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 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.

Design of Incidence Matrices With Limited Constellation Expansion in Massive Connectivity NOMA Systems

“…superimposition of the 3 MOLS(6), and 16, as shown at the bottom of the page, and its sparsity is shown in Figure 4. From Figure 4, we can visually observe that the ratio of the sparsity of SOMA(3,5) is 1 5 , which originates from the order n = 5 of this SOMA structure. Therefore, the higher the order of SOMA, the sparser the incidence matrix becomes.…”

“…Through a combinatorial design known as Simple Orthogonal Multi-Arrays (SOMA), our method is capable of producing larger dimension incidence matrices that are confirmed to be near equiangular tight frames (ETFs) by frame theory. ETFs have significant applications in signal processing and coding theory because of their robustness to noise and transmission losses [1]. ETFs are characterized by the fact that the coherence between any two distinct row or column vectors is equal to the Welch bound [2].…”

Sparsity-driven deterministic sampling strategy for coded aperture x-ray computed tomography

“…Hence, they provide stable sparse signal representation [9,23,24]. Therefore, if our objective is realised, our proposed DRMs, being close to ETFs also have full row rank, good condition number and low coherence, which are desired for CS-based reconstruction [6,26]. To achieve the stated objective, we view the deterministic row selection procedure from a full Radon matrix as a preconditioning method where the left preconditioner is a diagonal matrix consisting of nonzero diagonal entries corresponding to the selected row indices.…”

“…Further, minimizing the coherence or total coherence leads to an approximate tight frame property for the sensing matrix, which naturally implies stability in the recovery [4,23]. Two classes of optimization methods (greedy and convex) exist for solving (6) or its convex relaxation through the l 1 -norm. Among available solvers, OMP and TVAL3 are two popular ones.The OMP is a very simple solver [31], which at the first step starts the search by identifying a column of A possessing maximum correlation with y and thereafter it searches iteratively for the column in A that has maximum correlation with the current residual.…”