Nielsen, Peter Tolstrup

An expression is obtained for the expected duration of a search to find a givenL-ary sequence in a semi-infinite stream of randomL-ary data. The search time is found to be an increasing function of the lengths of the "bifices" of the pattern, where the term bifix denotes a sequence which is both a prefix and a suffix

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Downloaded from orbit.dtu.dk on: Dec 17, 2017On the expected duration of a search for a fixed pattern in random dataNielsen, 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). On the expected duration of a search for a fixed pattern in random data. I E E ETransactions on Information Theory, 19(5), 702-704.702 IEEE TRANSACTIONS ON INFORMATION THEORY, SEPTEMBER 1973 example, for a length-3 channel the pulse response is 1 + 20 + D2. The constant K2 for these channels and thus for all possible channels is overbounded by = 2l(l - 1). (13) IV., CONCLUSIONS An estimate of an upper bound on performance has been developed for the VA which is useful for estimating performance over channels for which only the pulse response energy and duration are known. time division multiplex), only one of which has been furnished with a recurrent synchronization pattern. The problem also relates to several other applications of unique words, for in- stance in digital burst communication, multiaccess systems, and various kinds of message switching. It is entirely possible to adapt Gilbert’s results to this some- what different problem and hence obtain a fairly complete statistical description of the search. The results, however, are generally intractable and useful only for numerical calculations. In spite of this, we shall see in the next section that the expected duration of a search depends on the structure of the pattern in a very simple manner, and we shall derive a closed-form expression REFERENCES for this expectation. [l] A. J. Viterbi, “Error bounds for convolutional codes and an asymp- r totically optimum decoding algorithm,” IEEE Trans. Inform. Theory II. THE SEARCH vol. IT-13, pp. 260-269, Apr. 1967. [2] G. D. Fomey, Jr., “Lower bounds on error probability in the presence of large intersymbol interference,” Let AL, L 2 2, denote an alphabet of L letters, and consider vol. COM-20, pp. 7677, Feb. 1972. IEEE Trans. Commun. (Corresp.), a semi-intinite sequence [3] -, “Maximum-likelihood sequence estimation of digital sequences in the presence of intersymbol interference,” IEEE Trans. Znform. d = [d1,d2,d,,. . . (1) Theory, vol. IT-l& pp. 363-378, May 1972. [4] F. R. Magee, Jr., and J. G. Proakis, “Adaptive maximum-likelihood sequence estimation for digital signahng in the presence of intersymbol of digits, where each digit di is selected independently and has interference,” IEEE Trans. Inform. Theory (Corresp.), vol. IT-19, pp. 120-124, Jan. 1973. probability L-l of being any of the L letters in A,. We shall call [S] A. J. Viterbi, “Convolutional codes and their performance in com- da random data stream. (Later we shall occasionally assume for munications systems,” IEEE Trans. Commun. Technol., vol. COM-19, pp. 7X-772, Oct. 1971. convenience that d is preceded by an initial digit do.) Finite data [6] C. R. Wyhe, Jr., Advanced Engineering Mathematics. New York: Strings Will be Written McGraw-Hill, 1966,. ch. 11. [7j M. E. Austin, “Decision feedback equalization for digital communica- tion over dispersive channels,” M.I.T. Lincoln Lab., Lexington, Mass., d [iJ> = Ldt + 1,. . * ,d,- 1 I (2) Tech. Rep. 437, Aug. 1967. for any i 2 0, where of course j 2 i + 1. Furthermore, let P = rPI,P2,***~P”l (3) On the Expected Duration of a Search for a Fixed Pattern be a fixed pattern of n digits each chosen from AL. Since we are concerned with the statistics of the first occur- in Random Data rence of the pattern p in the data stream d, we define the random P. TOLSTRUP NIELSEN variable x such that x = i if and only if Absfrucr-An expression is obtained for the expected duration of a search to 6nd a given L-ary sequence in a semi-infinite stream of random d[i,i+ n) =p d[j,j + 4 Z P, llj<i. (4) L-ary data. The search time is found to be an increasing function of the lengths of the “bifices” of the pattern, where the term bifix denotes a Hence x is just the number of positions examined before the sequence which is both a prefix and a suffix. pattern is found. As we shall see, the expectation E {x} is com- pletely determined by the lengths of the “bifices” of p which are I. INTR~DuC~~N defined as follows. The most common method for obtaining or maintaining word Definition: The sequence a = [a1,a2,. . .,a,,,] (1 I m < n) is a or frame synchronization in a digital communication system bifix ofp = [p1,p2,. . . ,pn] if utilizes the insertion of a fixed pattern at regular intervals into [PI,P29’ * * ,P,l = [Pn-?n+I,P”-m+2,‘~ -9Pnl = a (5) the transmitted sequence. Gilbert [l] (see also [2]) analyzed the constraints to be imposed on the data in applications where it is imperative to prevent the pattern from appearing between its intended locations. Usually, however, one does not care to constrain the data since the regularity with which the true pattern occurs makes it easily distinguishable from any spurious patterns; but in that event, Gilbert’s results may be used to calculate the probability of not seeing the pattern between two successive insertions. In this correspondence, we shall be concerned with the search for a specified pattern in a semi-infinite random sequence which has not been punctuated by any periodic insertions. This situa- i.e., if a is both the m-digit prefix and the m-digit suffix of p. We further define the bifix indicators hi of p, 1 5 i < n, in a manner such that hi = 1 if [p1,p2;. .,pl] is a bifix of p and hl = 0 otherwise. If hi = 0, 1 I i < n, we say that p is bifix- free. By way of convention we define ho = h, = 1. Example I: Consider A, = {O,l } and the six-digit pattern p = [l,O,l,l,O,l]. [l] and [l,O,l] are the only bifices of p, so that ho = h, = h3 = h, = 1 and h2 = h4 = h, = 0. The following conditional expectation will play a crucial role in what follows. Lemma I: For any n-digit pattern p, 1 I n < CO, tion may arise during acquisition of frame synchronization in E{x 1 d [O,n) = p} = L”. (6) systems where the synchronizer has the additional task of dis- criminating between a number of signals (possibly received in Proof: We define the indicator random variables zi, 1 I i < co,as Manuscript received July 5, 1972; revised February 20, 1973. The author is with the Institute of Circuit Theory and Telecommunica- tion, Technical University of Denmark, Lyngby, Denmark. z* = 1, ifd[i, i + n) = p (7) otherwise CORRESPONDENCE and for arbitrary N consider the sample mean W ith no constraints on the data stream, we have E{zi} = Pr {zl = 1) = Pr {d[i, i + n) = p} = L-” (9) so that from (8) E{s,} = L-“. (10) But from (7), it follows that Zi and Zj are statistically indepen- dent if Ii - jl 2  n. Moreover, we have always lZiZj[ I 1  SO that it follows from (8) that I $(2n - 1) + (L-“)2 (11) and hence that var {So> 5 $2n - 1). (12) Since the variance of s, vanishes as N approaches infinity, it follows that for any E > 0 and any 6 > 0 Pr L”-e<L<L”+c 1 I 21-a (13) SN for all suitably large N. But SN is just the fraction of the N positions 1,2, . . . ,N which contain the initial digit of length-n subsequences where p has been observed to occur. Thus sN-i is just the observed average distance between occurrences of p and (13) is merely the law of large numbers for this distance. The lemma then follows immediately. Although, as we have seen in the previous proof of Lemma 1, the expected distance between occurrences of the pattern p depends only on the length of the pattern; the expected distance x to the jirst occurrence of the pattern depends also on the structure of the pattern. It is somewhat surprising that, as we now proceed to show, the conditional expectation of x that was easily found by indicator random variable arguments can be utilized to obtain the desired unconditional expectation of x. We  shall obtain E(x) by an inductive argument, the basis of which will be as follows. Lemma 2: If the n-digit pattern p is bifix-free, then E(x) = 1 + L!’ - n. (14) Proof: Suppose that d [O,n) = p. Then d [i,n) # [pl,. . ., P,-~] for 1 I i < n since d [i,n) is the (n - i)-digit suffix of p and by hypothesis p has no bifices. Thus E{x 1 d[O,n) = p} = n - 1  + E(x) (15) since the search for the next occurrence of p when d [O,n) = p is equivalent to the search for the first occurrence of p in the random data stream [d,,d,+ l,dn+2,. . . . The lemma now follows from (15) and Lemma 1. 703 Theorem I: For an n-digit pattern p (n 2 l), with bifix indicators h,,h,,. . .,h,-l, E{x} = i h,,? - n  I=0 06) where, by convention, ho = h, = 1. Proof: We  proceed by induction on the number k of non- zero bifix indicators hi, 0  < i < n. Note that (16) reduces to (14) for k = 0, i.e., for a bifix-free pattern, so a basis has been established for the induction. Suppose that k > 0 and that (16) holds for all patterns with k - 1  or fewer nonzero bifix indicators. Let p1,p2,. . . ,pk be the bifices of p in order of increasing length. Since pk of length n’ < n is both a prefix and a suffix of p, as are also the shorter sequences PlrP2, ’ . ’ #k- 1, it follows that the latter sequences are all bifices of pk and the only bifices of pk. Letting x’ be the expected search duration to find pk in random data, we have then by the induction hypothesis E{x’} = $ h,L’ - n’. i=O Since we may consider the search for p as a search for& followed by a search for the entire pattern p (which latter search may terminate immediately since pk is a prefix of p), we may write E(x) = E{x’} + (E{x 1 d[l, n’ + 1) = pk} - 1). (18) We  next observe that E{x~d[O,n)=p}=E{x~d[l,n’+l)=pk}+n-n’-1 (19) since with the conditioning d [O,n) = p it is impossible for the pattern to begin in any position i, 1  I i < n - n’. This impos- sibility follows from the fact that otherwise d [i,n) would be both a prefix and a suffix of p of length n” = n - i which, since n’ < n” < n, would contradict the fact that n’ is the length of the longest bifix ofp. Equation (19) simply states that the search for p in the data stream d with the condition d [O,n) = p con- sists of n - n’ - 1 steps in which the search cannot possibly terminate, followed by a search for p in a random data stream in which the first n’ digits are known to equal pk. Invoking Lemma 1, we obtain from (19) E{x 1 d[l, n’ + 1) = pk} = 1 + C - n + n’ (20) and substitution of (17) and (20) into (18) then gives E{x} = f h,L’ + L” - n = e h,L’ - n (21) i=O i=o which establishes the theorem. Corollary: For any n-digit pattern p, the expected search to find p in random data satisfies L” - n + 1 5 E{x} < L”+’ - 1 -n L-l (22) with equality on the left if and only if p  is bifix-free and equality on the right if and only if p  consists of n repetitions of the same digit of AL. Inequality (22) follows immediately from (16) upon noting that hi = 0 for 0 < i < n if and only if p  is bifix-free and hi = 1 for 0 < i < n if and only if p  is a constant sequence. We  note also that the upper and lower bounds in (22) differ by a factor of only (slightly less than) two for the important case of a binary alphabet (L = 2) and that their ratio approaches 1 We  are now in a position to prove the main result of this correspondence. with increasing alphabet size. 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) 

On the expected duration of a search for a fixed pattern in random data

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On the expected duration of a search for a fixed pattern in random data

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