Frame Moments and Welch Bound with Erasures

Abstract

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 $n\ge m$. 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 $(F^{\prime }F{)}^2$. 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.