356 IEEE SIGNAL PROCESSING LETTERS, VOL. 22, NO. 3, MARCH 2015 Statistical Modeling of Spectrum Sensing Energy in Multi-Hop Cognitive Radio Networks Loredana Arienzo, Member, IEEE, and Daniele Tarchi, Senior Member, IEEE Abstract—The aim of this letter is to address the statistical modeling of the spectrum sensing energy consumption in cognitive radio networks. A Poisson point process has been shown to yield tractable and accurate results for the modeling of the interference in cognitive radio networks. We adopt this homogeneous stochastic process to develop an unified framework for deriving the energy consumption of the spectrum sensing in clustered cognitive radio networks. Furthermore, we extend the framework to multi-hop networks. The letter demonstrates that the spectrum sensing energy can be modeled as a Gamma-truncated distribution, as a function of the number of secondary users, their spatial density, and the number of hops of the cognitive radio network. Index Terms—Gamma distribution, multi-hop, Poisson point process, radio spectrum, sensing energy, statistical modeling. I. INTRODUCTION C OGNITIVE RADIO (CR) technology has been intro- duced with the aim of mitigating the wireless resource scarcity problem by dynamically changing the frequency spec- trum allocation. Spectrum sensing, i.e, the detection of unused spectrum bands, plays a fundamental role in CR networks (CRNs), and cooperative spectrum sensing can improve the detection performance of the primary signal [1] by taking advantage of diversity gains. However, such cooperation mech- anism introduces traffic overhead for the control signaling and data transmission which increases the energy consumption, especially in multi-hop networks [2], [3]. An energy-efficient CRN should: (1) improve the spectrum utilization; (2) minimize the interference to the primary communications; (3) minimize the energy consumption to the secondary communications. Most of the previous works on energy-efficient spectrum sensing focused on the detection performance of the spectrum sensing [4]–[6], while in [7] the focus is on the communication protocol. In literature some metrics have been proposed to evaluate the energy-efficiency of the spectrum sensing. In [8] the proposed metric is a function of the average number of transmitted bits and the average energy consumption in the spectrum sensing, while in [9] the metric has been defined as a function of the missed detection probability of the cooperative Manuscript received July 01, 2014; revised August 30, 2014; accepted September 11, 2014. Date of publication September 25, 2014; date of current version October 02, 2014. This work was supported in part by the European Commission (EC) under the CoRaSat project FP7 ICT STREP under Grant 316779. The associate editor coordinating the review of this manuscript and approving it for publication was Prof. Xin Wang. The authors are with the Department of Electrical, Electronic and Information Engineering, University of Bologna, Bologna 40136, Italy (e-mail: loredana. arienzo@unibo.it; daniele.tarchi@unibo.it). Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/LSP.2014.2360234 spectrum sensing and of the overall energy consumption in the spectrum sensing. In this letter, we firstly derive a model for the energy con- sumption of the sensing of the channels and then propose an approach to statistically model the energy consumption in clus- tered CRNs. At the best of our knowledge this is a first study that addresses the statistical modeling of the spectrum sensing energy. A Poisson point process (PPP) has been shown to yield tractable and accurate results for the modeling of the interfer- ence in CRNs [10]–[13]. We adopt an homogeneous PPP to de- velop an unified framework for deriving the energy consump- tion of the spectrum sensing. Moreover, we extend the frame- work to multi-hop CR networks. Our framework provides a strategy for choosing the best number of hops at the secondary network to optimize the spectrum sensing energy consumption for various radius of the clusters. This letter is organized as follows: Section II describes the network and energy model. In Section III we derive the statis- tical model of the spectrum sensing energy. Section IV discusses the performance of the proposed model, while in Section V, we draw the main conclusions. II. NETWORK AND ENERGY MODEL We consider a single primary transmitter, located at the origin, within a CRN. We assume that the secondary users (SUs) are spatially scattered according to an homogeneous PPP in a region [14]. The probability that SUs lie inside depends on the total area and is: (1) where is the set of SUs in the region , is the spatial density of the network (i.e., users per unit area), and is the radius of the area. In order to support end-to-end transmission from a CR source to a CR destination, we assume that the CRN is composed of clusters, each having sensing nodes, with ; among them, one assumes the role of cluster head (CH) within a coverage range equal to . Each SU senses the surrounding environment and transmits the sensing infor- mation to the related CH; the CHs, in turn, send the collected decision to a central receiver [15]. The energy consumption for transmitting bits of data be- tween a source and a destination at a distance with a given signal-to-noise ratio (SNR) at the receiver can be derived ex- ploiting the model in [16]: (2) where takes into account the overheads of the transmitter electronics and digital processing, takes into account the 1070-9908 © 2014 IEEE. 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