Frame Moments and Welch Bound with Erasures

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.