Low coherence unit norm tight frames

by   Somantika Datta, et al.

Equiangular tight frames (ETFs) have found significant applications in signal processing and coding theory due to their robustness to noise and transmission losses. ETFs are characterized by the fact that the coherence between any two distinct vectors is equal to the Welch bound. This guarantees that the maximum coherence between pairs of vectors is minimized. Despite their usefulness and widespread applications, ETFs of a given size N are only guaranteed to exist in R^d or C^d if N = d + 1. This leads to the problem of finding approximations of ETFs of N vectors in R^d or C^d where N > d+1. To be more precise, one wishes to construct a unit norm tight frame (UNTF) such that the maximum coherence between distinct vectors of this frame is as close to the Welch bound as possible. In this paper low coherence UNTFs in R^d are constructed by adding a strategically chosen set of vectors called an "optimal" set to an existing ETF of d+1 vectors. In order to do so, combinatorial objects called block designs are used. Estimates are provided on the maximum coherence between distinct vectors of this low coherence UNTF. It is shown that for certain block designs, the constructed UNTF attains the smallest possible maximum coherence between pairs of vectors among all UNTFs containing the starting ETF of d+1 vectors. This is particularly desirable if there does not exist a set of the same size for which the Welch bound is attained.


page 1

page 2

page 3

page 4


Algorithms for the Construction of Incoherent Frames Under Various Design Constraints

Unit norm finite frames are extensions of orthonormal bases with many ap...

On the Search for Tight Frames of Low Coherence

We introduce a projective Riesz s-kernel for the unit sphere S^d-1 and i...

Harmonic equiangular tight frames comprised of regular simplices

An equiangular tight frame (ETF) is a sequence of unit-norm vectors in a...

Doubly transitive equiangular tight frames that contain regular simplices

An equiangular tight frame (ETF) is a finite sequence of equal norm vect...

Frame Moments and Welch Bound with Erasures

The Welch Bound is a lower bound on the root mean square cross correlati...

A note on tight projective 2-designs

We study tight projective 2-designs in three different settings. In the ...

Finite Block Length Analysis on Quantum Coherence Distillation and Incoherent Randomness Extraction

We introduce a variant of randomness extraction framework in the context...

Please sign up or login with your details

Forgot password? Click here to reset