Frame Moments and Welch Bound with Erasures

Marina Haikin, Ram Zamir, Matan Gavish

Research output: Chapter in Book/Report/Conference proceedingConference contributionpeer-review

5 Scopus citations

Abstract

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.

Original languageAmerican English
Title of host publication2018 IEEE International Symposium on Information Theory, ISIT 2018
PublisherInstitute of Electrical and Electronics Engineers Inc.
Pages2057-2061
Number of pages5
ISBN (Print)9781538647806
DOIs
StatePublished - 15 Aug 2018
Event2018 IEEE International Symposium on Information Theory, ISIT 2018 - Vail, United States
Duration: 17 Jun 201822 Jun 2018

Publication series

NameIEEE International Symposium on Information Theory - Proceedings
Volume2018-June
ISSN (Print)2157-8095

Conference

Conference2018 IEEE International Symposium on Information Theory, ISIT 2018
Country/TerritoryUnited States
CityVail
Period17/06/1822/06/18

Bibliographical note

Publisher Copyright:
© 2018 IEEE.

Keywords

  • Analog coding
  • Code division multiple access
  • Equiangular tight frames
  • MANOVA distribution
  • Random matrix theory
  • Welch bound

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