Abstract: Sequences over the alphabet {-1,0,1} are called ternary sequences. Including the conventional ternary complementary sequences as special cases, the aperiodic ternary Z-complementary 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 Z-complementary sets are proposed. It is shown that aperiodic ternary Z-complementary 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 Z-complementary sets and their mates are given.
INTRODUCTION
An aperiodic ternary sequence set consists of k ternary sequences whose out-of-phase 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 frequency-division multiplexing is a method of transmitting data simultaneously over multiple equally-spaced carrier frequencies, using Fourier transform processing for modulation and demodulation. However, the lengths of the known binary complementary pair are 2a10b26c for all non-negative integers a, b and c, which are very limited. Quadriphase complementary pair of length 7, 9, 15 and 17 do not exist. Aperiodic binary Z-complementary sequences are presented firstly (Fan et al., 2007). A Z-complementary set is defined as a sequence set if the sum of out-of-phase aperiodic autocorrelation functions with a certain region around the in-phase 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 Z-complementary, set is reduced to a complementary set. Aperiodic quadriphase Z-complementary sequences are studied later (Li et al., 2010). Li et al. (2011) studied the existence problem of binary Z-complementary pairs with ZCZ widths 2, 3, 4, 5 and 6. An upper bound on zero correlation zones of odd-length binary Z-complementary pairs is proposed and a new construction of binary Z-complementary pairs is presented. Some constructions of Z-complementary sequences are given by Li et al. (2014) and Liu et al. (2014). In this study, however, interest focuses on the ternary Z-complementary sequences.
MATERIALS AND METHODS
The ternary sequence consists of 1, -1 and 0. Let, a = (a0, a1, ..., aN-1) and b = (b0, b1, ..., bN-1) be ternary sequences of length N. The aperiodic correlation function of a and b is given as:
| (1) |
generally, the aperiodic function of ternary sequences in the range 0≤τ≤N-1, i.e., in Eq. 2:
| (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 Ca (τ). When, a ≠ b, the above equation of the aperiodic correlation function of ternary sequences becomes aperiodic cross-correlation function (ACCF).
A ternary sequences set {ai, 1≤i≤P} of length N is called ternary Z-complementary sets (TZCS), which is denoted by
| (3) |
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 Z-complementary pair (TZCP) when, P = 2.
A ternary Z-complementary set {a1, a2, ..., ap} is called a ternary Z-complementary mate (mutually uncorrelated ternary Z-complementary set) of another ternary Z-complementary set {b1, b2, ..., bp} , if:
| (4) |
When, Z = N, the ternary Z-complementary mate becomes the conventional ternary complementary mate. A class of M ternary Z-complementary mates, each set with P ternary sequences of length N and zero relation zones Z, is denoted by
Elementary operations of TZCP: Transformations of individual four-level 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 = (a0, a1, ..., aN-I) be a ternary sequence of length N. The following transformations of individual ternary sequence preserve the aperiodic auto-correlation function:
| (i) | an→aN-n-1 (n = 0, 1, 2, ..., N-1), which is denoted by ā |
| (ii) | an→can (n = 0, 1, 2, ..., N-1, 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 a1, a2 be a Z-complementary pair of ternary sequences. The following operations also yield ternary Z-complementary pairs:
| (i) | Interchange, namely, (a1, a2)→(a2, a1) (ii) Reverse, namely, (a1, a2)→( ā1, a2) |
| (ii) | Scalar multiply, namely, (a1, a2)→(ca1, a2), while c generally equals 1 or -1 |
(iii) |
Negating elements with odd index in a1 and a2, denoted by |
The above operations are called to be elementary operations on ternary Z-complementary pair. It is clear that the above elementary operations on ternary Z-complementary pair keep ternary Z-complementary property, because all sequences in the set of ternary sequences denoted by {a1, ā1, - a1 - ā1} aperiodic auto-correlation function, so do those ternary sequences in the set {a2, ā2, - a2 - ā2} and
When a ternary Z-complementary pair can be obtained from another ternary Z-complementary pair by finite applications of elementary operations on the ternary Z-complementary pair, the two ternary Z-complementary pairs are said to be equivalent. All ternary Z-complementary 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 Z-complementary pairs hereafter. So, a set of nonequivalent generators describes the set of ternary Z-complementary pairs with same fixed length. Similarly, the elementary operations on ternary Z-complementary pair can be generalized to ternary Z-complementary sets.
Based on computer search, Table 1 presents a generator of ternary Z-complementary pairs for length N≤10 and in each case, its Z is N-1. Ternary Z-complementary pairs with ternary complementary ones are compared in Table 2.
As with ternary Z-complementary pairs, other TZCS can be obtained from a TZCS by elementary operations on TZCS.
Constructions of TZCS: Three constructions of quadriphase Z-complementary 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 Z-complementary 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 Z-complementary properties and have even better ternary Z-complementary properties.
Construction 1 of TZCS: Let,
is obtained by Eq. 5 and 6 below:
| Table 1: | Generators of ternary Z-complementry pairs |
| Table 2: | Comparing ternary Z-complementary pairs with complementary ones |
B2 k -1 = A2 k-1A2 k (k = 1,2,…,P)
| (5) |
B2 k = A2 k-1 (-A2 k ) (k = 1,2,…,P) |
(6) |
where, A2 k-1A2 k denotes the concatenation of the two ternary sequences A2 k-1 and A2 k. A TZCS of length N. The 2r can be formed from a starting TZCS of length N by using the recursive method r times.
Example 1: Let, {A1, A2, A3, A4} be a TZCS
Where:| A1 | = | {1, 1, 1, -1, 1} |
| A2 | = | {1, 1, 0, -1, 1} |
| A3 | = | {-1, -1, 0, 1, -1} |
| A4 | = | {-1, 1, -1, -1, -1} |
The sum of their aperiodic autocorrelation functions is {18, 0, 0, 0, 4}. Then, a new TZCS {B1, B2, B3, B4} is obtained by using Eq. 5 and 6:
Where:| B1 | = | {1, 1, 1, -1, 1, 1, 1, 0, -1, 1} |
| B2 | = | {1, 1, 1, -1, 1, -1, -1, 0, 1, -1} |
| B3 | = | {-1, -1, 0, 1, -1, -1, 1, -1, -1, -1} |
| B4 | = | {-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,
Construction 2 of TZCS: Given to be a ternary Z1-complementary pair (A1, A2) of length N1, its mate denoted by
| (7) |
| (8) |
where, Z = Z1 when, Z1<N1 or Z = Z1Z2 when, Z1 = N1 and ⊗ denotes the Kronecker product.
Example 2: Let {B1, B2, B3, B4} be a TZCS:
Where:| B1 | = | {1, 1, 1, -1, 1} |
| B2 | = | {1, 1, 0, -1, 1} |
| B3 | = | {-1, -1, 0, 1, -1} |
| B4 | = | {-1, 1, -1, -1, -1} |
The sum of their aperiodic autocorrelation functions is {18, 0, 0, 0, 4}. Let (A1, A2) a ternary complementary pair, where, A1 = {1, 1, -1} and A2 = {1, 0, 1}. Its mate is
| T1 | = | {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} |
T2 |
= | {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 } |
| T3 | = | {-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 } |
| T4 | = | {-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 (F10, F20, F30, …, F2P0) be an arbitrary TZCS with 2P ternary sequences, each of length N0, zero correlation zone Z0, a new TZCS with 2P ternary sequences, each of length Nn = 2nN0, zero correlation zone Zn = 2nZ0, can be constructed using Eq. 9 and 10:
| (9) |
| (10) |
where,
Example 3: Let {F10, F20, F30, F40} be a TZCS with Z = 6:
Where:| F10 | = | {1, 1, 1, 1, -1, 1, -1} |
| F20 | = | {1, 1, -1, -1, 1, 0, 0} |
| F30 | = | {1, 1, 1, 1, -1, 0, 1} |
| F40 | = | {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 {F11, F21, F31, F41} using Eq. 9 and 10:
Where:| F11 | = | {1, 1, 1, 1, 1, -1, 1, -1, -1, 1, 1, 0, -1, 0} |
| F21 | = | {1, -1, 1, -1, 1, 1, 1, 1, -1, -1, 1, 0, -1, 0} |
| F31 | = | {1, 1, 1, 0, 1, -1, 1, -1, -1, 1, 0, -1, 1, 1} |
| F41 | = | {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 Z-complementary 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 Z-complementary mates (Li et al., 2010).
Let
| (11) |
Construction 1 of TZCS mates: Similar to the theorem 13 (Tseng and Liu, 1972), the following construction of ternary Z-complementary mates was obtained immediately. Let
| (12) |
is also a matrix of ternary sequences, whose columns are new ternary Z-complementary mates, where,
Construction 2 of TZCS mates: Given a set of ternary Z-complementary mates
Construction 3 of TZCS mates: Similar to the theorem 12 (Tseng and Liu, 1972), the following construction of ternary Z-complementary mates was obtained immediately. Let
| (13) |
is also a matrix of ternary sequences whose columns are mates where,
RESULTS AND DISCUSSION
Aperiodic ternary Z-complementary pairs are better than aperiodic ternary complementary ones of the same length in terms of the number of them. Constructing methods of ternary Z-complementary 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 Z-complementary pairs is generally much longer than that of binary Z-complementary ones. For the fixed length, the maximum zero correlation zone of ternary Z-complementary pairs is the same as that of quadriphase Z-complementary pairs (Li et al., 2010). However, the maximum zero correlation zone of ternary Z-complementary pairs of length 6 is longer than that of four-level Z-complementary 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 Z-complementary 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 Z-complementary pairs are better than aperiodic ternary complementary ones of the same length in terms of the number of them. Constructing methods of ternary Z-complementary sets and their mates are presented here. However, searching the set of nonequivalent generators of ternary Z-complementary 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. S2jj2011-020), 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. SZD0802-09-1) and the National natural Science Foundation of China (NSFC, No. 61401369/U1433130).