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IEEE COMMUNICATIONS LETTERS, VOL. 18, NO. 2, FEBRUARY 2014

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Constructions of Z-Periodic Complementary Sequence Set with Flexible Flock Size Yubo Li, Chengqian Xu, Nan Jing, and Kai Liu Abstract—Based on orthogonal matrices, two constructions of Z-periodic complementary sequence set are introduced in this letter. The two proposed approaches can provide flexible choices for the flock size of the Z-periodic complementary sequence set, which cannot be obtained by known methods. The parameters of resultant Z-periodic complementary sequence NM1 NN1 sets are (M, N )P CSM and (M, N − 1)P CSM respectively, 2 2 which are all optimal when N1 < N − 1. The resultant ZPCSs can be potential applied in multi-carriers CDMA communication systems and MIMO channel estimation. Index Terms—Periodic complementary sequences; zero correlation zone (ZCZ); orthogonal matrices.

I. I NTRODUCTION OMPLEMENTARY sequence sets have been used in various areas such as channel estimation, spread spectrum communications, etc. [4]. Complementary sequence sets have ideal autocorrelation function (ACF) and cross-correlation function (CCF), they can improve spectrum efficiency and bit error rate (BER) performance of code division multiple access (CDMA) systems [2,3]. However, complementary sequences cannot support large number of users since the set size must not be larger than the flock size. To increase the set size of complementary sequence sets, Fan et al. exploited the idea of zero-correlation zone (ZCZ) to the generation of complementary sequences and introduced Zcomplementary binary sequences in [4]. Compared with the conventional complementary sequences, the Z-complementary sequences have fewer restrictions on length and have ideal autocorrelation and cross-correlation properties at a zone. The theoretical bound [5] shows that Z-complementary sets have much larger set size than conventional complementary sets. Li et al. extended the Z-complementary binary sequence set to quaternary Z-complementary sequence set (QZCS) [6]. Generalized pairwise complementary (GPC) sequences [7] of set-wise uniform IFWs were constructed on the basis of complete complementary sequences and generalized even shift orthogonal sequences. Based on the idea of generalized pairwise complementary (GPC) sequences in [7], Feng et al. proposed generalized pairwise Z-complementary (GPZ) sequences in [8], which includes Golay sequence as a special case [9]. Different from GPC and GPZ sequences which limit their flock size to be two, Z-complementary sets have larger flock size. Both periodic and aperiodic correlations are

C

Manuscript received September 4, 2013. The associate editor coordinating the review of this letter and approving it for publication was L. Dolocek. The authors are with the Department of Information Science and Engineering, Yanshan University, Qinhuangdao, HeBei, 066004 China (e-mail: {liyubo6316, cqxu, jingnan, liukai}@ysu.edu.cn). This work was supported by National Natural Science Foundation of China (61172094), Science and Technology Support Program of Qinhuangdao (201302A025) and The Ph.D. Foundation of Yanshan University (B788). Digital Object Identifier 10.1109/LCOMM.2013.121813.132021

favorable in channel estimation or spread spectrum communications. The Z-periodic complementary sequences (ZPCS) are optimal training sequences for single-carrier multi-antenna frequency-selective fading communication systems [10] and can also be potential spreading codes in multi-carriers CDMA systems to efficiently handle multi-path interference(MI) and multiple access interference (MAI) [11,12]. The details of how to use ZPCS in MIMO channel estimation are given in [10]. When it comes to the application of ZPCS in MCCDMA systems, Tu and Fan et al. [11] pointed out that one can adopt similar mechanism presented in [12]. ZPCSs with flexible flock size are more useful in applications. However, most of known constructions of ZPCS are based on PCS, the flock size of ZPCS obtained is equal to the flock size of the original PCS. Tu et al. [11] introduced a simple approach to generate Z-periodic complementary set by performing phase shift operation on periodic complementary sequence set. Feng et al. [13] constructed a class of Z-periodic complementary sequence sets with flexible ZCZ length based on PCS and interleaving operation. It is well known [4,5] that both the flock size of PCS and the number of PCS are strictly limited, which results in less number of ZPCSs and fixed flock size. This letter focuses on Z-periodic complementary set with flexible flock size, and it is organized as follows. In Section II, the useful notations and preliminaries are given. In Section III, two constructions of ZPCS with flexible flock size are presented, and the parameters of ZPCS are discussed in Section IV. Finally, concluding remark is given in Section V. II. P RELIMINARIES Given two complex sequences: a = (a(0), · · · , a(N − 1)) and b = (b(0), · · · , b(N − 1)). The periodic cross-correlation function between a and b is defined as: Ra,b (τ ) =

N −1 

a(t) · b∗ (t + τ ),

(1)

t=0

where b∗ (t + τ ) denotes complex conjugation of b(t + τ ) and t + τ is calculated modulo N . If a = b, then Ra,b (τ ) is called the periodic autocorrelation function, denoted by Ra (τ ). Let a = (a(0), · · · , a(N − 1)) be a complex sequence, then a is a perfect sequence if its autocorrelation satisfies:  Ea , τ = 0 (mod N ) Ra (τ ) = (2) 0, τ = 0 (mod N ), −1 2 where Ea = ΣN t=0 |a(t)| , namely, energy of the sequence a. “mod” denotes τ = 0 and τ = 0 is calculated modulo N . Let H = [hi,j ]T ×T denote a complex valued matrix with order T × T , where 0 ≤ i, j ≤ T − 1. If arbitrary two rows

c 2014 IEEE 1089-7798/14$31.00 

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IEEE COMMUNICATIONS LETTERS, VOL. 18, NO. 2, FEBRUARY 2014

of H satisfy: T −1 

 hi,k h∗j,k =

k=0

T, i = j 0,

i = j,

(3)

then H is called an orthogonal matrix. Given a set of sequences A = {A0 , A1 , · · · , AM−1 }, each sequence Am contains N subsequences with length m m m = L, such as: Am = {Am 0 , A1 , · · · , AN −1 }, An m m m (an (0), an (1), · · · , an (L − 1)). Suppose Am1 , Am2 ∈ A, where 0 ≤ m1 , m2 ≤ M − 1. m1 is called a periodic complementary sequence [1] if A RAm1 ,Am1 (τ ) =

N −1  n=0

n=0

=

0 < |τ | < L.

(4)

RAm1 ,Am2 (τ ) =

n=0

RAm1 ,Am2 (τ ) =

n=0

k=0 M 2 −1 

k

k

N M1 −1

a(t mod N ) · hm1 ,kM1 +(t mod M1 )

m2 (τ ) = 0, 0 ≤ τ < L. RAm 1 n ,An

(5)

⎧ N −1 m ⎪ ⎨ n=0 EAn 1 , τ = 0, m1 = m2 = 0, 0 < |τ | < Z, m1 = m2 ⎪ ⎩ 0, |τ | < Z, m1 = m2 , (6) where M denotes the set size, N is the flock size, L is the length of subsequences and Z is the zero-correlation zone. From the theoretical bound [5] of Z-periodic complemenL , there is: tary sequences, for a (M, Z)P CSN (7)

L , where Let Mo denote the theoretical bound of (M, Z)P CSN Mo = N · L/Z. If there is M = Mo , then it is called an optimal Z-periodic complementary sequence set.

III. C ONSTRUCTIONS OF Z-P ERIODIC C OMPLEMENTARY S ET A. Construction I Theorem 1: Given a complex valued orthogonal matrix with order M × M , H = [hi,j ]M×M , where |hi,j | = 1. Let a = (a(0), a(1), · · · , a(N − 1)) be a perfect sequence with length N . Set an integer M2 , where M = M1 M2 , gcd(N, M1 ) = 1. A Z-periodic complementary sequence set C = {C 0 , C 1 , · · · , C M−1 } is obtained by the following formulas: m }, 0 ≤ m ≤ M − 1, C m = {C0m , C1m , · · · , CM 2 −1

a(t1 mod N ) · a∗ ((t1 + τ ) mod N )

k=0 t1 =0 t2 =0

m2 (τ ) RAm 1 n ,An

M ≤ N · L/Z.

M −1 M 2 −1 N 1 −1   

· hm1 ,kM1 +(t2 mod M1 ) · h∗m2 ,kM1 +((t2 +τ ) mod M1 )

Am1 and Am2 are also called co-mates. If arbitrary two sequences in A are co-mates, then the sequence set A is a periodic complementary sequence set (PCS) [1]. The sequence set A is a zero-correlation zone periodic L [11] if complementary set (ZPCS) denoted by (M, Z)P CSN N −1 

RC m1 ,C m2 (τ )

· a∗ ((t + τ ) mod N ) · h∗m2 ,kM1 +((t+τ ) mod M1 ) =

Another sequence Am2 is called the mate of Am1 if N −1 

=

M 2 −1 

k=0 t=t1 M1 +t2 =0

EAm τ =0 1, n

0,

RC m1 ,C m2 (τ )

=

m1 (τ ) RAm 1 n ,An

 N −1

m m Ckm = (cm k (0), ck (1), · · · , ck (N M1 − 1)), 0 ≤ k ≤ M2 − 1, (9) (t) = a(t mod N ) · h , 0 ≤ t ≤ N M − 1. cm 1 m,kM1 +(t mod M1 ) k (10) N M1 . The sequence set C is a (M, N )P CSM 2 m1 m2 Proof: Let C , C ∈ C, the cross correlation is calculated as:

(8)

= Ra (τ mod N ) ·

M 2 −1 M 1 −1   k=0

t2 =0

·hm1 ,kM1 +(t2 mod M1 ) · h∗m2 ,kM1 +((t2 +τ ) mod M1 ) . (11)

Since N and M1 are relatively prime, the second and third equations of (11) are given. Suppose 0 < τ ≤ N − 1, since a is a perfect sequence, there is Ra (τ mod N ) = 0, which means RC m1 ,C m2 (τ ) = 0. Suppose τ = 0 and m1 = m2 , let hm1 and hm2 be two rows of the matrix H, then there is: RC m1 ,C m2 (0) = Ra (0) ·

M 2 −1 M 1 −1   k=0

t=0

hm1 ,kM1 +(t mod M1 ) h∗m2 ,kM1 +(t mod M1 )

= Ra (0) · Rhm1 ,hm2 (0).

(12)

Since H is an orthogonal matrix, there is Rhm1 ,hm2 (0) = 0. Then we have RC m1 ,C m2 (τ ) = 0. From the definition of N M1 ZPCS, we conclude that C is a (M, N )P CSM . 2 Example 1: Given a perfect sequence with length 7 as: a = (+ + 0 + 00−), and an orthogonal matrix H with order 8 × 8 as below: ⎡ + − − − + − ++ ⎤ ⎢ ⎢ ⎢ ⎢ ⎢ H =⎢ ⎢ ⎢ ⎢ ⎣

+ − − + − + +− + − + − + + −− + + − + + − −− + − + + − − −+ + + + − − − +− + + − − − + −+

⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦

+ + + + + + ++

where “ + ”, “ − ” represent “1” and “ − 1” respectively. Set the flock size M2 = 4, then a (8, 7)P CS414 is obtained from Theorem 1, (+ − 0 − 00 − − + 0 + 00+; − − 0 − 00 + − − 0 − 00+; + −0 − 00 − − + 0 + 00+; + + 0 + 00 − + + 0 + 00−)

LI et al.: CONSTRUCTIONS OF Z-PERIODIC COMPLEMENTARY SEQUENCE SET WITH FLEXIBLE FLOCK SIZE

(+ − 0 − 00 − − + 0 + 00+; − + 0 + 00 + + − 0 − 00−;

(+ + 0 + 00 − + + 0 + 00−; − + 0 + 00 + + − 0 − 00−;

where t3 = ((t1 + τ1 ) mod N1 )N2 + t0 + τ0 + (t1 + τ1 )/N1 . From the autocorrelation of a, the formula Ra (τ0 + (t1 + τ1 )/N1  + τ1 N2 ) = 0 is satisfied if and only if the following formula is satisfied:

+ −0 − 00 − − + 0 + 00+; − − 0 − 00 + − − 0 − 00+) (+ − 0 − 00 − − + 0 + 00+; + + 0 + 00 − + + 0 + 00−;

τ0 + (t1 + τ1 )/N1  = −τ1 N2 (mod N ).

− +0 + 00 + + − 0 − 00−; + − 0 − 00 − − + 0 + 00+) (+ − 0 − 00 − − + 0 + 00+; + − 0 − 00 − − + 0 + 00+; + +0 + 00 − + + 0 + 00−; − − 0 − 00 + − − 0 − 00+)

− −0 − 00 + − − 0 − 00+; − + 0 + 00 + + − 0 − 00−) (+ + 0 + 00 − + + 0 + 00−; + − 0 − 00 − − + 0 + 00+; − −0 − 00 + − − 0 − 00+; + − 0 − 00 − − + 0 + 00+) (+ + 0 + 00 − + + 0 + 00−; − − 0 − 00 + − − 0 − 00+; − +0 + 00 + + − 0 − 00−; − + 0 + 00 + + − 0 − 00−) (+ + 0 + 00 − + + 0 + 00−; + + 0 + 00 − + + 0 + 00−; + +0 + 00 − + + 0 + 00−; + + 0 + 00 − + + 0 + 00−).

B. Construction II Theorem 2: Given a complex valued orthogonal matrix with order M × M , H = [hi,j ]M×M , where |hi,j | = 1. Let a = (a(0), a(1), · · · , a(N − 1)) be a perfect sequence with length N , where N = N1 N2 . Set an integer M2 , where M = M2 N1 . A Z-periodic complementary sequence set C = {C 0 , C 1 , · · · , C M−1 } is obtained by the following formulas: C

m

=

{C0m , C1m , · · ·

m , CM }, 0 2 −1

≤ m ≤ M − 1,

m m Ckm = (cm k (0), ck (1), · · · , ck (N N1 − 1)),

M 2 −1 

RC 1 ,C k k k=0 N1 −1 M 2 −1 N  

=

k=0

m

t=0

m2

· a((t mod N1 )N2 + t/N1 ) (16)

Let t = t0 N1 + t1 , τ = τ0 N1 + τ1 , where 0 ≤ t1 , τ1 ≤ N1 − 1, 0 ≤ t0 , τ0 ≤ N − 1, then there is:

= ·

hm1 ,kN1 +t1 · h∗m2 ,kN1 +((t1 +τ1 ) mod N1 )

k=0 t1 =0 t0 =0 a(t1 N2 + t0 ) · a∗ (t3 )

= Ra (τ0 + (t1 + τ1 )/N1  + τ1 N2 ) ·

M 2 −1 N 1 −1   k=0

t1 =0

k=0 t1 =0

(19)

= 0.

From all above, the sequence set C generated in this paper is N N1 . This completes the ZPCS denoted by (M, N − 1)P CSM 2 proof. Example 2: Given a Hadamard matrix with order 8 × 8 as below: ⎡ + + + + + + ++ ⎤ ⎢ ⎢ ⎢ ⎢ ⎢ H =⎢ ⎢ ⎢ ⎢ ⎣

⎥ ⎥ ⎥ ⎥ ⎥ ⎥ + + + + − − −− ⎥ ⎥ + − + − − + −+ ⎥ ⎦ + + − − − − ++ + − + − + − +−

+ + − − + + −− + − − + + − −+

+ − − + − + +−

where “ + ”, “ − ” represent “1” and “ − 1” respectively. A perfect sequence with length 16, a = (0000012302020321), √ where “0”, √ “1”, “2” and “3” represent “1”, “ −1”, “ − 1” and “ − −1”. Set the flock size M2 = 2, and set N1 = N2 = 4, a ZPCS with parameters (8, 15)P CS264 is obtained from Theorem 2, (00000123020203210000123020203210000023010202210300 00301220201032; 000001230202032100001230202032100000

RC m1 ,C m2 (τ ) M −1 2 −1 N 1 −1 N   

hm1 ,kN1 +t1 · h∗m2 ,kN1 +t1

= Ra (0) · Rhm1 ,hm2 (0)

hm1 ,kN1 +(t mod N1 ) · h∗m2 ,kN1 +((t+τ ) mod N1 )

· a (((t + τ ) mod N1 )N2 + (t + τ )/N1 ).

M 2 −1 N 1 −1  

(14)

(τ )

∗

RC m1 ,C m2 (0)

(13)

RC m1 ,C m2 (τ )

hm1 ,kN1 +t1 · h∗m2 ,kN1 +((t1 +τ1 ) mod N1 ) , (17)

(18)

Since 0 ≤ t1 ≤ N1 − 1, 0 ≤ τ1 ≤ N1 − 1, there is (t1 + τ1 )/N1  ∈ {0, 1}. Suppose 0 ≤ τ0 ≤ N2 − 2, the following inequality is satisfied: 0 ≤ τ0 +(t1 +τ1 )/N1  ≤ N2 −1. Then formula (18) is satisfied if and only if τ0 = τ1 = 0. Thus there is Ra (τ0 +(t1 +τ1 )/N1 +τ1 N2 ) = 0 for 0 < τ ≤ N −N1 −1. It means RC m1 ,C m2 (τ ) = 0 for 0 < τ ≤ N − N1 − 1. Suppose τ0 = N2 − 1, there is N2 − 1 ≤ τ0 + (t1 + τ1 )/N1  ≤ N2 . The formula (18) is satisfied if and only if τ1 = N1 − 1, t1 ≥ 1. Then RC m1 ,C m2 (τ ) = 0 for τ0 = N2 − 1, 0 ≤ τ1 ≤ N1 − 2, thus we have RC m1 ,C m2 (τ ) = 0 for N − N1 ≤ τ ≤ N − 2. Suppose τ = 0, m1 = m2 , the cross-correlation of C m1 and C m2 is calculated as below:

= Ra (0) ·

cm k (t) = a((t mod N1 )N2 + t/N1 ) · hm1 ,kN1 +(t mod N1 ) , (15) where t/N1  is the floor function of t/N1 , i.e., it means the maximum integer that does not exceed t/N1 . The sequence N N1 . set C is a (M, N − 1)P CSM 2 m1 m2 Proof: Suppose C , C ∈ C. Their correlation is

=

203

2301020221030000301220201032) (02020321000001230202103222223012020221030000230102 02321022221230; 020203210000012302021032222230120202 2103000023010202321022221230) (00220101022003030022121220023232002223230220212100 22303020021010; 002201010220030300221212200232320022 2323022021210022303020021010)

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TABLE I C ONTRAST OF C ONSTRUCTIONS OF Z-P ERIODIC C OMPLEMENTARY S EQUENCE S ET Constructions Construction in [8] Construction in[11] Construction in[13] Theorem 1 Theorem 2

Parameters (M K, 2Z)P CS24NK L (N M  , Z)P CSM  2L (2N M , Z)P CSM NM1 (M, N )P CSM 2

NN1 (M, N − 1)P CSM 2

Parameters are optimal or not Not optimal Optimal when M = N Optimal when M = N Optimal

Flock size 2,Not flexible M ,Not flexible M , Not flexible M2 , Flexible

Conditions −− L = M  Z + r, 0 ≤ r ≤ Z − 1 L = M  Z + r, 0 ≤ r ≤ Z − 1 M = M1 M2 , gcd(N, M1 ) = 1

Optimal when N1 < N − 1

M2 , Flexible

N = N1 N2 , M = N 1 M 2

(02200303002201010220101022003030022021210022232302

V. C ONCLUSION

20323222001212; 022003030022010102201010220030300220

This letter presents two constructions for yielding Zperiodic complementary sequence sets with flexible flock sizes. Different from known methods which construct ZPCSs by using PCSs, the two presented approaches are based on orthogonal matrices. There are a large number of available orthogonal matrices. For example, there exist Hadamard matrices of all orders 2n × 2n , n ≥ 1. As a result, the two proposed constructions can produce quite rich Z-periodic complementary sequence sets for applications.

2121002223230220323222001212) (00000123020203210000123020203210000023010202210300 00301220201032; 222223012020210322223012020210322222 0123202003212222123002023210) (02020321000001230202103222223012020221030000230102 02321022221230; 202021032222230120203210000012302020 0321222201232020103200003012) (00220101022003030022121220023232002223230220212100 22303020021010; 220023232002212122003030022010102200 0101200203032200121202203232)

ACKNOWLEDGMENT

(02200303002201010220101022003030022021210022232302 20323222001212; 200221212200232320023232002212122002

The authors would like to thank anonymous referees for helpful suggestions that greatly improved the presentation quality of this letter.

0303220001012002101000223030).

R EFERENCES

IV. D ISCUSSION ON THE PARAMETERS OF ZPCS Table 1 lists the contrast of known constructions of ZPCS. From Table 1, the ZPCS constructed in [8] has fixed flock size 2 and it is not optimal. The ZPCS constructed in [11] or [13] has fixed flock size which is equal to the flock size of original PCS used. As a result, the resultant ZPCS is optimal only when the original PCS is optimal, that is M = N . Constructions presented in this letter can propose optimal ZPCSs with flexible flock size, which are more useful in applications. Remark 1: From Theorem 1, a ZPCS with parameters N M1 is obtained, the theoretical bound is cal(M, N )P CSM 2 culated as: Mo = M2 · N M1 /N  = M1 · M2 = M.

(20)

From the formula above, it can be concluded that the ZPCS constructed from Theorem 1 is optimal with respect to the theoretical bound. Remark 2: From Theorem 2, a ZPCS with parameters N N1 is obtained, the theoretical bound is (M, N − 1)P CSM 2 calculated as: Mo = M2 ·N N1 /(N −1) = M2 ·N1 +N1 /(N −1). (21) When N1 < N − 1,there is Mo = M2 · N1 = M , the ZPCS is optimal. When N1 ≥ N −1, there is Mo = M +N1 /(N −1). Since N = N1 · N2 , then N1 /(N − 1) = 1. There is Mo = M + 1, the set size of the resultant ZPCS is one less than the theoretical bound.

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