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The goal of spectrum sensing is to decide between the following two hypotheses

where x(t) is the signal received by secondary user and s(t) is primary users’s transmitted signal, n(t) is the additive white Gaussian noise (AWGN) and h is the amplitude gain of the channel. We also denote by γ the signal-to-noise ratio (SNR). We denote the output of integrator in Fig. 1 by Y which serves as decision statistic. Y may be shown to have the following distribution,

where X2 2TW and X2 2TW(2γ) denote central and non-central chi-square distributions respectively, each with 2TW degrees of freedom and a non-centrality parameter of 2γ for the latter distribution. For simplicity we assume that time-bandwidth product, TW, is an integer number which we denote by m. In a non-fading environment where h is deterministic, probabilities of detection and false alarm are given by the following formulas

where Γ(.) and Γ(., .) are complete and incomplete gamma functions respectively and Qm(., .) is the generalized Marcum Q-function defined as follows,

where Im−1(.) is the modified Bessel function of (m − 1)th order. The fundamental tradeoff between Pm = 1−Pd (probability of missed detection) and Pf has different implications in the context of dynamic spectrum-sharing. A high Pm would result in missing the presence of primary user with high probability which in turn increases interference to primary licensee. On the other hand, a high Pf would result in lowspectrum utilization since false alarms increase number of missed opportunities (white spaces). As expected, Pf is independent of γ since under H0 there is no primary signal present. On the other hand, when h is varying due to shadowing/fading, (1) gives probability of detection conditioned on the instantaneous SNR, γ. In this case, average probability of detection (which with an abuse of notation is denoted by Pd) may be derived by averaging (1) over fading statistics,

where fγ(x) is the probability distribution function (pdf) of SNR under fading. Performance of energy-detector for different values of average SNR and m may be characterized through complementary receiver operating characteristics (ROC) curves (plot of Pm vs. Pf ). In what follows we study performance under shadowing and Rayleigh fading.

Log-normal Shadowing

Empirical measurements suggest that medium-scale variations of the received-power, when represented in dB units, follow a normal distribution. In other words, the linear (as opposed to dB) channel gain may be modelled by a log-normal random variable, eX, where X is a zeromean Gaussian r.v. with variance σ2. Log-normal adowing is usually characterized in terms of its dB-spread, σdB, which is related to σ by σ = 0.1 ln(10)σdB. When γ is log-normally distributed due to shadowing, (3) may be evaluated numerically. Fig. 3 shows complementary ROC curves for three different dB-spreads. Average SNR,γ, and m are assumed to be 10 dB and 5 respectively. A plot for non-fading (pure AWGN) case is also provided for comparison.

Rayleigh Fading

Under Rayleigh fading, γ would have an exponential distribution. In this case, a closed-form formula for Pd may be obtained (after some manipulation) by substituting fγ(x) in (3)Fig. 4 provides plots of complementary ROC curve under AWGN and Rayleigh fading scenarios. γ and m are assumed to be 10 dB and 5 respectively. We observe that Rayleigh fading degrades performance of energy-detector significantly. Particularly, achieving Pm < 10−2 entails a probability of false-alarm greater than 0.9 which in turn results in poor spectrum utilization.

COLLABORATIVE SPECTRUM SENSING IN FADING CHANNELS

In order to improve performance of spectrum sensing, we allow different secondary users to collaborate by sharing their information. In order to minimize the communication overhead, users only share their final 1-bit decisions (H0 or H1) rather than their decision statistics. Let n denote the number of users collaborating. For simplicity we assume that all n users experience independent and identically distributed (iid) fading/shadowing with same average SNR. A fundamental result in distributed binary hypothesis testing is that when sensors are conditionallyindependent (as in our case), optimal decision rule for individual sensors is likelihood ratio test (LRT) . However, optimum individual thresholds are not necessarily equal and it is generally hard to derive them. We assume that all users employ energy-detection rather than LRT and use the same decision rule (i.e. same threshold λ). While these assumptions render our scheme sub-optimum, they facilitate analysis as well as practical implementation. A secondary user receives decisions from n−1 other users and decides H1 if any of the total n individual decisions is H1. This fusion rule is known as the OR-rule or 1-out-of-n rule . Probabilities of detection and false-alarm for the collaborative scheme (denoted by Qd and Qf respectively) may be written as follows,

where Pd and Pf are the individual probabilities of detection and false-alarm as defined by (3) and (2) respectively. It may be seen from (5) and (6) that compared to local sensing, this collaborative scheme increases probability of detection as well as probability of false-alarm . However, the net effect is an improvement in detection performance as seen in simulations. Fig. 5 and 6 show complementary ROC for different number of collaborating users under independent log-normal shadowing (σdB = 6 dB) and Rayleigh fading respectively. As before γ = 10 dB and m = 5. In both cases non-fading AWGN curve is shown for comparison. As seen in these figures, fusing decisions of different secondary users cancels deleterious impact of shadowing/fading effectively. Moreover, with increasing n, collaborative scheme is capable of outperforming AWGN local sensing (n = 1). This is due to the fact that for larger n, with high probability there will be a user with a channel better than that of the non-fading AWGN case.Fig. 7 shows Qd versus γ under iid Rayleigh fading for different number of collaborative spectrum sensors. For each curve, decision threshold, λ, is chosen such that Qf= 10−1. Time-bandwidth product, m, is set to 5 as before. Results indicate a significant improvement in terms of required average SNR for detection. n particular, for a probability of detection equal to 0.9, local spectrum sensing requires γ  16 dB while collaborative sensing with n = 10 only needs an verage SNR of 5 dB for individual users.