Frame Moments and Welch Bound with Erasures

The Welch (lower) Bound on the mean square cross correlation between n unit-norm vectors f-{1}, \ldots, f-{n} in the m dimensional space (\mathbb{R}^{m} or \mathbb{C}^{m}), for n\geq m, is a useful tool in the analysis and design of spread spectrum communications, compressed sensing and analog coding. Letting F=[f-{1}\vert \ldots\vert 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 present the erasure Welch bound and generalize it to the d-th order moment of the reduced frame, for d == 2, 3, 4. We provide simple, explicit formulae for the generalized bound, which interestingly is equal to the d-th moment of Wachter's classical MANOVA distribution plus a vanishing term (as n goes to infinity with \displaystyle \frac{m}{n} held constant). The bound holds with equality if (and for d = 4 only if) F is an Equiangular Tight Frame (ETF). Hence, our results offer a novel perspective on the superiority of ETFs over other frames, and provide explicit characterization for their subset moments.