In this paper, we address the problem of the estimation of a spatial field defined over a two-dimensional space with wireless sensor networks. We assume that the field is (spatially) bandlimited and that it is sampled by a set of sensors which are randomly deployed in a given geographical area. Further, we impose a total bandwidth constraint which forces the quantization error in the sensor-to-FC (Fusion Center) channels to depend on the actual number of sensors in the network. With these assumptions, we derive an analytical expression of the mean-square error (MSE) in the reconstructed random field and, on that basis, an approximate closed-form expression of the optimal sensor density which attains the best trade-off in terms of observation, sampling and quantization noises. The analysis is carried out both in Gaussian and Rayleigh-fading scenarios without transmit Channel State Information (CSI). For the latter scenario, we also derive an expression of the common and constant rate at which the observations must be quantized. Computer simulation results illustrate the dependency of the optimal operating point on the variance of the observation noise or the signal-to-noise ratio in the sensor-to-FC channels, as well as the scaling law of the reconstruction MSE (which is also derived analytically) for both scenarios.

We consider an environmental monitoring application where a scalar field (e.g., atmospheric pressure) is to be sensed by randomly distributed nodes. Each node takes a sample of a l-dimensional signal in its position and sends it to a collector entity through a wireless link. The latter performs signal reconstruction based on the received samples. Because of total bandwidth constraints, the transmission rate per node is limited. Hence, sensors are supposed to reduce the amount of information to be sent through quantization of the measurements. On the one hand, an increased amount of nodes deployed results in a more accurate sampling of the signal. On the other hand, it forces each sensor to perform looser quantization to cope with the smaller fraction of bandwidth allocated. From the theory of random sampling and by accounting for quantization, we compute the mean square error (MSE) in the reconstruction of the original signal. We then show the existence of an optimal deployment density that trades off between quantization error and aliasing error due to random sampling.

In this paper, the impact of edge effects on the performance of MAC protocols is evaluated. We consider a network where packets are distributed in finite space according to a 2-D PPP, while packets arrive in time according to a 1-D PPP. Each transmitter communicates with its own receiver a fixed distance away using either the ALOHA or CSMA protocol. The metric used for analysis is outage probability, which is the probability that the measured SINR falls below a predefined threshold for the duration of a packet. Approximate analytical expressions are derived for the outage probability of the different MAC protocols assuming a fading channel. Compared to unbounded regions, ALOHA and CSMA are shown to yield lower outage probabilities in finite regions, since the boundedness can be regarded as spatial filtering. Furthermore, the benefit seems to be greater for small outage probabilities, i.e., in the region of practical interest.

In this demo we present a solution for detailed coverage assessment and failures recovery in an IEEE 802.11 Wi-Fi network deployed in highly dynamical environment (e.g., where signal propagation is impacted by frequent topological changes). Monitoring of the electromagnetic energy received by an Access Point (AP) is performed by multiple battery-powered ZigBee-compliant devices dislocated in the environment. The key is a novel use of the Energy Detection Scan functionality implemented in the IEEE 802.15.4 standard. ZigBee devices wirelessly send their readings to a coordinator device which is in turn connected to a workstation. The latter serves as a i) configuration spot of the ZigBee network, ii) concentrator and data processor/visualizer, and iii) actuator. Such coordinating workstation can indeed detect and switch off an AP that is deemed ineffective (due to, e.g., damages or unfavorable propagation conditions), and start up one or more pre-deployed backup APs, so that coverage is preserved and clients can seamlessly keep on performing their activities. Hence, our scheme optimizes the performance of a Wi-Fi network by means of a low-cost, independent and self-maintaining mesh network, while strongly limits human intervention.

In this paper we present a mathematical model to study a multi-sink Wireless Sensor Network (WSN). Both sensors and sinks are assumed to be Poisson distributed in a given finite domain. Sinks send periodic queries, and each sensor transmits its sample to a sink, selected among those that are audible, thus creating a clustered network. Our aim is to describe how the Area Throughput, defined as the amount of samples per unit of time successfully transmitted to the sinks from the given area, depends on the density of sensors and the query interval. We jointly account for radio channel, Physical (PHY), Medium Access Control (MAC) and Network (NET) aspects (i.e., different network topologies, packet collisions, power losses and radio channel behaviour), and we compare the performance of two different simple data aggregation strategies. Performance is evaluated by varying the traffic offered to the network (i.e., the density of sensors deployed), the packet size, and, by considering IEEE 802.15.4 as a reference case, the number of Guaranteed Time Slots allocated, and the Superframe Order. The mathematical model shows how the Area Throughput can be optimized.

Channel fluctuations affecting links of ad-hoc and sensor networks show an evident spatial correlation, besides the random behavior. Nonetheless, the vast majority of models used in the literature assign edges between pairs of vertices of a graph according to either the deterministic disk model or some random connection model assuming i.i.d. fluctuations. We believe none of the approaches reflects the reality. In this paper we introduce a Correlated Random Connection Model (CRCM) which accounts for angular correlation, by means of a tunable parameter, in the fluctuations that affect two links sharing one of the endpoints. Assuming a constant average number of neighbors, we study the percolating properties of correlated footprints on random graphs by computing the relative size of the two largest components of the graph and the probability of the event of (almost) connectivity. We also compare it to the case of some non- probabilistic shapes of both theoretical and practical flavor. Our results show that the presence of correlation may be beneficial or detrimental, depending of whether one considers undirected or directed graphs, i.e., ultimately, on the application.

In this paper we present a mathematical approach to evaluate the area throughput and the energy consumption of a multi-sink Wireless Sensor Network (WSN). The WSN is organised into clusters, with one sink per cluster collecting data from sensors. A small variation of the Thomas point process is used to model sensors and sinks positions in the target area. We denote as area throughput the amount of samples per second successfully transmitted to the sinks. Both area throughput and energy consumption are strictly related to connectivity and MAC issues. The aim of this work is to devise a mathematical model that takes MAC and connectivity issues into account, under a common framework. We study the behavior of these two performance metrics when varying the target rate, defined as the maximum number of samples the network was deployed to deliver. Results show that a tradeoff between the area throughput and the energy consumption must be found. Finally, the impact of different sensors and sinks distributions on the area throughput is evaluated.

We discuss the correctness of one of the main results of about the probability density function of the distance between two audible nodes in an infinite 2-dimensional scenario. We prove that result is wrong and derive a more general expression, which is valid for an infinite d-dimensional area. It is worth noting that, although the results on the distribution of the distance between two nodes are wrong, the other results and in particular the fact that the number of audible nodes in an arbitrary area is a Poisson r.v. is correct.

In this paper we present a mathematical approach to evaluate the area throughput of a multi-sink wireless sensor network (WSN), where nodes transmit their packets to a sink, selected among many. Sensors and sinks are both Poisson distributed in a bounded domain. A carrier sensing multiple access (CSMA) based protocol is used by nodes to access the channel. We denote as area throughput the amount of samples per second successfully transmitted to the sinks. This performance metric is strictly related to both connectivity and MAC issues: it depends, in fact, on the probability that a given sensor node is not isolated and that it succeeds in transmitting its packet (i.e., the packet does not collide). The aim of this work is to devise a mathematical model that takes CSMA and connectivity issues into account under a joint approach. Through this model some network optimisation strategies could be derived. As an example, sensors could perform an aggregation procedure, responding sporadically to queries with a single packet composed of all samples taken since the previous transmission. Our model allows the evaluation of the optimum size of the packet that should be transmitted, so that the area throughput is maximised. Finally, the effects of the connectivity on the area throughput are evaluated.