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arXiv 1701.01550cs.ITmath.IT
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准互补序列集紧相关下界的渐近局部最优权向量设计

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

Zilong Liu, Yong Liang Guan, Wai Ho Mow

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摘要：

准互补序列集（QCSS）是指具有低的非平凡非周期自相关和互相关和的二维矩阵的集合。对于多载波码分多址应用，具有低相关和的大QCSS的可用性是期望的。广义Levenshtein界（GLB）是QCSS的最大非周期相关和的下界。GLB的边界表达式是权重向量$\mathbf{w}$的分数二次函数，并且用与QCSS相关联的三个附加参数来表示：集合大小$K$、信道数量$M$和序列长度$N$。已知仅当满足条件$M\geq2$和$K\geq \overline{K}+1$（其中$\overline{K}$是$M$和$N$的某个函数）时，更紧的GLB（与Welch界限相比）才是可能的。一个具有挑战性的研究问题是确定是否存在权重向量，该权重向量引起\textit{all}（不仅仅是\textit{some}）$K\geq \overline{K}+1$和$M\geq2$的更紧密的GLB，特别是对于大的$N$，即，这个条件是渐近必要的和充分的。为了实现这一点，我们\textit{analytically}优化GLB，它（一般）是非凸的，因为分子项是权重向量的不定二次函数。我们的核心思想是应用循环矩阵的频域分解（在分子项中）将非凸问题转化为凸问题。根据这种优化方法，我们得到了一个新的权重向量满足上述目标，并证明了它是一个局部极小的GLB在一定条件下。

英文摘要：

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.

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