We focus on a multidimensional field with uncorrelated spectrum and study the quality of the reconstructed signal when the field samples are irregularly spaced and affected by independent and identically distributed noise. More specifically, we apply linear reconstruction techniques and take the mean-square error (MSE) of the field estimate as a metric to evaluate the signal reconstruction quality. We find that the MSE analysis could be carried out by using the closed-form expression of the eigenvalue distribution of the matrix representing the sampling system. Unfortunately, such distribution is still unknown. Thus, we first derive a closed-form expression of the distribution moments, and we find that the eigenvalue distribution tends to the Marcenko-Pastur distribution as the field dimension goes to infinity. Finally, by using our approach, we derive a tight approximation to the MSE of the reconstructed field.
The goal of this work is to provide an analytical study on the reconstruction quality of a multidimensional physical field, with uncorrelated spectrum. The field samples are (i) irregularly spaced, since they are gathered by a randomly deployed sensor network and (ii) affected by i.i.d. noise. The sink node receives the field samples and runs the reconstruction algorithm in a centralized manner. Our major contributions with respect to previous work are as follows. [1.] Given a @math -dimensional problem formulation, we obtain analytical expressions for the moments of the eigenvalue distribution of the reconstruction matrix. Using the expressions of the moments, we show that the eigenvalue distribution tends to the Mar c enko-Pastur distribution @cite_26 as the field dimension @math . [2.] We apply our results to the study of the quality of a reconstructed field and derive a tight approximation to the MSE of the estimated field.
@cite_26
2060581589
We consider wireless sensor networks whose nodes are randomly deployed and, thus, provide an irregular sampling of the sensed field. The field is assumed to be bandlimited; a sink node collects the data gathered by the sensors and reconstructs the field by using a technique based on linear filtering. By taking the mean square error (MSE) as performance metric, we evaluate the effect of quasi-equally spaced sensor layouts on the quality of the reconstructed signal. The MSE is derived through asymptotic analysis for different sensor spatial distributions, and for two of them we are able to obtain an approximate closed form expression. The case of uniformly distributed sensors is also considered for the sake of comparison. The validity of our asymptotic analysis is shown by comparison against numerical results and it is proven to hold even for a small number of nodes. Finally, with the help of a simple example, we show the key role that our results play in the deployment of sensor networks. We consider the problem of obtaining a high quality estimates of band-limited sensor fields when sensor measurements are noisy and the nodes are irregularly deployed and subject to random motion. We consider the mean square error (MSE) of the estimate and we analytically derive the performance of several reconstruction estimation techniques based on linear filtering. For each technique, we obtain the mean value of the MSE, as well as its asymptotic expression in the case where the field bandwidth and the number of sensors grow to infinity, while their ratio is kept constant. Our results provide useful guidelines for the design of sensor networks when many system parameters have to be traded off. Distributed sampling and reconstruction of a physical field using an array of sensors is a problem of considerable interest in environmental monitoring applications of sensor networks. Our recent work has focused on the sampling of bandlimited sensor fields. However, sensor fields are not perfectly bandlimited but typically have rapidly decaying spectra. In a classical sampling set-up it is possible to precede the A D sampling operation with an appropriate analog anti-aliasing filter. However, in the case of sensor networks, this is infeasible since sampling must precede filtering. We show that even though the effects of aliasing on the reconstruction cannot be prevented due to the "filter-less" sampling constraint, they can be suitably controlled by oversampling and carefully reconstructing the field from the samples. We show using a dither-based scheme that it is possible to estimate non-bandlimited fields with a precision that depends on how fast the spectral content of the field decays. We develop a framework for analyzing non-bandlimited fields that lead to upper bounds on the maximum pointwise error for a spatial bit rate of R bits meter. We present results for fields with exponentially decaying spectra as an illustration. In particular, we show that for fields f(t) with exponential tails; i.e., F( spl omega ) < spl pi spl alpha sup - spl alpha | spl omega | , the maximum pointwise error decays as c2e sup -a sub 1 spl radic R+ c3 1 spl radic R (e sup -2a sub 1 spl radic R) with spatial bit rate R bits meter. Finally, we show that for fields with spectra that have a finite second moment, the distortion decreases as O((1 N) sup 2 3 ) as the density of sensors, N, scales up to infinity . We show that if D is the targeted non-zero distortion, then the required (finite) rate R scales as O (1 spl radic D log 1 D). In this paper we investigate the capability of large-scale sensor networks to measure and transport a two-dimensional field. We consider a data-gathering wireless sensor network in which densely deployed sensors take periodic samples of the sensed field, and then scalar quantize, encode and transmit them to a single receiver central controller where snapshot images of the sensed field are reconstructed. The quality of the reconstructed field is limited by the ability of the encoder to compress the data to a rate less than the single-receiver transport capacity of the network. Subject to a constraint on the quality of the reconstructed field, we are interested in how fast data can be collected (or equivalently how closely in time these snapshots can be taken) due to the limitation just mentioned. As the sensor density increases to infinity, more sensors send data to the central controller. However, the data is more correlated, and the encoder can do more compression. The question is: Can the encoder compress sufficiently to meet the limit imposed by the transport capacity? Alternatively, how long does it take to transport one snapshot? We show that as the density increases to infinity, the total number of bits required to attain a given quality also increases to infinity under any compression scheme. At the same time, the single-receiver transport capacity of the network remains constant as the density increases. We therefore conclude that for the given scenario, even though the correlation between sensor data increases as the density increases, any data compression scheme is insufficient to transport the required amount of data for the given quality. Equivalently, the amount of time it takes to transport one snapshot goes to infinity.