Quasi-complementary Sequence Sets Research Articles

New Correlation Bound and Construction of Quasi-Complementary Sequence Sets

New Correlation Bound and Construction of Quasi-Complementary Sequence Sets

Asymptotically Optimal and Near-Optimal Aperiodic Quasi-Complementary Sequence Sets Based on Florentine Rectangles

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 QCSSs, using Florentine rectangles. The CCCs and QCSSs are constructed over <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\mathbb {Z}_{N}$ </tex-math></inline-formula> , where <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$N\geq 2$ </tex-math></inline-formula> is any integer. The cross-correlation magnitude of any two of the constructed CCCs is upper bounded by <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$N$ </tex-math></inline-formula> . By combining the proposed CCCs, we propose asymptotically optimal and near-optimal QCSSs with new parameters.

Three New Constructions of Asymptotically Optimal Periodic Quasi-Complementary Sequence Sets With Small Alphabet Sizes

Quasi-complementary sequence sets (QCSSs) play an important role in multi-carrier code-division multiple-access (MC-CDMA) systems. They can support more users than perfect complementary sequence sets in MC-CDMA systems. It is desirable to design QCSSs with good parameters that are a trade-off of large set size, small periodic maximum magnitude correlation and small alphabet size. The main results are to construct new infinite families of QCSSs that all have small alphabet size and asymptotically optimal periodic maximum magnitude correlation. In this paper, we propose three new constructions of QCSSs using additive characters over finite fields. Notably, these QCSSs have new parameters and small alphabet sizes. Using the properties of characters and character sums, we determine their maximum periodic correlation magnitudes and prove that these QCSSs are asymptotically optimal with respect to the lower bound.

A Tighter Correlation Lower Bound for Quasi-Complementary Sequence Sets with Low Correlation Zone

Inspired by an idea due to Levenshtein, we apply the low correlation zone constraint in the analysis of the weighted mean square aperiodic correlation. Then we derive a lower bound on the measure for quasi-complementary sequence sets with low correlation zone (LCZ-QCSS). We discuss the conditions of tightness for the proposed bound. It turns out that the proposed bound is tighter than Liu-Guan-Ng-Chen bound for LCZ-QCSS. We also derive a lower bound for QCSS, which improves the Liu-Guan-Mow bound in general.

A Near-Optimal and Low-Complex Joint Multiuser Detection for QCSS-MC-CDMA System

Joint maximum likelihood (ML) multiuser detection for quasi-synchronous multicarrier code division multiple access (MC-CDMA) system has an excessive computational complexity, and it increases exponentially as the number of users and bits in each of the modulation symbols increases. Recently, sphere decoding (SD) has been adopted for multiuser detection whose complexity is polynomial in nature with the number of users, and it does not rely on the number of bits in the modulation. In this article, first we provide a construction of large size quasi-complementary sequence sets (QCSSs), and then a low-complexity optimum SD algorithm is proposed to detect the multiuser communication in QCSS-based MC-CDMA (QCSS-MC-CDMA) system. The complexity analysis has shown that the proposed SD-based QCSS-MC-CDMA receiver for multiuser communication provides lower complexity than that of ML-based QCSS-MC-CDMA receiver. We also show that peak to average power ratio of the proposed system is lesser than that of complete complementary code (CCC) based MC-CDMA (CCC-MC-CDMA) system. The proposed system overcomes the fractional delay problem which is noticed for CCC-MC-CDMA receiver. The bit error rate performance of the proposed SD-based QCSS-MC-CDMA system is compared with maximum ratio combining based MC-CDMA, ML-based MC-CDMA, frequency domain equalization based MC-CDMA, and CCC-MC-CDMA receivers over Rayleigh frequency selective fading channels. The work is validated through Monte Carlo simulation results.

A Generalized Construction of Multiple Complete Complementary Codes and Asymptotically Optimal Aperiodic Quasi-Complementary Sequence Sets

In recent years, complete complementary codes (CCCs) and quasi-complementary sequence sets (QCSSs) 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 generic construction of multiple sets of CCCs over $\mathbb {Z}_{N}$ , consisting of sequences of length $N$ , where $N\geq 3$ is an arbitrary odd integer. Interestingly, the maximum inter-set aperiodic cross-correlation magnitude of the proposed CCCs is upper bounded by $N$ . It turns out that the combination of the generated CCCs results in a new set of sequences to obtain asymptotically optimal and near-optimal aperiodic QCSSs. The proposed construction includes a recent optimal construction of QCSSs with prime length as a special case and leads to asymptotically optimal QCSSs with new flexible parameters.

Binary Complementary Sequence Set With Low Correlation Zone

Recently, quasi-complementary sequence sets (QCSSs) including low correlation zone complementary sequence sets (LCZ-CSSs), and low correlation complementary sequence sets (LC-CSSs) have raised lots of concerns. According to the theoretical bounds, the set size of a QCSS is much larger than that of a traditional perfect complementary sequence set (PCSS). As a result, it is believed that QCSSs have potential applications in multi-carrier code-division multiple-access (MC-CDMA) systems to support more users. As binary sequences are desired in practice because of easier realization, we consider the design of binary QCSSs with large set size in this letter. A construction of aperiodic binary LCZ-CSSs is proposed, which leads to aperiodic binary LCZ-CSSs with asymptotically optimal parameters. Additional, aperiodic binary LC-CSSs with asymptotically optimal parameters can also be derived from the proposed construction.

Constructions of Asymptotically Optimal Aperiodic Quasi-Complementary Sequence Sets

Mutually orthogonal complementary codes have found many practical applications in wireless communications owing to their perfect correlation properties. However, the set size of a mutually orthogonal complementary code set (MOCCS) is upper bounded by the number of constituent sequences in each complementary code. As a result, the number of users in the communication system is limited. To overcome this limitation, quasi-complementary sequence sets (QCSSs) were proposed, which include low correlation zone complementary sequence sets (LCZ-CSSs) and low correlation complementary sequence sets (LC-CSSs). In this paper, aperiodic quasi-complementary sequences are our main interest. Constructions of aperiodic LC-CSSs and aperiodic LCZ-CSSs over the complex roots of unity are proposed. With these constructions, asymptotically optimal aperiodic QCSSs can be obtained.

Constructions of Quasi-Complementary Sequence Sets Associated With Characters

An $(\boldsymbol {M},\boldsymbol {K},\boldsymbol {N},\delta _{\max })$ -quasi-complementary sequence set (QCSS) is referred to as a set of $\boldsymbol {M}$ 2-D matrices of order $\boldsymbol {K}\times \boldsymbol {N}$ with periodic tolerance $\delta _{\max }$ . In a multicarrier code-division multiple-access (MC-CDMA) communication system, the set size $\boldsymbol {M}$ of a QCSS is equal to the maximum number of users it can support, and the periodic tolerance $\delta _{\max }$ determines the interference performance. For the application of a QCSS, it is desirable that the set size should be as large as possible, and the periodic tolerance should be as small as possible. In this paper, a framework of a periodic QCSS is proposed from additive and multiplicative characters associated with a specific integer set. It is then discovered that the parameters of the obtained QCSS are determined by the employed integer set. From this discovery, new classes of periodic QCSSs with large set sizes and low periodic tolerances are constructed and associated with some known integer sets.

Two Constructions of Asymptotically Optimal Quasi-Complementary Sequence Sets

An $(M,K,N,\delta _{\mathrm {max}})$ -quasi-complementary sequence set (QCSS) is referred to as a set of $M$ two-dimensional matrices of size $K\times N$ with maximum periodic correlation magnitude $\delta _{\mathrm {max}}$ . It can be applied to a multi-carrier code-division multiple-access communication system to achieve low-interference performance. Compared with perfect complementary sequence sets, QCSSs with maximum periodic correlation magnitudes achieving or asymptotically achieving the correlation lower bounds have the advantage of supporting more users. In this paper, two constructions of periodic QCSSs from additive characters and multiplicative characters of finite fields are developed. In the first construction, new QCSSs with constituent sequence length $N=p$ are proposed by using cyclic classes, where $p$ is a prime. In the second construction, QCSSs with constituent sequence length $N=q-1$ and $N=r^{2}-1$ are presented by employing almost difference sets and special sets proposed by Katz, respectively, where $q>5$ is an odd prime power and $r$ is a prime power. Notably, the parameters of QCSSs derived from the second construction are flexible and have not been covered in the literature. In addition, all the proposed periodic QCSSs are asymptotically optimal with respect to the correlation lower bounds.

Construction of a Near-Optimal Quasi-Complementary Sequence Set from Almost Difference Set

Compared with the perfect complementary sequence sets, quasi-complementary sequence sets (QCSSs) can support more users to work in multicarrier CDMA communications. A near-optimal periodic QCSS is constructed in this paper by using an optimal quaternary sequence set and an almost difference set. With the changes of the values of parameters in the QCSS, the number of users supported by the subcarrier channels in CDMA system has an exponential growth.

Constructions of Asymptotically Optimal Quasi-Complementary Sequence Sets

Quasi-complementary sequence sets (QCSSs) have the advantage of supporting more users in multi-carrier CDMA systems. Recently, a generic construction of QCSSs from difference sets was developed by Liu et al. In this letter, the idea of Liu et al. is extended, and the constructions of QCSSs from both difference sets and almost difference set are proposed. These constructions produce QCSSs asymptotically achieving the theoretical bound. The QCSSs presented in this letter have new parameters not covered in the literature.

Asymptotically Locally Optimal Weight Vector Design for a Tighter Correlation Lower Bound of Quasi-Complementary Sequence Sets

A quasi-complementary sequence set (QCSS) refers to a set of two-dimensional matrices with low non-trivial aperiodic auto- and cross- correlation sums. For multicarrier code-division multiple-access applications, the availability of large QCSSs with low correlation sums is desirable. The generalized Levenshtein bound (GLB) is a lower bound on the maximum aperiodic correlation sum of QCSSs. The bounding expression of GLB is a fractional quadratic function of a weight vector $\mathbf{w}$ and is expressed in terms of three additional parameters associated with QCSS: the set size $K$, the number of channels $M$, and the sequence length $N$. It is known that a tighter GLB (compared to the Welch bound) is possible only if the condition $M\geq2$ and $K\geq \overline{K}+1$, where $\overline{K}$ is a certain function of $M$ and $N$, is satisfied. A challenging research problem is to determine if there exists a weight vector which gives rise to a tighter GLB for \textit{all} (not just \textit{some}) $K\geq \overline{K}+1$ and $M\geq2$, especially for large $N$, i.e., the condition is {asymptotically} both necessary and sufficient. To achieve this, we \textit{analytically} optimize the GLB which is (in general) non-convex as the numerator term is an indefinite quadratic function of the weight vector. Our key idea is to apply the frequency domain decomposition of the circulant matrix (in the numerator term) to convert the non-convex problem into a convex one. Following this optimization approach, we derive a new weight vector meeting the aforementioned objective and prove that it is a local minimizer of the GLB under certain conditions.

A Tighter Correlation Lower Bound for Quasi-Complementary Sequence Sets

Levenshtein improved the famous Welch bound on aperiodic correlation for binary sequences by utilizing some properties of the weighted mean square aperiodic correlation. Following Levenshtein's idea, a new correlation lower bound for quasi-complementary sequence sets (QCSSs) over the complex roots of unity is proposed in this paper. The derived lower bound is shown to be tighter than the Welch bound for QCSSs when the set size is greater than some value. The conditions for meeting the new bound with equality are also investigated.

Constructions of Optimal and Near-Optimal Quasi-Complementary Sequence Sets from Singer Difference Sets

Compared with the perfect complementary sequence sets, quasi-complementary sequence sets (QCSSs) have the advantage of supporting more users in multicarrier CDMA communications. Constructions for optimal and near-optimal periodic QCSSs are proposed in this paper by using the Singer difference sets and the existing optimal quaternary sequence sets. The maximum periodic correlation magnitude of the proposed optimal QCSS achieves the derived periodic correlation lower bound asymptotically. To the authors' best knowledge, such optimal QCSSs haven't been reported before.