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.