New Construction of Quaternary Low Correlation Zone ... - IEEE Xplore
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JOURNAL OF COMMUNICATIONS AND NETWORKS, VOL. 12, NO. 4, AUGUST 2010
New Construction of Quaternary Low Correlation Zone Sequence Sets from Binary Low Correlation Zone Sequence Sets Ji-Woong Jang, Sang-Hyo Kim, and Jong-Seon No Abstract: In this paper, using binary (N, M, L, ) low correlation zone (LCZ) sequence sets, we construct new quaternary LCZ sequence sets with parameters (2N, 2M, L, 2). Binary LCZ sequences for the construction should have period N ≡ 3 mod 4, L|N , and the balance property. The proposed method corresponds to a quaternary extension of the extended construction of binary LCZ sequence sets proposed by Kim, Jang, No, and Chung [1]. Index Terms: Correlation, low correlation zone (LCZ), pseudorandom, quasi-synchronous code division multiple access (QSCDMA), sequences.
of the construction, we construct quaternary LCZ sequence sets with parameters (2N, 2M, L, 2) from binary (N, M, L, ) LCZ sequence sets, where the binary LCZ sequences have period N ≡ 3 mod 4, L|N , and balance property. Using an optimal binary LCZ sequence set with parameters (N, M, L, 1), we can construct a quaternary LCZ sequence set with parameters (2N, 2M, L, 2), which is optimal with respect to Tang-FanMatsufuji bound [7]. II. PRELIMINARIES Let a(t) and b(t) be q-ary sequences of period N . Then their correlation function is defined as
I. INTRODUCTION In microcellular networks such as femtocell networks, the delay among the signals of multiple users can be maintained within a few chips. Quasi-synchronous code division multiple access (QS-CDMA) system is the system devised for such environment [2]. In order to suppress the inter-user interference from quasisynchronized signals, low correlation zone (LCZ) sequences have been used as signature sequences in the QS-CDMA systems [2]–[4]. Let S be a set of M sequences of period N . If the magnitude of correlation function between any two sequences in S takes the values less than or equal to within the range |τ | < L for the offset τ , then the sequence set is called LCZ sequence set with parameters (N, M, L, ). LCZ sequences sets were first constructed using GMW sequences for binary case [3] and for p-ary case [5]. Kim, Jang, No, and Chung introduced a construction method of quaternary LCZ sequences [6] from binary sequences with ideal autocorrelation property and the construction yields the first optimal LCZ sequence set with respect to Tang-Fan-Matsufuji bound [7]. Jang, No, Chung, and Tang also constructed an optimal pary LCZ sequence set [8]. In [1], Kim, Jang, No, and Chung proposed several design methods of LCZ sequence sets by manipulating sequences of the same alphabet. Using a similar mapping to the binary case Manuscript received March 10, 2009; approved for publication by Emanuele Viterbo, Division I Editor, November 27, 2009. Initial version of this work was presented at ISITA 2008. This paper was supported by Faculty Research Fund, Sungkyunkwan University, 2009. J.-W. Jang is with the LG Electronics, Gasan-dong, Geumcheon-gu 219-3, Seoul, Korea, email: [email protected]. S.-H. Kim is the corresponding author. He is with the School of Information and Communication Engineering, Sungkyunkwan University, Suwon, Gyeonggi-do 440-746, Korea, email: [email protected]. J.-S. No is with the Department of Electrical Engineering and Computer Science, Seoul National University, Seoul 151-742, Korea, email: [email protected].
Ra,b (τ ) =
N −1
ωqa(t)−b(t+τ )
t=0
where ωq is the complex primitive qth root of unity. The Ra,b (τ ) is called the autocorrelation function of a(t) if a(t) = b(t) and the cross-correlation function between a(t) and b(t), otherwise. Let N be a positive integer such that N = 3 mod 4 and P = 2N . Let ZP be the set of integers modulo P , i.e., ZP = {0, 1, · · ·, P − 1}. Let ai (t) be a binary sequence of period N with balance property. Let Dui be the characteristic set of ai (t − u), i.e., Dui = {t | ai (t − u) = 1, 0 ≤ t ≤ N − 1} = D0i + u where u ∈ ZN , D0i + u = {d + u | d ∈ D0i }, and “+” denotes addition modulo N . Binary sequences are said to be balanced if the occurrences of one in a period is the same as or once more than those of zero. From the balance of ai (t), it is clear that |Dui | =
N +1 N −1 i , |Du | = 2 2
i
where Du = ZN \Dui . Let u and v be positive integers and σ be the correlation value between ai (t − u) and ak (t − v) such that |σ| ≤ except for the inphase autocorrelation. Then it is easy to check
c 2010 KICS 1229-2370/10/$10.00
N +σ 1 + 4 2 N −σ k i |Du ∩ Dv | = 4 N −σ i k |Du ∩ Dv | = 4 N +σ 1 i k − . |Du ∩ Dv | = 4 2 |Dui ∩ Dvk | =
(1)
JANG et al.: NEW CONSTRUCTION OF QUATERNARY LOW CORRELATION ZONE...
If u = v and i = k, it is straightforward to check from the balance property that |Dui ∩ Dvk | = k
N +1 2
|Dui ∩ Dv | = 0 i
|Du ∩ Dvk | = 0 N −1 i k . |Du ∩ Dv | = 2 ∼ By the Chinese remainder theorem, we can represent ZP = Z2 ⊗ZN under the isomorphism φ : ζ −→ (ζ mod 2, ζ mod N ), where ⊗ denotes the direct product. For convenience, we use the notation ζ ∈ ZP interchangeably with (ζ mod 2, ζ mod N ) throughout the paper. III. CONSTRUCTION OF NEW QUATERNARY LCZ SEQUENCE SETS Using a binary (N, M, L, ) LCZ sequence set with period N ≡ 3 mod 4 and the balance property, we construct new quaternary LCZ sequence sets with parameters (2N, 2M, L, 2). Let L be a set of binary LCZ sequences with parameters (N, M, L, ) and balance property, where N ≡ 3 mod 4 and L|N . Furthermore, we assume that the correlation function B Rij (τ ) between any two binary sequences ai (t) and aj (t) in L has the absolute value smaller or equal to except for τ ≡ 0 mod L and τ = 0. Let Di be the characteristic set of the LCZ sequence ai (t) in L. Then quaternary LCZ sequence sets can be constructed as follows. Theorem 1: Let U1 be the set of M quaternary sequences of period 2N defined by ⎧ i ⎪ 0, if t ∈ {0} ⊗ D0 ⎪ ⎪ ⎪ i ⎨ 1, if t ∈ {1} ⊗ D0 + L u1,i (t) = i ⎪ ⎪ ⎪2, if t ∈ {0} ⊗ D0 ⎪ ⎩3, if t ∈ {1} ⊗ Di + L 0 and U2 be the set of M sequences of period 2N defined by ⎧ i ⎪ 0, if t ∈ {0} ⊗ D0 ⎪ ⎪ ⎪ ⎨1, if t ∈ {1} ⊗ Di + L 0 u2,i (t) = i ⎪ 2, if t ∈ {0} ⊗ D ⎪ 0 ⎪ ⎪ ⎩3, if t ∈ {1} ⊗ Di + L 0
for 0 ≤ i ≤ M − 1. We define the quaternary sequence set Q of period 2N and the cardinality 2M as for 0 ≤ i ≤ M − 1 u1,i (t), si (t) = u2,i−M (t), for M ≤ i ≤ 2M − 1. Then the quaternary sequence set Q = U1 ∪ U2 is a quaternary LCZ sequence set with parameters (2N, 2M, L, 2). Proof: Clearly, Q has 2M sequences and the period of the sequences is 2N . Therefore, it remains to show that the magnitude of the correlation value between any two sequences in Q is
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less than or equal to 2 except for the in-phase autocorrelation. Using D0i + L, for 0 ≤ i ≤ M − 1 i A0 = iM D0 + L, for M ≤ i ≤ 2M − 1 we simplify the constrcution of si (t) as ⎧ iM ⎪ 0, if t ∈ {0} ⊗ D0 ⎪ ⎪ ⎪ i ⎨ 1, if t ∈ {1} ⊗ A0 si (t) = iM ⎪ ⎪ ⎪2, if t ∈ {0} ⊗ D0 ⎪ ⎩3, if t ∈ {1} ⊗ Ai 0 where iM = i mod M . 2 Let τ = (τ1 , τ2 ), where τ1 ∈ Z2 and τ2 ∈ ZN . Then the correlation between si (t) and sk (t), 0 ≤ i, k ≤ 2M − 1 can be computed as Ri,k (τ ) =
2N −1 t=0
s (t)−sk (t+τ )
ω4 i
=
2N −1 t=0
s (t−τ )−sk (t)
ω4 i
iM kM i k = |{τ1 }⊗Dτ2 ∩{0}⊗D0 |+|{1 + τ1 }⊗Aτ2 ∩{1}⊗A0 | kM i k ∩{0}⊗D |+|{1 + τ }⊗A ∩{1}⊗A | + |{τ1 }⊗DτiM 1 τ2 0 0 2 iM i kM k + ω4 |{τ1 }⊗Dτ2 ∩{1}⊗A0 |+|{1 + τ1 }⊗Aτ2 ∩{0}⊗D0 | k kM i ∩{1}⊗A |+|{1 + τ }⊗A ∩{0}⊗D | + ω4 |{τ1 }⊗DτiM 1 0 τ 0 2 2 iM i kM k − |{τ1 }⊗Dτ2 ∩{0}⊗D0 |+|{1 + τ1 }⊗Aτ2 ∩{1}⊗A0 | kM k i ∩{0}⊗D |+|{1 + τ }⊗A ∩{1}⊗A | − |{τ1 }⊗DτiM 1 0 0 τ 2 2 iM k i − ω4 |{τ1 }⊗Dτ2 ∩{1}⊗A0 |+|{1 + τ1 }⊗Aτ2 ∩{0}⊗D0kM | kM k i ∩{1}⊗A |+|{1 + τ }⊗A ∩{0}⊗D | − ω4 {τ1 }⊗DτiM 1 0 0 τ 2 2 where ω4 is the fourth complex primitive root of unity, i.e., j = √ −1. If τ1 = 0, that is, τ ≡ 0 mod 2, then Ri,k (τ ) can be simplified as Ri,k (τ ) = i kM i k M kM i k ∩D |+|A ∩A | |Dτ2 ∩D0 |+|Aτ2 ∩A0 |+|DτiM τ 0 0 2 2 i i kM k M kM k iM − |Dτ2 ∩D0 |+|Aτ2 ∩A0 |+|Dτ2 ∩D0 |+|Aiτ2 ∩A0 | . In the case of τ1 = 1, that is, τ ≡ 1 mod 2, we have Ri,k (τ ) = i i kM k M ω4 |Dτ2 ∩Ak0 |+|Aτ2 ∩D0 |+|DτiM ∩A0 |+|Aiτ2 ∩D0kM | 2 i k i kM M k i ∩A |+|A ∩D | . − ω4 |Dτ2 ∩A0 |+|Aτ2 ∩D0kM |+|DτiM 0 0 τ2 2 Case 1) 0 ≤ i, k ≤ M − 1 ( i.e., si (t), sk (t) ∈ U1 ): In this case, Ai = Di + L and Ak = Dk + L and we should consider the following two sub-cases. i) τ ≡ 0 mod 2 (τ1 = 0) except for inphase autocorrelation;
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Table 1. List of binary LCZ sequence sets.
Long, Zhang, and Hu [3] Jang, No, Chung, and Tang [8] Tang and Udaya [9]
N 2 −1 2n − 1 2n − 1 n
In this sub-case, Ri,k (τ ) can be rewritten as i
k
i
k
+ |Dτi 2 ∩ D0k | + |(Dτi 2 + L) ∩ (D0k + L)| i
− |Dτ2 ∩ D0k | − |(Dτ2 + L) ∩ (D0k + L)| − |Dτi 2 ∩
k D0 |
− |(Dτi 2 + L) ∩
k (D 0
+ L)|.
B (−τ2 ) = σ1 . For τ2 ≡ 0 mod L, then |σ1 | ≤ and we Let Ri,k have
1 N + σ1 + 4 2 N − σ1 k k ∩ D0 | = |(Dτi 2 + L) ∩ (D0 + L)| = 4 N − σ1 i ∩ D0k | = |(D τ2 + L) ∩ (D0k + L)| = 4 1 N + σ1 k i k − . ∩ D0 | = |(D τ2 + L) ∩ (D 0 + L)| = 4 2
|Dτi 2 ∩ D0k | = |(Dτi 2 + L) ∩ (D0k + L)| = |Dτi 2 i
|Dτ2 i |Dτ2
Thus, Ri,k (τ ) reduces to 2σ1 and |Ri,k (τ )| = |2σ1 | ≤ 2 for 0 < |τ | < L. When τ = 0 and i = k, Ri,k (τ ) becomes the sum of two correlation functions at τ2 = 0 between ai (t) and ak (t), and ai (t + L) and ak (t + L). From the property of binary LCZ sequence set with parameters (N, M, L, ) [1], clearly |Ri,k (0)| ≤ 2. ii) τ ≡ 1 mod 2 (τ1 = 1); In this sub-case, Ri,k (τ ) can be rewritten as i i k Ri,k (τ ) = ω4 |Dτ2 ∩ (D0k + L)| + |(Dτ2 + L) ∩ D0 | k + ω4 |Dτi 2 ∩ (D0 + L)| + |(Dτi 2 + L) ∩ D0k | i k i − ω4 |Dτ2 ∩ (D0 + L)| + |(Dτ2 + L) ∩ D0k | k − ω4 |Dτi 2 ∩ (D0k + L)| + |(Dτi 2 + L) ∩ D0 | . B B Let Ri,k (L − τ2 ) = σ2 and Ri,k (−L − τ2 ) = σ3 . Applying the correlation values in (1), the correlation function can be computed as
Ri,k (τ ) = ω4 {−σ2 + σ3 } . Clearly, |Ri,k (τ )| ≤ 2 for τ ≡ 1 mod 2 and 0 < |τ | < L. Case 2) 0 ≤ i ≤ M − 1 and M ≤ k ≤ 2M − 1: kM
L (2 − 1)/(2m − 1) (2n − 1)/(2m − 1) (2n − 1)/(2m − 1)
1 1 1
n
In this sub-case, Ri,k (τ ) can be rewritten as
Ri,k (τ ) = |Dτ2 ∩ D0 | + |(Dτ2 + L) ∩ (D0 + L)| i
M < 2m − 1 2m − 1 m 2 −m−1
In this case, Ai0 = D0i + L and Ak0 = D0 + L. Similarly to case 1), we should consider the following two sub-cases. i) τ ≡ 0 mod 2 (τ1 = 0);
i
kM
i
Ri,k (τ ) = |Dτ2 ∩ D0 | + |(D τ2 + L) ∩ (D0kM + L)| kM
+ |Dτi 2 ∩ D0kM | + |(Dτi 2 + L) ∩ (D0 i
i
kM
− |Dτ2 ∩ D0kM | − |(Dτ2 + L) ∩ (D0 − |Dτi 2 ∩
kM D0 |
+ L)| + L)|
− |(Dτi 2 + L) ∩ (D0kM + L)|.
It is not difficult to prove that we have |(Dτi 2 + L) ∩ (D0kM + L)| = |Dτi 2 ∩ D0kM | kM
|(Dτi 2 + L) ∩ (D0 i
kM
+ L)| = |Dτi 2 ∩ D0 | i
|(Dτ2 + L) ∩ (D0kM + L)| = |Dτ2 ∩ D0kM | i
kM
|(Dτ2 + L) ∩ (D0
i
kM
+ L)| = |Dτ2 ∩ D0 |
and thus |Ri,k (τ )| = 0 for 0 ≤ |τ | < L. ii) τ ≡ 1 mod 2 (τ1 = 1); In this sub-case, Ri,k (τ ) can be rewritten as i kM i kM Ri,k (τ ) = ω4 |Dτ2 ∩ (D0 + L)| + |(Dτ2 + L) ∩ D0 | +ω4 |Dτi 2 ∩ (D0kM + L)| + |(Dτi 2 + L) ∩ D0kM | i i −ω4 |Dτ2 ∩ (D0kM + L)| + |(D τ2 + L) ∩ D0kM | kM kM −ω4 |Dτi 2 ∩ (D0 + L)| + |(Dτi 2 + L) ∩ D0 | . B B (L − τ2 ) = σ5 and Ri,k (−L − τ2 ) = σ6 . For Let Ri,k M M τ2 = 0 mod L, we have |σ5 | ≤ and |σ6 | ≤ . Applying the relationship (1) to these correlation functions, we find
Ri,k (τ ) = ω4 {σ5 + σ6 } , whose absolute value is smaller or equal to 2 for τ ≡ 1 mod 2 and 0 < |τ | < L. Case 3) M ≤ i, k ≤ 2M − 1: iM kM In this case, Ai0 = D0 + L and Ak0 = D0 + L. Similarly to case 1), it can be proved that |Ri,k (τ )| ≤ 2 for 0 ≤ |τ | < L. From case 1)–case 3), the sequence set Q is an LCZ sequence set with parameters (2N, 2M, L, 2). 2 Theorem 1 says that we can construct quaternary LCZ sequences sets with parameters (2N, 2M, L, 2) given that an optimal or suboptimal (N, M, L, 1) LCZ sequences set exists. If the employed binary LCZ sequence set with parameters (N, M, L, 1) is optimal, it is proven straightforwardly that the quaternary LCZ sequence set with parameters (2N, 2M, L, 2) defined in Theorem 1 is optimal with respect to Tang-FanMatsufuji bound [7] in the sense that its cardinality is the largest for the given length 2N and the correlation constraint 2.
JANG et al.: NEW CONSTRUCTION OF QUATERNARY LOW CORRELATION ZONE...
Binary LCZ sequences sets applicable to Theorem 1 are listed in Table 1. Note that an optimal binary LCZ sequence set introduced by Jang, No, Chung, and Tang [8] yields the construction of an optimal quaternary LCZ sequence set. There is an example of newly constructed quaternary LCZ sequence sets, even though it is not an optimal one. Example 2: Let a(t) be a binary Legendre sequence of period 31 constructed from quadratic residue in Z31 given by
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[7] [8] [9]
X. Tang, P. Fan, and S. Matsufuji, “Lower bounds on correlation of spreading sequence set with low or zero correlation zone,” Electron. Lett., vol. 36, no. 6, pp. 551–552, 2000. J.-W. Jang, J.-S. No, H. Chung, and X. Tang, “New sets of optimal p-ary low-correlation zone sequences,” IEEE Trans. Inf. Theory, vol. 53, no. 2, pp. 815–821, 2007. X. Tang and P. Udaya, “New construction of low correlation zone sequences from Hadamard matrices,” in Proc. IEEE ISIT, Adelaide, Austrailia, Sept. 4–9, 2005, pp. 482–486.
a(t) = (1110110111100010101110000100100). Let L be a binary LCZ sequence set with parameters (N, M, L, ) = (1023, 16, 33, 1) constructed using the Legendre sequence and the trace function from F210 to F25 by Theorem 8 in [8]. Since N = 1023 ≡ 3 mod 4, we can construct a quaternary LCZ sequence set Q with parameters (2N, 2M, L, 2) = (2046, 32, 33, 2) using the binary LCZ sequence set. IV. CONCLUDING REMARKS In this paper, we propose a simple construction of quaternary LCZ sequence sets with parameters (2N, 2M, L, 2) from binary LCZ sequence sets with parameters (N, M, L, ). Any binary LCZ sequence sets with low correlation value = 1 can be hired to derive new quaternary LCZ sequence sets with low correlation value = 2. The constructed quaternary LCZ sequence set becomes optimal with respect to Tang-Fan-Matsufuji bound [7] if the binary LCZ sequence set employed is optimal. Therefore, the proposed construction provides optimal quaternary LCZ sequence sets from the optimal binary LCZ sequence sets given in [8].
communication systems.
Ji-Woong Jang was born in 1976. He received the B.S., M.S., and Ph.D. degrees in Electrical Engineering and Computer Science from Seoul National University, Seoul, Korea, in 2000, 2002, and 2006, respectively. After Ph.D., he was a Senior Engineer at Samsung Electronics until June 2008. He was a postdoc. at UCSD from Aug. 2008 to July 2009. From Sept. 2009, he is a Senior Engineer at LG Electronics. His research interests includes pseudo-noise (PN) sequences, difference sets, cryptography, error correcting codes, cooperative communications and wireless
Sang-Hyo Kim received his B.S., M.S., and Ph.D. degrees in Electrical Engineering from Seoul National University, Seoul, Korea in 1998, 2000, and 2004, respectively. From 2004 to 2006, he was a senior engineer at Samsung Electronics. He visited University of Southern California as a visiting scholar from 2006 to 2007. In 2007, He joined the School of Information and Communication Engineering, Sungkyunkwan University, Suwon, Korea where he serves currently as an Assistant Professor. His research interests include error correcting codes, pseudo random sequences, cooperative communications, distributed source coding, etc.
REFERENCES [1] [2]
[3] [4]
[5] [6]
Y.-S. Kim, J.-W. Jang, J.-S. No, and H. Chung, “New design of lowcorrelation zone sequence sets,” IEEE Trans. Inf. Theory, vol. 52, no. 10, pp. 4607–4616, 2006. R. De Gaudenzi, C. Elia, and R. Viola, “Bandlimited quasi-synchronous CDMA: A novel satellite access technique for mobile and personal communication systems,” IEEE J. Sel. Areas Commun., vol. 10, no. 2, pp. 328–343, 1992. B. Long, P. Zhang, and J. Hu, “A generalized QS-CDMA system and the design of new spreading codes,” IEEE Trans. Veh. Technol., vol. 47, no. 4, pp. 1268–1275, 1998. J. S. Cha, S. Kameda, M. Yokoyama, H. Nakase, K. Masu, and K. Tsubouchi, “New binary sequences with zero-correlation duration for approximately synchronised CDMA,” Electron. Lett., vol. 36, no. 11, pp. 991–993, May 2000. X. Tang and P. Fan, “A class of pseudonoise sequences over GF (p) with low correlation zone,” IEEE Trans. Inf. Theory, vol. 47, no. 4, pp. 1644– 1649, 2001. S.-H. Kim, J.-W. Jang, J.-S. No, and H. Chung, “New constructions of quaternary low correlation zone sequences,” IEEE Trans. Inf. Theory, vol. 51, no. 4, pp. 1469–1477, 2005.
Jong-Seon No received the B.S. and M.S.E.E. degrees in Electronics Engineering from Seoul National University, Seoul, Korea, in 1981 and 1984, respectively, and the Ph.D. degree in Electrical Engineering from the University of Southern California, Los Angeles, in 1988. He was a Senior MTS with Hughes Network Systems, Germantown, MD, from February 1988 to July 1990. He was an Associate Professor with the Department of Electronic Engineering, Konkuk University, Seoul, from September 1990 to July 1999. He joined the Faculty of the Department of Electrical Engineering and Computer Science, Seoul National University, in August 1999, where he is currently a Professor. He served as a General Co-Chair for the IEICE ISITA 2006 and IEEE ISIT 2009. His research interests include error-correcting codes, sequences, cryptography, space-time codes, LDPC codes, network coding, compressed sensing, and wireless communication systems.
JOURNAL OF COMMUNICATIONS AND NETWORKS, VOL. 12, NO. 4, AUGUST 2010
New Construction of Quaternary Low Correlation Zone Sequence Sets from Binary Low Correlation Zone Sequence Sets Ji-Woong Jang, Sang-Hyo Kim, and Jong-Seon No Abstract: In this paper, using binary (N, M, L, ) low correlation zone (LCZ) sequence sets, we construct new quaternary LCZ sequence sets with parameters (2N, 2M, L, 2). Binary LCZ sequences for the construction should have period N ≡ 3 mod 4, L|N , and the balance property. The proposed method corresponds to a quaternary extension of the extended construction of binary LCZ sequence sets proposed by Kim, Jang, No, and Chung [1]. Index Terms: Correlation, low correlation zone (LCZ), pseudorandom, quasi-synchronous code division multiple access (QSCDMA), sequences.
of the construction, we construct quaternary LCZ sequence sets with parameters (2N, 2M, L, 2) from binary (N, M, L, ) LCZ sequence sets, where the binary LCZ sequences have period N ≡ 3 mod 4, L|N , and balance property. Using an optimal binary LCZ sequence set with parameters (N, M, L, 1), we can construct a quaternary LCZ sequence set with parameters (2N, 2M, L, 2), which is optimal with respect to Tang-FanMatsufuji bound [7]. II. PRELIMINARIES Let a(t) and b(t) be q-ary sequences of period N . Then their correlation function is defined as
I. INTRODUCTION In microcellular networks such as femtocell networks, the delay among the signals of multiple users can be maintained within a few chips. Quasi-synchronous code division multiple access (QS-CDMA) system is the system devised for such environment [2]. In order to suppress the inter-user interference from quasisynchronized signals, low correlation zone (LCZ) sequences have been used as signature sequences in the QS-CDMA systems [2]–[4]. Let S be a set of M sequences of period N . If the magnitude of correlation function between any two sequences in S takes the values less than or equal to within the range |τ | < L for the offset τ , then the sequence set is called LCZ sequence set with parameters (N, M, L, ). LCZ sequences sets were first constructed using GMW sequences for binary case [3] and for p-ary case [5]. Kim, Jang, No, and Chung introduced a construction method of quaternary LCZ sequences [6] from binary sequences with ideal autocorrelation property and the construction yields the first optimal LCZ sequence set with respect to Tang-Fan-Matsufuji bound [7]. Jang, No, Chung, and Tang also constructed an optimal pary LCZ sequence set [8]. In [1], Kim, Jang, No, and Chung proposed several design methods of LCZ sequence sets by manipulating sequences of the same alphabet. Using a similar mapping to the binary case Manuscript received March 10, 2009; approved for publication by Emanuele Viterbo, Division I Editor, November 27, 2009. Initial version of this work was presented at ISITA 2008. This paper was supported by Faculty Research Fund, Sungkyunkwan University, 2009. J.-W. Jang is with the LG Electronics, Gasan-dong, Geumcheon-gu 219-3, Seoul, Korea, email: [email protected]. S.-H. Kim is the corresponding author. He is with the School of Information and Communication Engineering, Sungkyunkwan University, Suwon, Gyeonggi-do 440-746, Korea, email: [email protected]. J.-S. No is with the Department of Electrical Engineering and Computer Science, Seoul National University, Seoul 151-742, Korea, email: [email protected].
Ra,b (τ ) =
N −1
ωqa(t)−b(t+τ )
t=0
where ωq is the complex primitive qth root of unity. The Ra,b (τ ) is called the autocorrelation function of a(t) if a(t) = b(t) and the cross-correlation function between a(t) and b(t), otherwise. Let N be a positive integer such that N = 3 mod 4 and P = 2N . Let ZP be the set of integers modulo P , i.e., ZP = {0, 1, · · ·, P − 1}. Let ai (t) be a binary sequence of period N with balance property. Let Dui be the characteristic set of ai (t − u), i.e., Dui = {t | ai (t − u) = 1, 0 ≤ t ≤ N − 1} = D0i + u where u ∈ ZN , D0i + u = {d + u | d ∈ D0i }, and “+” denotes addition modulo N . Binary sequences are said to be balanced if the occurrences of one in a period is the same as or once more than those of zero. From the balance of ai (t), it is clear that |Dui | =
N +1 N −1 i , |Du | = 2 2
i
where Du = ZN \Dui . Let u and v be positive integers and σ be the correlation value between ai (t − u) and ak (t − v) such that |σ| ≤ except for the inphase autocorrelation. Then it is easy to check
c 2010 KICS 1229-2370/10/$10.00
N +σ 1 + 4 2 N −σ k i |Du ∩ Dv | = 4 N −σ i k |Du ∩ Dv | = 4 N +σ 1 i k − . |Du ∩ Dv | = 4 2 |Dui ∩ Dvk | =
(1)
JANG et al.: NEW CONSTRUCTION OF QUATERNARY LOW CORRELATION ZONE...
If u = v and i = k, it is straightforward to check from the balance property that |Dui ∩ Dvk | = k
N +1 2
|Dui ∩ Dv | = 0 i
|Du ∩ Dvk | = 0 N −1 i k . |Du ∩ Dv | = 2 ∼ By the Chinese remainder theorem, we can represent ZP = Z2 ⊗ZN under the isomorphism φ : ζ −→ (ζ mod 2, ζ mod N ), where ⊗ denotes the direct product. For convenience, we use the notation ζ ∈ ZP interchangeably with (ζ mod 2, ζ mod N ) throughout the paper. III. CONSTRUCTION OF NEW QUATERNARY LCZ SEQUENCE SETS Using a binary (N, M, L, ) LCZ sequence set with period N ≡ 3 mod 4 and the balance property, we construct new quaternary LCZ sequence sets with parameters (2N, 2M, L, 2). Let L be a set of binary LCZ sequences with parameters (N, M, L, ) and balance property, where N ≡ 3 mod 4 and L|N . Furthermore, we assume that the correlation function B Rij (τ ) between any two binary sequences ai (t) and aj (t) in L has the absolute value smaller or equal to except for τ ≡ 0 mod L and τ = 0. Let Di be the characteristic set of the LCZ sequence ai (t) in L. Then quaternary LCZ sequence sets can be constructed as follows. Theorem 1: Let U1 be the set of M quaternary sequences of period 2N defined by ⎧ i ⎪ 0, if t ∈ {0} ⊗ D0 ⎪ ⎪ ⎪ i ⎨ 1, if t ∈ {1} ⊗ D0 + L u1,i (t) = i ⎪ ⎪ ⎪2, if t ∈ {0} ⊗ D0 ⎪ ⎩3, if t ∈ {1} ⊗ Di + L 0 and U2 be the set of M sequences of period 2N defined by ⎧ i ⎪ 0, if t ∈ {0} ⊗ D0 ⎪ ⎪ ⎪ ⎨1, if t ∈ {1} ⊗ Di + L 0 u2,i (t) = i ⎪ 2, if t ∈ {0} ⊗ D ⎪ 0 ⎪ ⎪ ⎩3, if t ∈ {1} ⊗ Di + L 0
for 0 ≤ i ≤ M − 1. We define the quaternary sequence set Q of period 2N and the cardinality 2M as for 0 ≤ i ≤ M − 1 u1,i (t), si (t) = u2,i−M (t), for M ≤ i ≤ 2M − 1. Then the quaternary sequence set Q = U1 ∪ U2 is a quaternary LCZ sequence set with parameters (2N, 2M, L, 2). Proof: Clearly, Q has 2M sequences and the period of the sequences is 2N . Therefore, it remains to show that the magnitude of the correlation value between any two sequences in Q is
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less than or equal to 2 except for the in-phase autocorrelation. Using D0i + L, for 0 ≤ i ≤ M − 1 i A0 = iM D0 + L, for M ≤ i ≤ 2M − 1 we simplify the constrcution of si (t) as ⎧ iM ⎪ 0, if t ∈ {0} ⊗ D0 ⎪ ⎪ ⎪ i ⎨ 1, if t ∈ {1} ⊗ A0 si (t) = iM ⎪ ⎪ ⎪2, if t ∈ {0} ⊗ D0 ⎪ ⎩3, if t ∈ {1} ⊗ Ai 0 where iM = i mod M . 2 Let τ = (τ1 , τ2 ), where τ1 ∈ Z2 and τ2 ∈ ZN . Then the correlation between si (t) and sk (t), 0 ≤ i, k ≤ 2M − 1 can be computed as Ri,k (τ ) =
2N −1 t=0
s (t)−sk (t+τ )
ω4 i
=
2N −1 t=0
s (t−τ )−sk (t)
ω4 i
iM kM i k = |{τ1 }⊗Dτ2 ∩{0}⊗D0 |+|{1 + τ1 }⊗Aτ2 ∩{1}⊗A0 | kM i k ∩{0}⊗D |+|{1 + τ }⊗A ∩{1}⊗A | + |{τ1 }⊗DτiM 1 τ2 0 0 2 iM i kM k + ω4 |{τ1 }⊗Dτ2 ∩{1}⊗A0 |+|{1 + τ1 }⊗Aτ2 ∩{0}⊗D0 | k kM i ∩{1}⊗A |+|{1 + τ }⊗A ∩{0}⊗D | + ω4 |{τ1 }⊗DτiM 1 0 τ 0 2 2 iM i kM k − |{τ1 }⊗Dτ2 ∩{0}⊗D0 |+|{1 + τ1 }⊗Aτ2 ∩{1}⊗A0 | kM k i ∩{0}⊗D |+|{1 + τ }⊗A ∩{1}⊗A | − |{τ1 }⊗DτiM 1 0 0 τ 2 2 iM k i − ω4 |{τ1 }⊗Dτ2 ∩{1}⊗A0 |+|{1 + τ1 }⊗Aτ2 ∩{0}⊗D0kM | kM k i ∩{1}⊗A |+|{1 + τ }⊗A ∩{0}⊗D | − ω4 {τ1 }⊗DτiM 1 0 0 τ 2 2 where ω4 is the fourth complex primitive root of unity, i.e., j = √ −1. If τ1 = 0, that is, τ ≡ 0 mod 2, then Ri,k (τ ) can be simplified as Ri,k (τ ) = i kM i k M kM i k ∩D |+|A ∩A | |Dτ2 ∩D0 |+|Aτ2 ∩A0 |+|DτiM τ 0 0 2 2 i i kM k M kM k iM − |Dτ2 ∩D0 |+|Aτ2 ∩A0 |+|Dτ2 ∩D0 |+|Aiτ2 ∩A0 | . In the case of τ1 = 1, that is, τ ≡ 1 mod 2, we have Ri,k (τ ) = i i kM k M ω4 |Dτ2 ∩Ak0 |+|Aτ2 ∩D0 |+|DτiM ∩A0 |+|Aiτ2 ∩D0kM | 2 i k i kM M k i ∩A |+|A ∩D | . − ω4 |Dτ2 ∩A0 |+|Aτ2 ∩D0kM |+|DτiM 0 0 τ2 2 Case 1) 0 ≤ i, k ≤ M − 1 ( i.e., si (t), sk (t) ∈ U1 ): In this case, Ai = Di + L and Ak = Dk + L and we should consider the following two sub-cases. i) τ ≡ 0 mod 2 (τ1 = 0) except for inphase autocorrelation;
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JOURNAL OF COMMUNICATIONS AND NETWORKS, VOL. 12, NO. 4, AUGUST 2010
Table 1. List of binary LCZ sequence sets.
Long, Zhang, and Hu [3] Jang, No, Chung, and Tang [8] Tang and Udaya [9]
N 2 −1 2n − 1 2n − 1 n
In this sub-case, Ri,k (τ ) can be rewritten as i
k
i
k
+ |Dτi 2 ∩ D0k | + |(Dτi 2 + L) ∩ (D0k + L)| i
− |Dτ2 ∩ D0k | − |(Dτ2 + L) ∩ (D0k + L)| − |Dτi 2 ∩
k D0 |
− |(Dτi 2 + L) ∩
k (D 0
+ L)|.
B (−τ2 ) = σ1 . For τ2 ≡ 0 mod L, then |σ1 | ≤ and we Let Ri,k have
1 N + σ1 + 4 2 N − σ1 k k ∩ D0 | = |(Dτi 2 + L) ∩ (D0 + L)| = 4 N − σ1 i ∩ D0k | = |(D τ2 + L) ∩ (D0k + L)| = 4 1 N + σ1 k i k − . ∩ D0 | = |(D τ2 + L) ∩ (D 0 + L)| = 4 2
|Dτi 2 ∩ D0k | = |(Dτi 2 + L) ∩ (D0k + L)| = |Dτi 2 i
|Dτ2 i |Dτ2
Thus, Ri,k (τ ) reduces to 2σ1 and |Ri,k (τ )| = |2σ1 | ≤ 2 for 0 < |τ | < L. When τ = 0 and i = k, Ri,k (τ ) becomes the sum of two correlation functions at τ2 = 0 between ai (t) and ak (t), and ai (t + L) and ak (t + L). From the property of binary LCZ sequence set with parameters (N, M, L, ) [1], clearly |Ri,k (0)| ≤ 2. ii) τ ≡ 1 mod 2 (τ1 = 1); In this sub-case, Ri,k (τ ) can be rewritten as i i k Ri,k (τ ) = ω4 |Dτ2 ∩ (D0k + L)| + |(Dτ2 + L) ∩ D0 | k + ω4 |Dτi 2 ∩ (D0 + L)| + |(Dτi 2 + L) ∩ D0k | i k i − ω4 |Dτ2 ∩ (D0 + L)| + |(Dτ2 + L) ∩ D0k | k − ω4 |Dτi 2 ∩ (D0k + L)| + |(Dτi 2 + L) ∩ D0 | . B B Let Ri,k (L − τ2 ) = σ2 and Ri,k (−L − τ2 ) = σ3 . Applying the correlation values in (1), the correlation function can be computed as
Ri,k (τ ) = ω4 {−σ2 + σ3 } . Clearly, |Ri,k (τ )| ≤ 2 for τ ≡ 1 mod 2 and 0 < |τ | < L. Case 2) 0 ≤ i ≤ M − 1 and M ≤ k ≤ 2M − 1: kM
L (2 − 1)/(2m − 1) (2n − 1)/(2m − 1) (2n − 1)/(2m − 1)
1 1 1
n
In this sub-case, Ri,k (τ ) can be rewritten as
Ri,k (τ ) = |Dτ2 ∩ D0 | + |(Dτ2 + L) ∩ (D0 + L)| i
M < 2m − 1 2m − 1 m 2 −m−1
In this case, Ai0 = D0i + L and Ak0 = D0 + L. Similarly to case 1), we should consider the following two sub-cases. i) τ ≡ 0 mod 2 (τ1 = 0);
i
kM
i
Ri,k (τ ) = |Dτ2 ∩ D0 | + |(D τ2 + L) ∩ (D0kM + L)| kM
+ |Dτi 2 ∩ D0kM | + |(Dτi 2 + L) ∩ (D0 i
i
kM
− |Dτ2 ∩ D0kM | − |(Dτ2 + L) ∩ (D0 − |Dτi 2 ∩
kM D0 |
+ L)| + L)|
− |(Dτi 2 + L) ∩ (D0kM + L)|.
It is not difficult to prove that we have |(Dτi 2 + L) ∩ (D0kM + L)| = |Dτi 2 ∩ D0kM | kM
|(Dτi 2 + L) ∩ (D0 i
kM
+ L)| = |Dτi 2 ∩ D0 | i
|(Dτ2 + L) ∩ (D0kM + L)| = |Dτ2 ∩ D0kM | i
kM
|(Dτ2 + L) ∩ (D0
i
kM
+ L)| = |Dτ2 ∩ D0 |
and thus |Ri,k (τ )| = 0 for 0 ≤ |τ | < L. ii) τ ≡ 1 mod 2 (τ1 = 1); In this sub-case, Ri,k (τ ) can be rewritten as i kM i kM Ri,k (τ ) = ω4 |Dτ2 ∩ (D0 + L)| + |(Dτ2 + L) ∩ D0 | +ω4 |Dτi 2 ∩ (D0kM + L)| + |(Dτi 2 + L) ∩ D0kM | i i −ω4 |Dτ2 ∩ (D0kM + L)| + |(D τ2 + L) ∩ D0kM | kM kM −ω4 |Dτi 2 ∩ (D0 + L)| + |(Dτi 2 + L) ∩ D0 | . B B (L − τ2 ) = σ5 and Ri,k (−L − τ2 ) = σ6 . For Let Ri,k M M τ2 = 0 mod L, we have |σ5 | ≤ and |σ6 | ≤ . Applying the relationship (1) to these correlation functions, we find
Ri,k (τ ) = ω4 {σ5 + σ6 } , whose absolute value is smaller or equal to 2 for τ ≡ 1 mod 2 and 0 < |τ | < L. Case 3) M ≤ i, k ≤ 2M − 1: iM kM In this case, Ai0 = D0 + L and Ak0 = D0 + L. Similarly to case 1), it can be proved that |Ri,k (τ )| ≤ 2 for 0 ≤ |τ | < L. From case 1)–case 3), the sequence set Q is an LCZ sequence set with parameters (2N, 2M, L, 2). 2 Theorem 1 says that we can construct quaternary LCZ sequences sets with parameters (2N, 2M, L, 2) given that an optimal or suboptimal (N, M, L, 1) LCZ sequences set exists. If the employed binary LCZ sequence set with parameters (N, M, L, 1) is optimal, it is proven straightforwardly that the quaternary LCZ sequence set with parameters (2N, 2M, L, 2) defined in Theorem 1 is optimal with respect to Tang-FanMatsufuji bound [7] in the sense that its cardinality is the largest for the given length 2N and the correlation constraint 2.
JANG et al.: NEW CONSTRUCTION OF QUATERNARY LOW CORRELATION ZONE...
Binary LCZ sequences sets applicable to Theorem 1 are listed in Table 1. Note that an optimal binary LCZ sequence set introduced by Jang, No, Chung, and Tang [8] yields the construction of an optimal quaternary LCZ sequence set. There is an example of newly constructed quaternary LCZ sequence sets, even though it is not an optimal one. Example 2: Let a(t) be a binary Legendre sequence of period 31 constructed from quadratic residue in Z31 given by
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[7] [8] [9]
X. Tang, P. Fan, and S. Matsufuji, “Lower bounds on correlation of spreading sequence set with low or zero correlation zone,” Electron. Lett., vol. 36, no. 6, pp. 551–552, 2000. J.-W. Jang, J.-S. No, H. Chung, and X. Tang, “New sets of optimal p-ary low-correlation zone sequences,” IEEE Trans. Inf. Theory, vol. 53, no. 2, pp. 815–821, 2007. X. Tang and P. Udaya, “New construction of low correlation zone sequences from Hadamard matrices,” in Proc. IEEE ISIT, Adelaide, Austrailia, Sept. 4–9, 2005, pp. 482–486.
a(t) = (1110110111100010101110000100100). Let L be a binary LCZ sequence set with parameters (N, M, L, ) = (1023, 16, 33, 1) constructed using the Legendre sequence and the trace function from F210 to F25 by Theorem 8 in [8]. Since N = 1023 ≡ 3 mod 4, we can construct a quaternary LCZ sequence set Q with parameters (2N, 2M, L, 2) = (2046, 32, 33, 2) using the binary LCZ sequence set. IV. CONCLUDING REMARKS In this paper, we propose a simple construction of quaternary LCZ sequence sets with parameters (2N, 2M, L, 2) from binary LCZ sequence sets with parameters (N, M, L, ). Any binary LCZ sequence sets with low correlation value = 1 can be hired to derive new quaternary LCZ sequence sets with low correlation value = 2. The constructed quaternary LCZ sequence set becomes optimal with respect to Tang-Fan-Matsufuji bound [7] if the binary LCZ sequence set employed is optimal. Therefore, the proposed construction provides optimal quaternary LCZ sequence sets from the optimal binary LCZ sequence sets given in [8].
communication systems.
Ji-Woong Jang was born in 1976. He received the B.S., M.S., and Ph.D. degrees in Electrical Engineering and Computer Science from Seoul National University, Seoul, Korea, in 2000, 2002, and 2006, respectively. After Ph.D., he was a Senior Engineer at Samsung Electronics until June 2008. He was a postdoc. at UCSD from Aug. 2008 to July 2009. From Sept. 2009, he is a Senior Engineer at LG Electronics. His research interests includes pseudo-noise (PN) sequences, difference sets, cryptography, error correcting codes, cooperative communications and wireless
Sang-Hyo Kim received his B.S., M.S., and Ph.D. degrees in Electrical Engineering from Seoul National University, Seoul, Korea in 1998, 2000, and 2004, respectively. From 2004 to 2006, he was a senior engineer at Samsung Electronics. He visited University of Southern California as a visiting scholar from 2006 to 2007. In 2007, He joined the School of Information and Communication Engineering, Sungkyunkwan University, Suwon, Korea where he serves currently as an Assistant Professor. His research interests include error correcting codes, pseudo random sequences, cooperative communications, distributed source coding, etc.
REFERENCES [1] [2]
[3] [4]
[5] [6]
Y.-S. Kim, J.-W. Jang, J.-S. No, and H. Chung, “New design of lowcorrelation zone sequence sets,” IEEE Trans. Inf. Theory, vol. 52, no. 10, pp. 4607–4616, 2006. R. De Gaudenzi, C. Elia, and R. Viola, “Bandlimited quasi-synchronous CDMA: A novel satellite access technique for mobile and personal communication systems,” IEEE J. Sel. Areas Commun., vol. 10, no. 2, pp. 328–343, 1992. B. Long, P. Zhang, and J. Hu, “A generalized QS-CDMA system and the design of new spreading codes,” IEEE Trans. Veh. Technol., vol. 47, no. 4, pp. 1268–1275, 1998. J. S. Cha, S. Kameda, M. Yokoyama, H. Nakase, K. Masu, and K. Tsubouchi, “New binary sequences with zero-correlation duration for approximately synchronised CDMA,” Electron. Lett., vol. 36, no. 11, pp. 991–993, May 2000. X. Tang and P. Fan, “A class of pseudonoise sequences over GF (p) with low correlation zone,” IEEE Trans. Inf. Theory, vol. 47, no. 4, pp. 1644– 1649, 2001. S.-H. Kim, J.-W. Jang, J.-S. No, and H. Chung, “New constructions of quaternary low correlation zone sequences,” IEEE Trans. Inf. Theory, vol. 51, no. 4, pp. 1469–1477, 2005.
Jong-Seon No received the B.S. and M.S.E.E. degrees in Electronics Engineering from Seoul National University, Seoul, Korea, in 1981 and 1984, respectively, and the Ph.D. degree in Electrical Engineering from the University of Southern California, Los Angeles, in 1988. He was a Senior MTS with Hughes Network Systems, Germantown, MD, from February 1988 to July 1990. He was an Associate Professor with the Department of Electronic Engineering, Konkuk University, Seoul, from September 1990 to July 1999. He joined the Faculty of the Department of Electrical Engineering and Computer Science, Seoul National University, in August 1999, where he is currently a Professor. He served as a General Co-Chair for the IEICE ISITA 2006 and IEEE ISIT 2009. His research interests include error-correcting codes, sequences, cryptography, space-time codes, LDPC codes, network coding, compressed sensing, and wireless communication systems.
