Low coherence unit norm tight frames

Abstract

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 ${\mathbb{R}}^d$ or ${\mathbb{C}}^d$ if $N=d+1$. This leads to the problem of finding approximations of ETFs of N vectors in ${\mathbb{R}}^d$ or ${\mathbb{C}}^d$ 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 ${\mathbb{R}}^d$ 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.

Keywords:

• Block designs
• coherence
• equiangular frames
• tight frames
• Welch bound

AMS Subject Classifications:

• 42C15
• 94Axx

Acknowledgements

The authors would like to thank the anonymous reviewer for insightful comments and useful suggestions that greatly helped to improve the quality of the paper.

Disclosure statement

No potential conflict of interest was reported by the authors.

Notes

1 In an infinite dimensional space, the notion of a frame is far more subtle [28,29] and will not be needed here.

2 The results can be easily generalized to any d-dimensional Hilbert space $\mathcal{H}$ since $\mathcal{H}$ would be isomorphic to ${\mathbb{R}}^d$ or ${\mathbb{C}}^d.$

3 Particular block designs known as Steiner systems have been used to construct equiangular tight frames [18].

Additional information

Funding

This material is based upon work supported by the National Science Foundation [award number CCF-1422252].