Nielsen, Peter Tolstrup

A bifix of anL-aryn-tuple is a sequence which is both a prefix and a suffix of thatn-tuple. The practical importance of bifix-free patterns is emphasized, and we devise a systematic way of generating all such sequences and determine their number

Online Research Database In Technology

                       General rights Copyright and moral rights for the publications made accessible in the public portal are retained by the authors and/or other copyright owners and it is a condition of accessing publications that users recognise and abide by the legal requirements associated with these rights.  • Users may download and print one copy of any publication from the public portal for the purpose of private study or research. • You may not further distribute the material or use it for any profit-making activity or commercial gain • You may freely distribute the URL identifying the publication in the public portal   If you believe that this document breaches copyright please contact us providing details, and we will remove access to the work immediately and investigate your claim.     Downloaded from orbit.dtu.dk on: Dec 17, 2017A note on bifix-free sequencesNielsen, Peter TolstrupPublished in:I E E E Transactions on Information TheoryPublication date:1973Document VersionPublisher's PDF, also known as Version of recordLink back to DTU OrbitCitation (APA):Nielsen, P. T. (1973). A note on bifix-free sequences. I E E E Transactions on Information Theory, 19(5), 704-706.704 IEEE TRANSACTIONS ONINFORMATIONTHEORY,SEPTEMBER 1973 III. CONCLUSION We have derived an expression for the expected duration of a search for a fixed pattern in a semi-infinite random sequence. It was demonstrated that the presence of bifices in the pattern tends to delay its occurrence in random data. The analysis employed the use of indicator random variables with a somewhat novel twist, and it is anticipated that this approach may prove useful in other contexts as well. ACKNOWLEDGMENT Thanks are due to Prof. J. L. Massey, who assisted this work through many helpful discussions and who coined the word “bifix” to facilitate the statement of the results. REFERENCES [l] E. N. Gilbert, “Synchronization of binary messages,” IRE Trans. Inform. Theory, vol. IT-6, pp. 470-477, Sept. 1960. [2] J. J. Stiffler, Theory of Synchronous Communications. Englewood Cliffs, N.J.: Prentice-Hall, 1971, sect. 12.4. A Note on Bifix-Free Sequences P. TOLSTRUP NIELSEN Abstruct-A bifix of an L-ary n-tuple is a sequence which is both a prefix and a suffix of that n-tuple. The practical importance of bifix-free patterns is emphasized, and we devise a systematic way of generating all such sequences and determine their number. I. UNIQUE PATTERNS Much attention has been given to the problem of selecting from the set of all L-ary n-tuples a few patterns which are particularly suitable for use as synchronization patterns in digital communications or for other similar applications. Whenever such a unique pattern is to be inserted into random data in order to provide an easily recognizable reference point for the receiver, it is essential to avoid accidental imitation of the pattern by data- pattern overlaps. When noise is a major consideration, this requirement imposes certain bounds on the shape of the pattern correlation function. The Barker sequences [l ] constitute a classical example of such patterns with “perfect” correlation properties. A more application-oriented treatment of the problem may be found in [2]. When noise is absent the shape of the correlation function is S,(n) before the search is initiated. In this correspondence we count the number of elements in S,(n) and devise a systematic way of generating the elements for any values of L 2 2 and II 2 2. II. THE NUMBER OF BIFIX-FREE PATTERNS We shall write n-tuples in the form P = h,P2,**.J4tl, PI E A, (1) where AL (L 5: 2) is an alphabet of L letters. We define the bifx indicators hl ofp, 1 I i < n, in a manner such that hl = 1 if [P~,P~,. . . ,pi] is a bifix of p, i.e., if [PL,P2,. . * ,PJ = [P”--l+L,Pn-1+2,. . *,P,l (2) and hi = 0 otherwise. p is bitix-free if and only if hl = 0, lIi<n. The following property was hinted at by Artom [3], who referred to bifix-free sequences as being “valid” for synchro- nization purposes. In this context, however, we shall need the strongest statement possible. Lemma 1: A necessary and sufficient condition for p to be bifix-free is that hi = 0, (Here and hereafter 1. ] denotes the integer part of the argument.) Proof: Necessity is obvious. To prove sufficiency, we assume that (3) holds and that for some Z, [n/2] < I < II, we have h, = 1. Then p has the bifix b = [p1,p2,. + .,pt]; but since b is both a prefix and a suffix of p and its length exceeds half the length of p it follows that the sequence b’ = [p,- t + 1,. . . ,pl ] of length 21 - n < 1 must be a bifix of b and, therefore, also of p. By repeating this argument we eventually conclude that p must have a bifix of length less than or equal to [n/2 J, contradicting (3), and the proof is complete. Given a bifix-free pattern p of even length, we now proceed to demonstrate how to construct longer bifix-free patterns by in- serting extra digits in the “middle” of p. Letting p1 = [p1,p2, . . .,pnlz] and pz = [pPnlZ+L;. .,P,] we may write P = P1p2. Next, we define the (n + 1)-digit pattern p’ and the (n + 2)- digit pattern p” as PI = PLnLPz (4) P” = Pl~L7c2P2 (5) immaterial if only the pattern is-“bifix”-free. By a bifix we shall where nr and n2 are single letters from A,. understand a sequence which is both a prefix and a suffix of the Lemma 2: For any choice of nL E AL and 7c2 o A, the follow- pattern. A bifix-free pattern of arbitrary length greater than one ing implications are true: may always be obtained in the form aa-abb-b where a and b denote two distinct letters from the alphabet. Such patterns, however, generally have rather poor correlation properties which leave them unsuitable for use in noisy systems. Fortunately, the set S&) of L-ary bifix-free n-tuples is always rich, and for large values of n contain far more patterns than those mentioned previously. In fact, the search for patterns with outstanding correlation characteristics may frequently be restricted to en- compass only elements of S,(n). Since the search for long pat- terns may be very time consuming, even on large computers, significant savings may in some cases be achieved by generating a) p’ is bifix-free o p is bifix-free; b) p” is bifix-free => p is bifix-free. (Note that the second implication is undirectional.) If p is bifix-free, p” will have a bifix if and only if both of the following conditions are satisfied: i) bw21 = [nd~l ii) [pz,. . * ,Pn,nl = [P*,Z+L,‘. .,Pn-11, n 2 4 only. Proof: Let the bifix indicators of p’ and p” be hi’, i = 1,2,. . .,n, and hjb, i = 1,2,. . ., n + 1, respectively. From the constructions (4) and (5) we clearly have Manuscript received July 5! 1972; revised February 20, 1973. The author is with the Instttute for Circuit Theory and Telecommunica- hi’ = hi” = hi = 0, tion, Technical University of Denmark, Lyngby, Denmark. Isis;= [FJ < [f$J (6) CORRESPONDENCE 705 TABLE I SYSTEMATIC LISTINGOF ALL BIFIX-FREE BINARY 12-TUPLES (2 i n 5 6) n=2 n=3 ?l=4 n=.5 n=6 10 100 110 100000 1Olooo 101100 1100 11000 lloooo 11100 110100 lllooo 111100 1110 3 11010 110010 11110 111010 111110 Only half of the complete list is shown; the other half results from an interchange of O’s and 1’s. TABLE II L V, 2 0.267786 : 0.556979 0.687748 :, 0.760064 0.805577 7 0.836743 8 0.859378 and the first part of the lemma then follows immediately from Lemma 1. Furthermore, we note that because of (6) p” will have a bifix if and only if h&+ 1 = 1. The lemma states the necessary and sufficient condition for this to happen. Equipped with Lemma 2, we can now show how to generate in a systematic manner all bifix-free sequences of arbitrary length IZ. First, we list all L(L - 1) bifix-free patterns of length 2. This set forms the basis of a recursion in which each bifix-free pattern of even length n gives birth to L distinct bifix-free (n + I)-digit patterns as shown in (4), and to either L2 or L2 - 1 (depending on whether condition ii) of Lemma 2 is satisfied or not) distinct bifix-free patterns of length n + 2 in accordance with (5). The procedure is illustrated in Table I for binary patterns. If u, denotes the number of bifix-free n-tuples for a fixed alphabet size L, we immediately see that u"+l = Lu,, n even, n22 (7) but in order to get a similar expression for u,,, we must de- termine how many of the u, n-tuples branch into only L2 - 1 rather than L2 (n + 2)-tuples. To this end, consider the (42 + I)-dkitpatternp* = [P~,PZ,... -P,,~z-I,P,,~z,P,,~(~ 2 4) with bifix indicators hi*, i = 1,2,. . a, n/2. Supposing that p satisfies condition ii) of Lemma 2, we see that hi* = hi, i = 1,2, * . *, n/2, and we conclude from Lemma 1 that p is bifix-free if and only if p* is bifix-free. Hence the number of bifix-free patterns p which branch into only L2 - 1 (n + 2)-tuples must equal u,,,~+~, so that %+2 = L2u, - %/,,l, n even, n 2 4. (8) Theorem 1: The number, u,,, of n-digit bifix-free patterns satisfies u = L%-1 - &l/2, 12 even n Lu,- 1, n odd (9) with u,, = 1. The sequence vo,v1,v2,~ . . , where v, g L-Q,, is monoton- ically nonincreasing and it converges to a constant, urn, which satisfies 0, 2 1 - L-l - L-2 > 0. (10) Proof: The first part of the theorem follows immediately from a rearrangement of (7) and (8) along with a direct verifica- tion for the smallest values of n. As to the second, part, we rewrite (9) in terms of the v,,‘s, &l-l - L-“13j v, = n/23 n even (11) VIP-l> n odd where ve = 1. Assigning to n a fixed even value, say n = 2m, we obtain for m 2 1 V 2m = VZrn-1 - V2m-b1 = V2m v2m+2 = v2m+1 - = l&-~ - 2 t&-l - L-mu, L-(m+%,+l L-“v, - L-(m+%,+l v,(L--m + L-‘m+l’). (12) Generally, for any m 1 1 and q 2 0 v2m+2q = v2m-1 - j$o L-(m+j)vm+j 2 l&-1 - L-mu, i L-j j=0 _ L-“v L - L-q = v2m-1 mL-l L-iTt+l 2 t&-l - v, - . L-l (13) Setting m = 2, we have v2,,,-i = v,,, = 1 - L-l so that for any n 2 4 we have v, 2 (1 - L-1) (1 - &) = 1 - L-l - L-2. (14) The sequence vo,vl,v2,~ . . must converge since it has been shown to be nonincreasing and to satisfy the aforementioned inequality (14). Equation (10) then follows immediately. While (10) provides only a relatively coarse bound on v,, (13) may be utilized to evaluate v, with any desired degree of ac- curacy. Observe from (13) that L-m fJ2m - %a - L-l I v, I vznt (1% where the size of the interval within which v, is bounded vanishes exponentially with m. Hence only a few applications of the recursion (11) suffice to determine v, with high accuracy. Some numerical results are listed in Table II for alphabet sizes L I 8. Table II shows the tightest six-significant-digit lower bounds on the fraction of bifix-free L-ary sequences of arbi- trarily large length n. III. CONCLUSION In situations where a computerized search is needed to find a long unique word satisfying certain requirements, it will often be allowable to restrict the search to include only bifix-free patterns. From the results in Table II we note that the saving in computer time thus obtainable may be significant, particularly in the binary case where the set to be searched is then reduced by 706 IEEETRANSACTIONS ONINFORMATION THEORY,SEPTEMBER 1973 almost 75 percent. The algorithm presented in this correspondence for generating bifix-free patterns is well suited for computer programming. REFERENCES [l] R. H. Barker, “Group synchronization of binary digital systems,” in Communication Theory, W. Jackson, Ed. New York: Academic Press, 1953, p. 213. [2] M. W. Williard, “Optimum code patterns for PCM synchronization,” in Proc. Nat. T&m. Conf., session 5-5, May 1962. 131 A. Artom, “Choice of prefix in self-synchronizing codes,” IEEE Trans. Commun. (Corresp.), vol. COM-20, pp. 253-254, Apr. 1972. Bounds on Rate-Distortion Functions for Stationary Sources and Context-Dependent Fidelity Criteria BARRY M. LEINER AND ROBERT M. GRAY Abstract-A class of lower bounds to ratedistortion functions of stationary sources with context-dependent fidelity criteria is derived by mapping the source and distortion measure into an equivalent restricted- transition stationary source with a single-letter fidelity criterion, and then applying the composite bound. This approach is seen to yield bounds which, although sometimes quite loose, apply to general stationary sources and context-dependent fidelity criteria. Two examples are presented. Rate-distortion theory is usually restricted to single-letter fidelity criteria due to the complexity of more general cases. In many cases, however, the fidelity of the reproduction depends on the context of the message. To take context into account, Shannon [l] introduced local distortion measures as follows. Let the information source produce a sequence of letters {Xi} from an alphbeat Ax which is reproduced as the sequence {Xi} from an alphabet Ag. A local distortion measure of span g is then any function p9(xl,x2,. . a,.~,; R,,R,, . . .,$,) & p,&; f) of source and reproduction sequences of length g such that pg( .) 2 0. The distortion between n-tuples is then given by n-g+1 &,- * * 1 &k+g- 1 ). When g 2 2, the family of distortion measures {p,: n 2 g} is said to be context dependent. For simplicity assume the source is discrete; let Qx,(x) be the source probability mass function and define R,,(o) by where R,“(D) = inf Z(X; S) P,Eb, Manuscript received October 31, 1972 ; revised March 9, 1973. This work was supported in part by the National Science Foundation under Grant GK-31630 and by the Joint Services Program at Stanford Electronics Laboratories, Stanford, Calif., under U.S. Navy Contract N00014-67-A- 0112-0044. B. M. Leiner is with the Department of Electrical Engineering, Stanford University, Stanford, Calif., and GTE-Sylvania, Inc., Mountain View, Calif. R. M. Gray is with the Department of Electrical Engineering, Stanford University, Stanford, Calif. 94305. and Z(X; d) is the average mutual information between the source n-tuple and the output of the “test channel” p,, Z(X; 8) = E log Pn(2 I X) [ 1 4m and the sums are taken over A,” = {x: Qx,(x) > 0) and A?,’ the available reproducing alphabet. The rate-distortion function for the source X and context-dependent distortion measure generated by pg is defined by R,(D) = lim n-lJ&,(D). “-Pm The appropriate source coding theorem and converse have been proved when the source is stationary ergodic [l], [2], [5]. When the fidelity criterion is context dependent, the rate- distortion function R,(D) is extremely difficult to calculate in general Thus far the only calculation of a rate-distortion func- tion for a context-dependent fidelity criterion has been a bound for a class of distortion measures called modular distortion measures. This bound is tight in the case of an equiprobable memoryless source [3]. Berger [6] and Goblick [7] have sug- gested using a map of the original source to a source with an expanded alphabet and single-letter distortion measure (g = 1) to evaluate the rate-distortion function for context-dependent fidelity criteria. In what follows, the equivalence of these two rate-distortion functions is established and a lower bound is obtained using a recent result for single-letter distortion measures. This bound is quite general, since it applies to any finite local distortion measure and discrete-time stationary source. Unfor- tunately, however, the bound is often quite loose, particularly when the context distortion matrix is sparse. The equivalent source is described in terms of successive g-tuples of the source X. Define the source {Uk} by uk = (xkyxk+ 1, ’ ’ * ,Xk+g-l). Since each letter of the source U is a g-tuple, define uk = (uk,r,L&, ’ ’ ‘,uk,g). Clearly uk,i = uk-l,i+l; and U, and uk+ i share g - 1 components. A block U,, from the source U will have probability QX,+g-,(uI,1,“2,1,u3,1,’ * ‘rUn,19U”,2,’ ’ ‘d’h,g-l), = i if u~,~ = u~+~,~-~, k = 1,. ~0, n - 1, l<ilg; 0, otherwise. The equivalent source U has restricted transitions since it is not true that Qu21r,I(u2 ( ul) > 0 for all ul, u2 E Au, = Aq. Defin- ing the distortion measure on U in terms of the original X sequence mapped into U results in a single-letter distortion mea- sure, since =n -’ kgl Pg(Xk,Xk+l,“‘,Xk+g-1; ~kr”‘vRk+g-d where d(uk,fik) is the per-letter distortion measure on U induced by the span-g distortion measure defined on X. To calculate a rate-distortion function, it is necessary to specify the available 

A note on bifix-free sequences

http://orbit.dtu.dk/ws/files/4640005/Tolstrup.pdf

A note on bifix-free sequences

Abstract

Similar works

Full text

Available Versions

Online Research Database In Technology