 1 1 0 0 0 1 0

2 .2 .3

R em a rk s

W hen an LFSM is used aa the generator in BIST, it is sequenced through a num ber of states w ith each sta te serving as a test vector. Table 2.1 shows the test vectors produced by th e LFSR(I) and LFSR(II) of F igure 2.1. and LHC.A of Figure 2.2. The test vectors produced cover all possible nonzero states, beginning

C H A P T E R 2. L I N E A R FI NI TE S T A T E M A C H IN E S 15

T a b le 2.1 Test vectors produced by th e LFSR(I). LFSR (II). and LHCA

Ti me L F S R ( I ) L F S R { I I ) L HC. A Ti me L F S R ( I ) L F S R ( I I ) LHC. A 0 00001 00001 00001 16 00110 00111 10001 1 10100 10000 00010 17 00011 00011 11010 2 01010 01000 00111 18 10101 10001 00011 3 00101 00100 01011 19 11110 11000 00101 4 10110 10010 11001 20 01111 01100 01100 •5 01011 01001 00110 21 10011 10110 10010 6 10001 10100 01001 22 11101 11011 11111 7 11100 11010 11110 23 11010 IIIOI 01111 8 01110 01101 01101 24 01101 OHIO 10111 9 00111 00110 10000 25 10010 10111 10011 10 10111 10011 11000 26 01001 01011 11101 11 11111 11001 00100 27 10000 10101 01000 12 11011 11100 01110 28 01000 01010 11100 13 11001 11110 10101 29 00100 00101 01010 14 11000 11111 10100 30 00010 00010 11011 15 01100 01111 10110 31 00001 00001 00001

from a nonzero state. Thus, they are all m axim um length cycles (with length 2” — 1 = 2" — 1 = 31). Moreover, since the LFSR(I), LFSR (II). and LHC.A. have the same prim itive characteristic polynomial, they produce the sam e output stream in each bit position ap art from the appropriate phase shift, flowing from each single cell of th e generators, e . g. . starting at 0 marked for s ta te in Table 2.1. we can see

th a t three sequences on Si for the LFSR(I). LFSR(II) and LHC.A are identical. The calculation of th e phase-shift between the bit stream s can be found in [Bar92a].

T he LFSR and LHC.A associated w ith prim itive polynomials are more desirable because they generate all possible nonzero test vectors, beginning from any nonzero state. T here has been much recent work concerning LHCA {e.g.. [SSMM90. SS90. DGD'*'90. CM91. Bar92a. CM96, Cat95]). In general, we minimize the hardware cost of an LFSR im plem entation by using m inim al weight prim itive polynomials. Such polynom ials for th e LFSR of degree through 500 are contained in [BMS87. Bar92b].

C H A P T E R 2. L I N E A R FINIT E S T A T E M A C H IN E S 16

wherecis the m inim al cost LHCA of degree through 500 axe available in [ZMM9L CZ95] (Also see A ppendix A).

The algorithm of determ ining whether a given n-degree LHCA has m axim um length cycle is as follows.

(a) C om pute th e characteristic polynomial of th e LHCA using the recurrence relation in [SSMM90]:

(b) Check if the characteristic polynomicd is prim itive: if so. the LHC.A has m axim um length cycle.

For each degree, we first generate all of the LHCA with a single rule 150 cell. If this is not successful, we then generate all of the LHC.A with a pair of rule 150 cells. For each degree, the search is stopped at the first LHC.A w ith maxim um length cycle. This search has never failed, meaning th a t for each degree up to 500. there is an LHC.A with m axim um length cycle th at has either one or two rule 150 cells. More details can be found in [ZMM91. CZ95].

2.3

G en era l P ro p erties for T ra n sitio n s

For sequential faults, the actual test vector sequence produced by a generator is critical since a fault requires a pair of vectors for stim ulation. We are thus concerned with the transitions produced by the generator. Moreover, as noted above, it is not sufficient to ju st consider transitions of n-cell vectors, but we must also look at transitions of Ar-cell substate vectors for k < n. We staxt by defining these A:-cell substate vectors, and the corresponding transitions.

D e fin itio n 2.2 For a given n-cell LFSM state vector s = ( s i.s o s*). Sp € {0. 1}. 1 < p < n. a k-cell substate vector w o f s is defined by

C H A P T E R 2. L I N E A R F I N I T E S T A T E M AC H IN ES 17

and a transition corresponding to w is defined as

( ( S j j . 5 , 2 5 ,|^ ) . ("Sij . •S,2 ... ) )

-where I < < i; < n fo r I < j < I < k.

For notational convenience, w is used to denote a substate vector of s w ith cells which are not in w . VVe count one transition even if (5,,.5^2...5,*) = {sf[.sf^ sf^) because it simplifies the derivation of a general equation to evaluate th e num ber of transitions for a given su b state vector. T h a t is. for a Ar-cell su b sta te vector, we use

2k X 2^ as its maximum num ber of transitions, instead of 2* x (2^ — 1).

E x z u n p le 2 .2 For the three LFSM s in Table 2.1, i f s = ( s j . 53. 53. 54. 55) = (00111) and w = (52. 53. 54), then the transition corresponding to w is

which is

( (5 0.5 3.5 4). (5J .5J .5J ) ) ,

((O il). (O il)) fo r the L F S R { I ) because 5"'' = (10111). ((O il). (001)) f o r the L F S R ( I I ) because 5"^ = (00011). ((O il). ( 101)) f o r the LHC.A because 5"^ = (01011).

For any particular substate vector, we can count the total num ber o f transitions fo r the given substate vector fo r the LFSM . For example, fo r w = (5 3.5 4) fo r the

LFSR( I ) in Table 2.1, we have the following transitions: ((00). (10)). ((10).(01)). ((O l).(lO )). ((10).(11)). ( ( ll) .( O l) ) . ((01), (00)). ( ( l l ) . ( l l ) ) . ((00). (00)). giving a total o f 8 transitions. Obviously, the maximum number o f transitions in this case is

16 (four possible choices fo r the first vector o f the pair, and fo u r fo r the second), so this generator only produces half o f the maximum possible num ber o f transitions fo r this substate. The complete list o f all the transitions fo r 2-cell substate vectors fo r

C H A P T E R 2. L I N E A R F I N I T E S T A T E M ACH INE S 18

T a b le 2.2 An exam ple for th e num ber of the transitions.

Substate Vector LFSR(I) LFSR(II) LHCA

( ■Sl , S2) 8 8 8 (•Si- ■S3) 1 6 1 6 1 6 ( S 1 . S 4 ) 1 6 1 6 1 6 ( a i . ' S s ) 8 1 6 1 6 (■S2- -Ss) 1 6 8 1 6 ( S2, S4) 1 6 1 6 1 6 (•S2- -Ss) 1 6 1 6 1 6 (■S3- S 4 ) 8 8 1 6 (■S3- S 5 ) 1 6 1 6 1 6 (■S4, S 5 ) 8 8 8

As shown in Table 2.2, if we consider different A:-ceII su b sta te vectors, there are some differences between the num bers of distinct tran sitio n s generated by the LHC.A. and the LFSR. Here we briefly review the m ethod, based on evaluating the rank of a specific subm atrix of the transition m atrix, in [FM91] and propose a different approach to select a Ar-cell substate vector w which can achieve 2~^ distinct

transitions for 1 < Ar < [n /2 J. First we give an exam ple using th e transition matrix to evaluate the num ber of distinct transitions.

E x a m p le 2.3 Let s = (s i.S o .S3, 54,^ 5) be the state vector. We want to know the number o f distinct transitions fo r a 2-cell substate vector w = (S1.S 3). By defi­ nitions. we let w = (s2.S4.S5) and w+ = ). For the L F S R ( f l ) shown in

C H A P T E R 2. L I N E A R F I N I T E S T A T E M A C H IN E S 19

Figure 2.1(b), we have the following next state fun ctio n fo r w

= (51. 53) 0 0 1 0 0 1 I 0 0 0

\

/

Therefore, based on Theorem I in [F.\I9l], fo r each value o f (61.^ 3). there are 2~ distinct next values (5]^.3 3 ), and the total number o f distinct transitions fo r the

given substate vector is 2}'2~ = 2^. □

For convenience. Theorem I in [FM91] is stated as follows.

T h e o re m 2.1 [FM91] For a k-cell substate vector w with next state function writ­ ten in Eqn (2.2), 2'’ distinct transitions occur from every k-cell substate vector, where r = rank{Tü^), while the whole state vectors goes through all possible 2" states.

T h e o re m 2.2 [ZBM92] ( u p p e r a n d lo w er b o u n d s ) Consider any n-cell LFSM vector generator with a maxim um length cycle. Let (F{k) be the maximal number o f

C H A P T E R 2. L I N E A R F I N I T E S T A T E M A C H IN E S 20

distinct transitions corresponding to a k-cell substate vector w . VVe have 2 ^ . l < A : < [ n / 2 l , upper bound: IFik) < <

\ n !2] < k < n .

I < k < n. lower bound: F ’{k) >

k = n.

Based on the above theorem , it can be seen th a t for any fc-cell su b state vector w of an n-cell LFSM sta te vector s w ith k = [n /2 j. if n is odd. the mtiximum possible num ber of distinct transitions produced by w is 2^*. and if n is even, the m axim um possible num ber of distinct transitions produced by w is 2^^ — 1 = 2" — 1 because the all zeros state is not included in the sequence produced by th e LFSM. W ithout loss of generality, when n is even and k = [n/2J. we assum e th at w can produce 2*^' transitions in the best case. i.e.. the corresponding T-^ has rank k.

It is proposed in [FM91] th a t transition coverage for k = ^n/2j could be used as a universal m etric of transition capability of an LFSM because of the following reasons, .\ssum e a A:-cell substate vector w of s = ( s i.s o Sn) can produce 2"^’ transitions. Then it follows th at

(a ) .\n y m-cell su bstate vector x of w can produce 2'"* transitions:

( b ) .-\ny m-cell substate vector x, which includes w . of s can produce at least 2"^ transitions.

In th e next section, we derive th e num ber of different A:-cell su b state vectors, which have 2"^ transition capability, with k < [n/2J for th e LH C .\ and LFSR in order to com pare the performance of the LHC.A. and LFSR concerning testin g faults requiring a pair of vectors. .A key to determ ining whether a given fc-cell su b state vector w has 2^*^ transition capability is to check if the corresponding Tût has rank k. However, to avoid com puting th e rank of 7 ^ , we introduce the idea of a p artn er set and prove th a t th e cardinality of this set gives us the rank of TV.

C H A P T E R 2. L I N E A R F I N I T E S T A T E M A C H IN E S 21

D efin itio n 2.3 Let w be a substate vector o f a state vector s = •^n) an n-cell LFSM . The next state s f corresponding to 3; in w is given by

6; := «j. fo r som e subset A, Ç

^n}-{.\ote that the subset A, fo r a specific Si depends on the transition m atrix f o r the given LFSM. ) .All such Sj. which are not in w . are eligible partners fo r s,.

E x a m p le 2 .4 For the LHC.A shown in Figure 2.2. let w = ( ^ 4)- The next state functions fo r S2 and 54 are given by

5^ = + •33. s f = 33 + 54 + S5.

That is. k'2 = {5i .^ 2--33} and A4 = {^3.^ 4,.$3}. B y definition 2.3. the eligible

partners are and S3 fo r so- and S3 and 55 fo r 64. O

In o th er words, partners are those states th a t are not in w . b u t have an im m ediate effect on the next sta te function corresponding to the s ta te in w .

Before defining a p artn er set. we review a m axim al m atching problem in a bi­ p artite graph[CLR90]. Let G = (V. E) be a b ip artite graph w ith V partitio n ed as -V U V' (Each edge of E has the form ( x . y ) with x £ X an d y € V').

(a) A matching of G is a subset of E such th at no two edges share a com m on vertex in X or Y .

(b ) A maxim al matching in G is one th a t m atches as m any vertices in X as possible w ith vertices in V .

C H A P T E R 2. L I N E A R FINITE S T A T E M A C H IN E S 22

D efin itio n 2.4 Let w be a k-cell substate vector o f a state vector and G = (V'. E) be a bipartite graph with V partitioned as X C Y . where

V = { s i , ^ 2 . . . . , 5 „ } .

X = { a, I j, is in w }, V' = { Sj I Sj is in w }.

E = { (si.Sj) I Si € X . S j 6 V. and Sj is an eligible p a rtn er o f Si }.

I f \ I . M Ç E . is a maximal matching o f 0 . then ps{w) = { 1(si-Sj) € \ f } is a

partner set o f w and |p5(w )| = \M\.

F igu re 2.3 An exam ple for the m axim al m atching and p artn er set.

E x a m p le 2 .5 For the w = (S2. 64) given in example 2-4. a corresponding bipar­ tite graph G is shown in Figure 2.3 with X = } = {s1. s 3. j 5}. and E = {(.So. 5i). (s2, S3). ( S4. S3). (S4. S 5 ) } . The possible maximal matchings fo r this

example are { ( s 2 - S i). ( S 4 , S 3 ) } , { ( s 2 , s i ) . ( S 4 . S 5 ) } , and { ( S 2 . S 3 ) , ( S 4 . S 5 ) } . Thus, the

corresponding partner sets are ( s i. S3}, {si, S5} . and {S3. S 3 } . □

Note th a t for a given w . the choice of partners m ay not be unique, which m ay result in different p artn er sets.

T h e o r e m 2 .3 Let w be a k-cell substate vector o f a state vector s fo r an n-cell LHC.A. or LF SR with a maximum length cycle. |p s(w )| = k i f and only if rank{TyF} = k. i.e.. w can produce 2^^ distinct transitions.

C H A P T E R 2. L I N E A R F I NI T E S T A T E M AC H IN ES 23

P r o o f. (<^=) It is obvious.

(=>) The trick of the proof is to find th a t k columns in are linearly indepen­ dent. Here, we show the key to the proof for the LHCA. For the LFSR. th e proof is sim ilar.

Consider an n-cell LHC.A with an n x n state transition m atrix T in Eqn (2.1) and a Ar-cell su b state vector w. For simplicity, if an element of T is one. we call it a 1-element: otherw ise, a 0-elem ent.

Because of th e nature of the sta te transition m atrix T of the LH C .\. for the {n — k) X k subm atrix of T. we have the following:

(a) 7 ^ is given by = (Fu7(1)) 0 0 {Tw[2)) 0 0

w/here ! < ? < " . has kg columns and K ~

(b) Each colum n/row in 7 ^ has at most two 1-elements. which m ust be adjacent:

(c) If an elem ent f,j. I < i < n — k and I < j < k. in 7 ^ is a 1-elem ent, then any elem ent (1 < i and j < f < k) or (/ < i' < n — k and I < j ' < j ) . m ust be a 0-elem ent. It implies th at any two columns/rows in which have two 1-elem ents are not identical.

Moreover since |p5(w)l = k, it guarantees th at

(d) Each colum n of TV has at least one 1-element: and

C H A P T E R 2. L I N E A R F IN IT E S T A T E M A C H IN E S 24

VVe can conclude from properties (b)-(e) th at 1 < <7 < u. m ust be with one of four stru ctu res as follows (w ithout loss of com prehensibility, all 0-elem ents are not displayed). Ti = T . = I ^ f l \ I _{I I} ••• 1 _{. To = I} 1 I I < 1 I I 1 ••• I . T, = ( l ) . 1 I N ote th at 1 1 0 1 I 0 1 1 and

are special cases of Ti. To. and T3 respectively.

It is straightforw ard to see that all columns in Ts7(,) are linearly independent. Consequently, all columns in TV are linearly independent. ■

Xote th a t Theorem 2.3 does not hold for a générai LFSM.

E x a m p le 2 .6 For a 5-cell LFSR(II), as shown in Figure 2.1(b). by Theorem 2.3. we select all 2-cell substate vectors, reported in Table 2.3. which produce = 16 transitions. B y referring to Tables 2.2 and 2.3, we can see that the selections in Table 2.3 are correct and no other 2-cell substate vector can be chosen to achieve 16

C H A P T E R 2. L I N E A R FI NI TE S T A T E M A C H I N E S 25

T able 2 . 3 An exam ple of selecting cells and partners.

w p s { w ) Notes

(51.53) {S5.S0} S5 is a partner o f and S2 is ^sa's

(51.34) {55.53} 55 is a partner of 5i and 53 is 54"s

(51.55) {53,54} 53 is a partner of 5i and 54 is 55‘s

(52.54) {51,53} 5i is a partner of 53 and 53 is 54 s

(50.55) {51.54} 5i is a partner of 5o and 54 is 55's

(53.53) {50,54} 5o is a partner of 53 and 54 is 55's

In Theorem 2.3 we only consider a special case, in which each elem ent in w

has its own partner. In fact, we can easily extend the theorem to a general case:

|p5(w)| = r (i. e.. there exist at most r o f k elem ents in w . each having its own

partner), if and only if rank(Tüj) = r. where r < k.

2 .4

A lg o r ith m s for D e te r m in in g P a rtn er S e ts

If we want to evaluate the number of transitions for a given A:-cell substate vector w

o f a sta te vector s for an n-cell LFSR or LHC.A. with a maxim um length cycle, we only

need to construct a partner set ps(w). and then evaluate the number of transitions

2*'*"''. where r = |p5(w)|. Since the LHC.A and LFSR have straightforward next sta te functions, it is easy to design algorithms which determ ine the partner set for

the given su bstate vector w for them.

We first consider an n-cell LHC.A with a m axim um length cycle. By observing

the next state function in Definition 2.1. we can see the following cases are a key to

designing a algorithm for determining the ps(w).

(a ) If 51 is in w , only 5? may be used as a partner of Si:

C H A P T E R 2. LI NE A R F I N I T E S T A T E MACHINES 26

F i g u r e 2.4 An algorithm for determ ining a partner set

ps(w)

C = { i I s, is in w }: D = 0: for p = 1 to n do if p € C th e n b eg in if (p = 1) or (p — 1 € D and

p ^

'J {p. p + 1}

in w or Sp_i is a p artn er of then Sp+i may be used as a p artner of Sp: otherwise. Sp_i is chosen as a partner of Spi

(c) If Sn is in w . only s„_i may be used as a partner of

Formalizing the above cases, we describe the algorithm for th e LHC.A. in Figure 2.4.

For an n-cell LFSR(II) w ith a m axim um length cycle, an algorithm for determ in­ ing a partner set ps(w ) is shown in Figure 2.5. In a sim ilar fashion, th e algorithm for th e LFSR(I) can be easily designed.

E x a m p le 2.7 Consider an n~cell LHCA with a maximum length cycle, where n is even. He want to derive a partner set ps(w) fo r w = ( s i. S3, . . . . s„_i ). i.e.. w includes all odd cells and k = n /2 . B y Algorithm 1, we get p s(w ) = {s^, s„}.

C H A P T E R 2. L I N E A R FINITE S T A T E M A C H I N E S 'i

F ig u re 2.5 An algorithm for determ ining a p a rtn e r set ps(w ) for an LFSR(II).

/ * L e t P{ x ) = j ' * + C n —1-T” * + • • • + C îX ^ + CiJ*^ + 1 * / C = { i 1 s, i s in w }: F = { n — j | 1 < ( < n a n d Q ! 5 1 i n P ( j ) } U { n }: D = { p — i . p \ p E C a n d p 7^ 1 }: if (1 G C) th en b eg in D = D U { 1 }: if 3( 1 < i < n and i

^

=

U {

}:

i.e.. ps(w ) includes all even cells and |p s(w )| = k . By Theorem 2.3. we conclude that the given w has the maximum transition capability. This is a simple approach to prove Theorem -5 in [ZBM92]. In fact, all o f the theorems fo r transition properties, given in [ZBM92] can be easily proved by the algorithms and theorems presented in

C h a p te r 3

T reinsitions o f L H C A a n d L F S R

In this chapter, we use a com binatorial approach to derive th e num ber of different Ar-cell substate vectors, which have 2^^ transition capability, w ith k < |_n/2j for th e LHC.A. and LFSR.

N ote th at a deep understanding of th e m aterial presented in this chapter is useful in selecting a set of cells such th a t th e corresponding test vector sequence covers th e m axim al transition.

3.1

T ran sition s o f L H C A

Consider an n-cell LHCA with a m axim um length cycle, as shown in Figure 3. 1.

F ig u r e 3.1 .A simplified stru ctu re of LHC.A.

Si $2 S3 Sn-2 Sn-1 Sn

C H A P T E R 3. T R A N S I T I O N S O F LHCA A N D LFSR 29

T h eorem 3.1 Let f \ ( n ) be the total number o f distinct k-cell substate vectors which produce 2^* transitions with k = [n/2J fo r the n-cell LH C A. We have

2 /i(n — 2). even n and n > 4.

(3.1) 2 /i(n — 2) + f i { n — 3). odd n and n > -5.

/ i ( « ) =

/ i ( l ) = I. /i(2 ) = 2. /i(3 ) = 3

P roof. The key to the proof is to determ ine how m any selections there are for w such th a t each elem ent in w hcis its own partner, i.e.. |p s(w )| = k. For n = 1.2. and 3. it is obvious.

W hen n is even zmd n > 4. if we select k cells for w . each must have a unique partn er from th e rem aining k cells. T hat is.

(a) If Si is included in w . sg m ust be used as a partn er of Si. There are / i ( n — 2) ways to choose k — 1 cells from {S3.S4. ---- s„}:

(b) If s-2 is included in w . Si m ust be used as a partner of so. There are f i { n — 2)

ways to choose ^ — 1 cells from {53. S4. ---- s„}.

Therefore, for the case of even n. th e total num ber of selections is 2 /i(n — 2). W hen n is odd and n > 5 . if we select k cells for w . we need k cells as partners of w and have one cell which is n either included in w nor a partner. T hat is.

(a) If Si is included in w . sg m ust be used as a p artn er of Si. Then we consider selecting [n /2 j — 1 elem ents from {S3.S4, -s„} such th a t each has its own partner. The to tal num ber of selections is / i( n — 2):

(b) If S2 is included in w and si is counted as a p artn er of so. the total num ber of selections is f i { n — 2) — f i [ n — 3), where f i [ n — 3) is the num ber of selections if So is in w and S3 is a p a rtn e r of sg;

(c) If Si is neither included nor a p artn er of any cell in w . there are / i ( n — I) = 2/1 (n — 3) selections.

C H A P T E R 3. T R A N S I T I O N S OF LH CA A N D LFSR 30

Hence, the to tal number of selections for the case of odd n is 2f i ( n — 2) + f i ( n — 3).

/ i ( ” ) =

It is easy to solve the recurrence relation in Eqn (3.1) as follows.

even n. ( [ n / 2j + 2 )2( " - ^ ) /\ o d d n .

T h e o r e m 3 .2 Let f2{n, k) be the total number o f distinct k-cell substate vectors

which produce 2"* transitions with k < [n/2J fo r the n-cell LHCA. We have

'2f2{n — 2. k — 1 ) "h /aC^ k) — /^ (n — 3. k — I ). k <. [_n/2j.

/i(n ). A : = [ n / 2J.

f 2 ( n . k ) =

/ o l n .l ) = n.

P r o o f . It is similar to the proof of Theorem 3.1.

3.2

T ran sition s o f L F S R (I)

Consider an n-cell LFSR(I) with a m axim um length cycle and assume^ th at each cell s,. 1 < / < n. gets feedback from cell s„. as shown in Figure 3.2.

F ig u r e 3 .2 .A. simplified structure of LFSR(I).

Si Sa S3 Sn-2 Sn-1 Sn

HVe use this assum ption to determ ine an upper bound on the number o f selection s for the LFSR(I) because there is no general equation for any L FSR (I) concerning the num ber o f different selections.

C H A P T E R 3. T R A N S I T I O N S OF LHCA A N D LFSR 31

T h eorem 3.3 Let f i { n ) be the total number o f distinct k-cell substate vectors which produce 2^^' transitions with k = [n/2J fo r the n-cell LFSR( I ) . IVe have

f i ( n — 2) + I. even n and n > 4.

(3.2) f i { n — 2) + f i { ^ — 1 ) [n/2J — 1. odd n and n ^ 5.

/ i ( «) =

/ i ( l ) = 1. /i(2 ) = 2. fii'3) = 3.

P r o o f . T he key to th e proof is to select k cells to construct w such th at each elem ent in w has its own partner, i.e.. |/)s(w)( = k. For n = 1.2. and 3. it is obvious.

W hen n is even and n > 4. each cell should be considered as either an elem ent in w or a p artn er of an elem ent in w.

(a) If Si is included in w . Sn must be used as a p artn er of Si. We only have one way to select |_n/2J — 1 cells from {sg. S3 5„_i} in order to produce 2*^ transitions, i.e.. we select w = ( s i . S3 ^n -i) such th a t each elem ent in w has its own partner:

(b) If S ) is included in w . Si m ust be used ais a partn er of Si because if s„ is used as a p artn er of s^. there does not exist a proper selection. The num ber of selecting [n/2J — I cells from {S3.S4...s„} is f i { n — 2).

Therefore, th e total num ber of selections for the case of even n is / i ( n — 2) + 1. W hen n is odd and n > 0. we have one cell which is neither included in w nor a partner.

(a) If Si is included in w , s„ must be used aa a p artn er of Si. with the result th a t S2. S3 s„_i cannot consider s„ as a p artner. Thus there are [n/2J selections in this case:

( b ) If S3 is included in w and s% is counted as a p artn er of so. there are / i ( n — 2 ) — I selections, w here the -1 term corresponds to a selection of using s„ as a p a rtn e r of So:

C H A P T E R 3. T R A N S I T I O N S OF LHCA A N D LFSR 32

(c) If is neither included not a p artn er of any cell in w . there are f i ( n — I) selections.

Thus th e total num ber of selections for the case of odd n is / i ( n — 2) + / i ( n — I ) +

[n /2 j - I. ■

The recurrence relation in Eqn (3.2) can be solved as follows:

n/ 2 + 1. even n.

(n^ + 3 )/4 , odd n .

T h eorem 3.4 Let f ^ i n . k ) be the total number o f distinct k-cell substate vectors which produce 2^* transitions with k < [n/2J fo r the n-cell LFSR(I ) . We have

g(n — 2. k — I ) — — 3. k — I ) + / 2( ^ ~ 2, fc — l) + / 2( ^ ~ l.A:). k < [n /2 J. / i ( n ) . k = [ n /2\. f i { n ) = f i i n . k ) = < / 2( n. I ) = n.

where g{ n. k ) is defined as follows:

g ( n . k ) = <

p(n — 2. /l — I ) + ^ ( n — I ./l). A: < [ n / 2 J .

1. k = |_n/2j and even n.

[n/2J + 1, k = [n/2J and odd n.

g(n. I) = n — I.

P r o o f . Sim ilar to the proof of Theorem 3.3.

3 .3

T ra n sitio n s o f L F S R (II)

Consider an n-cell LFSR(II) w ith a m axim um length cycle and assume" th a t cell receives feedback from cells S2, S3, . . . , s„, as shown in Figure 3.3.

"The assum ption is used to d eterm in e an upper bound on the num ber o f selection s for the LFSR (II) because there is no genereii equation for any L FSR (II).