# BILBO registers with nonlinear feedback

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BILBO registers with nonlinear feedbackN.P. CagigalProf. S. BrachoIndexing terms: Feedback, Nonlinear systems, Digital circuits, LogicAbstract: Built-in logic block observation(BILBO) has become one of the most widelyaccepted techniques for self-testing of complexdigital ICs. This technique is based on groupingthe storage elements of the circuit in the two regis-ters which give this technique its name. A BILBOregister has four functional modes: with each ofthe stages acting as independent registers; as agenerator of pseudorandom sequences; asanalyser of multiple-input signatures; and reset ofall stages. For a BILBO register to perform thesefour functions correctly the feedback from its stateto its first stage must satisfy certain conditions.First, the pseudorandom sequence that it gener-ates must be of maximum length, i.e. it must runthrough all possible states except zero. Secondly,when it is acting as a parallel signature analyserthe number of stages must be large enough for theerror escape probability to be negligible. Normallythe feedback of a BILBO register is linear. Wepropose the use of nonlinear feedback, throughExclusive-Nor gates, which conserves the proper-ties of linear feedback and provides the advantagethat the fourth functional mode, reset, serves as aseed for the other modes. Additional logic is notrequired and a seed does not have to be intro-duced from an external source.1 IntroductionThe idea of design for testability emerged in the late1960s in response to the situation created by the highdegree of complexity of ICs. Soon afterwards, the idea ofself-testing appeared, although it was more limited innature than the former concept [1].By the end of the 1970s this movement had gainedmomentum and a great many contributions on this topicwere published. One of the most widely accepted of thesetechniques is built-in logic block observation (BILBO) [2,3], based on the self-stimulation of the circuit by pseudo-random sequences and compacting the output test datasequence in a digital signature analyser.Many variations on the original idea have appeared,but all of them start by grouping the memory elements ofthe circuits in BILBO registers, which give the techniqueits name.In contrast with the linear feedback used until now, wepropose using BILBO registers with nonlinear feedback.Paper 5459G (E10, E3, C2), first received 4th August 1986 and inrevised form 16th February 1987The authors are with the Departmento de Electricidad y Electronica,Facultad de Ciencias, Universidad de Santander, Santander, SpainNonlinear feedback results in the invariance of the fourfunctional modes and eliminates the need for seeds in twoof these modes.2 Built-in logic block observationA digital circuit having built-in logic block observation ischaracterised by the fact that its storage elements aregrouped in two registers called BILBO registers. Thismeans that the circuit can be considered as four blocks:two sequential blocks (the BILBO registers) and twocombinational blocks. In the test phase the circuit has aspecial functional mode (see Fig. 1). If suitable circuitcontrol values are used, one BILBO register will act as apseudorandom generator which stimulates one com-binational block, and the other BILBO register will actas a multiple-input signature analyser which compactsthe response of the circuit to the stimulus from the otherBILBO register. The other combinational block can betested by repeating this operation with the functions ofthe registers reversed.Trandomsequencegeneratoroo 5O 0tCD TX 11digitalsignatureanalysero a,co I!O Q(CD *ZdigitalsignatureanalyserrandomsequencegeneratorFig. 1 Tests of combinational circuits using BILBO registersa Circuit 1b Circuit 2The storage elements of the circuit are grouped in theBILBO registers (Figs. 2-5) which are controlled by twolines, Bo and Bx, that determine their functional modes.If Bo = 0 and Bt = 1, the registers function as inde-pendent flip-flops.If Bo = 1 and Bx = 0, the registers function as arandom sequence generator. If the feedback function isappropriated [4, 7], the sequence generated has a periodp = 2N 1, for N stages. With linear feedback this meansIEE PROCEEDINGS, Vol. 134, Pt. G, No. 4, AUGUST 1987 205that the register runs through all possible states except00 ... 0, which is a prohibited state and will be called y 0 .Furthermore, the BILBO register can make a quasi-systematic sweep of the states with much less logic than acounter.F i g . 2 BILBO registerQi As the register has N stages, the maximum period will bep = 2N 1. Let p be expressed as a product of powers ofdistinct primes:p = kVk?~-k? (10)where k(, e{ e N and kt < ki+l.Form the set p, of maximum divisors of p:(11)Pn ~Under these conditions the general theorem proved inReference 7 can be extended to the case that we are con-cerned with here.Theorem 1: Given an initial state Y(t) # Yo, a shift regis-ter of N stages has a period p = 2N - 1 if and only ifY(t + p) = Y(t) and Y(t + p,)# Y(t), i = 1, 2 , . . . , n.Corollary 2: Given any initial state such that Y(t)#Y0,and if p 2N 1 is prime, a feedback shift-register of Nstages has a period p if and only if Y(t + p) = Y(t).With these results and the expression for the stateY(t + K) at an instant K based on a generic state Y(t) wecan establish the following results.Theorem 3: A shift register of N stages fedback byEx-Nor gates has a maximum-length sequence if andonly if Bp+1 = B and BPi+l#B, i = l , 2 n.Proof: If Bp+X =B and #" + 1 # # then Y(t + 1 + p) =Y(t + 1) and Y{t + \ + p,)# F(f + 1). Making the changeof variable t + 1 = t we obtain F(t + p) = ^(r) and Y(t+ p.)# y(f). Thus, the conditions of Theorem 1 are satis-fied; therefore, the shift register has a maximum-lengthsequence.Choose Y(t) as the state in which each stage has a zeroexcept for the ith stage and the pseudostage N + 1, thatisY(t) = (0, . . . , 1, . . . , 0, 1) i = 1, 2 , . . . , N, N + 1We represent B and BK by B = (fc(l),. j) and BK =(fc( ),.,). Then Y(t + 1) and K(t + X) will be given by:Y(t + K) = (b(K)uib(K)UN + 1,If / = N + 1, i.e. each stage has a zero except the pseudo-stage, the fact that Y(t + 1) = Y(t + K) leads tob(\)j N+l = fr(K)y Ar+1. This implies that for any other ithe equality Y{t + 1) = F(f + /C) leads to fc(l),.,_,. = W),,,-,7 = 1, 2, . . . , N, N + 1. So, the ith columns of matrices Band BK are equal and therefore B = /?*.If K(t + \)#Y(t + K), it must hold for the state inwhich every stage, except the pseudostage, contains azero. Therefore, at least the columns N + l are differentin Band BK. So B#BK.If these results are applied to K = p + I and K =Pi + 1 the theorem is proved.Corollary 4: If p = 2N 1 is prime, a shift register with Nstages fedback by Ex-Nor gates has a maximum-lengthsequence if and only if Bp+1 = B.3.1.2 Algorithm: With these results we can establish thefollowing algorithm for determining whether a set offeedback coefficients generates a maximum-lengthsequence when the signal is fedback through Ex-Norgates:(a) Given the feedback coefficients of a shift register ofN stages, calculate its matrix B(b) If p is a prime number go to (c); otherwise calcu-late its set of divisors p,-(c) Calculate Bp+l. If Bp+l # # go to (/); otherwise goon(d) If p is a prime number go to (e); if not, calculateBPi + i if gPi+1 _ g for s o m e ,-s g 0 t 0 (y-). otherwise go on(e) The feedback coefficients are such that thesequence generated by this register has maximum length.Go to (g)(/) The feedback coefficients are such that thesequence generated by this register does not havemaximum length. Go to (g)(g) End.3.1.3 Analogy with linear feedback shift registers: Thenumber of feedback connections must be even for both/x,[4, 7], and f2, because if not, both 00 ... 0 and 11 ... 1would be prohibited states and the register would nothave a maximum-length sequence. Accordingly, we canestablish the following theorems.Theorem 5: A BILBO register acting as a pseudorandomsequence generator which has a feedback function fx gen-erated by a set of connections C = {cl5 c2, . . . , cN} pro-duces maximum-length sequences if and only if this sameregister produces maximum-length sequences with a feed-back function f2 generated by the same set C.Proof: Suppose that ft generates a maximum-lengthsequence and assume any state different from the prohibi-ted state in the register is fedback by f2. Let this state beY(t) = {Ql(t),Q2(t),...,QN(t)}The register fedback by/ t will reach the state= {Q\(t),Q'2(t),...,Q'N(t)}because the sequence generated in the register fedback by/ i is of maximum length, and because Y(t) is differentfrom the prohibited state in / 2 , as its complementarystate is the prohibited state for j ^ .If the number of nonzero elements in the set CY ={ciQi(t\ c2Q2(t), ...,cNQN(t)} is even (odd), then a T ('0')is generated by/ 2 .If the number of elements of the set CY(t) is even(odd), then, as the number of nonzero elements of C iseven, the number of nonzero elements of the set CY(t) ={ciQ\(t), c2Q'2(t), . . . , cNQ'N(t)} will be even (odd) and '0'('0') will be generated by/x.The reverse can be proved in an analogous way.Corollary 6: Consider a BILBO register acting as a gen-erator of pseudorandom sequences with maximumlength. If this register has a feedback function /x which isgenerated by a set of connections C = {c1? c2, . . . , cN}and the initial state is a nonprohibited state Y(t) = {Q^0,2, > QN}> it generates the complementary sequence ofthe one which is generated by the same register when it isfedback by a function f2 with the same set of feedbackconnections and initial state Y(t) = {Q\, Q'2,..., Q'N}.This result means that all the properties demonstratedfor linear feedback registers for generating pseudoran-IEE PROCEEDINGS, Vol. 134, Pt. G, No. 4, AUGUST 1987 207dom sequences, such as the three randomness propertiesproposed and proved by Golomb [4], can be applieddirectly to shift registers which are fedback throughEx-Nor gates.3.2 Multiple-input signature analyserWhen the register is acting as a digital signature analyser,Bo = Bi = 1, exprs. 1 can be written in the followingform:/: Qt{t + 1) = Zt(t) 0where * e* c2Q2(t) * * cNQN(t)(12)Theorem 7: Consider a BILBO register acting as a multi-ple signature analyser. If this register is fedback by afunction fx which is generated by a set of connectionsC = {cu c2, . . . , CJV} and the initial state is Y{t) = {QuQi> > QN}I ^ produces the complementary signature ofthe one which is produced by the same register when it isfedback by a function f2 with the same set of fedbackconnections C and initial state vector Y(t) = {Q\, Q'2,...,Q'N}-Proof: Under the conditions mentioned above and forZ^t) 0, expr. 12 can be reduced to those in eqns. 4 and5, which have complementary values. Let these values beQfb{t + 1) and Q'fb(t + 1), respectively, where Qfb is thefunction generated by the feedback.Then, exprs. 12 can be written as(14)which clearly produce complementary results.For any other stage i, 1 < i + 1 ^ N we have(15)(16)which, as in the previous case, produce complementarysequences.Corollary 8: A BILBO register acting as a multiple-inputsignature analyser, which has a feedback function fu hasthe same error escape probability as it would have if ithad a feedback function f2 generated by the same set ofconnections C.4 ConclusionsWe have studied shift registers with nonlinear feedbackacting as BILBO registers and the following results havebeen obtained.(a) The feedback connections of a BILBO registeracting as a random number generator can be calculated,without needing to know the connections of its linearlyfedback analogue, by means of an algorithm similar tothe one presented in Reference 7 for linear feedback(b) When acting as a random number generator it hasthe randomness properties established by Golomb [4](c) For a given set of feedback connections it producesthe sequence which is complementary to the one pro-duced by its analogue with linear feedback(d) As a signature analyser it produces the comple-mentary signature, for a given set of feedback connec-tions, of the one produced by its analogue with linearfeedback(e) The error escape probability is the same as that ofa register with linear feedback(/) In addition, the reset mode for a BILBO registerwith feedback through Ex-Nor gates allows a seed to beintroduced without requiring additional logic and/orextra time for it to be introduced externally.5 References1 BENOWITZ, N., CALHOUN, D.F., ALDERSON, G.E., BAUER,J.E., and JOECKEL, C.T.: 'An advanced fault isolation system fordigital logic', IEEE Trans., 1975, C-24, (5), pp. 489-4972 KONEMANN, B., MUCHA, J., and ZWIEHOFF, G.: 'Built-in logicblock observation techniques'. Proceedings of 1979 SemiconductorTest Conference, Cherry Hill, NJ, USA 25th-27th October 1979,pp. 37-413 KONEMANN, B., MUCHA, J., and ZWIEHOFF, G.: 'Built-in testsfor complex digital integrated circuits', IEEE J. Solid-State Circuits,1980, SC-15, (3), pp. 315-3194 GOLOMB, S.W.: 'Shift register sequences' (Holden-Day, San Fran-cisco, 1967)5 FROHWEK, R.A.: 'Signature analysis: a new digital service method'.Hewlett-Packard Application Note 222-2, pp. 9-156 NADING, H.J.: 'Signature analysis-concepts, examples and guide-lines', Hewlett-Packard J., May 1977, pp. 15-217 CAGIGAL, N.P., and BRACHO, S.: 'Algorithmic determination oflinear-feedback in a shift register for pseudorandom binary sequencegeneration', IEE Proc. G, Electron. Circuit. & Syst., 1986, 133, (4),pp. 191-194Niceto P. Cagigal received his MScdegree in physical sciences, in 1981, fromthe University of Santander, Spain. Since1981 he has been an Assistant Professorat the University of Santander. From1983 to 1984 he was a scientific associateat the European Centre for NuclearResearch, CERN, Geneva. At present heis researching in design for testability andself-test.Salvador Bracho received his MSc degreein 1967, and his PhD degree in physicalsciences in 1969, both from the Universityof Sevilla, Spain. From 1967 to 1972 hewas an assistant Professor at the Uni-versity of Sevilla. Since 1973 he has beenHead of the Department of Electronics atthe University of Santander, Spain, wherehe was appointed Professor of Electronicsin 1973. His research interest are in thedesign and test of digital circuits andCAD systems for VLSI.208 IEE PROCEEDINGS, Vol. 134, Pt. G, No. 4, AUGUST 1987