Frame Moments and Welch Bound with Erasures

The Welch (lower) Bound on the 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, is a useful tool in the analysis and design of spread spectrum communications, compressed sensing and analog coding. 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 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 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.