Simulation Results

We simulated a wireless network with N nodes randomly distributed within a unit square. By means of the distance r_con we derived a connectivity matrix in such a way that a connectivity exists within a circle of radius r_con around each node. We modelled the local measurements as random values taken from a Gaussian distribution with variance σ_ω2. In Figure 2.20, we present the behaviour of bounds K_s and K_ns

calculated using eqs. (2.90) and (2.91) respectively as a function of the density of nodes per unit area averaged over 100 randomly generated topologies. The local estimatesω_nare simulated as normally distributed random variables with meanµ= 0 and variance σ_ω = 0.02. It has been shown that the influence of fading directly impacts the convergence of the distributed estimation. In Figure 2.20, the values of

K_s and K_ns decrease when the connectivity is increased. By setting r_con = 1.4, the networks always form fully connected graphs; this case presents the lowest value ofK_s, as can be seen in Figure 2.13. We can also see that if one of the local measurements has a high reliability (high SNR) and differs significantly from others nodes, this

10 20 30 40 50 60 70 80 90 100 10−4 10−3 10−2 10−1 100 number of nodes,N K K s for rcon = 0.5 K s for rcon = 0.4 K s for rcon = 0.6 K ns for rcon = 0.4 K ns for rcon = 0.5 K ns for rcon = 0.6

Figure 2.20: Dependance of bounds K_s and K_ns on node density.

frequency will not be locked and the local mean field amplitude r will have a lower value and some significant variations in amplitude. The change in the mean field r

below a pre-established threshold may be used in CR systems to indicate that one of the nodes senses a strong signal in a certain frequency band which is not visible for the other nodes. This can be very useful in solving the hidden terminal problem in CR networks. In the analysis of the convergence properties we do not consider the influence of fading, because the sin(_·) term disappears when multiplying by the row vector cT = 1T_ND_C in equation (2.86). However we are investigating this for further studies.

Chapter 3

Sequential Analysis Detection in

Cognitive Radio Networks

Most of the existing spectrum sensing schemes found in the literature are based on fixed sample size detectors, which have a preset and fixed sensing time. In this chapter, we present some novel results based on the work of Abraham Wald [29], who showed that a detector based on a sequential detection requires less average sensing time than a fixed size detector. We show that, in general, it is possible to achieve the same performance as other fixed sample based techniques using as few as half of the samples on average for low SNR scenarios. We then assess the impact of non-coherent detection with signals detected using sequential analysis, and we use the Wald test as a new of cooperative approach for sensing. This is addressed as an optimal fusion rule for distributed Wald detectors, and its performance is assessed. Later on, we present a novel methodology to evaluate the cumulants of the sample random distribution in sequential analysis. We use this to present a modification of the Dual Sequential Ratio Test algorithm used for Primary User (PU) detection in Cognitive Radio (CR) networks. In the considered scenario, the Secondary Users (SUs) utilize the sequential ratio test to sense the wireless channel looking for the presence of a transmitting PU. A Fusion Centre (FC) gathers the decisions from the SUs in order to perform a sequential ratio test and achieve a final verdict. Collection instance by the FC is optimized such that the total time to make a decision is minimized. This allows for better energy usage from the SUs along with reliable and fast detection of the PU.

3.1

Introduction

There exist several spectrum sensing algorithms aimed to solve the hypotheses testing problem explained in the previous chapters. In CR networks it is also of major importance to be able to detect the presence of PU as fast as possible, since the time of decision has great impact on the overall throughput of the system [60]. It is shown in [29, 31] that Wald’s Sequential Probability Ratio Test (SPRT) results in a savings of about fifty percent in the average number of observations in comparison to other well-known techniques such, as the Neyman-Pearson (NP) test. In order to improve performance and avoid hidden terminal problems, cooperative sensing can also be used [11]. There exist several papers in the literature focused on reducing the power consumption in distributed CR sensing scenarios. A distributed scheme is proposed in [61] which groups SUs into clusters and assigns one specific user as a cluster head, which gathers the spectrum sensing results from the other users in the cluster and forwards the local result to a FC. The energy savings comes from some SUs now sending their decision to their cluster head and not to the FC. In [62], a sleeping and censoring scheme for distributed networks is proposed in order to minimize energy consumption. In this case, each radio stops sensing while in sleep mode. The results from each user are sent to the FC only when they are within a reliable energy region defined by two thresholds which are to be optimized. While an energy savings occurs in [61] and [62], nothing is said about how fast their detection scheme performs. In [63], the authors use a Bayesian formulation to detect abrupt changes in multiple on-off processes. While they use a modification of the cumulative sum algorithm (CUSUM) for multiple channels, they do not address with the distribution approach. Moreover,a priori information about PU activity must be assumed. In [64], a random access-based reporting order control scheme is proposed for cooperative sensing. In this scenario, the local test statistics are reported to the FC in descending order of magnitude. They gain in time of detection by reducing the reporting time to the FC. In [65], the authors propose a Dual SPRT to perform collaborative spectrum sensing. Here, the SPRT is used at both the SU front-end and the FC in charge of taking the final decision about the presence of the PU. The FC is required to gather information

about the decisions made by SUs at a particular instance. However because of the nature of sequential analysis, those decisions are not always available at the same time. If the FC waits to gather information until all SUs have made a decision, the accuracy of the sensing improves but the overall throughput decreases. On the other hand, if the FC makes a decision with very little information, the reliability of its final decision is compromised. For this reason, we suggest that decisions from SUs are only transmitted after an optimal time τ₀ in order to decrease power usage and provide sufficient number of decisions to the FC. In [66], a threshold broadcast scheme is proposed for collaborative quickest spectrum sensing. The authors obtain a reduction in the detection delay in comparison to schemes using random broadcast. Our approach differs from theirs, as we use a FC to make a final decision and do not consider limited communication slots.