
Research Article


Constructions of Ternary Zcomplementary Sequences


Xudong Li,
Jing Wang
and
Xianhua Niu


ABSTRACT

Sequences over the alphabet {1,0,1} are called ternary sequences. Including the conventional ternary complementary sequences as special cases, the aperiodic ternary Zcomplementary sequences are brought forward and may be used as an alternative of ternary complementary sequences in many engineering applications. The elementary transformations on ternary sequences and elementary operations on ternary Zcomplementary sets are proposed. It is shown that aperiodic ternary Zcomplementary pairs are better than aperiodic ternary complementary ones of the same length in terms of the number of them. In the end, constructions of ternary Zcomplementary sets and their mates are given.





Received:
December 18, 2015; Accepted: February 29, 2016;
Published: June 15, 2016 

INTRODUCTION
An aperiodic ternary sequence set consists of k ternary sequences whose outofphase aperiodic autocorrelations sum to zero. When k = 2, the aperiodic ternary complementary set is called an aperiodic ternary complementary pair. Aperiodic ternary complementary pairs include binary complementary ones as special cases, which are called Golay complementary pairs, were originally considered by Golay in connection with his study of infrared spectrometry (Golay, 1961). Tseng and Liu (1972) investigated aperiodic binary complementary sets of k sequences with k>2. Sivaswamy (1978) studied multiphase complementary sets. Gavish and Lempel (1994) introduced ternary complementary Sequences. Complementary sequences were subsequently used in radar, synchronization, channel estimation and so on (Yuan et al., 2008; Spasojevic and Georghiades, 2001). In particular, modern application of Golay complementary pairs is in multicarrier communications, which have recently attracted much attention in wireless applications. Orthogonal frequencydivision multiplexing is a method of transmitting data simultaneously over multiple equallyspaced carrier frequencies, using Fourier transform processing for modulation and demodulation. However, the lengths of the known binary complementary pair are 2^{a}10^{b}26^{c} for all nonnegative integers a, b and c, which are very limited. Quadriphase complementary pair of length 7, 9, 15 and 17 do not exist. Aperiodic binary Zcomplementary sequences are presented firstly (Fan et al., 2007). A Zcomplementary set is defined as a sequence set if the sum of outofphase aperiodic autocorrelation functions with a certain region around the inphase position is zero. Such a region is called Zero Correlation Zone (ZCZ). When the width of ZCZ is equal to the sequence length, such a Zcomplementary, set is reduced to a complementary set. Aperiodic quadriphase Zcomplementary sequences are studied later (Li et al., 2010). Li et al. (2011) studied the existence problem of binary Zcomplementary pairs with ZCZ widths 2, 3, 4, 5 and 6. An upper bound on zero correlation zones of oddlength binary Zcomplementary pairs is proposed and a new construction of binary Zcomplementary pairs is presented. Some constructions of Zcomplementary sequences are given by Li et al. (2014) and Liu et al. (2014). In this study, however, interest focuses on the ternary Zcomplementary sequences.
MATERIALS AND METHODS
The ternary sequence consists of 1, 1 and 0. Let, a = (a_{0}, a_{1}, ..., a_{N1}) and b = (b_{0}, b_{1}, ..., b_{N1}) be ternary sequences of length N. The aperiodic correlation function of a and b is given as:
generally, the aperiodic function of ternary sequences in the range 0≤τ≤N1, i.e., in Eq. 2:
When, a = b is considered, the above equation of the aperiodic correlation function of ternary sequences becomes aperiodic autocorrelation function (AACF), which is denoted by C_{a} (τ). When, a ≠ b, the above equation of the aperiodic correlation function of ternary sequences becomes aperiodic crosscorrelation function (ACCF). A ternary sequences set {a_{i}, 1≤i≤P} of length N is called ternary Zcomplementary sets (TZCS), which is denoted by , if:
where, Z is the length of ZCZ. Above equation of TZCS includes the conventional ternary complementary set as a special case when Z = N. The TZCS becomes a ternary Zcomplementary pair (TZCP) when, P = 2. A ternary Zcomplementary set {a_{1}, a_{2}, ..., a_{p}} is called a ternary Zcomplementary mate (mutually uncorrelated ternary Zcomplementary set) of another ternary Zcomplementary set {b_{1}, b_{2}, ..., b_{p}} , if:
When, Z = N, the ternary Zcomplementary mate becomes the conventional ternary complementary mate. A class of M ternary Zcomplementary mates, each set with P ternary sequences of length N and zero relation zones Z, is denoted by , where, M is an integer larger than one.
Elementary operations of TZCP: Transformations of individual fourlevel sequence were given by Li and Hao (2010). Similarly, individual ternary sequence transformations are defined in Lemma 1 to provide the basis for the study of TZCS.
Lemma 1: Let a = (a_{0}, a_{1}, ..., a_{NI}) be a ternary sequence of length N. The following transformations of individual ternary sequence preserve the aperiodic autocorrelation function: (i)  a_{n}→a_{Nn1} (n = 0, 1, 2, ..., N1), which is denoted by ā 
(ii) 
a_{n}→ca_{n} (n = 0, 1, 2, ..., N1, c ε {±1}), which is denoted by ca 
Above transformations of individual ternary sequence was defined as elementary transformations on ternary sequences based on aperiodic autocorrelation function. When a ternary sequence can be obtained from another ternary sequence via the successive application of the elementary transformations, the two ternary sequences are proved to be equivalent based on aperiodic autocorrelation function.
Theorem 1: Let a_{1}, a_{2} be a Zcomplementary pair of ternary sequences. The following operations also yield ternary Zcomplementary pairs: (i) 
Interchange, namely, (a_{1}, a_{2})→(a_{2}, a_{1}) (ii) Reverse, namely, (a_{1}, a_{2})→( ā_{1}, a_{2}) 
(ii) 
Scalar multiply, namely, (a_{1}, a_{2})→(ca_{1}, a_{2}), while c generally equals 1 or 1 
(iii) 
Negating elements with odd index in a_{1} and a_{2}, denoted by , namely, (a_{1}, a_{2})→ 
The above operations are called to be elementary operations on ternary Zcomplementary pair. It is clear that the above elementary operations on ternary Zcomplementary pair keep ternary Zcomplementary property, because all sequences in the set of ternary sequences denoted by {a_{1}, ā_{1},  a_{1}  ā_{1}} aperiodic autocorrelation function, so do those ternary sequences in the set {a_{2}, ā_{2},  a_{2}  ā_{2}} and is deduced from .
When a ternary Zcomplementary pair can be obtained from another ternary Zcomplementary pair by finite applications of elementary operations on the ternary Zcomplementary pair, the two ternary Zcomplementary pairs are said to be equivalent. All ternary Zcomplementary pairs which are equivalent form an equivalent class. Any element chosen from the equivalence class is defined as a generator of the equivalence class and we also call it a generator of ternary Zcomplementary pairs hereafter. So, a set of nonequivalent generators describes the set of ternary Zcomplementary pairs with same fixed length. Similarly, the elementary operations on ternary Zcomplementary pair can be generalized to ternary Zcomplementary sets.
Based on computer search, Table 1 presents a generator of ternary Zcomplementary pairs for length N≤10 and in each case, its Z is N1. Ternary Zcomplementary pairs with ternary complementary ones are compared in Table 2. As with ternary Zcomplementary pairs, other TZCS can be obtained from a TZCS by elementary operations on TZCS. Constructions of TZCS: Three constructions of quadriphase Zcomplementary sets (Li et al., 2010) similarly are applicable to TZCS. Besides, three recursive methods of constructing TZCS are presented and they are obtained by modifying or improving the original methods of quadriphase Zcomplementary sets (Li et al., 2010). Different from those conventional construction methods using quadriphase complementary sets as a basic starter, the proposed ones are based on TZCS. The constructed TZCS preserve the original ternary Zcomplementary properties and have even better ternary Zcomplementary properties.
Construction 1 of TZCS: Let, be a TZCS with 2P ternary sequences, each of length N. Then a new TZCS with 2P ternary sequences of length 2N, which is denoted by:
is obtained by Eq. 5 and 6 below:
Table 1:  Generators of ternary Zcomplementry pairs 

Table 2: 
Comparing ternary Zcomplementary pairs with complementary ones 

B_{2 k 1} = A_{2 k1}A_{2 k} (k = 1,2,…,P)
 (5) 
B_{2 k } = A_{2 k1} (A_{2 k} ) (k = 1,2,…,P) 
(6) 
where, A_{2 k1}A_{2 k} denotes the concatenation of the two ternary sequences A_{2 k1} and A_{2 k}. A TZCS of length N. The 2^{r} can be formed from a starting TZCS of length N by using the recursive method r times. Example 1: Let, {A_{1}, A_{2}, A_{3}, A_{4}} be a TZCS Where:
A_{1} 
= 
{1, 1, 1, 1, 1} 
A_{2} 
= 
{1, 1, 0, 1, 1} 
A_{3} 
= 
{1, 1, 0, 1, 1} 
A_{4} 
= 
{1, 1, 1, 1, 1} 
The sum of their aperiodic autocorrelation functions is {18, 0, 0, 0, 4}. Then, a new TZCS {B_{1}, B_{2}, B_{3}, B_{4}} is obtained by using Eq. 5 and 6:
Where:
B_{1} 
= 
{1, 1, 1, 1, 1, 1, 1, 0, 1, 1} 
B_{2} 
= 
{1, 1, 1, 1, 1, 1, 1, 0, 1, 1} 
B_{3} 
= 
{1, 1, 0, 1, 1, 1, 1, 1, 1, 1} 
B_{4} 
= 
{1, 1, 0, 1, 1, 1, 1, 1, 1, 1} 
The sum of the new sequence aperiodic autocorrelation functions is {36, 0, 0, 0, 8, 0, 0, 0, 0, 0}. Namely, is formed from by using the recursive method once.
Construction 2 of TZCS: Given to be a ternary Z_{1}complementary pair (A_{1}, A_{2}) of length N_{1}, its mate denoted by and a ternary Z_{2}complementary set with 2P ternary sequences of length N_{2}, which is denoted by , a new TZCS with 2P sequences of length 2N_{1}N_{2} which is denoted by can be constructed by Eq. 7 and 8:
where, Z = Z_{1} when, Z_{1}<N_{1} or Z = Z_{1}Z_{2} when, Z_{1} = N_{1} and ⊗ denotes the Kronecker product. Example 2: Let {B_{1}, B_{2}, B_{3}, B_{4}} be a TZCS: Where:
B_{1} 
= 
{1, 1, 1, 1, 1} 
B_{2} 
= 
{1, 1, 0, 1, 1} 
B_{3} 
= 
{1, 1, 0, 1, 1} 
B_{4} 
= 
{1, 1, 1, 1, 1} 
The sum of their aperiodic autocorrelation functions is {18, 0, 0, 0, 4}. Let (A_{1}, A_{2}) a ternary complementary pair, where, A_{1} = {1, 1, 1} and A_{2} = {1, 0, 1}. Its mate is , where A_{1}' = {1, 0, 1}, A_{2}' = {1, 1, 1}. Then new TZCS is obtained {T_{1}, T_{2}, T_{3}, T_{4}} by 7 and 8:
Where:
T_{1} 
= 
{1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1,1, 0, 1, 1, 0, 1, 0, 0, 0, 1, 0, 1, 1, 0, 1} 
T_{2} 
= 
{1, 0, 1, 1, 0, 1, 1, 0, 1, 1, 0, 1, 1, 0, 1, 1, 1, 1, 1, 1, 1, 0, 0, 0, 1, 1, 1, 1, 1, 1 } 
T_{3} 
= 
{1, 1, 1, 1, 1, 1, 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 0, 1, 1, 0, 1, 1, 0, 1, 1, 0, 1 } 
T_{4} 
= 
{1, 0, 1, 1, 0, 1, 0, 0, 0, 1, 0, 1, 1, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 } 
The sum of the new sequence aperiodic autocorrelation functions is {90, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 20, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0}.
Construction 3 of TZCS: Let (F_{1}^{0}, F_{2}^{0}, F_{3}^{0}, …, F_{2P}^{0}) be an arbitrary TZCS with 2P ternary sequences, each of length N_{0}, zero correlation zone Z_{0}, a new TZCS with 2P ternary sequences, each of length N_{n} = 2^{n}N_{0}, zero correlation zone Z_{n} = 2^{n}Z_{0}, can be constructed using Eq. 9 and 10:
where, is formed by negating the ternary sequence and denotes the bitinterleaved operation between ternary sequences .
Example 3: Let {F_{1}^{0}, F_{2}^{0}, F_{3}^{0}, F_{4}^{0}} be a TZCS with Z = 6:
Where:
F_{1}^{0} 
= 
{1, 1, 1, 1, 1, 1, 1} 
F_{2}^{0} 
= 
{1, 1, 1, 1, 1, 0, 0} 
F_{3}^{0} 
= 
{1, 1, 1, 1, 1, 0, 1} 
F_{4}^{0} 
= 
{1, 0, 1, 1, 1, 1, 1} 
The sum of their aperiodic autocorrelation functions is {24, 0, 0, 0, 0, 0, 1}. Then, a new TZCS is obtained with Z = 12, which is denoted by {F_{1}^{1}, F_{2}^{1}, F_{3}^{1}, F_{4}^{1}} using Eq. 9 and 10:
Where:
F_{1}^{1} 
= 
{1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 1, 0} 
F_{2}^{1} 
= 
{1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 1, 0} 
F_{3}^{1} 
= 
{1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1} 
F_{4}^{1} 
= 
{1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1} 
The sum of the new sequence autocorrelation functions is {48, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 0}.
Constructions of TZCS Mates: Three recursive constructing methods of quadriphase Zcomplementary mates similarly are applicable to TZCS mates (Li et al., 2010). Three recursive methods of constructing TZCS mates are given in this section and they are obtained by modifying the original methods of quadriphase Zcomplementary mates (Li et al., 2010). Let be a class of M ternary Zcomplementary mates, each set with p ternary sequences of the same length N and zero relation zones Z, denoted by Eq. 11 the matrix form as follow:
Construction 1 of TZCS mates: Similar to the theorem 13 (Tseng and Liu, 1972), the following construction of ternary Zcomplementary mates was obtained immediately. Let be a matrix of ternary sequences whose columns are ternary Zcomplementary mates, then Eq. 12:
is also a matrix of ternary sequences, whose columns are new ternary Zcomplementary mates, where, denotes the matrix whose mpth entry is the concatenation of the mpth entry of  and the mpth entry of . In general, can be constructed by repeated applications of the Eq. 12, r times from .
Construction 2 of TZCS mates: Given a set of ternary Zcomplementary mates , a new set of ternary Zcomplementary mates was obtained by employing Eq. 9 and 10, r times.
Construction 3 of TZCS mates: Similar to the theorem 12 (Tseng and Liu, 1972), the following construction of ternary Zcomplementary mates was obtained immediately. Let be a matrix of ternary sequences whose columns are mates, then Eq. 13:
is also a matrix of ternary sequences whose columns are mates where, denotes the matrix whose mpth entry is the interleaved sequence of the mpth entry of  and the mpth entry of . In general, from , can be constructed by using the Eq. 13, r times.
RESULTS AND DISCUSSION Aperiodic ternary Zcomplementary pairs are better than aperiodic ternary complementary ones of the same length in terms of the number of them. Constructing methods of ternary Zcomplementary sets and their mates are given. Compared with Table 1 (Fan et al., 2007), Table 1 in this study, indicates that the maximum zero correlation zone of ternary Zcomplementary pairs is generally much longer than that of binary Zcomplementary ones. For the fixed length, the maximum zero correlation zone of ternary Zcomplementary pairs is the same as that of quadriphase Zcomplementary pairs (Li et al., 2010). However, the maximum zero correlation zone of ternary Zcomplementary pairs of length 6 is longer than that of fourlevel Zcomplementary ones (Li et al., 2010). Constructions of TZCS and their mates are obtained by modifying or improving the original methods (9, 10, 11, 15). Besides, other synthesizing methods of TZCS and their mates are presented here.
CONCLUSION Aperiodic ternary Zcomplementary sequences are investigated in this study. The notions of elementary transformations on ternary sequences and elementary operations on TZCS are put forward. The results show that aperiodic ternary Zcomplementary pairs are better than aperiodic ternary complementary ones of the same length in terms of the number of them. Constructing methods of ternary Zcomplementary sets and their mates are presented here. However, searching the set of nonequivalent generators of ternary Zcomplementary pairs of longer length remains an open problem. ACKNOWLEDGMENT
This study was supported by the Key Natural Science Fund of Sichuan Education Department (Grant No. 13ZA0031), the Key Scientific research fund of Xihua University (Grant No. R1222628), Open Research Fund of Key Laboratory of Xihua University (Grant No. S2jj2011020), Distinguished Young Fund of the School of Mathematics and Computer Engineering of Xihua University, the National Key Technology R and D program (No. 2011BAH26B00), the International Cooperation Project of Sichuan Province (No. 2009HH0009) and the Fund of Key Disciplinary of Sichuan Province (No. SZD0802091) and the National natural Science Foundation of China (NSFC, No. 61401369/U1433130).

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