text: 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.
  @cite_6: 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.
  @cite_3: 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.
summary: We point out that our main contribution with respect to previous work on signal sampling and reconstruction is the probabilistic approach we adopt to analyze the quality level of a signal reconstructed from a set of irregular, noisy samples. Our analysis, however, applies to sampling systems where the field reconstruction is performed in a centralized manner. Finally, we highlight that our previous work @cite_3 assumes that sensors are uniformly distributed over the spatial observation interval and may be displaced around a known average location. The effects of noisy measures and jittered positions are analyzed when linear reconstruction techniques are employed. However, only the unidimensional case is studied and semi-analytical derivations of the MSE of the reconstructed field are obtained. In @cite_6 , instead, sensors are assumed to be fixed, and the objective is to evaluate the performance of a linear reconstruction technique in the presence of quasi-equally spaced sensor layouts.