框架力矩与带删除的Welch界限
Frame Moments and Welch Bound with Erasures
摘要：
对于$n\geq m$，Welch界限是$m$维空间（$\mathbb{R} ^m$或$\mathbb{C} ^m$）中$n$单位范数向量$f_1,...,f_n$之间的均方根互相关的下限。让$F = [f_1|...|f_n]$表示$m$ × $n$框架矩阵，Welch界可以被看作是$F$的二阶矩的下界，即在平方格拉姆矩阵$(F'F)^2$的迹上.我们考虑一个擦除设置，其中一个减少的框架，组成的随机子集的伯努利选择的向量，是感兴趣的。我们扩展Welch界到这个设置，并提出了{ \em擦除Welch界}的预期值的减少帧的革兰氏矩阵。有趣的是，这个界限推广到$F$的$d$阶矩。我们提供了简单的，明确的公式的广义界$d=2,3,4$，这是总和的$d$ -th时刻Wachter的经典MANOVA分布和消失项（$n$去无穷大与$\frac{m}{n}$保持常数）。当$F$是一个等角紧标架（ETF）时（且对于$d = 4$仅当$F$是一个等角紧标架时），该界相等成立.我们的研究结果提供了一个新的视角，在各种应用中，包括扩频通信，压缩传感和模拟编码的ETF的优越性超过其他帧。
英文摘要：
The Welch Bound is a lower bound on the root mean square cross correlation between $n$ unit-norm vectors $f_1,...,f_n$ in the $m$ dimensional space ($\mathbb{R} ^m$ or $\mathbb{C} ^m$), for $n\geq m$. Letting $F = [f_1|...|f_n]$ denote the $m$-by-$n$ frame matrix, the Welch bound can be viewed as a lower bound on the second moment of $F$, namely on the trace of the squared Gram matrix $(F'F)^2$. We consider an erasure setting, in which a reduced frame, composed of a random subset of Bernoulli selected vectors, is of interest. We extend the Welch bound to this setting and present the {\em erasure Welch bound} on the expected value of the Gram matrix of the reduced frame. Interestingly, this bound generalizes to the $d$-th order moment of $F$. We provide simple, explicit formulae for the generalized bound for $d=2,3,4$, which is the sum of the $d$-th moment of Wachter's classical MANOVA distribution and a vanishing term (as $n$ goes to infinity with $\frac{m}{n}$ held constant). The bound holds with equality if (and for $d = 4$ only if) $F$ is an Equiangular Tight Frame (ETF). Our results offer a novel perspective on the superiority of ETFs over other frames in a variety of applications, including spread spectrum communications, compressed sensing and analog coding.
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