Frame Moments and Welch Bound with Erasures
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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 (R ^m or C ^m), for n≥ 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 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 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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