Equiangular tight frames

Malozemov, V. N.; Pevnyi, A. B.

doi:10.1007/s10958-009-9366-6

Cited by 18 publications

(22 citation statements)

References 17 publications

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“…For the case where the sparsity level of the signal is greater than two, we provide a lower bound on the worst-case performance. If the number m of measurements allowed is greater than or equal to √ N , then the Grassmannian line packing frame will be an equiangular uniform tight frame (see, e.g., [26]- [33]) and the maximal worst-case SNR can be expressed in terms of the Welch bound. Numerical examples presented in Section 7 show that Grassmannian line packing frames provide better worst-case performance than matrices with i.i.d.…”

Section: Problem Statement and Main Contributionsmentioning

confidence: 99%

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Measurement design for detecting sparse signals

Zahedi

Pezeshki

Chong

2012

Physical Communication

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We consider the problem of testing for the presence (or detection) of an unknown sparse signal in additive white noise. Given a fixed measurement budget, much smaller than the dimension of the signal, we consider the general problem of designing compressive measurements to maximize the measurement signal-to-noise ratio (SNR), as increasing SNR improves the detection performance in a large class of detectors. We use a lexicographic optimization approach, where the optimal measurement design for sparsity level k is sought only among the set of measurement matrices that satisfy the optimality conditions for sparsity level k−1. We consider optimizing two different SNR criteria, namely a worst-case SNR measure, over all possible realizations of a k-sparse signal, and an average SNR measure with respect to a uniform distribution on the locations of the up to k nonzero entries in the signal. We establish connections between these two criteria and certain classes of tight frames. We constrain our measurement matrices to the class of tight frames to avoid coloring the noise covariance matrix. For the worst-case problem, we show that the optimal measurement matrix is a Grassmannian line packing for most-and a uniform tight frame for allsparse signals. For the average SNR problem, we prove that the optimal measurement matrix is a uniform tight frame with minimum sum-coherence for most-and a tight frame for all-sparse signals.

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Section: Problem Statement and Main Contributionsmentioning

confidence: 99%

“…Remark 2: Methods for constructing uniform tight frames with frame elements that have a coherence µ * is equivalent to optimal Grassmannian packings of one-dimensional subspaces, or Grassmannian line packings (see, e.g., [25]- [33]). We will say more about this point later in the paper.…”

Section: Sparsity Level K =mentioning

confidence: 99%

Measurement design for detecting sparse signals

Zahedi

Pezeshki

Chong

2012

Physical Communication

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“…In other cases where N and m do not satisfy the condition (19) or (20), the bound (16) suggests that we use an optimal Grassmannian packing where the k largest angles among angles between column pairs of the matrix C * H are as close to the angle α as possible. This is, however, a very difficult problem since even finding optimal Grassmannian packings for different values of N and m is still an open problem.…”

Section: Equiangular Uniform Tight Frames and Grassmannian Packingsmentioning

confidence: 99%

Robust measurement design for detecting sparse signals: Equiangular uniform tight frames and grassmannian packings

Zahedi

Pezeshki

Chong

2010

Proceedings of the 2010 American Control Conference

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Detecting a sparse signal in noise is fundamentally different from reconstructing a sparse signal, as the objective is to optimize a detection performance criterion rather than to find the sparsest signal that satisfies a linear observation equation. In this paper, we consider the design of lowdimensional (compressive) measurement matrices for detecting sparse signals in white Gaussian noise. We use a lexicographic optimization approach to maximize the worst-case signal-tonoise ratio (SNR). More specifically, we find an optimal solution for a k-sparse signal among optimal solutions subject to sparsity level k − 1. We show that for all sparse signals, columns of the optimal measurement matrix must form a uniform tight frame. For 2-sparse signals, the smallest angle among angles between element pairs of this frame must be maximized. In this case, the optimal solution matrix is an optimal Grassmannian packing. For k-sparse signals where k > 2, the largest angle among such angles must be as close to the maximum smallest angle as possible. We show that under certain conditions, columns of the optimal measurement matrix form an equiangular uniform tight frame. For this case, we derive an expression for the maximal SNR in the worst-case scenario, as a function of the signal dimension and the number of measurements.

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“…To further analyze (9) we note that for an arbitrary P × Q matrix, its coherence is bounded by μ ≥ √ Q−P P (Q−1) , a bound known as the Welch bound [10]. Moreover, this bound is achieved only if all pairs of columns have the same magnitude inner product, which is then equal to this lower bound (such matrices are called equiangular tight frames (ETF) [11]). In the case of no overlap (p = m = √ M ), C and Φ0 can be designed freely.…”

Section: Coherence Analysismentioning

confidence: 99%

Sparsity order estimation for single snapshot compressed sensing

Römer

Lavrenko

Galdo

et al. 2014

2014 48th Asilomar Conference on Signals, Systems and Computers

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In this paper we discuss the estimation of the spar-sity order for a Compressed Sensing scenario where only a single snapshot is available. We demonstrate that a specific design of the sensing matrix based on Khatri-Rao products enables us to transform this problem into the estimation of a matrix rank in the presence of additive noise. Thereby, we can apply existing model order selection algorithms to determine the sparsity order. The matrix is a rearranged version of the observation vector which can be constructed by concatenating a series of non-overlapping or overlapping blocks of the original observation vector. In both cases, a Khatri-Rao structured measurement matrix is required with the main difference that in the latter case, one of the factors must be a Vandermonde matrix. We discuss the choice of the parameters and show that an increasing amount of block overlap improves the sparsity order estimation but it increases the coherence of the sensing matrix. We also explain briefly that the proposed measurement matrix design introduces certain multilinear structures into the observations which enables us to apply tensor-based signal processing, e.g., for enhanced denoising or improved sparsity order estimation. © 2014 IEEE

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Equiangular tight frames

Cited by 18 publications

References 17 publications

Measurement design for detecting sparse signals

Measurement design for detecting sparse signals

Robust measurement design for detecting sparse signals: Equiangular uniform tight frames and grassmannian packings

Sparsity order estimation for single snapshot compressed sensing

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