Design of universal multicoset sampling patterns for compressed sensing of multiband sparse signals

Jiménez, María Elena Domínguez; González–Prelcic, Nuria; Vazquez-Vilar, Gonzalo; López-Valcarce, Roberto

doi:10.1109/icassp.2012.6288630

Cited by 26 publications

(9 citation statements)

References 9 publications

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“…This theorem permits one to build entire suites of In fact this result has been strengthened in [23]. To fix notations, the base sets we will consider will be for some…”

Section: The Full Spark Condition Ismentioning

confidence: 98%

The spark of Fourier matrices: Connections to vanishing sums and coprimeness

Achanta

Biswas

Dasgupta

et al. 2017

Digital Signal Processing

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We consider conditions under which L rows of an N point DFT matrix form a matrix with spark L+1, i.e. a matrix with full spark. A matrix has spark N +1 if all L columns are linearly independent. This has application in compressed sensing for MRI and synthetic aperture radar, where measurements are under sampled Fourier measurements, and the observation matrix comprises certain rows of the DFT matrix. It is known that contiguous rows of the DFT matrix render full spark and that from such a base set one can build a suite of other sets of rows that maintain full spark. We consider an alternative base set of the form {0, 1, • • • , K} \ {n}, and derive conditions on K, n and the prime factors of N , under which full spark is retained. We show that such a matrix has full spark iff there are no K distinct N-th roots of unity whose n-products form a vanishing sum, and leverage recent characterizations of vanishing sums of N-th roots of unity to establish the stated conditions.

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“…This theorem permits one to build entire suites of In fact this result has been strengthened in [23]. To fix notations, the base sets we will consider will be for some…”

Section: The Full Spark Condition Ismentioning

confidence: 98%

The spark of Fourier matrices: Connections to vanishing sums and coprimeness

Achanta

Biswas

Dasgupta

et al. 2017

Digital Signal Processing

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“…In fact, the latter provides optimal solutions for most values of L below 60 (see Table I). Further alternatives include [57].…”

Section: B Dense Samplersmentioning

confidence: 99%

Compression Limits for Random Vectors with Linearly Parameterized Second-Order Statistics

Romero

López-Valcarce

Leus

2015

IEEE Trans. Inform. Theory

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Abstract-The class of complex random vectors whose covariance matrix is linearly parameterized by a basis of Hermitian Toeplitz (HT) matrices is considered, and the maximum compression ratios that preserve all second-order information are derived -the statistics of the uncompressed vector must be recoverable from a set of linearly compressed observations. This kind of vectors arises naturally when sampling widesense stationary random processes and features a number of applications in signal and array processing.Explicit guidelines to design optimal and nearly optimal schemes operating both in a periodic and non-periodic fashion are provided by considering two of the most common linear compression schemes, which we classify as dense or sparse. It is seen that the maximum compression ratios depend on the structure of the HT subspace containing the covariance matrix of the uncompressed observations. Compression patterns attaining these maximum ratios are found for the case without structure as well as for the cases with circulant or banded structure. Universal samplers are also proposed to compress unknown HT subspaces. I. PRELIMINARIES Consider the problem of estimating the second-order statistics of a zero-mean random vector x ∈ C L from a set of K linear observations collected in the vector y ∈ C K given bywhereΦ ∈ C K×L is a known matrix and several realizations of y may be available. This matrix may be referred to as the compression matrix, measurement matrix or sampler, where compression is achieved by setting K < L (typically K L). The covariance matrix Σ = E xx H contains the second-order statistics of x and is assumed to be a linear combination of the Hermitian Toeplitz (HT) matrices in a given set This problem arises in inference operations over the secondorder statistics of a random vector with a Toeplitz covariance matrix. Operating on the compressed observations y entails multiple advantages due to their smaller dimension. In fact, many research efforts in the last decades have been pointed towards designing compression methods and reconstruction algorithms that allow for sampling rate reductions. While most efforts have been focused on reconstructing x, there were also important advances when only the second-order statistics of this vector are of interest. This paper is concerned with problems of the second kind.The compression ratio ρ = L/K measures how much x is compressed. The maximum compression ratio remains an open problem in many cases of interest; and most existing results rely on the usage of specific reconstruction algorithms (see Sec. I-D). This paper presents a general and unifying framework built on abstract criteria where the maximum compression ratio is defined and computed for most relevant settings. The proofs involved in this theory are constructive, resulting in several methods for designing optimal compression matrices. A. Covariance Matching FormulationThe prior information restricts the structure of Σ, thus determining how much x can be compressed. When no information at all is a...

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“…Discarding the samples whose indices are not present in I must entail no loss of information, at least in the noiseless case. This is the philosophy used in [5] and [6] for sampling of multi-band signals.…”

Section: A Relation To Compressive Samplingmentioning

confidence: 99%

Compressive covariance sampling

Romero

Leus

2013

2013 Information Theory and Applications Workshop (ITA)

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Abstract-Most research efforts in the field of compressed sensing have been pointed towards analyzing sampling and reconstruction techniques for sparse signals, where sampling rates below the Nyquist rate can be reached. When only second-order statistics or, equivalently, covariance information is of interest, perfect signal reconstruction is not required and rate reductions can be achieved even for non-sparse signals. This is what we will refer to as compressive covariance sampling. In this paper, we will study minimum-rate compressive covariance sampling designs within the class of non-uniform samplers. Necessary and sufficient conditions for perfect covariance reconstruction will be provided and connections to the well-known sparse ruler problem will be highlighted. I. PROBLEM STATEMENTConsider the problem of estimating the covariance matrix Σ of a random vector x with components x[n] when it is known that Σ is a linear combination of the matrices in the set S. This problem has a long history and a wide range of applications [1], [2]. Now consider a modification of the same problem where only a subset of the samples x[n] is available, i.e., when the observations are y[m] = x[n m ] for some I = {n 0 , n 1 , . . .} ⊂ Z. Intuitively, if the grid defined by I is dense enough, then this estimation problem, which we label as compressive covariance sampling, can still be solved.More specifically, the above sample selection produces a transformation of the problem: the vector y collecting the compressed observations y[m], has a covariance matrixΣ, which is a linear combination of the matrices inS. The coefficients in the linear combination are in both problems the same so that solving the latter amounts to solving the former. In this paper we address the optimal design of the index set I so that the coefficients can be estimated, i.e., we look for sets of indices with a minimum number of elements guaranteeing that these parameters remain statistically identifiable. A. Relation to Compressive SamplingRecent interest in sampling signals below their Nyquist rate owes to compressive sampling (or compressed sensing) [3], which has motivated a great deal of research efforts in the last few years. In compressive sampling we are interested in reducing the number of samples by focusing on a special family of signals referred to as sparse. These signals are intrinsically redundant, so that this reduction does not entail any information loss if properly done.Non-uniform sampling [4] is a particular case of compressive sampling. Mathematically, this process can be described as first acquiring a continuous-index signal x(t) at rate 1/T , resulting in the sequence x[n] = x(nT ), and then downsampling x[n] to obtain y[m] according to a set of indices I, in a periodic or non-periodic fashion (periodic non-uniform sampling is also known as multi-coset sampling).Either in the context of compressive sampling (the nonuniform sampling version) or in the context of what we call compressive covariance sampling, the design of the set I has ...

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Design of universal multicoset sampling patterns for compressed sensing of multiband sparse signals

Cited by 26 publications

References 9 publications

The spark of Fourier matrices: Connections to vanishing sums and coprimeness

The spark of Fourier matrices: Connections to vanishing sums and coprimeness

Compression Limits for Random Vectors with Linearly Parameterized Second-Order Statistics

Compressive covariance sampling

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