Perturbation of Regular Sampling in Shift-Invariant Spaces for Frames

Zhao, Ping; Zhao, Chun; Casazza, Peter G.

doi:10.1109/tit.2006.881704

Cited by 27 publications

(24 citation statements)

References 15 publications

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“…Also, note that, in the product in (19), each factor depends on a single random vector, . Since and is invertible then, by defining we have and (21) In the last term of (21) we removed the subscript from the argument of the average operator, since the distribution of does not depend on .…”

Section: Discussionmentioning

confidence: 99%

“…ii) Signal reconstruction in sensor networks. Sensor networks, whose nodes sample a physical field, like air temperature, light intensity, pollution levels or rain falls, typically represent an example of quasi-equally spaced sampling [2], [7], [19], [20]. Indeed, often sensors are not regularly deployed in the area of interest due to terrain conditions and deployment practicality and, thus, the physical field is not regularly sampled in the space domain.…”

Section: Applicationsmentioning

confidence: 99%

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Asymptotic Analysis of Multidimensional Jittered Sampling

Nordio

Chiasserini

Viterbo

2010

IEEE Trans. Signal Process.

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Abstract-We study a class of random matrices that appear in several communication and signal processing applications, and whose asymptotic eigenvalue distribution is closely related to the reconstruction error of an irregularly sampled bandlimited signal. We focus on the case where the random variables characterizing these matrices are d-dimensional vectors, independent, and quasi-equally spaced, i.e., they have an arbitrary distribution and their averages are vertices of a d-dimensional grid. Although a closed form expression of the eigenvalue distribution is still unknown, under these conditions we are able i) to derive the distribution moments as the matrix size grows to infinity, while its aspect ratio is kept constant, and ii) to show that the eigenvalue distribution tends to the Marčenko-Pastur law as d 0 ! 1. These results can find application in several fields, as an example we show how they can be used for the estimation of the mean square error provided by linear reconstruction techniques.

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Section: Discussionmentioning

confidence: 99%

Section: Applicationsmentioning

confidence: 99%

Asymptotic Analysis of Multidimensional Jittered Sampling

Nordio

Chiasserini

Viterbo

2010

IEEE Trans. Signal Process.

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“…The existence of stable perturbed sampling sets and formulas has been studied in spaces of band-limited functions (see [12] and references therein), in wavelet spaces [5,15] and in shift invariant spaces [6,14,25] but the resulting sampling formulas are complicated. We mention that certain estimates on the maximum perturbation have been established, based on decay assumptions on the generator function φ [6,25].…”

Section: Introductionmentioning

confidence: 99%

“…We mention that certain estimates on the maximum perturbation have been established, based on decay assumptions on the generator function φ [6,25]. In [19, Theorem 3.2] a perturbation formula for non-necessarily shift-invariant spaces was established.…”

Section: Introductionmentioning

confidence: 99%

Perturbed sampling formulas and local reconstruction in shift invariant spaces

Atreas

2011

Journal of Mathematical Analysis and Applications

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Let V φ be a closed subspace of L 2 (R) generated from the integer shifts of a continuous function φ with a certain decay at infinity and a non-vanishing property for the functionIn this paper we determine a positive number δ φ so that the set {n + δ n } n∈Z is a set of stable sampling for the space V φ for any selection of the elements δ n within the ranges ±δ φ . We demonstrate the resulting sampling formula (called perturbation formula) for functions f ∈ V φ and also we establish a finite reconstruction formula approximating f on bounded intervals. We compute the corresponding error and we provide estimates for the jitter error as well.

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“…The work in Ganesan et al [2004] investigates this issue showing how irregular spatiotemporal sampling affects the performance of sensor networks. Other interesting studies can be found in Zhao et al [2006] and Early and Long [2001], just to name a few, which address the perturbations of regular sampling in shift-invariant spaces ] and the reconstruction of irregularly sampled images in presence of measure noise [Early and Long 2001].…”

Section: Introductionmentioning

confidence: 99%

The impact of quasi-equally spaced sensor topologies on signal reconstruction

Nordio

Chiasserini

Viterbo

2010

ACM Trans. Sen. Netw.

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A wireless sensor network with randomly deployed nodes can be used to provide an irregular sampling of a physical field of interest. We assume that a sink node collects the data gathered by the sensors and uses a linear filter for the reconstruction of a bandlimited scalar field defined over a d -dimensional domain. Sensors' locations are assumed to be known at the sink node, up to a certain position error. We then take the mean square error (MSE) of the reconstructed field as performance metric, and evaluate the effect of both uniform and quasi-equally spaced sensor layouts on the quality of the reconstructed field. We define a parameter that provides a measure of the regularity of the sensors deployment, and, through asymptotic analysis, we derive the MSE in the case of different sensor spatial distributions. For two of them, an approximate closed form expression is obtained. We validate our analysis through numerical results, and we show that an excellent match exists between analysis and simulation even for a small number of sensors.

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Perturbation of Regular Sampling in Shift-Invariant Spaces for Frames

Cited by 27 publications

References 15 publications

Asymptotic Analysis of Multidimensional Jittered Sampling

Asymptotic Analysis of Multidimensional Jittered Sampling

Perturbed sampling formulas and local reconstruction in shift invariant spaces

The impact of quasi-equally spaced sensor topologies on signal reconstruction

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