Recursive construction of optimal frequency‐hopping sequence sets
2016; Institution of Engineering and Technology; Volume: 10; Issue: 9 Linguagem: Inglês
10.1049/iet-com.2015.0864
ISSN1751-8636
AutoresShanding Xu, Xiwang Cao, Guangkui Xu,
Tópico(s)Wireless Communication Networks Research
ResumoIET CommunicationsVolume 10, Issue 9 p. 1080-1086 Research ArticlesFree Access Recursive construction of optimal frequency-hopping sequence sets Shanding Xu, Corresponding Author Shanding Xu sdxzx11@163.com School of Mathematical Science, Nanjing University of Aeronautics and Astronautics, Nanjing, Jiangsu, People's Republic of China Department of Mathematics and Physics, Nanjing Institute of Technology, Nanjing, Jiangsu, People's Republic of ChinaSearch for more papers by this authorXiwang Cao, Xiwang Cao School of Mathematical Science, Nanjing University of Aeronautics and Astronautics, Nanjing, Jiangsu, People's Republic of China State Key Laboratory of Information Security, Institute of Information Engineering, Chinese Academy of Sciences, Beijing, People's Republic of ChinaSearch for more papers by this authorGuangkui Xu, Guangkui Xu School of Mathematical Science, Nanjing University of Aeronautics and Astronautics, Nanjing, Jiangsu, People's Republic of China School of Mathematical Science, Huainan Normal University, Huainan, Anhui, People's Republic of ChinaSearch for more papers by this author Shanding Xu, Corresponding Author Shanding Xu sdxzx11@163.com School of Mathematical Science, Nanjing University of Aeronautics and Astronautics, Nanjing, Jiangsu, People's Republic of China Department of Mathematics and Physics, Nanjing Institute of Technology, Nanjing, Jiangsu, People's Republic of ChinaSearch for more papers by this authorXiwang Cao, Xiwang Cao School of Mathematical Science, Nanjing University of Aeronautics and Astronautics, Nanjing, Jiangsu, People's Republic of China State Key Laboratory of Information Security, Institute of Information Engineering, Chinese Academy of Sciences, Beijing, People's Republic of ChinaSearch for more papers by this authorGuangkui Xu, Guangkui Xu School of Mathematical Science, Nanjing University of Aeronautics and Astronautics, Nanjing, Jiangsu, People's Republic of China School of Mathematical Science, Huainan Normal University, Huainan, Anhui, People's Republic of ChinaSearch for more papers by this author First published: 01 June 2016 https://doi.org/10.1049/iet-com.2015.0864Citations: 6AboutSectionsPDF ToolsRequest permissionExport citationAdd to favoritesTrack citation ShareShare Give accessShare full text accessShare full-text accessPlease review our Terms and Conditions of Use and check box below to share full-text version of article.I have read and accept the Wiley Online Library Terms and Conditions of UseShareable LinkUse the link below to share a full-text version of this article with your friends and colleagues. Learn more.Copy URL Share a linkShare onFacebookTwitterLinkedInRedditWechat Abstract In this study, first the authors present a simplified representation of the Peng–Fan bounds on the periodic Hamming correlation of frequency-hopping sequence (FHS) sets, which may also be used to check the optimality of an FHS set with respect to the Peng–Fan bounds. Second, they propose a recursive construction of FHS sets from the known ones using some injective functions and the Chinese remainder theorem. It generalises the previous construction of optimal FHSs and FHS sets with composite lengths employing a given function. Without the limit of the specific function, their construction can produce new optimal FHSs and FHS sets that cannot be produced by the earlier construction. By choosing appropriate injective functions and known optimal FHSs and FHS sets, infinitely many new optimal FHSs and FHS sets can be recursively obtained. 1 Introduction Frequency-hopping multiple access (FHMA) plays an important role in mobile communication, military radio communication, modern radar and sonar echolocation systems. Frequency-hopping sequence (FHS) is an integral part of FHMA systems in which each user is represented by a sequence of hopping frequencies. To discriminate their own signals from the others and reduce multiple access collisions by simultaneous transmission, generally we have to minimise the maximum of Hamming out-of-phase autocorrelation and cross-correlation of the set of FHSs. Furthermore, it is also preferred to have more FHSs to accommodate a great quantity of users. As a consequence, it is very desired to find FHSs with low Hamming correlation (HC), large family size, long period and small available frequencies simultaneously. However, these parameters are closely connected with each other as an inseparable whole. In fact, they are subjected to some theoretic bounds [1-4]. These bounds become standards of evaluating the performance of the sequences. An FHS (set) with corresponding HC meeting one of these bounds with equality is called optimal with respect to this bound. As far as the design of FHSs is concerned, it is of particular interest to construct optimal FHSs meeting these bounds. As a result, both algebraic and combinatorial constructions leading to FHSs reaching these bounds have been proposed in the literature [1, 3-27]. There are two measures on the periodic HC of an FHS set which are both important indicators for performance of the employed FHSs: one is the maximum periodic HC (MHC) [1, 2] and the other is the average periodic HC (AHC) [3]. The AHC among FHSs measures the average error performance of the FHMA systems, whereas the MHC represents their worst-case performance. For the above two correlation measures, traditionally, it is the MHC of FHSs that has received the most attention. During the past decades, numerous FHSs and FHS sets (see [1, 4-7, 9-13, 15-21, 23-25, 27], and references therein) with optimal MHC [1, 2] have been reported. Nevertheless, the study of the AHC of the sequences is paid little attention. Only a few constructions for FHS sets with optimal AHC [3] have been known in the literature [1, 3, 5, 8, 14, 18, 20, 22, 23, 26]. Furthermore, some previously known FHS sets with optimal MHC do not have good AHC properties, as summarised in Table 1. For more details, please see [18]. At present, there are merely several known constructions [1, 5, 18, 20, 23] for FHS sets having optimal MHC and optimal AHC concurrently. Thus, it is meaningful to construct as many FHS sets with both optimal MHC and optimal AHC simultaneously as possible. Table 1. Related known constructions of the optimal FHS sets References (L, M, λ; υ) Constraints Characters MHC AHC [1] (pn − 1, pk, pn−k; pk) 1 ≤ k ≤ n pn−k − 1 L(LM − υ) > aυ (LM − 1) optimal optimal [5, 23] (p2, p, p; p) — p υ|L but L ≠ υ optimal optimal [7] (p, e, f; e + 1) p = ef + 1 and f ≤ e f − 1 L(LM − υ) > aυ (LM − 1) optimal not optimal [9] gcd(2, n) = 1 L(LM − υ) < aυ (LM − 1) optimal not optimal [10, 12] (q − 1, e, f; e + 1) q = ef + 1 and f + 1 ≤ e f − 1 L(LM − υ) > aυ (LM − 1) optimal not optimal [11] e|(q − 1) and gcd(e, n) = 1 L(LM − υ) < aυ (LM − 1) optimal not optimal [17] 1 ≤ k ≤ n, e|(q − 1) and gcd(e, n) = 1 L(LM − υ) < aυ (LM − 1) optimal not optimal [18] (p, e, f + 1; e) p = ef + 1, e is odd or f is odd and f L(LM − υ) > aυ (LM − 1) optimal optimal [18] k > 1, k|(p1 − 1) and e1, …, er ≥ 1 k υ|L but L ≠ υ optimal optimal [18, 20] (p2 − p, p, p; p) – p − 1 υ|L but L ≠ υ near-optimal optimal [21] is not a prime, pi = efi + 1, 1 ≤ i ≤ r, e > 1, f1 > 1 e − 1 L(LM − υ) > aυ (LM − 1) optimal not optimal [25] e + 1) q = ef + 1 and k(f + 1) ≤ e kf − 1 L(LM − υ) > aυ (LM − 1) optimal not optimal The contribution of this paper is two-fold. First, we simplify the representation of the Peng–Fan bounds on the periodic HC of FHS sets, which makes it more convenient to check the optimality of an FHS set with respect to the Peng–Fan bounds, as shown in Table 1. Second, We give a recursive construction for FHS sets of period nL for two positive integers n and L with gcd(n, L) = 1 by means of some injective functions and the Chinese remainder theorem (CRT), which can help us to get more FHS sets with both optimal MHC and optimal AHC simultaneously, as summarised in Table 2. It can be viewed as a generalisation of Construction A in [24]. Though our construction does not give new parameters, it can generate infinitely many new optimal FHSs and FHS sets by choosing appropriate injective functions. Table 2. Parameters of some new FHS sets with optimal MHC and optimal AHC (L, M, λ; υ) Constraints MHC AHC Lempel–Greenberger (np, e, f + 1;ne) p = ef + 1, e is odd or f is odd and , p < p1 optimal optimal optimal (n(pm − 1), pk, pm−k; npk) 1 ≤ k ≤ m, gcd(n, pm − 1) = 1 and pm ≤ p1 optimal optimal optimal (np2, p, p; np) gcd(n, p) = 1 and p2 < p1 optimal optimal optimal (n(p2 − p), p, p; np) gcd(n, p2 − p) = 1 and p2 − p < p1 near-optimal optimal optimal The rest of this paper is organised as follows. In Section 2, we give some preliminaries that will be used in the sequel, and review some bounds on the MHC and AHC. In Section 3, we introduce a simplified representation of the Peng–Fan bounds, which is helpful for determining the optimality of an FHS set with respect to the Peng–Fan bounds. In Section 4, a new generic algebraic construction for FHS sets of period nL is proposed and the properties of these FHS sets are also analysed. Finally, Section 5 concludes this paper. 2 Preliminaries For convenience, we introduce the following notation in this paper: q is a power of an odd prime p; p1, …, pr are odd primes where p1 < · · · < pr; 〈x〉y is the least non-negative residue of x modulo y for two positive integers x and y; is the smallest integer ≥z; is the largest integer ≤z; is the residue class ring modulo m for a positive integer m; and is the set consisting of all elements in relatively prime to m. Let be an alphabet of υ available frequencies. A sequence X = {x0, x1, …, xL−1} is called an FHS of period L over if for 0 ≤ t < L. Given any two sequences X = {x0, x1, …, xL−1} and Y = {y0, y1, …, yL−1} of period L over , the periodic HC HX, Y is defined by where h[a, b] = 1 if a = b and 0 otherwise, and the addition operation in the subscript is performed modulo L. If X = Y, it is called the periodic Hamming autocorrelation of X, denoted by HX(τ) for short. The maximum periodic Hamming out-of-phase autocorrelation H(X) of X and the maximum periodic Hamming cross-correlation H(X, Y) for two distinct FHSs X and Y are defined, respectively, by Throughout this paper, let (L, υ, λ) denote an FHS X of period L over an alphabet of size υ with λ = H(X). In this case, we say that the sequence X has parameters (L, υ, λ). To estimate the measurement of a single FHS X, Lempel and Greenberger in 1974 established the first bound of H(X) as follows. Lemma 1 (The Lempel–Greenberger bound) [1].For any FHS X of period L over an alphabet of size υ, we have where b = 〈L〉υ. The Lempel–Greenberger bound can also be rewritten by the following lemma. Lemma 2 [6].For an (L, υ, λ) FHS X where . An (L, υ, λ) FHS X is called optimal if the Lempel–Greenberger bound in Lemma 1 or Lemma 2 is met with equality. Let be the set of M FHSs of period L over an alphabet of size υ, the MHC of is defined by Henceforth, we use (L, M, λ; υ) to denote an FHS set containing M FHSs of period L over an alphabet of size υ with , and we say that the set has parameters (L, M, λ; υ). In 2004, Peng and Fan developed the following bound on by incorporating the parameter M. Lemma 3 (The Peng–Fan bounds) [2].Let be an (L, M, λ; υ) FHS set. Define . Then (1)and (2) An (L, M, λ; υ) FHS set is called optimal with respect to the Peng–Fan bounds if one of the Peng–Fan bounds in Lemma 3 is met with equality. In this case, we say that has optimal MHC. Roughly speaking, an (L, M, λ; υ) FHS set is said to have near-optimal MHC if is bigger than the right-hand side of inequation (1) or inequation (2) by one. Another important performance indicator of the FHS set is the AHC defined as follows. Definition 1 [8].Let be the set of M FHSs of period L over an alphabet of size υ, we call The sum of all periodic Hamming out-of-phase autocorrelation and all periodic Hamming cross-correlation of , respectively, and call The average periodic Hamming out-of-phase autocorrelation and cross-correlation of , respectively. In 2008, Peng et al. established the following theoretical bound on the AHC of an FHS set. Lemma 4 (The Peng–Niu–Tang–Chen bound) [3].Let be the set of M FHSs of period L over an alphabet of size υ. Then (3) An FHS set is said to have optimal AHC if the pair satisfies (3) with equality. For each and FHS X = {x0, x1, …, xL−1} in , we define and Definition 2 [18].An FHS set is said to be uniformly distributed if for any . Lemma 5 [26].Let be the set of M FHSs of period L over an alphabet of size υ. Then, has optimal AHC if and only if is uniformly distributed. 3 Simplified representation of the Peng–Fan bounds To make it more convenient to determine the optimality of an FHS set with respect to the Peng–Fan bounds, we simplify the representation of the Peng–Fan bounds as follows. Theorem 1.For an (L, M, λ; υ) FHS set (4)and (5)where and . Proof.We only give the proof for inequation (4) since the proof for inequation (5) is similar to it. We assume that L = aυ + b, where 0 ≤ b < υ. Obviously, a ≥ 1. Therefore Let Then (6)Now we distinguish the following three cases to discuss (6).Case 1: b = 0, a = 1 and M = 1. Then (A − B)/C = −1. Therefore Case 2: b = 0, a ≠ 1 or M ≠ 1. Then −1 < (A − B)/C < 0. Hence Case 3: 0 < b < υ. Then 0 < A < C and 0 < B < C. Thus, |(A − B)/C| < 1. For this case, the discussion is divided into two subcases.Case 3.1: 0 < b < υ and 0 < (A − B)/C < 1. In this case, we have We next show the sufficient and necessary condition of 0 < (A − B)/C < 1. Note that (7)Clearly, if L(LM − υ) > aυ (LM − 1), then b ≠ 0. Therefore, 0 < b < υ and 0 < (A − B)/C < 1 if and only if L(LM − υ) > aυ (LM −1).Case 3.2: 0 < b < υ and −1 < (A − B)/C ≤ 0. In this case, we have With a slight modification of the proof of inequation (7), we can get that −1 < (A − B)/C ≤ 0 if and only if L(LM − υ) ≤ aυ (LM − 1). Hence the assertion is proved. Remark 1. As we have seen in Theorem 1, if M = 1 and L > υ, then L(L − υ) ≤ aυ (L − 1). Therefore, Lemma 2 is a special case of Theorem 1. If υ |L but L ≠ υ, then L = aυ and L(LM − υ) = aυ (aυ M − υ) ≤ aυ (aυ M − 1) = aυ (LM − 1). Hence λ ≥ a. It is exactly a special case of Theorem 1. An (L, M, λ; υ) FHS set is said to have optimal MHC if one of the Peng–Fan bounds in Theorem 1 is met with equality. Furthermore, an (L, M, λ; υ) FHS set is said to have near-optimal MHC if is bigger than the right-hand side of inequation (4) or inequation (5) by one. In Corollary 2.4 of [27], Bao and Ji gave a simplified version of the Peng–Fan bounds (2). However, they mentioned without a detailed proof that in Page 3, line 11. This argument is not true in general since is not always valid for any real numbers a and b with −1 < b ≤ 0. Now we list the related known constructions of the FHS sets with (near-) optimal MHC in Table 1, whose optimality can be judged from Theorem 1. 4 Recursive construction of FHS sets In this section, we will propose a recursive construction of the FHS sets based on the known optimal FHS sets. Before presenting our construction, we give a necessary lemma below. Lemma 6.Let be an FHS set over the alphabet with , 0 ≤ i < M. For any , let Then {Ak : 0 ≤ k < υ} is a partition of . Proof.This lemma is obvious, so we omit the proof. Using the CRT, it is easy to get that any integer x in can be uniquely represented as when m and n are relatively prime. By using the CRT and appropriate functions, we can extend an FHS set to another one as follows. Definition 3.Consider an (L, M, λ; υ) FHS set over , where . Let q = pe for an odd prime p and a positive integer e with gcd(p, L) = 1. For each k with 0 ≤ k < υ, the set Ak is defined as Lemma 6 and max 0≤k<υ{|Ak|} ≤ p − 1. For fixed positive integer a with gcd(a, p) = 1, if g is a function from into and is injective for every k with 0 ≤ k < υ, then an FHS set over with for 0 ≤ i < M is defined as where t0 = 〈t〉q, t1 = 〈t〉L. Theorem 2.For the FHS set in the above definition, we have H(Yi) = H(Xi) for each i with 0 ≤ i < M; H(Yi, Yj) = H(Xi, Xj) for any two distinct integers i and j with 0 ≤ i, j < M; ; and for every and every . Proof. By the CRT, for any , we can write τ = (τ0, τ1), where τ0 = 〈τ〉q and τ1 = 〈τ〉L. Then (8) Now we distinguish the following two cases to discuss the calculation of the periodic HC between Yi and Yj.Case 1: i = j, τ1 = 0 and τ0 ≠ 0. By (8), we have where we apply the fact that 〈aτ0g(j, t1)〉q ≠ 0 since 1 ≤ g(j, t1) ≤ p − 1 and gcd(a, p) = 1.Case 2:i ≠ j or τ1 ≠ 0. To compute , we only need to consider the case , which is equivalent to that there exists such that (i, t1) ∈ Ak and (j, 〈t1 + τ1〉L) ∈ Ak. If this is the case, note that is injective, then 1 ≤ g(i, t1) ≠ g(j, 〈t1 + τ1〉L) ≤ p − 1. It then follows that Thus, we have Summarising the results in all cases shows that (9)Then the assertion follows from (9): Applying (1), we arrive at the conclusion at once. For every and every , we have Since 1 ≤ g(i, t1) ≤ p − 1 and gcd(a, p) = 1. Corollary 1.By means of our method, Chung's Construction A in [24] is equivalent to the simple case: a = 1 and g(i, t1) = ηi(t1), where ηi is the function given by Proof.Clearly, the function g satisfies that is injective for every k with 0 ≤ k < υ. Corollary 2.Let be an (L, M, λ; υ) FHS set over , where . Let q = pe for an odd prime p and a positive integer e, where p satisfies gcd(p, L) = 1. For each k with 0 ≤ k < υ, the set Ak is defined as Lemma 6 and max 0≤k<υ{|Ak|} ≤ p − 1. For any and , define (10)For fixed , construct an FHS set over with for 0 ≤ i < M, defined as where t0 = 〈t〉q, t1 = 〈t〉L. Then the FHS set has parameters (qL, M, λ; qυ). Proof.Note that the number g(i, t1) in Corollary 2 is equal to the number of appearances of the symbol in the following Clearly, for any 0 ≤ i < M and any 0 ≤ t1 < L. Furthermore, we have that g(i, t1) = g(j, t2) if and only if i = j and t1 = t2, when . Corollary 3.Let be an (L, M, λ; υ) FHS set over , where . Let q = pe for an odd prime p and a positive integer e, where p satisfies LM < p. Then there exists an (qL, M, λ; qυ) FHS set over . Proof.Clearly, there exists the function g which carries injectively into since LM < p. Theorem 3.Consider an (L, M, λ; υ) FHS set over , where . For a positive integer r, let for 1 ≤ i ≤ r, where p1, …, pr are odd primes with p1 < · · · < pr and e1, …, er are positive integers. For each k with 0 ≤ k < υ, the set Ak is defined as Lemma 6 and max 0≤k<υ{|Ak|} ≤ p1 − 1. Assume that n = q1, …, qr satisfies gcd(n, L) = 1. Then there exists an (nL, M, λ; nυ) FHS set as over . Proof.Note that p1 < · · · < pr and max 0≤k aυ (nLM − 1), where ; and has optimal AHC if and only if has optimal AHC. Proof. Owing to , we get (1) by Lemma 2 and Theorem 2. The conclusion follows from and Theorem 1. Applying Lemma 5 and Theorem 2, we arrive at the conclusion. □ Remark 2.On the basis of the known FHS sets with both optimal MHC and optimal AHC in [1, 5, 18, 20, 23], some new optimal FHS sets obtained by recursively applying Definition 3 are listed in Table 2. Here, p, p1, …, pr are odd primes with p1 < · · · < pr, . Since Definition 3 can be recursively applied to any existing FHS sets, it is expected that there exist some more classes of optimal FHS sets which can be obtained from our construction but are not listed there. Finally, we use the following example to elucidate the construction in Section 4. Example 1.Let be the (9, 3, 3; 3) FHS set over given by Theorem 2 in [5], where π(·) and g(·) are an identity mapping and a zero mapping on , respectively. Then If g(i, t1) is defined as Corollary 2, 0 ≤ i ≤ 2, 0 ≤ t1 ≤ 8, then Let a = 1 and q = p = 11 in Corollary 2. Then the FHSs of the new FHS set are The periodic Hamming out-of-phase autocorrelation of Yi for 0 ≤ i ≤ 2 is For 0 ≤ i, j ≤ 2 with i ≠ j, the periodic Hamming cross-correlation Yi and Yj is The MHC of the FHS set is 3; the average periodic Hamming out-of-phase autocorrelation and cross-correlation are 2.6939 and 2.6667, respectively. Obviously, the FHS set has optimal MHC and optimal AHC. Moreover, each Xi is an optimal (9, 3, 3) FHS. According to Theorem 2, each Yi in is an optimal (99, 33, 3) FHS and is an (99, 3, 3; 33) FHS set with optimal MHC and optimal AHC. In a similar way, we can obtain infinitely many optimal FHS sets. 5 Conclusion In this paper, we simplify the representation of the Peng–Fan bounds, which makes it more convenient to check the optimality of FHS sets with respect to the Peng–Fan bounds. Furthermore, we give a recursive construction for FHS sets by means of some injective functions and the CRT. It not only includes the aforementioned Construction A in [24] as a special case but also gives more new optimal FHSs and FHS sets due to the choice of the injective functions. In a similar manner, we can also extend Construction B and combination of Constructions A and B in [24] to much more general cases. 6 Acknowledgments The authors are grateful to the anonymous referees for their helpful comments and suggestions. 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