Reconstruction of irregularly sampled discrete-time bandlimited signals with unknown sampling locations

The purpose of this paper is to develop methods that can reconstruct a bandlimited discrete-time signal from an irreg- ular set of samples at unknown locations. We define a solution to the problem using first a geometric and then an algebraic point of view. We find the locations of the irregular set of samples by treating the problem as a combinatorial optimization problem. We employ an exhaustive method and two descent methods: the random search and cyclic coordinate methods. The numerical simulations were made on three types of irregular sets of locations: random sets; sets with jitter around a uniform set; and periodic nonuniform sets. Furthermore, for the periodic nonuniform set of locations, we de- velop a fast scheme that reduces the computational complexity of the problem by exploiting the periodic nonuniform structure of the sample locations in the DFT

Bandlimited Spatial Field Sampling with Mobile Sensors in the Absence of Location Information

Sampling of physical fields with mobile sensor is an emerging area. In this context, this work introduces and proposes solutions to a fundamental question: can a spatial field be estimated from samples taken at unknown sampling locations? Unknown sampling location, sample quantization, unknown bandwidth of the field, and presence of measurement-noise present difficulties in the process of field estimation. In this work, except for quantization, the other three issues will be tackled together in a mobile-sampling framework. Spatially bandlimited fields are considered. It is assumed that measurement-noise affected field samples are collected on spatial locations obtained from an unknown renewal process. That is, the samples are obtained on locations obtained from a renewal process, but the sampling locations and the renewal process distribution are unknown. In this unknown sampling location setup, it is shown that the mean-squared error in field estimation decreases as $O(1/n)$ where $n$ is the average number of samples collected by the mobile sensor. The average number of samples collected is determined by the inter-sample spacing distribution in the renewal process. An algorithm to ascertain spatial field's bandwidth is detailed, which works with high probability as the average number of samples $n$ increases. This algorithm works in the same setup, i.e., in the presence of measurement-noise and unknown sampling locations.Comment: Submitted to IEEE Trans on Signal Processin

Slepian functions and their use in signal estimation and spectral analysis

It is a well-known fact that mathematical functions that are timelimited (or spacelimited) cannot be simultaneously bandlimited (in frequency). Yet the finite precision of measurement and computation unavoidably bandlimits our observation and modeling scientific data, and we often only have access to, or are only interested in, a study area that is temporally or spatially bounded. In the geosciences we may be interested in spectrally modeling a time series defined only on a certain interval, or we may want to characterize a specific geographical area observed using an effectively bandlimited measurement device. It is clear that analyzing and representing scientific data of this kind will be facilitated if a basis of functions can be found that are "spatiospectrally" concentrated, i.e. "localized" in both domains at the same time. Here, we give a theoretical overview of one particular approach to this "concentration" problem, as originally proposed for time series by Slepian and coworkers, in the 1960s. We show how this framework leads to practical algorithms and statistically performant methods for the analysis of signals and their power spectra in one and two dimensions, and on the surface of a sphere.Comment: Submitted to the Handbook of Geomathematics, edited by Willi Freeden, Zuhair M. Nashed and Thomas Sonar, and to be published by Springer Verla

Performance of Linear Field Reconstruction Techniques with Noise and Uncertain Sensor Locations

We consider a wireless sensor network, sampling a bandlimited field, described by a limited number of harmonics. Sensor nodes are irregularly deployed over the area of interest or subject to random motion; in addition sensors measurements are affected by noise. Our goal is to obtain a high quality reconstruction of the field, with the mean square error (MSE) of the estimate as performance metric. In particular, we analytically derive the performance of several reconstruction/estimation techniques based on linear filtering. For each technique, we obtain the MSE, as well as its asymptotic expression in the case where the field number of harmonics and the number of sensors grow to infinity, while their ratio is kept constant. Through numerical simulations, we show the validity of the asymptotic analysis, even for a small number of sensors. We provide some novel guidelines for the design of sensor networks when many parameters, such as field bandwidth, number of sensors, reconstruction quality, sensor motion characteristics, and noise level of the measures, have to be traded off

Scalar and vector Slepian functions, spherical signal estimation and spectral analysis

It is a well-known fact that mathematical functions that are timelimited (or spacelimited) cannot be simultaneously bandlimited (in frequency). Yet the finite precision of measurement and computation unavoidably bandlimits our observation and modeling scientific data, and we often only have access to, or are only interested in, a study area that is temporally or spatially bounded. In the geosciences we may be interested in spectrally modeling a time series defined only on a certain interval, or we may want to characterize a specific geographical area observed using an effectively bandlimited measurement device. It is clear that analyzing and representing scientific data of this kind will be facilitated if a basis of functions can be found that are "spatiospectrally" concentrated, i.e. "localized" in both domains at the same time. Here, we give a theoretical overview of one particular approach to this "concentration" problem, as originally proposed for time series by Slepian and coworkers, in the 1960s. We show how this framework leads to practical algorithms and statistically performant methods for the analysis of signals and their power spectra in one and two dimensions, and, particularly for applications in the geosciences, for scalar and vectorial signals defined on the surface of a unit sphere.Comment: Submitted to the 2nd Edition of the Handbook of Geomathematics, edited by Willi Freeden, Zuhair M. Nashed and Thomas Sonar, and to be published by Springer Verlag. This is a slightly modified but expanded version of the paper arxiv:0909.5368 that appeared in the 1st Edition of the Handbook, when it was called: Slepian functions and their use in signal estimation and spectral analysi

On the Estimation of Nonrandom Signal Coefficients from Jittered Samples

This paper examines the problem of estimating the parameters of a bandlimited signal from samples corrupted by random jitter (timing noise) and additive iid Gaussian noise, where the signal lies in the span of a finite basis. For the presented classical estimation problem, the Cramer-Rao lower bound (CRB) is computed, and an Expectation-Maximization (EM) algorithm approximating the maximum likelihood (ML) estimator is developed. Simulations are performed to study the convergence properties of the EM algorithm and compare the performance both against the CRB and a basic linear estimator. These simulations demonstrate that by post-processing the jittered samples with the proposed EM algorithm, greater jitter can be tolerated, potentially reducing on-chip ADC power consumption substantially.Comment: 11 pages, 8 figure