The previous constructions of quadrature amplitude modulation (QAM) Golay complementary sequences (GCSs) were generalized as $4^q $-QAM GCSs of length $2^{m}$ by Li \textsl{et al.} (the generalized cases I-III for $q\ge 2$) in 2010 and Liu \textsl{et al.} (the generalized cases IV-V for $q\ge 3$) in 2013 respectively. Those sequences are presented as the combination of the quaternary standard GCSs and compatible offsets. By providing new compatible offsets based on the factorization of the integer $q$, we proposed two new constructions of $4^q $-QAM GCSs, which have the generalized cases I-V as special cases. The numbers of the proposed GCSs (including the generalized cases IV-V) are equal to the product of the number of the quaternary standard GCSs and the number of the compatible offsets. For $q=q_{1}\times q_{2}\times \dots\times q_{t}$ ($q_k>1$), the number of new offsets in our first construction is lower bounded by a polynomial of $m$ with degree $t$, while the numbers of offsets in the generalized cases I-III and IV-V are a linear polynomial of $m$ and a quadratic polynomial of $m$, respectively. In particular, the numbers of new offsets in our first construction is seven times more than that in the generalized cases IV-V for $q=4$. We also show that the numbers of new offsets in our two constructions is lower bounded by a cubic polynomial of $m$ for $q=6$. Moreover, our proof implies that all the mentioned GCSs over QAM in this paper can be regarded as projections of Golay complementary arrays of size $2\times2\times\cdots\times2$.
We generalize the three-stage process for constructing and enumerating Golay array and sequence pairs given in 2008 by Frank Fiedler et al. [A multi-dimensional approach to the construction and enumeration of Golay complementary sequences, Journal of Combinatorial Theory, Series A 115 (2008) 753-776] to $4^{q}$-QAM constellation based on para-unitary matrix method, which partly solves their open questions. Our work not only includes the main part of known results of Golay complementary sequences over $4^{q}$-QAM based on Boolean functions and standard Golay sequence pairs over QPSK, but also g…
We extend the paraunitary (PU) theory for complementary pairs to comple- mentary sets and complete complementary codes (CCC) by proposing a new PU construction. A special, but very important case of complementary sets (and CC- C), based on standard delays, is analyzed in details and a new 'Radix-M generator' (RM-G) is presented. The RM-G can be viewed as a generalization of the Boolean generator for complementary pairs. An efficient correlator for standard complemen- tary sets and CCC is also presented. Finally, examples of polyphase, QAM and hexagonal PU sets of three sequences are given.
In wireless communication systems, there are many stages for signal transmission. Among them, mapping and demapping convert a sequence of bits into a sequence of complex numbers and vice versa. This operation is performed by a system of constellations~ -- by a set of labeled points on the complex plane. Usually, the geometry of the constellation is fixed, and constellation points are uniformly spaced, e.g., the same quadrature amplitude modulation (QAM) is used in a wide range of signal-to-noise ratio (SNR). By eliminating the uniformity of constellations, it is possible to achieve greater val…
In this paper, a recent method to construct complementary sequence sets and complete complementary codes by Hadamard matrices is deeply studied. By taking the algebraic structure of Hadamard matrices into consideration, our main result determine the so-called $δ$-linear terms and $δ$-quadratic terms. As a first consequence, a powerful theory linking Golay complementary sets of $p$-ary ($p$ prime) sequences and the generalized Reed-Muller codes by Kasami et al. is developed. These codes enjoy good error-correcting capability, tightly controlled PMEPR, and significantly extend the range of codin…
A new method to construct $q$-ary complementary sequence (or array) sets (CSSs) and complete complementary codes (CCCs) of size $N$ is introduced in this paper. An algorithm on how to compute the explicit form of the functions in constructed CSS and CCC is also given. A general form of these functions only depends on a basis of functions from $\Z_N$ to $\Z_q$ and representatives in the equivalent class of Butson-type Hadamard matrices. Surprisingly, all the functions fill up a larger number of cosets of a linear code, compared with the existing constructions. From our general construction, its…
In recent years, complementary sequence sets have found many important applications in multi-carrier code-division multiple-access (MC-CDMA) systems for their good correlation properties. In this paper, we propose a construction, which can generate multiple sets of complete complementary codes (CCCs) over $\mathbb{Z}_N$, where $N~(N\geq 3)$ is a positive integer of the form $N=p_0^{e^0}p_1^{e^1}\dots p_{n-1}^{e^{n-1}}$, $p_0<p_1<\cdots<p_{n-1}$ are prime factors of $N$ and $e_0,e_1,\cdots,e_{n-1}$ are non-negative integers. Interestingly, the maximum inter-set aperiodic cross-correlation magni…
Due to the zero nontrivial aperiodic correlation of complete complementary code (CCCs), it is used in asynchronous multi carrier code division multiple access (MC-CDMA) communication to provide zero interference performance. However, there is no CCC for some lengths. In this case, if the system pays more attention to the correlation, it can use mutually orthogonal complementary sequence sets (MOCSSs), and if the system pays more attention to the set size, it can use Z-complementary code sets (ZCCSs). In this paper, we propose a direct construction of $q$-ary $(q^{v+1},q,q^m,q^{m-v})$-ZCCS. In …
This letter presents a direct construction of cross Z-complementary sequence sets (CZCSSs), whose aperiodic correlation sums exhibit zero correlation zones at both the front-end and tail-end shifts. CZCSS can be regarded as an extension of the symmetrical Z-complementary code set (SZCCS). The available construction of SZCCS has a limitation on the set size, with a maximum set size of 8. The proposed generalized Boolean function based construction can generate CZCSS of length in the form of non-power-of-two with variable set size $2^{n+1}$, where each code has $2^{n+1}$ constituent sequences. T…
Previously, we have presented a framework to use the para-unitary (PU) matrix-based approach for constructing new complementary sequence set (CSS), complete complementary code (CCC), complementary sequence array (CSA), and complete complementary array (CCA). In this paper, we introduce a new class of delay matrices for the PU construction. In this way, generalized Boolean functions (GBF) derived from PU matrix can be represented by an array of size $2\times 2 \times \cdots \times 2$. In addition, we introduce a new method to construct PU matrices using block matrices. With these two new ingred…
A new method to construct $q$-ary complementary sequence sets (CSSs) and complete complementary codes (CCCs) of size $N$ is proposed by using desired para-unitary (PU) matrices. The concept of seed PU matrices is introduced and a systematic approach on how to compute the explicit forms of the functions in constructed CSSs and CCCs from the seed PU matrices is given. A general form of these functions only depends on a basis of the functions from $\Z_N$ to $\Z_q$ and representatives in the equivalent class of Butson-type Hadamard (BH) matrices. Especially, the realization of Golay pairs from the…
The construction of complementary sets (CSs) of sequences with different set size and sequence length become important due to its practical application for OFDM systems. Most of the constructions of CSs, based on generalized Boolean functions (GBFs), are of length $2^α$ ($α$ is a natural number). Recently some works have been reported on construction of CSs having lengths non-power of two, i.e., in the form of $2^{m-1}+2^v$ ($m$ is natural number, $0\leq v <m $), $N+1$ and $N+2$, where $N$ is a length for which $q$-ary complementary pairs exist. In this paper, we propose a construction of CSs …
Quasi-complementary sequence sets (QCSSs) can be seen as a generalized version of complete complementary codes (CCCs), which enables multicarrier communication systems to support more users. The contribution of this work is two-fold. First, we propose a systematic construction of Florentine rectangles. Secondly, we propose several sets of CCCs and QCSS, using Florentine rectangles. The CCCs and QCSS are constructed over $\mathbb{Z}_N$, where $N\geq2$ is any integer. The cross-correlation magnitude of any two of the constructed CCCs is upper bounded by $N$. By combining the proposed CCCs, we pr…
Five Constructions of Asymptotically Optimal Aperiodic Doppler Resilient Complementary Sequence Sets with New Parameters
School of Mathematics and Statistics, Fujian Normal University（福建师范大学数学与统计学院）；Key Laboratory of Analytical Mathematics and Applications (Ministry of Education), Fujian Normal University（福建师范大学分析数学与应用教育部重点实验室）；School of Mathematics and Statistics, Zhengzhou University（郑州大学数学与统计学院）
In this paper we define a new transform on (generalized) Boolean functions, which generalizes the Walsh-Hadamard, nega-Hadamard, $2^k$-Hadamard, consta-Hadamard and all $HN$-transforms. We describe the behavior of what we call the root- Hadamard transform for a generalized Boolean function $f$ in terms of the binary components of $f$. Further, we define a notion of complementarity (in the spirit of the Golay sequences) with respect to this transform and furthermore, we describe the complementarity of a generalized Boolean set with respect to the binary components of the elements of that set.
The contributions of this paper are twofold: Firstly, we introduce a novel class of sequence pairs, called "cross Z-complementary pairs (CZCPs)", each displaying zero-correlation zone (ZCZ) properties for both their aperiodic autocorrelation sums and crosscorrelation sums. Systematic constructions of perfect CZCPs based on selected Golay complementary pairs (GCPs) are presented. Secondly, we point out that CZCPs can be utilized as a key component in designing training sequences for broadband spatial modulation (SM) systems. We show that our proposed SM training sequences derived from CZCPs lea…
The sequence pairs of length $2^{m}$ projected from complementary array pairs of Type-II of size $\mathbf{2}^{(m)}$ and mixed Type-II/III and of size $\mathbf{2}^{(m-1)}\times2$ are complementary sequence pairs Type-II and Type-III respectively. An exhaustive search for binary Type-II and Type-III complementary sequence pairs of small lengths $2^{m}$ ($m=1,2,3,4$) shows that they are all projected from the aforementioned complementary array pairs, whose algebraic normal forms satisfy specified expressions. It's natural to ask whether the conclusion holds for all $m$. In this paper, we proved t…
In this study, we propose two schemes for uplink control channels based on non-contiguous complementary sequences (CSs) where the peak-to-average-power ratio (PAPR) of the resulting orthogonal frequency division multiplexing (OFDM) signal is always less than or equal to 3 dB. To obtain the proposed schemes, we extend Golay's concatenation and interleaving methods by considering extra upsampling and shifting parameters. The proposed schemes enable a flexible non-contiguous resource allocation in frequency, e.g., an arbitrary number of null symbols between the occupied resource blocks (RBs). The…
Zero correlation zone (ZCZ) sequences and Golay sequences are two kinds of sequences with different preferable correlation properties. It was shown by Gong \textit{et al.} and Chen \textit{et al.} that some Golay sequences also possess a large ZCZ and are good candidates for pilots in OFDM systems. Known Golay sequences with ZCZ reported in the literature have a limitation in the length which is the form of a power of 2. One objective of this paper is to propose a construction of Golay complementary pairs (GCPs) with new lengths whose periodic autocorrelation of each of the Golay sequences and…
A major drawback of orthogonal frequency division multiplexing (OFDM) systems is their high peak-to-mean envelope power ratio (PMEPR). The PMEPR problem can be solved by adopting large codebooks consisting of complementary sequences with low PMEPR. In this paper, we present a new construction of polyphase complementary sets (CSs) using generalized Boolean functions (GBFs), which generalizes Schmidt's construction in 2007, Paterson's construction in 2000 and Golay complementary pairs (GCPs) given by Davis and Jedwab in 1999. Compared with Schmidt's approach, our proposed CSs lead to lower PMEPR…
The correlation properties of sequences form a focal point in the design of multiple access systems of communications. Such a system must be able to serve a number of simultaneous users while keeping interference low. A popular choice for the set of sequences to deploy is the quasi-complementary sequence set (QCSS). Its large set size enables the system to accommodate a lot of users. The set has low nontrivial correlation magnitudes within a zone around the origin. This keeps undue interference among users under control. A QCSS performs better than the perfect complementary sequence set (PCSS)…
In this study, we propose a flexible construction of complementary sequences (CSs) that can contain zero-valued elements. To derive the construction, we use Boolean functions to represent a polynomial generated with a recursion. By applying this representation to recursive CS constructions, we show the impact of construction parameters such as sign, amplitude, phase rotation used in the recursion on the elements of the synthesized CS. As a result, we extend Davis and Jedwab's CS construction by obtaining independent functions for the amplitude and phase of each element of the CS, and the seed …
Golay complementary sequences have been put a high value on the applications in orthogonal frequency-division multiplexing (OFDM) systems since its good peak-to-mean envelope power ratio(PMEPR) properties. However, with the increase of the code length, the code rate of the standard Golay sequences suffer a dramatic decline. Even though a lot of efforts have been paid to solve the code rate problem for OFDM application, how to construct large classes of sequences with low PMEPR is still difficult and open now. In this paper, we propose a new method to construct $q$-ary Golay complementary set o…
The one-dimensional (1-D) Golay complementary set (GCS) has many well-known properties and has been widely employed in engineering. The concept of 1-D GCS can be extended to the two-dimensional (2-D) Golay complementary array set (GCAS) where the 2-D aperiodic autocorrelation of constituent arrays sum to zero except for the 2-D zero shift. The 2-D GCAS includes the 2-D Golay complementary array pair (GCAP) as a special case when the set size is 2. In this paper, 2-D generalized Boolean functions are introduced and novel constructions of 2-D GCAPs, 2-D GCASs, and 2-D Golay complementary array m…
In this paper, two modulation schemes based on complementary sequences (CSs) are proposed for uplink control channels in unlicensed bands. These schemes address high peak-to-average-power ratio (PAPR) under non-contiguous resource allocation in the frequency domain and reduce the maximum PAPR to 3 dB. The first scheme allows the users to transmit a small amount of uplink control information (UCI) such as acknowledgment signals and does not introduce a trade-off between PAPR and co-channel interference (CCI). The second scheme, which enables up to 21 UCI bits for a single user or 11 UCI bits fo…
In this study, we propose partitioned complementary sequences (CSs) where the gaps between the clusters encode information bits to achieve low peak-to-average-power ratio (PAPR) orthogonal frequency division multiplexing (OFDM) symbols. We show that the partitioning rule without losing the feature of being a CS coincides with the non-squashing partitions of a positive integer and leads to a symmetric separation of clusters. We analytically derive the number of partitioned CSs for given bandwidth and a minimum distance constraint and obtain the corresponding recursive methods for enumerating th…
This paper introduces a novel class of code sets, called "symmetrical Z-complementary code sets (SZCCSs)" , whose aperiodic auto- and cross- correlation sums exhibit zero-correlation zones (ZCZs) at both the front-end and tail-end of the entire correlation window. Three constructions of (optimal) SZCCSs based on general Boolean functions are presented. As a second major contribution, we apply SZCCSs to design optimal training sequences for broadband generalized spatial modulation (GSM) systems over frequency-selective channels. Key words: Complementary code set, channel estimation, training se…
In this paper, we propose a direct construction of a novel type of code set, which has combined properties of complete complementary code (CCC) and zero-correlation zone (ZCZ) sequences and called it complete complementary-ZCZ (CC-ZCZ) code set. The code set is constructed by using multivariable functions. The proposed construction also provides Golay-ZCZ codes with new lengths, i.e., prime-power lengths. The proposed Golay-ZCZ codes are optimal and asymptotically optimal for binary and non-binary cases, respectively, by \emph{Tang-Fan-Matsufuzi} bound. Furthermore, the proposed direct constru…
This paper is devoted to sequences and focuses on designing new two-dimensional (2-D) Z-complementary array pairs (ZCAPs) by exploring two promising approaches. A ZCAP is a pair of 2-D arrays, whose 2-D autocorrelation sum gives zero value at all time shifts in a zone around the $(0,0)$ time shift, except the $(0,0)$ time shift. The first approach investigated in this paper uses a one-dimensional (1-D) Z-complementary pair (ZCP), which is an extension of the 1-D Golay complementary pair (GCP) where the autocorrelations of constituent sequences are complementary within a zero correlation zone (…
We present a general method for constructing radar transmit pulse trains and receive filters for which the radar point-spread function in delay and Doppler (radar cross-ambiguity function) is essentially free of range sidelobes inside a Doppler interval around the zero-Doppler axis. The transmit and receive pulse trains are constructed by coordinating the transmission of a pair of Golay complementary waveforms across time according to zeros and ones in a binary sequence $P$. In the receive pulse train filter, each waveform is weighted according to an element from another sequence $Q$. We show …
This paper presents an efficient construction of two-dimensional (2D) complete complementary codes (CCCs) for their modern application as omnidirectional precoding matrices in massive MIMO systems to attain enhanced cell coverage. Unlike the traditional 1D CCCs, little progress has been made on efficient and systematic constructions of the 2D counterpart. In contrast to the existing recursive constructions with the aid of various sequence operations, certain 1D seed sequences or 2D arrays, we propose to use 2D generalized Boolean functions for direct synthesis of 2D CCCs. Simulation results sh…
Complementary set sequences (CSSs) are useful for dealing with the high peak-to-average power ratio (PAPR) problem in orthogonal frequency division multiplexing (OFDM) systems. In practical OFDM transmission, however, certain sub-carriers maybe reserved and/or prohibited to transmit signals, leading to the so-called \emph{spectral null constraint} (SNC) design problem. For example, the DC sub-carrier is reserved to avoid the offsets in D/A and A/D converter in the LTE systems. While most of the current research focus on the design of low PAPR CSSs to improve the code-rate, few works address th…
Zero correlation zone (ZCZ) sequences and Golay complementary sequences are two kinds of sequences with different preferable correlation properties. Golay-ZCZ sequences are special kinds of complementary sequences which also possess a large ZCZ and are good candidates for pilots in OFDM systems. Known Golay-ZCZ sequences reported in the literature have a limitation in the length which is the form of a power of 2. In this paper, we propose two constructions of Golay-ZCZ sequence sets with new parameters which generalize the constructions of Gong et al. (IEEE Transaction on Communications 61(9),…
Certain applications require the use of signals that combine both the capability to operate with low signal-to-noise ratios and the ability to support multiple users without interference. In the case where many users have very different signal-to-noise ratios, it is necessary to consider coding schemes that can be used in a multi-user environment but with different noise immunity levels. Traditional detection systems based on the correlation function and coding sequences have significant limitations in satisfying both objectives, since the cross-correlation between coded signals corresponding …
Cross Z-complementary pairs (CZCPs) are a special kind of Z-complementary pairs (ZCPs) having zero autocorrelation sums around the in-phase position and end-shift position, also having zero cross-correlation sums around the end-shift position. It can be utilized as a key component in designing optimal training sequences for broadband spatial modulation (SM) systems over frequency selective channels. In this paper, we focus on designing new CZCPs with large cross Z-complementary ratio $(\mathrm{CZC}_{\mathrm{ratio}})$ by exploring two promising approaches. The first one of CZCPs via properly ca…
One objective of this paper is to propose a novel class of sequence pairs, called "Quasi-orthogonal Z-complementary pairs (QOZCPs)", each depicting Z-complementary property for their aperiodic auto-correlation sums and also have a zero correlation zone when their aperiodic cross-correlation is considered. Construction of QOZCPs based on Successively Distributed Algorithms under Majorization Minimization (SDAMM) is presented. Another objective of this paper is to apply the proposed QOZCPs in fully polarimetric radar systems and analyse the corresponding ambiguity functions. It turns out that QO…
