THE PEAK SIDELOBE LEVEL OF RANDOM BINARY SEQUENCES KAI-UWE SCHMIDT. n u 1. j=0

Transcription

1 THE PEAK SIDELOBE LEVEL OF RANDOM BINARY SEQUENCES KAI-UWE SCHMIDT Abstract. Let A = (a 0, a,..., a ) be draw uiformly at radom from {, +} ad defie u M(A ) = max 0<u< a ja j+u for >. j=0 It is proved that M(A )/ log coverges i probability to 2. This settles a problem first studied by Moo ad Moser i the 960s ad proves i the affirmative a recet cojecture due to Alo, Litsy, ad Shput. It is also show that the expectatio of M(A )/ log teds to 2.. Itroductio Cosider a biary sequece A = (a 0, a,..., a ) of legth, amely a elemet of {, +}. Defie the aperiodic autocorrelatio at shift u of A to be C u (A) = u j=0 ad defie the peak sidelobe level of A as a j a j+u for u {0,,..., } M(A) = max 0<u< C u(a) for >. Biary sequeces with small autocorrelatio at ozero shifts have a wide rage of applicatios i digital commuicatios, icludig sychroisatio ad radar (see [6], for example). Let µ() be the miimum of M(A) take over all 2 biary sequeces A of legth. By a parity argumet, it is see that µ() ad it is kow that µ() = for {2, 3, 4, 5, 7,, 3} (biary sequeces attaiig the miimum are ofte called Barker sequeces). It is a classical problem to decide whether µ() > for all > 3. Although deep methods have bee developed [4], [3], this problem is still ope; the curretly smallest udecided case arises for > [8]. It is cojectured that µ() grows Date: 2 February 20 (revised 2 December 203). 200 Mathematics Subject Classificatio. Primary: 05D40; Secodary: 94A55, 60F0. The author was supported by Germa Research Foudatio uder Research Fellowship SCHM 2609/-.

2 2 KAI-UWE SCHMIDT as, perhaps like. We refer to Tury [5] ad Jedwab [7] for excellet surveys o this problem. I this paper, we will be cocered with the asymptotic behaviour, as, of M(A) for almost all biary sequeces A of legth. This problem was first studied by Moo ad Moser [2]. Let A be a radom biary sequece of legth, by which we mea that A is draw uiformly at radom from {, +}. I other words, each of the sequece elemets of A takes o each of the values ad + idepedetly with probability /2. Util ow, the best kow bouds are (.) lim Pr [ ɛ < M(A ) log < 2 + ɛ ] = for all ɛ > 0. The upper boud is due to Mercer []. I fact, Mercer proved a weaker result but poited out i a fial remark [, p. 670] that his proof establishes the above upper boud. The lower boud was proved by Alo, Litsy, ad Shput [2], i respose to umerical evidece provided by Dmitriev ad Jedwab [4]. The authors of [2] also cojectured that the lower boud ca be improved to 2 ɛ. The aim of this paper is to prove this cojecture ad therefore to establish the limit distributio, as, of M(A )/ log. I particular, we prove the followig. Theorem.. Let A be a radom biary sequece of legth. The, as, M(A ) 2 i probability log ad E [ M(A ) ] log 2. Alo, Litsy, ad Shput [2] already observed that, as a cosequece of McDiarmid s iequality (Lemma 3.), M(A ) is cocetrated aroud its expected value, but could oly show that (.2) lim if E [ M(A ) ] log. Their proof cosiders C u (A ) oly for u /2 ad crucially relies o the fact that C u (A ) ad C v (A ) are idepedet wheever /2 u < v <. Our method cosiders C u (A ) also for u < /2. I particular, by a careful estimatio of the momets of C u (A )C v (A ) for 0 < u < v <, we will show that the lower boud (.2) ca be improved to 2, which together with (.) establishes the secod part of Theorem.. The first part of Theorem. the follows from McDiarmid s iequality. As poited out i [2], give a biary sequece A = (a 0, a,..., a ) of legth, the quatity M(A) is related to the more geeral rth-order correlatio measure S r (A), which was defied by Mauduit ad Sárközy [9]

3 to be S r (A) := max 0 u <u 2 < <u r< RANDOM BINARY SEQUENCES 3 max 0 k u r k a j+u a j+u2 a j+ur for r. Alo, Kohayakawa, Mauduit, Moreira, ad Rödl [] established that, give a radom biary sequece A of legth, the for all r 2, [ lim Pr 2 5 < S r(a ) log ( ) < ] 3 + ɛ = for all ɛ > 0. r Sice, for every biary sequece A, we have M(A) S 2 (A), Theorem. implies that for r = 2 the lower boud ca be improved from 2/5 to ɛ. j=0 2. Prelimiary Results The mai results of this sectio are the followig. Give a radom biary sequece A of legth, Propositio 2.2 gives a lower boud for (2.) Pr [ C u (A ) 2 log ] for small u. This result ca also be cocluded from [2]. However, the proof preseted here is cosiderably simpler ad more direct. Propositio 2.7 gives a upper boud for (2.2) Pr [ C u (A ) 2 log C v (A ) 2 log ] for 0 < u < v <. These bouds will be the crucial igrediets to prove the mai result of this paper. 2.. To boud (2.), we shall eed the followig refiemet of the cetral limit theorem. Lemma 2. (Cramér [3, Thm. 2]). Let X 0, X,... be idetically distributed mutually idepedet radom variables satisfyig E[X 0 ] = 0 ad E[X0 2] = ad suppose that there exists T > 0 such that E[e tx 0 ] < for all t < T. Write Y k = X 0 + X + + X k ad let Φ be the distributio fuctio of a ormal radom variable with zero mea ad uit variace. If θ k > ad θ k /k /6 0 as k, the Pr [ ] Y k θ k k. 2Φ( θ k ) Propositio 2.2. Let A be a radom biary sequece of legth > 2 ad let u be a iteger satisfyig u log. The for all sufficietly large. Pr [ C u (A ) 2 log ] 5 log

4 4 KAI-UWE SCHMIDT Proof. Write A = (a 0, a,..., a ). It is well kow that the u products a 0 a u, a a +u,..., a u a are mutually idepedet. A proof of this fact was give by Mercer [, Prop..]. Hece C u (A ) is a sum of u mutually idepedet radom variables, each takig each of the values ad + with probability /2. Notice that E[e ta 0a u ] = cosh(t) ad, settig 2 log ξ = u, we fid that ξ /( u) /6 0 sice u Lemma 2. to coclude, as, log. (2.3) Pr [ C u (A ) 2 log ] 2Φ( ξ ), We ca therefore apply where Φ is the distributio fuctio of a stadard ormal radom variable. It is well kow (see [5, Thm..2.3], for example) that ( ) 2π z z 2 e z2 /2 Φ( z) e z2 /2 for z > 0, 2π z so that, sice u, as, 2Φ( ξ ) e π log Usig u log, we coclude u log. e u log e log log log e as. It the follows from (2.3) that for all α > e π ad all sufficietly large we have Pr [ C u (A ) 2 log ] α log. The lemma follows sice 5 > e π We ow tur to the derivatio of a upper boud for (2.2). It will be coveiet to defie the otio of a eve tuple as follows. Defiitio 2.3. A tuple (x, x 2,..., x 2m ) is eve if there exists a permutatio σ of {, 2,..., 2m} such that x σ(2i ) = x σ(2i) for each i {, 2,..., m}. For example, (, 3,, 4, 3, 4) is eve, while (2,,, 2,, 3) is ot eve. I the ext two lemmas we will prove two results about eve tuples, which we the use to estimate momets of C u (A )C v (A ). Recall that, for positive iteger k, the double factorial (2k )!! = (2k)! = (2k )(2k 3) 3 k! 2k is the umber of ways to arrage 2k objects ito k uordered pairs.

5 RANDOM BINARY SEQUENCES 5 Lemma 2.4. Let m ad q be positive itegers ad let R be the set of eve tuples i { (x, x 2,..., x 2q ) : x i Z, 0 x i < m }. The R (2q )!! m q. Proof. There are (2q )!! ways to arrage x, x 2,..., x 2q ito q uordered pairs ad to each of these q pairs we assig a value of {0,,..., m }. I this way we costruct all elemets of R at least oce, which proves the lemma. Lemma 2.5. Let u, v, ad be itegers satisfyig 0 < u, v < ad u v. Write I = {, 2,..., 2q} ad let t be a iteger satisfyig 0 t < q. Let S be the subset of { (xi, x i + u, y i, y i + v) i I : x i, y i Z, 0 x i < u, 0 y i < v } cotaiig all eve elemets (x i, x i + u, y i, y i + v) i I such that (x i ) i J is ot eve for all (2q 2t)-elemet subsets J of I. The S (8q )!! 2q (t+)/3. Proof. We will costruct a set of tuples that cotais S as a subset. Arrage the 8q variables (2.4) x, x + u,..., x 2q, x 2q + u, y, y + v,..., y 2q, y 2q + v ito 4q uordered pairs (a, b ), (a 2, b 2 ),..., (a 4q, b 4q ) such that there are at most q t pairs (x i, x j ). This ca be doe i at most (8q )!! ways. We formally set a i = b i for all i {, 2,..., 4q}. If this assigmet does ot yield a cotradictio, the we call the arragemet of (2.4) ito 4q pairs cosistet. For example, if there are pairs of the form (x i, y j ) ad (x i + u, y j + v), the the arragemet is ot cosistet sice u v by assumptio. Now, for every cosistet arragemet, pairs of the form (x i, x j ) or (y i, y j ) determie the value of aother pair (amely, (x i +u, x j +u) or (y i +v, y j +v), respectively). O the other had, for every cosistet arragemet, pairs ot of the form (x i, x j ), (y i, y j ), (x i + u, x j + u), or (y i + v, y j + v) determie the value of at least two other pairs. For example, if there exists the pair (x i, y j ), the x i +u ad y j +v must lie i differet pairs. Therefore, sice there are at most q t pairs of the form (x i, x j ) ad at most q pairs of the form (y i, y j ), for each cosistet arragemet, at most 2 (4q 2t 2) + 3 (2t + 2) = 2q 3 (t + ) of the variables x,..., x 2q, y,..., y 2q ca be chose idepedetly. We assig to each of these a value of {0,,..., }. I this way, we costruct a set of at most (8q )!! 2q (t+)/3 tuples that cotais S as a subset, as required.

6 6 KAI-UWE SCHMIDT We ow use Lemmas 2.4 ad 2.5 to boud momets of C u (A )C v (A ). Lemma 2.6. Let p ad h be itegers satisfyig 0 h < p ad let A be a radom biary sequece of legth. The, for 0 < u < v <, [ (Cu E (A )C v (A ) ) ] 2p 2p[ (2p )!! ] ) 2( + (8p)8h /3 + (8p)4p (h+)/3. Proof. Write I = {, 2,..., 2p} ad let T be the set cotaiig all eve tuples of { (xi, x i + u, y i, y i + v) i I : x i, y i Z, 0 x i < u, 0 y i < v }. Writig A = (a 0, a,..., a ), we have [ (Cu E (A )C v (A ) ) ] 2p [ ( u ) 2p ( v ) ] 2p = E a i a i+u a j a j+v = u i=0 i,...,i 2p =0 v j,...,j 2p =0 j=0 E [ ] a i a i +u a i2p a i2p +ua j a j +v a j2p a j2p +v (2.5) = T sice a 0, a,..., a are mutually idepedet, E[a j ] = 0, ad a 2 j = for all j {0,,..., }. We defie the followig subsets of T. () T cotais all elemets (x i, x i +u, y i, y i +v) i I of T such that (x i ) i I ad (y i ) i I are eve. (2) T 2 cotais all elemets (x i, x i +u, y i, y i +v) i I of T such that (x i ) i I or (y i ) i I is ot eve ad (x i ) i J ad (y i ) i K are eve for some (2p 2h)-elemet subsets J ad K of I. (3) T 3 cotais all elemets (x i, x i +u, y i, y i +v) i I of T such that either (x i ) i J is ot eve for all (2p 2h)-elemet subsets J of I or (y i ) i K is ot eve for all (2p 2h)-elemet subsets K of I. It is immediate that T, T 2, ad T 3 partitio T, so that (2.6) T = T + T 2 + T 3. We ow boud the cardialities of T, T 2, ad T 3. The set T. Usig Lemma 2.4, we have the crude estimate (2.7) T [ (2p )!! ] 2 2p. The set T 2. Let (x i, x i + u, y i, y i + v) i I be a elemet of T 2. The there exist (2p 2h)-elemet subsets J ad K of I such that (x i ) i J ad (y i ) i K are eve ad (2.8) (x i ) i I\J or (y i ) i I\K

7 RANDOM BINARY SEQUENCES 7 is ot eve. Sice (x i ) i J ad (y i ) i K are eve, (x i, x i + u, y j, y j + v) i J, j K is eve. Sice (x i, x i + u, y i, y i + v) i I is also eve, it follows that (2.9) (x i, x i + u, y j, y j + v) i I\J, j I\K is eve as well. There are ( ) ( 2p 2h subsets J ad 2p 2h) subsets K. By Lemma 2.4, for each such J ad K, there are at most (2p 2h )!! p h eve tuples (x i ) i J satisfyig 0 x i < for each i J ad at most (2p 2h )!! p h eve tuples (y i ) i K satisfyig 0 y i < for each i K. By Lemma 2.5 applied with t = 0 ad by iterchagig u ad v ad (x i ) i I\J ad (y i ) i I\K if ecessary, the umber of eve tuples i {0,,..., } 8h of the form (2.9) such that oe of the tuples i (2.8) is ot eve is at most (8h )!! 2h /3. Therefore, [( ] 2p 2 T 2 2 2h /3 (8h )!! )(2p 2h )!! p h 2h (2.0) 2p /3[ (2p )!! ] 2 (8p) 8h, usig very crude bouds. The set T 3. By Lemma 2.5 applied with t = h ad by iterchagig u ad v ad (x i ) i I ad (y i ) i I if ecessary, (2.) T 3 2 2p (h+)/3 (8p )!! 2p (h+)/3 (8p) 4p. Now the lemma follows by combiig (2.5), (2.6), (2.7), (2.0), ad (2.). Lemma 2.6 is ow used to prove the desired upper boud for (2.2). Propositio 2.7. Let A be a radom biary sequece of legth ad write λ = 2 log. The, for 0 < u < v < ad all sufficietly large, Pr [ C u (A ) λ C v (A ) λ ] Proof. Let (X, X 2 ) be a radom vector takig values i R R ad let p be a positive iteger. The by Markov s iequality, for θ, θ 2 > 0, Pr [ X θ X 2 θ 2 ] E [ (X X 2 ) 2p] (θ θ 2 ) 2p. Let h be a arbitrary iteger satisfyig 0 h < p. Applicatio of Lemma 2.6 gives (2.2) Pr [ C u (A ) λ C v (A ) λ ) [(2p )!!] 2 (2 log ) 2p [ + K (, p, h) + K 2 (, p, h) ],

8 8 KAI-UWE SCHMIDT where K (, p, h) = /3 (8p) 8h ad K 2 (, p, h) = (h+)/3 (8p) 4p. We apply (2.2) with p = log ad h = 7 log log, so that for all sufficietly large we have h < p, as assumed. By Stirlig s approximatio 2πk k k e k k! 3πk k k e k, we have We also have ad [(2p )!! ] 2 (2 log ) 2p 3p2p e 2p 3e2 (log ) 2p 2. K (, p, h) K (, log, 7 log log ) = 36(log log )(log 8+log log ) 3 log = O( /4 ) as K 2 (, p, h) K 2 (, log, 6 log log ) = 3 +4 log log log = O( log log ) as. Substitute ito (2.2) to obtai the claimed result, usig 3e 2 < Proof of Mai Theorem We require the followig result, which is a cosequece of Azuma s iequality for martigales. Lemma 3. (McDiarmid [0]). Let X 0, X,..., X be mutually idepedet radom variables takig values i a set S. Let f : S R be a measurable fuctio ad suppose that f satisfies f(x) f(y) c wheever x ad y differ oly i oe coordiate. Defie the radom variable Y = f(x 0, X,..., X ). The, for θ 0, Pr [ Y E[Y ] θ ] 2e 2θ2 c 2. Give a radom biary sequece A = (a 0, a,..., a ) of legth, we will apply Lemma 3. with X j = a j for j {0,,..., } ad u f(x 0, x,..., x ) = max x j x j+u, 0<u< so that M(A ) = f(a 0, a,..., a ). We ca take c = 4 i Lemma 3. ad obtai the followig corollary. j=0

9 RANDOM BINARY SEQUENCES 9 Corollary 3.2. Let A be a radom biary sequece of legth. The, for θ 0, Pr [ M(A ) E[M(A )] θ ] 2e θ2 8. We ow prove the secod part of Theorem.. Theorem 3.3. Let A be a radom biary sequece of legth. The, as, E [ M(A ) ] log 2. Proof. By the triagle iequality ad the uio boud we have, for all ɛ > 0, [ [ E M(A ) ] Pr ] 2 > ɛ log [ [ E M(A ) ] Pr M(A ] [ ) > log log 2 ɛ M(A ) + Pr 2 > log 2 ]. ɛ By Corollary 3.2 ad the upper boud of (.), the two terms o the righthad side ted to zero as, hece E [ M(A ) ] (3.) lim sup 2. log Let δ > 0 ad defie the set (3.2) N(δ) = { > : E [ M(A ) ] log < } 2 δ. We claim that the size of N(δ) is fiite for all choices of δ, which together with (3.) will prove the theorem. The proof of the claim is based o a idea developed i [2]. Let > 2 ad write W = ad λ = 2 log. The { u Z : u } log Pr [ ] [ M(A ) λ Pr max C ] u(a ) λ u W Pr [ ] C u (A ) λ Pr [ ] C u (A ) λ C v (A ) λ u W u,v W u<v by the Boferroi iequality. By Propositios 2.2 ad 2.7, (3.3) Pr [ ] W 2 M(A ) λ W 5(log ) (log ) 3 2 (log ) 2 0(log )

10 0 KAI-UWE SCHMIDT log for all sufficietly large, usig 2 3 log W for > 2. Now, by the defiitio (3.2) of N(δ), for all N(δ) we have λ > E[M(A )], so that we ca apply Corollary 3.2 with θ = λ E[M(A )] to give, for all N(δ), Pr [ ] M(A ) λ 2e 8 (λ E[M(A)])2. Compariso with (3.3) yields, for all sufficietly large N(δ), which implies 0(log ) 3 2 E [ M(A ) ] log 2 2e 8 (λ E[M(A)])2, 2 log log + 8 log 20. log From the defiitio (3.2) of N(δ) it the follows that N(δ) has fiite size for all δ > 0, as required. Usig Corollary 3.2, it is ow straightforward to prove the first part of Theorem.. Corollary 3.4. Let A be a radom biary sequece of legth. The, as, M(A ) 2 i probability. log Proof. By the triagle iequality ad the uio boud we have, for all ɛ > 0, [ M(A ) Pr ] 2 log > ɛ [ M(A ) Pr E [ M(A ) ] ] [ > log log 2 ɛ E [ M(A ) ] + Pr ] 2 log > 2 ɛ. By Corollary 3.2 ad Theorem 3.3, the two terms o the right-had side ted to zero as, which proves the corollary. Ackowledgemet I would like to thak Joatha Jedwab for may valuable discussios ad Joatha Jedwab ad Daiel J. Katz for their careful commets o this paper. Refereces [] N. Alo, Y. Kohayakawa, C. Mauduit, C. G. Moreira, ad V. Rödl, Measures of pseudoradomess: Typical values, Proc. Lod. Math. Soc 95 (2007), o. 3, [2] N. Alo, S. Litsy, ad A. Shput, Typical peak sidelobe level of biary sequeces, IEEE Tras. Iform. Theory 56 (200), o., [3] H. Cramér, Sur u ouveau théorème-limite de la théorie des probabilités, Actualités Sci. Idust. 736 (938), 5 23.

11 RANDOM BINARY SEQUENCES [4] D. Dmitriev ad J. Jedwab, Bouds o the growth rate of the peak sidelobe level of biary sequeces, Adv. Math. Commu. (2007), [5] R. Durrett, Probability: Theory ad examples, 4th ed., Cambridge Uiversity Press, 200. [6] S. W. Golomb ad G. Gog, Sigal desig for good correlatio: For wireless commuicatio, cryptography, ad radar, Cambridge Uiversity Press, Cambridge, [7] J. Jedwab, What ca be used istead of a Barker sequece?, Cotemp. Math. 46 (2008), [8] K. H. Leug ad B. Schmidt, New restrictios o possible orders of circulat Hadamard matrices, Des. Codes Cryptogr. 64 (202), o. -2, [9] C. Mauduit ad A. Sárközy, O fiite pseudoradom biary sequeces I: Measure of pseudoradomess, the Legedre symbol, Acta Arith. 82 (997), o. 4, [0] C. McDiarmid, O the method of bouded differeces, Surveys i Combiatorics (J. Siemos, ed.), Lodo Math. Soc. Lectures Notes Ser. 4, Cambridge Uiv. Press, Cambridge, 989, pp [] I. D. Mercer, Autocorrelatios of radom biary sequeces, Combi. Probab. Comput. 5 (2006), o. 5, [2] J. W. Moo ad L. Moser, O the correlatio fuctio of radom biary sequeces, SIAM J. Appl. Math. 6 (968), o. 2, [3] B. Schmidt, Cyclotomic itegers ad fiite geometry, J. Amer. Math. Soc. 2 (999), o. 4, [4] R. J. Tury, Character sums ad differece sets, Pacific J. Math. 5 (965), o., [5], Sequeces with small correlatio, Error Correctig Codes (Hery B. Ma, ed.), Wiley, New York, 968. Departmet of Mathematics, Simo Fraser Uiversity, 8888 Uiversity Drive, Buraby BC V5A S6, Caada. Curret address: Faculty of Mathematics, Otto-vo-Guericke Uiversity, Uiversitätsplatz 2, 3906 Magdeburg, Germay. address: kaiuwe.schmidt@ovgu.de