# BILBO registers with nonlinear feedback

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- BILBO registers with nonlinear feedback N.P. Cagigal Prof. S. Bracho Indexing terms: Feedback, Nonlinear systems, Digital circuits, Logic Abstract: Built-in logic block observation (BILBO) has become one of the most widely accepted techniques for self-testing of complex digital ICs. This technique is based on grouping the storage elements of the circuit in the two regis- ters which give this technique its name. A BILBO register has four functional modes: with each of the stages acting as independent registers; as a generator of pseudorandom sequences; as analyser of multiple-input signatures; and reset of all stages. For a BILBO register to perform these four functions correctly the feedback from its state to its first stage must satisfy certain conditions. First, the pseudorandom sequence that it gener- ates must be of maximum length, i.e. it must run through all possible states except zero. Secondly, when it is acting as a parallel signature analyser the number of stages must be large enough for the error escape probability to be negligible. Normally the feedback of a BILBO register is linear. We propose the use of nonlinear feedback, through Exclusive-Nor gates, which conserves the proper- ties of linear feedback and provides the advantage that the fourth functional mode, reset, serves as a seed for the other modes. Additional logic is not required and a seed does not have to be intro- duced from an external source. 1 Introduction The idea of design for testability emerged in the late 1960s in response to the situation created by the high degree of complexity of ICs. Soon afterwards, the idea of self-testing appeared, although it was more limited in nature than the former concept [1]. By the end of the 1970s this movement had gained momentum and a great many contributions on this topic were published. One of the most widely accepted of these techniques 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 data sequence in a digital signature analyser. Many variations on the original idea have appeared, but all of them start by grouping the memory elements of the circuits in BILBO registers, which give the technique its name. In contrast with the linear feedback used until now, we propose using BILBO registers with nonlinear feedback. Paper 5459G (E10, E3, C2), first received 4th August 1986 and in revised form 16th February 1987 The authors are with the Departmento de Electricidad y Electronica, Facultad de Ciencias, Universidad de Santander, Santander, Spain Nonlinear feedback results in the invariance of the four functional modes and eliminates the need for seeds in two of these modes. 2 Built-in logic block observation A digital circuit having built-in logic block observation is characterised by the fact that its storage elements are grouped in two registers called BILBO registers. This means that the circuit can be considered as four blocks: two sequential blocks (the BILBO registers) and two combinational blocks. In the test phase the circuit has a special functional mode (see Fig. 1). If suitable circuit control values are used, one BILBO register will act as a pseudorandom generator which stimulates one com- binational block, and the other BILBO register will act as a multiple-input signature analyser which compacts the response of the circuit to the stimulus from the other BILBO register. The other combinational block can be tested by repeating this operation with the functions of the registers reversed. T random sequence generator o o 5 O 0t CD TX 11 digital signature analyser o a, co I! O Q( CD *Z digital signature analyser random sequence generator Fig. 1 Tests of combinational circuits using BILBO registers a Circuit 1 b Circuit 2 The storage elements of the circuit are grouped in the BILBO registers (Figs. 2-5) which are controlled by two lines, 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 a random sequence generator. If the feedback function is appropriated [4, 7], the sequence generated has a period p = 2N â 1, for N stages. With linear feedback this means IEE PROCEEDINGS, Vol. 134, Pt. G, No. 4, AUGUST 1987 205
- that the register runs through all possible states except 00 ... 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 a counter. F i g . 2 BILBO register Qi
- As the register has N stages, the maximum period will be p = 2N â 1. Let p be expressed as a product of powers of distinct 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 in Reference 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 if Y(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 â 2N â 1 is prime, a feedback shift-register of N stages has a period p if and only if Y(t + p) = Y(t). With these results and the expression for the state Y(t + K) at an instant K based on a generic state Y(t) we can establish the following results. Theorem 3: A shift register of N stages fedback by Ex-Nor gates has a maximum-length sequence if and only 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 change of 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-length sequence. Choose Y(t) as the state in which each stage has a zero except for the ith stage and the pseudostage N + 1, that is Y(t) = (0, . . . , 1, . . . , 0, 1) i = 1, 2 , . . . , N, N + 1 We represent B and BK by B = (fc(l),. j) and BK = (fc( Â£),â¢.,â¢). Then Y(t + 1) and K(t + X) will be given by: Y(t + K) = (b(K)uiÂ®b(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 to b(\)j N+l = fr(K)y Ar+1. This implies that for any other i the equality Y{t + 1) = F(f + /C) leads to fc(l),.,_,. = W),,,-, 7 = 1, 2, . . . , N, N + 1. So, the ith columns of matrices B and BK are equal and therefore B = /?*. If K(t + \)#Y(t + K), it must hold for the state in which every stage, except the pseudostage, contains a zero. Therefore, at least the columns N + l are different in 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 â 1 is prime, a shift register with N stages fedback by Ex-Nor gates has a maximum-length sequence if and only if Bp+1 = B. 3.1.2 Algorithm: With these results we can establish the following algorithm for determining whether a set of feedback coefficients generates a maximum-length sequence when the signal is fedback through Ex-Nor gates: (a) Given the feedback coefficients of a shift register of N 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 go on (d) If p is a prime number go to (e); if not, calculate BPi + 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 the sequence generated by this register has maximum length. Go to (g) (/) The feedback coefficients are such that the sequence generated by this register does not have maximum length. Go to (g) (g) End. 3.1.3 Analogy with linear feedback shift registers: The number of feedback connections must be even for both/x, [4, 7], and f2, because if not, both 00 ... 0 and 11 ... 1 would be prohibited states and the register would not have a maximum-length sequence. Accordingly, we can establish the following theorems. Theorem 5: A BILBO register acting as a pseudorandom sequence 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 same register produces maximum-length sequences with a feed- back function f2 generated by the same set C. Proof: Suppose that ft generates a maximum-length sequence and assume any state different from the prohibi- ted state in the register is fedback by f2. Let this state be Y(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 different from the prohibited state in / 2 , as its complementary state 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 is even, 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 maximum length. If this register has a feedback function /x which is generated by a set of connections C = {c1? c2, . . . , cN} and the initial state is a nonprohibited state Y(t) = {Q^ 0,2, â¢ â¢ â¢ > QN}> it generates the complementary sequence of the one which is generated by the same register when it is fedback by a function f2 with the same set of feedback connections and initial state Y(t) = {Q\, Q'2,..., Q'N}. This result means that all the properties demonstrated for linear feedback registers for generating pseudoran- IEE PROCEEDINGS, Vol. 134, Pt. G, No. 4, AUGUST 1987 207
- dom sequences, such as the three randomness properties proposed and proved by Golomb [4], can be applied directly to shift registers which are fedback through Ex-Nor gates. 3.2 Multiple-input signature analyser When the register is acting as a digital signature analyser, Bo = Bi = 1, exprs. 1 can be written in the following form: /: Qt{t + 1) = Zt(t) 0 where * e * c2Q2(t) * â¢â¢â¢ * cNQN(t) (12) Theorem 7: Consider a BILBO register acting as a multi- ple signature analyser. If this register is fedback by a function fx which is generated by a set of connections C = {cu c2, . . . , CJV} and the initial state is Y{t) = {Qu Qi> â¢â¢â¢â¢> QN}I ^ produces the complementary signature of the one which is produced by the same register when it is fedback by a function f2 with the same set of fedback connections C and initial state vector Y(t) = {Q\, Q'2,..., Q'N}- Proof: Under the conditions mentioned above and for Z^t) â 0, expr. 12 can be reduced to those in eqns. 4 and 5, which have complementary values. Let these values be Qfb{t + 1) and Q'fb(t + 1), respectively, where Qfb is the function 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 complementary sequences. Corollary 8: A BILBO register acting as a multiple-input signature analyser, which has a feedback function fu has the same error escape probability as it would have if it had a feedback function f2 generated by the same set of connections C. 4 Conclusions We have studied shift registers with nonlinear feedback acting as BILBO registers and the following results have been obtained. (a) The feedback connections of a BILBO register acting as a random number generator can be calculated, without needing to know the connections of its linearly fedback analogue, by means of an algorithm similar to the one presented in Reference 7 for linear feedback (b) When acting as a random number generator it has the randomness properties established by Golomb [4] (c) For a given set of feedback connections it produces the 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 linear feedback (e) The error escape probability is the same as that of a register with linear feedback (/) In addition, the reset mode for a BILBO register with feedback through Ex-Nor gates allows a seed to be introduced without requiring additional logic and/or extra time for it to be introduced externally. 5 References 1 BENOWITZ, N., CALHOUN, D.F., ALDERSON, G.E., BAUER, J.E., and JOECKEL, C.T.: 'An advanced fault isolation system for digital logic', IEEE Trans., 1975, C-24, (5), pp. 489-497 2 KONEMANN, B., MUCHA, J., and ZWIEHOFF, G.: 'Built-in logic block observation techniques'. Proceedings of 1979 Semiconductor Test Conference, Cherry Hill, NJ, USA 25th-27th October 1979, pp. 37-41 3 KONEMANN, B., MUCHA, J., and ZWIEHOFF, G.: 'Built-in tests for complex digital integrated circuits', IEEE J. Solid-State Circuits, 1980, SC-15, (3), pp. 315-319 4 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. 9-15 6 NADING, H.J.: 'Signature analysis-concepts, examples and guide- lines', Hewlett-Packard J., May 1977, pp. 15-21 7 CAGIGAL, N.P., and BRACHO, S.: 'Algorithmic determination of linear-feedback in a shift register for pseudorandom binary sequence generation', IEE Proc. G, Electron. Circuit. & Syst., 1986, 133, (4), pp. 191-194 Niceto P. Cagigal received his MSc degree in physical sciences, in 1981, from the University of Santander, Spain. Since 1981 he has been an Assistant Professor at the University of Santander. From 1983 to 1984 he was a scientific associate at the European Centre for Nuclear Research, CERN, Geneva. At present he is researching in design for testability and self-test. Salvador Bracho received his MSc degree in 1967, and his PhD degree in physical sciences in 1969, both from the University of Sevilla, Spain. From 1967 to 1972 he was an assistant Professor at the Uni- versity of Sevilla. Since 1973 he has been Head of the Department of Electronics at the University of Santander, Spain, where he was appointed Professor of Electronics in 1973. His research interest are in the design and test of digital circuits and CAD systems for VLSI. 208 IEE PROCEEDINGS, Vol. 134, Pt. G, No. 4, AUGUST 1987