Extrapolating a band-limited function from its samples taken in a finite interval

Landau, H. J.

doi:10.1109/tit.1986.1057205

Cited by 39 publications

(41 citation statements)

References 8 publications

(11 reference statements)

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“…Then the mathematical problem is to find conditions under which f can be reconstructed completely from its samples f (x n ). This problem is almost completely understood thanks to the work of Duffin-Schaeffer, Beurling, Malliavin, Landau, Pavlov and others [5,2,3,12,13,14,15,17]. Their work has provided deep insights into sets of uniqueness, Riesz bases, and sets of stable sampling.…”

Section: Karlheinz Gröchenigmentioning

confidence: 99%

Irregular sampling, Toeplitz matrices, and the approximation of entire functions of exponential type

Gröchenig

1999

Math. Comp.

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Abstract. In many applications one seeks to recover an entire function of exponential type from its non-uniformly spaced samples. Whereas the mathematical theory usually addresses the question of when such a function in L 2 (R) can be recovered, numerical methods operate with a finite-dimensional model. The numerical reconstruction or approximation of the original function amounts to the solution of a large linear system.We show that the solutions of a particularly efficient discrete model in which the data are fit by trigonometric polynomials converge to the solution of the original infinite-dimensional reconstruction problem. This legitimatizes the numerical computations and explains why the algorithms employed produce reasonable results. The main mathematical result is a new type of approximation theorem for entire functions of exponential type from a finite number of values. From another point of view our approach provides a new method for proving sampling theorems.

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Section: Karlheinz Gröchenigmentioning

confidence: 99%

Irregular sampling, Toeplitz matrices, and the approximation of entire functions of exponential type

Gröchenig

1999

Math. Comp.

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“…Accordingly, we can apply to it all results known for band-limited functions, in particular, the noteworthy property that these functions are analytical, i.e., such that their exact knowledge in a finitelength interval allows, in principle, their extrapolation over the entire domain of definition, having infinite extension [4][5][6][7][8][9], thus potentially prospecting the achievement of unlimited angular resolution. As extensively discussed in these references, however, the implementation aspects, such as the need to sample the field by means of a sensor array, with consequent loss of information due to the finiteness of the spatial sampling frequency, and the errors inherent in the analog-to-digital conversion and in the array implementation, along with other limiting factors, make the above extrapolation process reliable only within a limited range in the vicinity of the array.…”

Section: Introductionmentioning

confidence: 99%

“…In the literature, the mentioned techniques have been addressed with reference to different categories of problems. On the one hand, super-resolution algorithms have been studied extensively in the past in conjunction with ingenious extrapolation schemes to improve the discrimination capability of specific instrumentation, such as optical sensors or spectrum analyzers, in response to one-or two-dimensional band-limited signals observed in truncated intervals [4][5][6][7][8][9]. In these references the interest of researchers is mainly focused on the extrapolation schemes and the relevant performance limits, with no reference to their possible applications to beamforming issues.…”

Section: Introductionmentioning

confidence: 99%

Enhancing resolution properties of array antennas via field extrapolation: application to MIMO systems

Reggiannini

2015

EURASIP J. Adv. Signal Process.

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This paper is concerned with spatial properties of linear arrays of antennas spaced less than half wavelength. Possible applications are in multiple-input multiple-output (MIMO) wireless links for the purpose of increasing the spatial multiplexing gain in a scattering environment, as well as in other areas such as sonar and radar. With reference to a receiving array, we show that knowledge of the received field can be extrapolated beyond the actual array size by exploiting the finiteness of the interval of real directions from which the field components impinge on the array. This property permits to increase the performance of the array in terms of angular resolution. A simple signal processing technique is proposed allowing formation of a set of beams capable to cover uniformly the entire horizon with an angular resolution better than that achievable by a classical uniform-weighing half-wavelength-spaced linear array. Results are also applicable to active arrays. As the above approach leads to arrays operating in super-directive regime, we discuss all related critical aspects, such as sensitivity to external and internal noises and to array imperfections, and bandwidth, so as to identify the basic design criteria ensuring the array feasibility.

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“…As well-known, this is an ill-posed problem widely studied in literature with reference to the case of bandlimited signals which are known only in a finite time interval [12][13][14][15][16][17][18][19][20][21].…”

Section: Introductionmentioning

confidence: 99%

“…As explicitly stated in [21], when the samples are error affected, an accurate extrapolation of the signal is possible for at most a bounded distance beyond the observation interval. Accordingly, since physical measurements can never be perfect, this implies that only few samples external to the observation interval can be reliably estimated.…”

Section: Introductionmentioning

confidence: 99%

An Effective Technique for Reducing the Truncation Error in the Near-Field-Far-Field Transformation With Plane-Polar Scanning

D’Agostino¹,

Ferrara²,

Gennarelli³

et al. 2007

PIER

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Abstract-An effective approach is proposed in this paper for estimating the near-field data external to the measurement region in the plane-polar scanning. It relies on the nonredundant sampling representations of the electromagnetic field and makes use of the singular value decomposition method for the extrapolation of the outside samples. It is so possible to reduce in a significant way the error due to the truncation of the measurement zone thus increasing the farfield angular region of good reconstruction. The comparison of such an approach, based on the optimal sampling interpolation expansions, with an existing procedure using the cardinal series has highlighted that the proposed technique works better. Some numerical tests are reported for demonstrating its effectiveness.

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Extrapolating a band-limited function from its samples taken in a finite interval

Cited by 39 publications

References 8 publications

Irregular sampling, Toeplitz matrices, and the approximation of entire functions of exponential type

Irregular sampling, Toeplitz matrices, and the approximation of entire functions of exponential type

Enhancing resolution properties of array antennas via field extrapolation: application to MIMO systems

An Effective Technique for Reducing the Truncation Error in the Near-Field-Far-Field Transformation With Plane-Polar Scanning

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