## 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 language | American English |
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Title of host publication | 2018 IEEE International Symposium on Information Theory, ISIT 2018 |

Publisher | Institute of Electrical and Electronics Engineers Inc. |

Pages | 2057-2061 |

Number of pages | 5 |

ISBN (Print) | 9781538647806 |

DOIs | |

State | Published - 15 Aug 2018 |

Event | 2018 IEEE International Symposium on Information Theory, ISIT 2018 - Vail, United States Duration: 17 Jun 2018 → 22 Jun 2018 |

### Publication series

Name | IEEE International Symposium on Information Theory - Proceedings |
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Volume | 2018-June |

ISSN (Print) | 2157-8095 |

### Conference

Conference | 2018 IEEE International Symposium on Information Theory, ISIT 2018 |
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Country/Territory | United States |

City | Vail |

Period | 17/06/18 → 22/06/18 |

### Bibliographical note

Funding Information:∗This work has been partially supported by the Israeli Science Foundation grants no. 676/15, 1523/16.

Publisher Copyright:

© 2018 IEEE.

## Keywords

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