counting with feedback shift registers by means of a jump technique

2
1962 Correspondence 285 2n consistent canonical forms for n-variable n soiie n-stage FSR (not necessarily linear). The switching functions. Eqs. (2a)-(2c) show the re- n proof is in fact extremely simple and the result sult of some consistent complementations on - l is given here. the canonical forms of (la)-(lc). 2 It should perhaps be mentioned that a search and that this cannot be improved. The latter of the literature revealed only one claim to a f (z) = ho 1 + hi z' (2a) bound is replaced by proof of the existence of cycles of any length up is(Y, rpae by ho 1 + /zl z + h2 at + y' (2b)to 2n in the class of n-stage FSR's; this was by n ) n Radchenko,3 who claims to have proved the f(X, Av, z) = ho 1 + hi- z' +1 he y + h83 yz' n-1 n + theorem in his dissertation.4'5 + h4-Xh. + ho5 X'Z' + h6-X'Y 12] 2J Theorem + 17 X'yz'. (2c) for odd n; it imnproves the former botund il Given a linear n-stage binary FSR produc- the senise that ing a maximal length cycle (2- 1) and any The class of consistent canonical forms was integer k satisfying 1<k<2n-1, then by a studied by Akers3 in which he drew the analogy n slight modification of the given linear FSR, a to Taylor series expansions and showed that n < 2n-1 for n > 2, new FSR may be obtained which produces a the coefficients ki could actually be defined in - self-starting cycle of length k (self-starting ex- terms of a partial-difference operator. 2 cept from the zero state). It is the purpose of the present note to Note: The existence of linear n-stage binary point out that the restriction to consistent and FSR's producing maximal length cycles has complementation is unnecessary; that is, in a F n been proved by Elspas2 for all n. given form a variable may occur comple- lim 2-n+1 n -O mented in some terms and uncomplemented in n- =*| Proof others, but nonetheless the form is canonical. 2 We first show that there exists on the maxi- The expressions in (3a) and (3b) are examples MARTIN COHN mal length cycle a state XJk, say, from which a of inconsistent modulo-2 canonical forms. Sperry Rand Res. Ctr. jump may be made so that a cycle of length k Sudbury, Mass. is produced. f(y, z) = ho I + h * z + /h 2 i -Y + - 3 h YZ' (3a) Denote by T the nXn transition matrix2 associated with the given FSR. Then the state -I(x, y, z) = lio I It,A- hz - h2 y - h3 Y'Z' Xjk+il following Xjk on the maximal length + -hg x' + h5-XZ' + 116. X'Y cycle is given by + h7 X'y'Z- (3b) Xjk+1 = TXjk. (1) In order that the cycle of length 2n -1 be- The proof that all inconsistent forms are Counting with Feedback Shift come a cycle of length k, it is necessary that the canonical is given in (letail in Cohn;4 in outline, state following Xjk on the modified cycle is it is as follows. The proof begins with the ob- Registers by Means of Xj0±2nk; i.e., we must jump from Xj5 to servation that the set of n-variable switching a Jump Technique* Xj*F2'f-k. If it is possible to make this jump functions un(ler modulo-2 addition forms a 2-- then the following relation must hold dimensional vector space over the field of Ac recent paper by Heath and Gribble de- t h integers mo(lulo 2. Since all the forms under scribes a method of obtaining binary feedback F consideration are of the sum-of-products type, shift registers (FSR's) to count to any number functions are expressetl as linear combinations, up to 127. Their technique is to take a linear Xsk+2sLe = Xi,+, 3 . (2) Thus if a form is canonical it defines a basis for FSR producing a maximal length (2"-1) the vector space of n-variable switching func- cycle and to modify it so that the necessary tions; coniversely, every basis defines a canoni- number of states in the cy cle are jumped over. (where eD denotes addition modulo two) since cal form. But the set of terms which defines an Defined precisely this jumping technique is all digits must be the same except the first, inconsistent form can be shown to be related to as follows. The nonzero states of a binary n- which is reversed. the terms in the completely uncomplemented stage FSR are represented by binary n vectors, But we also have from the maximal length form [e.g., (la)-(lc)] by a nonsingular linear which we denote by Xj (j= 1, 2, - * ., 2 - 1), cycle transformation. Since the uncomplemented (Elspas2). We number the Xj so that the suf- form is known to be canonical1-3 each incon- fixes denote the sequence of states in the maxi- Xik+2n-JC = T2i-kXik. (3) sistent form is dlefined bv a basis an(d is there- mum length cycle of the given FSR; thus From (1)-(3) we obtain fore canonical. This completes the proof. Xfr follows XX in this cycle and is obtained A combinatorial argument can be used to from XJ by moving the digits of Xj down one prove that for n variables there are 2v-2N place (losing the nth digit) and placing a one inconsistent forms, where N=2n2n- . The inl- or a zero as the first digit, depending upon the n2nk 1( troduction of these forms not only extends the logic of the FSR. If we now wish to produce a A = class of consistent forms, but can be used to set jump from some particular Xj, then the jump _ _ a bound on the number of terms actually re- must be made to that state which differs from where 1 is the unit matrix of order n. Hence quired to write an arbitrary switching function Xet only in its first digit, which Is reversed, from (4), X1k is uniquely determined, provided as a modulo-2 sum of products. The following stt rmwihtejm smd,FRsmy the matrix result is proved:5 For any switching function of state from which the jump is made, FSRs may to 1T D n variables there exists a canonical form (possi- be designed to count to any number up to 127. [Tnnki e 1,]T bly inconsistent) in which at least 24-l of the This method of generating different length has an inverse (i.e., is nonsingular: modulo coefficients ) are zero. This lbound does not cycles prompted the question of whether an t a.r hold for the class of consistent forms. analytic proof could be devised, based upon ) Furthermore, it is conjectured by S. Even this method, to show that a count up to any and the w7riter that the upper bound of 2 f-i number k <2n -1 could alw7ays be obgtained by 3 A. N. Radchenko, and V. I. Filipov, "Shift registers with logical feedback and their use as counting and nonvanishing terms can be low^eredl to coding devices," Automation and Remote Control, vol. 20, pp. i467-i473; November, 1959. *Received December 27, i96i1 revised manuscript 4A. N. Radchenko, "Code Rings and Their Use in received, February 5, i962. Contactiess Coding Devices," Ph.D. dissertation, Uni- 3S. Akers, Jr., "On a theocy of Boolean functions," l'F. G. Heath and M. W. Gribble, "sChain codes and versity of Leningrad, U.S.S.R.; i955. (In Russian.) J. SIAM, vol. 7, pp. 487-498; December, i959. their electronic applications," Proc. lEE, vol. 105 C, 5It has been pointed out to the authors that the follow. 4M. Cohn, "Switchingf Function Canonical Forms pp. 50-57, March, i96i; Monograph No. 392 M, July, lug report, which they have not seen, also contains acon- over Integer Fields," Ph.D. Dissertation, Harvard Uni- i960. structive proof: S. W. Golomb, L. R. Welch, R. M Gold- versity, Cambridge, Mass., ch. .3, pp. i-4; December, 2 B. Elspas, "The theory of autonomous linear sequen- stein, "Cycles from Non-linear Shift Registers," Jet 1960. tial networks," IRE TRANS. ON CIRCUIT THEORY, vol. Propulsion Lab., Calif. Inst. Tech., Pasadena, Calif., 6Ibid., ch. 3, pp. 5-6. CT-6, pp. 45-60; March, 1959. Prog. Rept. No. 20-389; August, 5959.

Upload: r-d

Post on 16-Mar-2017

212 views

Category:

Documents


0 download

TRANSCRIPT

Page 1: Counting with Feedback Shift Registers by Means of a Jump Technique

1962 Correspondence 285

2n consistent canonical forms for n-variable n soiie n-stage FSR (not necessarily linear). Theswitching functions. Eqs. (2a)-(2c) show the re- n proof is in fact extremely simple and the resultsult of some consistent complementations on - l is given here.the canonical forms of (la)-(lc). 2 It should perhaps bementioned that a search

and that this cannot be improved. The latter of the literature revealed only one claim to af (z) = ho 1 + hi z' (2a) bound is replaced by proof of the existence of cycles of any length up

is(Y,rpae byho 1 + /zl z + h2 at + y' (2b)to 2n in the class of n-stage FSR's; this was byn ) n Radchenko,3 who claims to have proved the

f(X, Av, z) = ho 1 +hi- z' +1he y + h83yz' n-1 n + theorem in his dissertation.4'5

+ h4-Xh. + ho5 X'Z' + h6-X'Y 12] 2J Theorem

+ 17 X'yz'. (2c) for odd n; it imnproves the former botund il Given a linear n-stage binary FSR produc-the senise that ing a maximal length cycle (2- 1) and any

The class of consistent canonical forms was integer k satisfying 1<k<2n-1, then by astudied by Akers3 in which he drew the analogy n slight modification of the given linear FSR, ato Taylor series expansions and showed that n < 2n-1 for n > 2, new FSR may be obtained which produces athe coefficients ki could actually be defined in - self-starting cycle of length k (self-starting ex-terms of a partial-difference operator. 2 cept from the zero state).

It is the purpose of the present note to Note: The existence of linear n-stage binarypoint out that the restriction to consistent and FSR's producing maximal length cycles hascomplementation is unnecessary; that is, in a F n been proved by Elspas2 for all n.given form a variable may occur comple- lim 2-n+1 n -Omented in some terms and uncomplemented in n-=*| Proofothers, but nonetheless the form is canonical. 2 We first show that there exists on the maxi-The expressions in (3a) and (3b) are examples MARTIN COHN mal length cycle a state XJk, say, from which aof inconsistent modulo-2 canonical forms. Sperry Rand Res. Ctr. jump may be made so that a cycle of length k

Sudbury, Mass. is produced.f(y, z) = ho I + h * z +/h 2i -Y +- 3 hYZ' (3a) Denote by T the nXn transition matrix2

associated with the given FSR. Then the state-I(x, y, z) = lio I It,A-hz - h2 y - h3 Y'Z' Xjk+il following Xjk on the maximal length

+-hgx' + h5-XZ' + 116.X'Y cycle is given by

+ h7 X'y'Z- (3b) Xjk+1 = TXjk. (1)In order that the cycle of length 2n -1 be-

The proof that all inconsistent forms are Counting with Feedback Shift come a cycle of length k, it is necessary that thecanonical is given in (letail in Cohn;4 in outline, state following Xjk on the modified cycle isit is as follows. The proof begins with the ob- Registers by Means of Xj0±2nk; i.e., we must jump from Xj5 toservation that the set of n-variable switching a Jump Technique* Xj*F2'f-k. If it is possible to make this jumpfunctions un(ler modulo-2 addition forms a 2-- then the following relation must holddimensional vector space over the field of Ac recent paper by Heath and Gribble de- t hintegers mo(lulo 2. Since all the forms under scribes a method of obtaining binary feedback Fconsideration are of the sum-of-products type, shift registers (FSR's) to count to any numberfunctions are expressetl as linear combinations, up to 127. Their technique is to take a linear Xsk+2sLe = Xi,+,3 . (2)Thus if a form is canonical it defines a basis for FSR producing a maximal length (2"-1)the vector space of n-variable switching func- cycle and to modify it so that the necessarytions; coniversely, every basis defines a canoni- number of states in the cycle are jumped over. (where eD denotes addition modulo two) sincecal form. But the set of terms which defines an Defined precisely this jumping technique is all digits must be the same except the first,inconsistent form can be shown to be related to as follows. The nonzero states of a binary n- which is reversed.the terms in the completely uncomplemented stage FSR are represented by binary n vectors, But we also have from the maximal lengthform [e.g., (la)-(lc)] by a nonsingular linear which we denote by Xj (j= 1, 2, - * ., 2 - 1), cycletransformation. Since the uncomplemented (Elspas2). We number the Xj so that the suf-form is known to be canonical1-3 each incon- fixes denote the sequence of states in the maxi- Xik+2n-JC = T2i-kXik. (3)sistent form is dlefined bv a basis an(d is there- mum length cycle of the given FSR; thus From (1)-(3) we obtainfore canonical. This completes the proof. Xfr followsXX in this cycle and is obtained

A combinatorial argument can be used to from XJ by moving the digits of Xj down oneprove that for n variables there are 2v-2N place (losing the nth digit) and placing a one

inconsistent forms, where N=2n2n- . The inl- or a zero as the first digit, depending upon the n2nk 1(

troduction of these forms not only extends the logic of the FSR. If we now wish to produce a A =class of consistent forms, but can be used to set jump from some particular Xj, then the jump _ _

a bound on the number of terms actually re- must be made to that state which differs from where 1 is the unit matrix of order n. Hencequired to write an arbitrary switching function Xet only in its first digit, which Is reversed, from (4), X1k is uniquely determined, providedas a modulo-2 sum of products. The following

stt rmwihtejm smd,FRsmythe matrix

result is proved:5 For any switching function of state from which the jump is made, FSRs mayto1T D

n variables there exists a canonical form (possi- be designed to count to any number up to 127. [Tnnki e 1,]Tbly inconsistent) in which at least 24-l of the This method of generating different length has an inverse (i.e., is nonsingular: modulocoefficients

)

are zero. This lbound does not cycles prompted the question of whether an t a.r

hold for the class of consistent forms. analytic proof could be devised, based upon )

Furthermore, it is conjectured by S. Even this method, to show that a count up to anyand the w7riter that the upper bound of 2 f-i number k<2n-1 could alw7ays be obgtained by 3 A. N. Radchenko, and V. I. Filipov, "Shift registers

with logical feedback and their use as counting andnonvanishing terms can be low^eredl to coding devices," Automation and Remote Control, vol. 20,

pp. i467-i473; November, 1959.*Received December 27, i96i1 revised manuscript 4A. N. Radchenko, "Code Rings and Their Use in

received, February 5, i962. Contactiess Coding Devices," Ph.D. dissertation, Uni-3S. Akers, Jr., "On a theocy of Boolean functions," l'F. G. Heath and M. W. Gribble, "sChain codes and versity of Leningrad, U.S.S.R.; i955. (In Russian.)

J. SIAM, vol. 7, pp. 487-498; December, i959. their electronic applications," Proc. lEE, vol. 105 C, 5It has been pointed out to the authors that the follow.4M. Cohn, "Switchingf Function Canonical Forms pp. 50-57, March, i96i; Monograph No. 392 M, July, lug report, which they have not seen, also contains acon-

over Integer Fields," Ph.D. Dissertation, Harvard Uni- i960. structive proof: S. W. Golomb, L. R. Welch, R. M Gold-versity, Cambridge, Mass., ch. .3, pp. i-4; December, 2 B. Elspas, "The theory of autonomous linear sequen- stein, "Cycles from Non-linear Shift Registers," Jet1960. tial networks," IRE TRANS. ON CIRCUIT THEORY, vol. Propulsion Lab., Calif. Inst. Tech., Pasadena, Calif.,

6Ibid., ch. 3, pp. 5-6. CT-6, pp. 45-60; March, 1959. Prog. Rept. No. 20-389; August, 5959.

Page 2: Counting with Feedback Shift Registers by Means of a Jump Technique

286 IRE TRANSACTIONS ON ELECTRONIC COMPUTERS April

Now T is nonsingular (Elspas2) and 0 1 is made to [1, 0, 1, 0]. The original and new se-1 quences are shown in Table I.

[T2ff- -1,3ln] 1 I The logic which produced the maximal

must be nonsingular for, if not, the equation L 0 length FSR was

T2n-k-IX = X (5) HenceC

This must now be modified towould have a nonzero solution. But all states X

i ms_(other than the zero state) lie on the maximallength cycle, and so the only k for which (5) XC D =AB.CDis true is L which may be simplified to

k = 0 (modulo 2n - 1). L 0

Hence for any k satisfying 1<k <2n-1, and instead of proceeding to [0, 0, 1, 0], a jump C D+CD ±ABCDX k is uniquely defined by (4) as where + and - denote logical OR and AND,

respectively.[ _H~_ P. R. BRYANT

Xj0=T-l[T2-§-'@1ff]-ll° |- (6) / -% The General Electric Co., Ltd.-Yk= Til[T2 ki1 0D lnj1 (6) Telecommun. Res. Labs.

I ~~~~~~~~~~~~~ZEROL STATE Hirst Research Centre

Wembley, EnglandNote that this gives Xik as the first column of F. G. HEATHtthe matrix Computer Dept.

Ferranti Ltd.T-i[m~~~~-k--I0lnP. ~~~~(a) Manchester, England

Having obtained the Xjk from which the R. D. KILLICKttjump is to be made (to count up to k), the logic x General Electric Co., Ltd.of the original linear FSR has only to be modi- _, jk+ Internl. Systems Control, Ltd.fied to recognize this state and insert the ap- Wembley, Englandpropriate change in the digit in order to pro-

ZERO

duce the required count.' JUMP STATE . t Formerly with Dept. Elec. Engrg., University ofThe cycle is self-starting (except from the \ l Manchester.

zero state) as is obvious from Fig.( which tt Formerly with Telecommun. Res. Labs., Hirst Re-illustrates the effect of the jump procedure.

serhCete

A program has been written for a digital -k+2-kcomputer (the I.C.T. 1301) which will obtainthe jump state for any count k<224-2. The (b)maximum time taken for any count is 55 sec, Fig. I-State diagrams of feedback shift registers. (a)and that for k= 24 _-2 takes 35 sec. Producing a maximal length cycle. (b) After carrying Hysteresis-Free Tunnel-Diodeout a jump procedure.

Amplitude Comparator*Example Conventional amplitude comparators do

Given the 4-stage FSR counting up to 15 not permit an arbitrarily close realization ofshown in Fig. 2, to produce a 4-stage FSR A B c D the ideal transfer characteristic illustrated incounting up to 9, the transition matrix T for Fig. 1. As a matter of fact, an electrically stablethis FSR is2 circuit can never exactly attain this type of

rO 01 1~ 0/ characteristic unless stability in enforced by0 0

1

some type of latching mechanism. Tunnel di-

T==I1 0 0 0 odes have all the properties necessary to allowT

1 0 0 Fig. 2-A linear four-stage feedback shift register an arbitrarily close realization of the ideal com-LO 0 1 0] producing a maximal length cycle. parator characteristic. The circuit discussed is

remarkably simple and exhibits a high degreeWe have 2l= 16, k=9, therefore 2n-k-1=6 of thermal stability, besides performing the

TABLE I comparator function at a high switching speed1 0 1 01 typical of tunnel diodes.0 1 0 1 Original Maximal Length Cycle To construct the tunnel-diode amplitude

T6 = 10 1 1 1 1 comparator, cascade two tunnel diodes (Fig.o 00 1 2) whose current-voltage characteristics are of

Li 1 0 0 o 000 the type shown in Fig. 3. The insert on the

Thus o l o o right of this figure defines the notation used.T h 1 0 Observe that the current crest of the bottom1 0 01 0 0000 1 0 ] 1 10 ti diode is assumed smaller than that of the upper0 0 0 1 0 1 1 0 diode; conversely, the current valley of the

[T6 (D 14]= |-° 1 0 1 bottom diode is assumed larger than that of the0 0 1 l 0 1 0 upper diode. Utilizing two identical tunnel

Ll I 0 11 1 0o diodes, this relationship between the current

[T6 0 K 1 0 i i1 1 0 1 1 0 0 1 2 diod~~~~~~ext e cha a ctberi to t h valued ofsh nt nthere is or -Hence New Cycle: Self-Starting and of LJength exrmscnb9bandb hnigtebt0100] 0010 tom diode wlith abattery-resistor network (Fig.01 l o l o e_ o o l o ~~~~4).Because of the curvature of the tunnel-

1|] 0 001 1 1 000 an|L Io series with the tunnel-diode pair (Fig. 2) IS100 I 011°t1 1011o1] ~ ~~~0 0 10 - * Received October 2, 1961;,revised received, December

giving, as the first column of the inverse 6, 1961.