Frame Moments and Welch Bound with Erasures

TL;DR

This paper extends the Welch bound to settings with random erasures, providing explicit formulas for higher moments and highlighting the optimality of Equiangular Tight Frames in various applications.

Contribution

It introduces the erasure Welch bound for reduced frames with random erasures and generalizes it to higher moments, with explicit formulas for moments d=2,3,4.

Findings

The erasure Welch bound is derived for the expected Gram matrix of reduced frames.

Explicit formulas for the bounds at moments d=2,3,4 are provided.

Equiangular Tight Frames achieve equality in the bounds, demonstrating their optimality.

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 $n\geq m$. 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.