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Wireless Personal Communications

, Volume 100, Issue 4, pp 1845–1857 | Cite as

Performance of ED Based Spectrum Sensing Over α–η–μ Fading Channel

  • Sandeep Kumar
Article

Abstract

Internet of things contains the hefty number of devices communicating with each other; give rise to the problem of spectrum scarcity. Cognitive radio has emerged as the promising solution to this problem. Spectrum sensing is the important function of cognitive radio and energy detector is the most popular technique used for spectrum sensing. In this paper, the performance of energy detector (ED) over α–η–μ fading channel has been analyzed. The analytical expressions for average probability of detection and average area under the receiver operating characteristics curve (AUC) are derived for the generalized fading channel in terms of the bivariate Fox H-function. The closed-form mathematical expressions for the average probability of detection for cooperative spectrum sensing as well as square law selection diversity reception are derived. The implication of the system parameters on the performance of ED is studied in terms of complimentary receiver operating characteristics and AUC. It is shown that the performance of ED can be improved when cooperation and diversity are employed. The derived results are generic and can be directly used for the performance analysis of η–μ and α–μ fading channels and their special cases. Monte-Carlo simulations are incorporated for validating the accuracy of the derived results.

Keywords

Generalised fading Cognitive radio Receiver operating characteristic Bivariate Fox H-function Cooperative spectrum sensing Diversity reception 

1 Introduction

Internet of things is one of the demanding technologies and is probable to be an indispensable fragment of the next generation 5G technology. Primarily, these devices act as a source of interference to the primary users in the absence of appropriate collision detection technique. Secondly large amount of bandwidth is required for the devices to interact with each other so spectrum scarcity has become the biggest problem. Cognitive radio (CR) has emerged as a promising solution to the scarcity of the spectrum problem and to provide collision free communication. Spectrum sensing is the important function of cognitive radio and energy detector is the most popular technique used for spectrum sensing. Because of the low implementation complexity and low computational cost, it is most studied spectrum sensing technique in literature. Energy detector (ED) works on the principle of non-coherent detection and prior knowledge of the signal is not required at the receiver. For detecting the presence of a signal, the received signal is sampled over a time interval and its energy is compared with the threshold energy. The decision statistic is based on the assumption that it’s a binary hypothesis problem and follows the Chi square distribution. Several works are present in literature related to the performance evaluation of ED for different fading models. The performance of ED in terms of complimentary receiver operating characteristic (CROC) was studied in [1] while in terms of area under the receiver operating characteristic curve (AUC) was addressed in [2] over Nakagami-m fading channel. Closed-form analytical expression for average probability of detection (\(\bar{P}_{d}\)) over α–η generalized fading channel was derived in [3]. In [4], the analysis focuses on ED-based spectrum sensing over κ–μ and extreme κ–μ fading channel and the analytic expressions for \(\bar{P}_{d}\) for both channels were derived. In [5], ED performance over generalized fading channels was studied using moment generating function (MGF) based approach. The mathematical expressions for \(\bar{P}_{d}\) and average AUC (\(\bar{A}\)) over η–μ fading channel has been analysed in [6, 7]. In addition to the impact of the multipath fading, the wireless channel is also affected by the shadowing, and several works are present in literature related to that. The detailed analysis of an ED over composite Rician/gamma fading channels over different diversity techniques was performed in [8]. Cooperative spectrum sensing [9] over κ–μ shadowed fading channel was analysed in [9] while the effect of MRC diversity was studied in [10]. Simplified expressions for the performance parameters for gamma shadowed generalized fading channel was derived in [11] using mixture gamma (MG) distribution.

The α–η–μ is a general fading distribution used to represent small-scale fading. Non-linearity and inhomogeneity effect of the channel is included in this distribution and hence best fits with the experimental data. The α–η–μ can be easily transformed to other well-known distributions like Rayleigh, Exponential, Nakagami-m, Nakagami-q, Weibull, and η–μ. By putting α = 2 in α–η–μ distribution we can easily obtain η–μ distribution as a special case. Recently the performance of wireless communication system over α–η–μ distribution was studied. The expressions for the probability distribution function (PDF) of the received signal envelope and its kth moment was derived in [12]. The effect of non-linearity on the channel capacity and entropy over α–η–μ distribution was investigated in [13, 14] while the closed-form expression for error probability of different modulation techniques were derived in [15, 16]. The estimator of the α–η–μ distribution based on the maximum likelihood (ML) method was proposed in [17, 18] by using different methods.

To the best of author’s knowledge the performance analysis of ED over α–η–μ fading channel is not present in literature. Motivated by this fact we have derived the closed-form expressions for \(\bar{P}_{d}\) and \(\bar{A}\) in terms of the bivariate Fox H-function. The analytical expressions for average probability of detection for cooperative spectrum sensing and square law selection (SLS) diversity reception over the said channel are presented. The effect of system parameters on the performance of ED is also demonstrated.

The outline of the paper is as follows. The system and channel model is described in Sect. 2. In Sect. 3, the closed-form analytical expression for the average probability of detection is presented. Section 4 presents the analysis of the average AUC. Section 5 gives results and discussion followed by conclusion in Sect. 6.

2 System and Channel Model

Consider a narrowband signal \(x\left( t \right)\) is detected at the received and passed through a band pass filter (BPF). The output of the band pass filter (BPF) is forwarded to a squaring device followed by an integrator. ED compares the energy of the n number of samples of the signal with threshold (\(\lambda\)) and detects the presence of unknown signal in it. So, the received signal can be formulated by binary hypothesis with \(H_{0}\) (signal is not present) and \(H_{1}\) (signal is present) as [9]
$$x\left( t \right) = \left\{ {\begin{array}{*{20}l} {n\left( t \right);} \hfill & {H_{0} } \hfill \\ {h\,*s\left( t \right) + n\left( t \right);} \hfill & {H_{1} } \hfill \\ \end{array} } \right.$$
(1)
where \(h\) is the channel gain, \(s\left( t \right)\) denotes unknown deterministic signal and \(n\left( t \right)\) is additive white Gaussian noise (AWGN). The expressions for probability of detection (\(P_{d}\)) and probability of false alarm (\(P_{f}\)) for AWGN channel is given by [1]
$$P_{f} \left( \lambda \right) = \frac{{\varGamma \left( {n,\,\lambda /2} \right)}}{\varGamma \left( n \right)}$$
(2)
$$P_{d} \left( {\gamma ,\,\lambda } \right) = Q_{n} \left( {\sqrt {2\gamma } ,\,\sqrt \lambda } \right)$$
(3)
where \(n\) is defined as \(n = TW\). \(T\) is the time interval of the integrator and \(W\) is the bandwidth of the BPF. \(\varGamma \left( {.\,,\,.} \right)\) is the upper incomplete Gamma function, \(\varGamma \left( . \right)\) is the gamma function, \(Q_{n} \left( {.\,,\,.} \right)\) is the nth order generalized Marcum \(Q\)-function, and \(\gamma\) is the received SNR. Generalized Marcum \(Q\)-function is defined as [4].
$$Q_{n} \left( {\sqrt {2\gamma } ,\sqrt \lambda } \right) = exp\left( { - \gamma } \right)\sum\limits_{m = 0}^{\infty } {\frac{{\gamma^{m} }}{m!}} \frac{{\varGamma \left( {m + n,{\lambda \mathord{\left/ {\vphantom {\lambda 2}} \right. \kern-0pt} 2}} \right)}}{{\varGamma \left( {m + n} \right)}}$$
(4)
The PDF of the received SNR (\(\gamma\)) following α–η–μ distribution is given as [16].
$$f_{\gamma } \left( \gamma \right) = \frac{{\sqrt \pi \alpha \mu^{{\left( {\mu + {1 \mathord{\left/ {\vphantom {1 2}} \right. \kern-0pt} 2}} \right)}} h^{\mu } \gamma^{{\left( {{\alpha \mathord{\left/ {\vphantom {\alpha 2}} \right. \kern-0pt} 2}} \right)\left( {\mu + {1 \mathord{\left/ {\vphantom {1 2}} \right. \kern-0pt} 2}} \right) - 1}} }}{{\varGamma \left( \mu \right)H^{{\left( {\mu - {1 \mathord{\left/ {\vphantom {1 2}} \right. \kern-0pt} 2}} \right)}} \bar{\gamma }^{{\left( {{\alpha \mathord{\left/ {\vphantom {\alpha 2}} \right. \kern-0pt} 2}} \right)\left( {\mu + {1 \mathord{\left/ {\vphantom {1 2}} \right. \kern-0pt} 2}} \right)}} }}\exp \left( { - \frac{{2\mu h\gamma^{{\left( {{\alpha \mathord{\left/ {\vphantom {\alpha 2}} \right. \kern-0pt} 2}} \right)}} }}{{\bar{\gamma }^{{\left( {{\alpha \mathord{\left/ {\vphantom {\alpha 2}} \right. \kern-0pt} 2}} \right)}} }}} \right)I_{{\left( {\mu - {1 \mathord{\left/ {\vphantom {1 2}} \right. \kern-0pt} 2}} \right)}} \left( {\frac{{2\mu H\gamma^{{\left( {{\alpha \mathord{\left/ {\vphantom {\alpha 2}} \right. \kern-0pt} 2}} \right)}} }}{{\bar{\gamma }^{{\left( {{\alpha \mathord{\left/ {\vphantom {\alpha 2}} \right. \kern-0pt} 2}} \right)}} }}} \right),\quad \gamma \ge 0$$
(5)
where \(\alpha\) is non-linearity parameter and \(\mu\) is non-homogeneity parameter of the propagation medium. The value of \(h\) and \(H\) is defined in 2 formats. In format 1, \(h = {{\left( {1 + \eta } \right)^{2} } \mathord{\left/ {\vphantom {{\left( {1 + \eta } \right)^{2} } {4\eta }}} \right. \kern-0pt} {4\eta }}\) and \(H = {{\left( {1 - \eta^{2} } \right)} \mathord{\left/ {\vphantom {{\left( {1 - \eta^{2} } \right)} {4\eta }}} \right. \kern-0pt} {4\eta }}\) where \(0 < \eta < \infty\). In format 2, \(h = {1 \mathord{\left/ {\vphantom {1 {\left( {1 - \eta^{2} } \right)}}} \right. \kern-0pt} {\left( {1 - \eta^{2} } \right)}}\) and \(H = {\eta \mathord{\left/ {\vphantom {\eta {\left( {1 - \eta^{2} } \right)}}} \right. \kern-0pt} {\left( {1 - \eta^{2} } \right)}}\) where \(- 1 < \eta < 1\). \(\bar{\gamma }\) is the average received SNR while \(I_{v} \left( . \right)\) is the modified Bessel’s function of first type and vth order.

3 Average Probability of Detection

3.1 Single User Spectrum Sensing

For fading channel the channel gain varies, the average probability of detection, \(\overline{{P_{d} }} (\lambda )\) can be obtained by averaging the probability of detection given in (3) as
$$\overline{{P_{d} }} (\lambda ) = E[P_{d} \left( {\gamma ,\lambda } \right)] = \int\limits_{0}^{\infty } {P_{d} \left( {\gamma ,\lambda } \right)f_{\gamma } \left( \gamma \right)} d\gamma$$
(6)
Putting (3) and (5) into (6) and using (4) yields
$$\begin{aligned} \overline{{P_{d} }} (\lambda ) & = \frac{{\sqrt \pi \alpha \mu^{{\left( {\mu + {1 \mathord{\left/ {\vphantom {1 2}} \right. \kern-0pt} 2}} \right)}} h^{\mu } }}{{\varGamma \left( \mu \right)H^{{\left( {\mu - {1 \mathord{\left/ {\vphantom {1 2}} \right. \kern-0pt} 2}} \right)}} \bar{\gamma }^{{\left( {{\alpha \mathord{\left/ {\vphantom {\alpha 2}} \right. \kern-0pt} 2}} \right)\left( {\mu + {1 \mathord{\left/ {\vphantom {1 2}} \right. \kern-0pt} 2}} \right)}} }}\sum\limits_{k = 0}^{\infty } {\frac{{\varGamma \left( {n + k,{\lambda \mathord{\left/ {\vphantom {\lambda 2}} \right. \kern-0pt} 2}} \right)}}{{\left( {k!} \right)\varGamma \left( {n + k} \right)}}} \\ & \quad \times \,\int\limits_{0}^{\infty } {\gamma^{{\left( {{\alpha \mathord{\left/ {\vphantom {\alpha 2}} \right. \kern-0pt} 2}} \right)\left( {\mu + {1 \mathord{\left/ {\vphantom {1 2}} \right. \kern-0pt} 2}} \right) + k - 1}} \exp \left( { - \,\gamma } \right)\exp \left( { - \,\frac{{2\mu h\gamma^{{\left( {{\alpha \mathord{\left/ {\vphantom {\alpha 2}} \right. \kern-0pt} 2}} \right)}} }}{{\bar{\gamma }^{{\left( {{\alpha \mathord{\left/ {\vphantom {\alpha 2}} \right. \kern-0pt} 2}} \right)}} }}} \right)I_{{\left( {\mu - {1 \mathord{\left/ {\vphantom {1 2}} \right. \kern-0pt} 2}} \right)}} \left( {\frac{{2\mu H\gamma^{{\left( {{\alpha \mathord{\left/ {\vphantom {\alpha 2}} \right. \kern-0pt} 2}} \right)}} }}{{\bar{\gamma }^{{\left( {{\alpha \mathord{\left/ {\vphantom {\alpha 2}} \right. \kern-0pt} 2}} \right)}} }}} \right)} d\gamma \\ \end{aligned}$$
(7)
By rearranging the terms and doing some mathematical manipulation equation (7) can be written as
$$\begin{aligned} & \overline{{P_{d} }} (\lambda ) = \frac{{\sqrt \pi \alpha \mu^{{\left( {\mu + {1 \mathord{\left/ {\vphantom {1 2}} \right. \kern-0pt} 2}} \right)}} h^{\mu } }}{{\varGamma \left( \mu \right)H^{{\left( {\mu - {1 \mathord{\left/ {\vphantom {1 2}} \right. \kern-0pt} 2}} \right)}} \bar{\gamma }^{{\left( {{\alpha \mathord{\left/ {\vphantom {\alpha 2}} \right. \kern-0pt} 2}} \right)\left( {\mu + {1 \mathord{\left/ {\vphantom {1 2}} \right. \kern-0pt} 2}} \right)}} }}\sum\limits_{k = 0}^{\infty } {\frac{{\varGamma \left( {n + k,{\lambda \mathord{\left/ {\vphantom {\lambda 2}} \right. \kern-0pt} 2}} \right)}}{{\left( {k!} \right)\varGamma \left( {n + k} \right)}}} \\ & \quad\quad \times \,\int\limits_{0}^{\infty } {\gamma^{{\left( {{\alpha \mathord{\left/ {\vphantom {\alpha 2}} \right. \kern-0pt} 2}} \right)\left( {\mu + {1 \mathord{\left/ {\vphantom {1 2}} \right. \kern-0pt} 2}} \right) + k - 1}} \exp \left( { - \,\gamma } \right)\exp \left( { - \frac{{2\mu \left( {h - H} \right)\gamma^{{\left( {{\alpha \mathord{\left/ {\vphantom {\alpha 2}} \right. \kern-0pt} 2}} \right)}} }}{{\bar{\gamma }^{{\left( {{\alpha \mathord{\left/ {\vphantom {\alpha 2}} \right. \kern-0pt} 2}} \right)}} }}} \right)\exp \left( { - \,\frac{{2\mu H\gamma^{{\left( {{\alpha \mathord{\left/ {\vphantom {\alpha 2}} \right. \kern-0pt} 2}} \right)}} }}{{\bar{\gamma }^{{\left( {{\alpha \mathord{\left/ {\vphantom {\alpha 2}} \right. \kern-0pt} 2}} \right)}} }}} \right)I_{{\left( {\mu - {1 \mathord{\left/ {\vphantom {1 2}} \right. \kern-0pt} 2}} \right)}} \left( {\frac{{2\mu H\gamma^{{\left( {{\alpha \mathord{\left/ {\vphantom {\alpha 2}} \right. \kern-0pt} 2}} \right)}} }}{{\bar{\gamma }^{{\left( {{\alpha \mathord{\left/ {\vphantom {\alpha 2}} \right. \kern-0pt} 2}} \right)}} }}} \right)} d\gamma \\ \end{aligned}$$
(8)
With the help of [19, 8.4.3.1], [19, 8.4.22.3] and [19, 8.3.2.21], (8) can be written as
$$\begin{aligned} & \overline{{P_{d} }} (\lambda ) = \frac{{\alpha \mu^{{\left( {\mu + {1 \mathord{\left/ {\vphantom {1 2}} \right. \kern-0pt} 2}} \right)}} h^{\mu } }}{{\varGamma \left( \mu \right)H^{{\left( {\mu - {1 \mathord{\left/ {\vphantom {1 2}} \right. \kern-0pt} 2}} \right)}} \bar{\gamma }^{{\left( {{\alpha \mathord{\left/ {\vphantom {\alpha 2}} \right. \kern-0pt} 2}} \right)\left( {\mu + {1 \mathord{\left/ {\vphantom {1 2}} \right. \kern-0pt} 2}} \right)}} }}\sum\limits_{k = 0}^{\infty } {\frac{{\varGamma \left( {n + k,{\lambda \mathord{\left/ {\vphantom {\lambda 2}} \right. \kern-0pt} 2}} \right)}}{{\left( {k!} \right)\varGamma \left( {n + k} \right)}}} \\ & \quad\quad \times \int\limits_{0}^{\infty } {\gamma^{{\left( {{\alpha \mathord{\left/ {\vphantom {\alpha 2}} \right. \kern-0pt} 2}} \right)\left( {\mu + {1 \mathord{\left/ {\vphantom {1 2}} \right. \kern-0pt} 2}} \right) + k - 1}} \exp \left( { - \,\gamma } \right)H_{0}^{1} \,_{1}^{0} \,\left[ {\left. {\frac{{2\mu \left( {h - H} \right)\gamma^{{\left( {{\alpha \mathord{\left/ {\vphantom {\alpha 2}} \right. \kern-0pt} 2}} \right)}} }}{{\bar{\gamma }^{{\left( {{\alpha \mathord{\left/ {\vphantom {\alpha 2}} \right. \kern-0pt} 2}} \right)}} }}} \right|\,_{(0\,,1)\,} } \right]H_{1}^{1} \,_{2}^{1} \,\left[ {\left. {\frac{{4\mu H\gamma^{{\left( {{\alpha \mathord{\left/ {\vphantom {\alpha 2}} \right. \kern-0pt} 2}} \right)}} }}{{\bar{\gamma }^{{\left( {{\alpha \mathord{\left/ {\vphantom {\alpha 2}} \right. \kern-0pt} 2}} \right)}} }}} \right|\,\begin{array}{*{20}c} {\left( {\frac{1}{2},1} \right)} & {} \\ {\left( {\mu - \frac{1}{2},1} \right)} & {\left( {\frac{1}{2} - \,\mu ,1} \right)} \\ \end{array} } \right]} d\gamma \\ \end{aligned}$$
(9)
Recalling the definition of Fox-H function [19, 8.3.1.1] and doing some mathematical manipulation, (9) can be rewritten as
$$\begin{aligned} \overline{{P_{d} }} (\lambda ) & = \frac{{\alpha \mu^{{\left( {\mu + {1 \mathord{\left/ {\vphantom {1 2}} \right. \kern-0pt} 2}} \right)}} h^{\mu } }}{{\varGamma \left( \mu \right)H^{{\left( {\mu - {1 \mathord{\left/ {\vphantom {1 2}} \right. \kern-0pt} 2}} \right)}} \bar{\gamma }^{{\left( {{\alpha \mathord{\left/ {\vphantom {\alpha 2}} \right. \kern-0pt} 2}} \right)\left( {\mu + {1 \mathord{\left/ {\vphantom {1 2}} \right. \kern-0pt} 2}} \right)}} }}\sum\limits_{k = 0}^{\infty } {\frac{{\varGamma \left( {n + k,{\lambda \mathord{\left/ {\vphantom {\lambda 2}} \right. \kern-0pt} 2}} \right)}}{{\left( {k!} \right)\varGamma \left( {n + k} \right)}}\frac{1}{{\left( {2\pi i} \right)^{2} }}\int\limits_{0}^{\infty } {\gamma^{{\left( {{\alpha \mathord{\left/ {\vphantom {\alpha 2}} \right. \kern-0pt} 2}} \right)\left( {\mu + {1 \mathord{\left/ {\vphantom {1 2}} \right. \kern-0pt} 2} - s - t} \right) + k - 1}} \exp \left( { - \,\gamma } \right)} d\gamma } \\ & \quad \times \,\int_{L1} {\int_{L2} {\frac{{\varGamma \left( s \right)\varGamma \left( {\mu - \frac{1}{2} + t} \right)\varGamma \left( {\frac{1}{2} - t} \right)}}{{\varGamma \left( {\frac{1}{2} + \mu - t} \right)}}\left( {\frac{{2\mu \left( {h - H} \right)\gamma^{{\left( {{\alpha \mathord{\left/ {\vphantom {\alpha 2}} \right. \kern-0pt} 2}} \right)}} }}{{\bar{\gamma }^{{\left( {{\alpha \mathord{\left/ {\vphantom {\alpha 2}} \right. \kern-0pt} 2}} \right)}} }}} \right)^{ - s} \left( {\frac{{4\mu H\gamma^{{\left( {{\alpha \mathord{\left/ {\vphantom {\alpha 2}} \right. \kern-0pt} 2}} \right)}} }}{{\bar{\gamma }^{{\left( {{\alpha \mathord{\left/ {\vphantom {\alpha 2}} \right. \kern-0pt} 2}} \right)}} }}} \right)^{ - t} } } dsdt \\ \end{aligned}$$
(10)
where \(L_{1}\) and \(L_{2}\) are two suitable contours. The integral in 1st part of equation can be solved using [20, 3.326.2] as
$$\begin{aligned} \overline{{P_{d} }} (\lambda ) & = \frac{{\alpha \mu^{{\left( {\mu + {1 \mathord{\left/ {\vphantom {1 2}} \right. \kern-0pt} 2}} \right)}} h^{\mu } }}{{\varGamma \left( \mu \right)H^{{\left( {\mu - {1 \mathord{\left/ {\vphantom {1 2}} \right. \kern-0pt} 2}} \right)}} \bar{\gamma }^{{\left( {{\alpha \mathord{\left/ {\vphantom {\alpha 2}} \right. \kern-0pt} 2}} \right)\left( {\mu + {1 \mathord{\left/ {\vphantom {1 2}} \right. \kern-0pt} 2}} \right)}} }}\sum\limits_{k = 0}^{\infty } {\frac{{\varGamma \left( {n + k,{\lambda \mathord{\left/ {\vphantom {\lambda 2}} \right. \kern-0pt} 2}} \right)}}{{\left( {k!} \right)\varGamma \left( {n + k} \right)}}\frac{1}{{\left( {2\pi i} \right)^{2} }}} \\ & \quad \times \,\int_{L1} {\int_{L2} {\frac{{\varGamma \left( s \right)\varGamma \left( {\mu - \frac{1}{2} + t} \right)\varGamma \left( {\frac{1}{2} - t} \right)\varGamma \left( {\frac{\alpha }{2}\left( {\mu + \frac{1}{2} - s - t} \right) + k} \right)}}{{\varGamma \left( {\frac{1}{2} + \mu - t} \right)}}\left( {\frac{{2\mu \left( {h - H} \right)}}{{\bar{\gamma }^{{\left( {{\alpha \mathord{\left/ {\vphantom {\alpha 2}} \right. \kern-0pt} 2}} \right)}} }}} \right)^{ - s} \left( {\frac{4\mu H}{{\bar{\gamma }^{{\left( {{\alpha \mathord{\left/ {\vphantom {\alpha 2}} \right. \kern-0pt} 2}} \right)}} }}} \right)^{ - t} } } dsdt \\ \end{aligned}$$
(11)
Using [21, 2.57] and after some mathematical manipulation the final expression of \(\overline{{P_{d} }} (\lambda )\) can be written as
$$\begin{aligned} \overline{{P_{d} }} (\lambda ) & = \frac{{\alpha \mu^{{\left( {\mu + {1 \mathord{\left/ {\vphantom {1 2}} \right. \kern-0pt} 2}} \right)}} h^{\mu } }}{{\varGamma \left( \mu \right)H^{{\left( {\mu - {1 \mathord{\left/ {\vphantom {1 2}} \right. \kern-0pt} 2}} \right)}} \bar{\gamma }^{{\left( {{\alpha \mathord{\left/ {\vphantom {\alpha 2}} \right. \kern-0pt} 2}} \right)\left( {\mu + {1 \mathord{\left/ {\vphantom {1 2}} \right. \kern-0pt} 2}} \right)}} }}\sum\limits_{k = 0}^{\infty } {\frac{{\varGamma \left( {n + k,{\lambda \mathord{\left/ {\vphantom {\lambda 2}} \right. \kern-0pt} 2}} \right)}}{{\left( {k!} \right)\varGamma \left( {n + k} \right)}}} \\ & \quad \times \,H_{1,0:0,1:1,2}^{0,1:1,0:1,1} \,\,\left[ {\left. {\begin{array}{*{20}c} {\frac{{2\mu \left( {h - H} \right)}}{{\bar{\gamma }^{{\left( {{\alpha \mathord{\left/ {\vphantom {\alpha 2}} \right. \kern-0pt} 2}} \right)}} }}} \\ {\frac{4\mu H}{{\bar{\gamma }^{{\left( {{\alpha \mathord{\left/ {\vphantom {\alpha 2}} \right. \kern-0pt} 2}} \right)}} }}} \\ \end{array} } \right|\,\begin{array}{*{20}c} {\left( {1 - \frac{\alpha }{2}\left( {\mu + \frac{1}{2}} \right) - k;\frac{\alpha }{2},\frac{\alpha }{2}} \right): - ;\left( {\frac{1}{2},1} \right)} \\ { - :\left( {\mu - \frac{1}{2},1} \right);\left( {\frac{1}{2} - \mu ,1} \right)} \\ \end{array} } \right]. \\ \end{aligned}$$
(12)

3.2 Cooperative Spectrum Sensing

Sometimes due to deep fading, the channel condition is so destructive that the cognitive user is not able to do the spectrum sense efficiently. In that case cooperative SS is used as a means of improving the detection performance [22]. It exploits the spatial diversity among SUs to alleviate the effects of shadowing and multipath. The probabilities of false alarm, \(Q_{f}\) and average probability of detection, \(\bar{Q}_{d}\) for a cooperative spectrum sensing scheme with m collaborative SUs are given in [23] as
$$Q_{f} = 1 - \left( {1 - P_{f} } \right)^{m}$$
(13)
$$\bar{Q}_{d} = 1 - \left( {1 - \bar{P}_{d} } \right)^{m}$$
(14)
Putting (12) in (14) and we can evaluate the final expression of \(\bar{Q}_{d}\) as
$$\bar{Q}_{d} = 1 - \,\left( \begin{aligned} 1 - \frac{{\alpha \mu^{{\left( {\mu + {1 \mathord{\left/ {\vphantom {1 2}} \right. \kern-0pt} 2}} \right)}} h^{\mu } }}{{\varGamma \left( \mu \right)H^{{\left( {\mu - {1 \mathord{\left/ {\vphantom {1 2}} \right. \kern-0pt} 2}} \right)}} \bar{\gamma }^{{\left( {{\alpha \mathord{\left/ {\vphantom {\alpha 2}} \right. \kern-0pt} 2}} \right)\left( {\mu + {1 \mathord{\left/ {\vphantom {1 2}} \right. \kern-0pt} 2}} \right)}} }}\sum\limits_{k = 0}^{\infty } {\frac{{\varGamma \left( {n + k,{\lambda \mathord{\left/ {\vphantom {\lambda 2}} \right. \kern-0pt} 2}} \right)}}{{\left( {k!} \right)\varGamma \left( {n + k} \right)}}} \hfill \\ \times \,H_{1,0:0,1:1,2}^{0,1:1,0:1,1} \,\,\left[ {\left. {\begin{array}{*{20}c} {\frac{{2\mu \left( {h - H} \right)}}{{\bar{\gamma }^{{\left( {{\alpha \mathord{\left/ {\vphantom {\alpha 2}} \right. \kern-0pt} 2}} \right)}} }}} \\ {\frac{4\mu H}{{\bar{\gamma }^{{\left( {{\alpha \mathord{\left/ {\vphantom {\alpha 2}} \right. \kern-0pt} 2}} \right)}} }}} \\ \end{array} } \right|\,\begin{array}{*{20}c} {\left( {1 - \,\frac{\alpha }{2}\left( {\mu + \frac{1}{2}} \right) - k;\frac{\alpha }{2},\frac{\alpha }{2}} \right): - ;\left( {\frac{1}{2},1} \right)} \\ { - :\left( {\mu - \frac{1}{2},1} \right);\left( {\frac{1}{2} - \mu ,1} \right)} \\ \end{array} } \right] \hfill \\ \end{aligned} \right)^{m} .$$
(15)

3.3 SLS Diversity reception

Diversity technique is used to overcome the effect of fading and SLS is the most commonly used diversity reception scheme due to its simplicity. The output of L independent and identically distributed (i.i.d.) branches SLS combiner can be given as \(y_{SLS} = \hbox{max} (y_{1} ,y_{2} ,{ \ldots }y_{L} )\), where \(y_{L}\) is the decision statistics of the Lth branch. The average probability of detection and probability of false alarm for an SLC diversity receiver is given by [1] as
$$\bar{P}_{d}^{SLS} (\lambda ) = 1 - \,\prod\limits_{l = 1}^{L} {\int\limits_{0}^{\infty } {\left[ {1 - P_{d} \left( {\gamma_{l} ,\lambda } \right)f_{{\gamma_{l} }} \left( {\gamma_{l} } \right)d\gamma_{l} } \right]} }$$
(16)
$$P_{f}^{SLS} \left( \lambda \right) = 1 - \,\prod\limits_{l = 1}^{L} {\int\limits_{0}^{\infty } {\left[ {1 - \frac{{\varGamma \left( {n,\,\lambda /2} \right)}}{\varGamma \left( n \right)}} \right]} }$$
(17)
The closed-form expression of \(P_{f}^{SLS} \left( \lambda \right)\) can be evaluate by putting (12) in (16) as
$$\bar{P}_{d}^{SLS} (\lambda ) = 1 - \,\prod\limits_{l = 1}^{L} {\int\limits_{0}^{\infty } {\left[ \begin{aligned} &1 - \,\frac{{\alpha \mu^{{\left( {\mu + {1 \mathord{\left/ {\vphantom {1 2}} \right. \kern-0pt} 2}} \right)}} h^{\mu } }}{{\varGamma \left( \mu \right)H^{{\left( {\mu - {1 \mathord{\left/ {\vphantom {1 2}} \right. \kern-0pt} 2}} \right)}} \bar{\gamma }_{l}^{{\left( {{\alpha \mathord{\left/ {\vphantom {\alpha 2}} \right. \kern-0pt} 2}} \right)\left( {\mu + {1 \mathord{\left/ {\vphantom {1 2}} \right. \kern-0pt} 2}} \right)}} }}\sum\limits_{k = 0}^{\infty } {\frac{{\varGamma \left( {n + k,{\lambda \mathord{\left/ {\vphantom {\lambda 2}} \right. \kern-0pt} 2}} \right)}}{{\left( {k!} \right)\varGamma \left( {n + k} \right)}}} \hfill \\ &\times \,H_{1,0:0,1:1,2}^{0,1:1,0:1,1} \,\,\left[ {\left. {\begin{array}{*{20}c} {\frac{{2\mu \left( {h - H} \right)}}{{\bar{\gamma }_{l}^{{\left( {{\alpha \mathord{\left/ {\vphantom {\alpha 2}} \right. \kern-0pt} 2}} \right)}} }}} \\ {\frac{4\mu H}{{\bar{\gamma }_{l}^{{\left( {{\alpha \mathord{\left/ {\vphantom {\alpha 2}} \right. \kern-0pt} 2}} \right)}} }}} \\ \end{array} } \right|\,\begin{array}{*{20}c} {\left( {1 - \frac{\alpha }{2}\left( {\mu + \frac{1}{2}} \right) - k;\frac{\alpha }{2},\frac{\alpha }{2}} \right): - ;\left( {\frac{1}{2},1} \right)} \\ { - :\left( {\mu - \frac{1}{2},1} \right);\left( {\frac{1}{2} - \mu ,1} \right)} \\ \end{array} } \right] \hfill \\ \end{aligned} \right]} } .$$
(18)

4 Average Area Under the ROC curve

It is difficult to comparison the performance of two energy detectors based on visual perception of their ROC when their ROC curves cross each other. In that situation, AUC is the single figure of merit that provides a better understanding as to what factors it affects the performance of the ED. In [2], it was indicated that the average AUC shows the probability of adopting the appropriate decision at the detector than the inappropriate decision. The value of AUC ranging between 0.5 and 1 as the detection threshold varies from 0 to \(\infty\). For a fading channel, the average AUC can be obtained by averaging the AUC by the distribution of \(\gamma\) and is given by [2] as
$$\begin{aligned} \bar{A} & = \int\limits_{0}^{\infty } {f_{\gamma } \left( \gamma \right)} d\gamma - \sum\limits_{n = 0}^{u - 1} {\frac{1}{{2^{n} (n!)}}} \int\limits_{0}^{\infty } {\gamma^{n} \exp \left( { - \frac{\gamma }{2}} \right)f_{\gamma } \left( \gamma \right)} d\gamma \\ & \quad + \,\sum\limits_{n = 1 - u}^{u - 1} {\frac{{\varGamma \left( {u + n} \right)}}{{2^{u + n} \varGamma \left( u \right)}}} \int\limits_{0}^{\infty } {\exp \left( { - \gamma } \right){}_{1}\tilde{F}_{1} \left( {u + n;1 + n;\frac{\gamma }{2}} \right)f_{\gamma } \left( \gamma \right)} d\gamma \\ \end{aligned}$$
(19)
The above equation (19) can be written as \(\bar{A} = I_{1} - I_{2} + I_{3}\), where \(I_{1} = 1\). Using [19, 8.4.3.1], [19, 8.4.22.3] and [19, 8.3.2.21], \(I_{2}\) can be written as
$$\begin{aligned} & I_{2} = \frac{{\alpha \mu^{{\left( {\mu + {1 \mathord{\left/ {\vphantom {1 2}} \right. \kern-0pt} 2}} \right)}} h^{\mu } }}{{\varGamma \left( \mu \right)H^{{\left( {\mu - {1 \mathord{\left/ {\vphantom {1 2}} \right. \kern-0pt} 2}} \right)}} \bar{\gamma }^{{\left( {{\alpha \mathord{\left/ {\vphantom {\alpha 2}} \right. \kern-0pt} 2}} \right)\left( {\mu + {1 \mathord{\left/ {\vphantom {1 2}} \right. \kern-0pt} 2}} \right)}} }}\sum\limits_{n = 0}^{u - 1} {\frac{1}{{2^{n} (n!)}}} \\ & \quad \times \,\int\limits_{0}^{\infty } {H_{0}^{1} \,_{1}^{0} \,\left[ {\left. {\frac{{2\mu \left( {h - H} \right)\gamma^{{\left( {{\alpha \mathord{\left/ {\vphantom {\alpha 2}} \right. \kern-0pt} 2}} \right)}} }}{{\bar{\gamma }^{{\left( {{\alpha \mathord{\left/ {\vphantom {\alpha 2}} \right. \kern-0pt} 2}} \right)}} }}} \right|\,_{(0\,,1)\,} } \right]H_{1}^{1} \,_{2}^{1} \,\left[ {\left. {\frac{{4\mu H\gamma^{{\left( {{\alpha \mathord{\left/ {\vphantom {\alpha 2}} \right. \kern-0pt} 2}} \right)}} }}{{\bar{\gamma }^{{\left( {{\alpha \mathord{\left/ {\vphantom {\alpha 2}} \right. \kern-0pt} 2}} \right)}} }}} \right|\,\begin{array}{*{20}c} {\left( {\frac{1}{2},1} \right)} & {} \\ {\left( {\mu - \frac{1}{2},1} \right)} & {\left( {\frac{1}{2} - \mu ,1} \right)} \\ \end{array} } \right]\gamma^{{\frac{\alpha }{2}\left( {\mu + {\raise0.7ex\hbox{$1$} \!\mathord{\left/ {\vphantom {1 2}}\right.\kern-0pt} \!\lower0.7ex\hbox{$2$}}} \right) + n - 1}} } \exp \left( { - \,\frac{\gamma }{2}} \right)\,d\gamma \\ \end{aligned}$$
(20)
With the help of [20, 3.326.2], (20) can be simplified as
$$ \begin{aligned} & I_{2} = \frac{{\alpha \mu^{{\left( {\mu + {1 \mathord{\left/ {\vphantom {1 2}} \right. \kern-0pt} 2}} \right)}} h^{\mu } }}{{\varGamma \left( \mu \right)H^{{\left( {\mu - {1 \mathord{\left/ {\vphantom {1 2}} \right. \kern-0pt} 2}} \right)}} \bar{\gamma }^{{\left( {{\alpha \mathord{\left/ {\vphantom {\alpha 2}} \right. \kern-0pt} 2}} \right)\left( {\mu + {1 \mathord{\left/ {\vphantom {1 2}} \right. \kern-0pt} 2}} \right)}} }}\sum\limits_{n = 0}^{u - 1} {\frac{1}{{2^{n} (n!)}}} \frac{1}{{0.5^{{^{{\frac{\alpha }{2}\left( {\mu + {\raise0.7ex\hbox{$1$} \!\mathord{\left/ {\vphantom {1 2}}\right.\kern-0pt} \!\lower0.7ex\hbox{$2$}}} \right) + n}} }} }}\frac{1}{{\left( {2\pi i} \right)^{2} }} \\ & \quad \times \,\int_{L1} {\int_{L2} {\frac{{\varGamma \left( s \right)\varGamma \left( {\mu - \frac{1}{2} + t} \right)\varGamma \left( {\frac{1}{2} - t} \right)\varGamma \left( {\frac{\alpha }{2}\left( {\mu + \frac{1}{2} - s - t} \right) + n} \right)}}{{\varGamma \left( {\frac{1}{2} + \mu - t} \right)}}\left( {\frac{{2\mu \left( {h - H} \right)}}{{\left( {0.5\bar{\gamma }} \right)^{{\left( {{\alpha \mathord{\left/ {\vphantom {\alpha 2}} \right. \kern-0pt} 2}} \right)}} }}} \right)^{ - s} \left( {\frac{4\mu H}{{\left( {0.5\bar{\gamma }} \right)^{{\left( {{\alpha \mathord{\left/ {\vphantom {\alpha 2}} \right. \kern-0pt} 2}} \right)}} }}} \right)^{ - t} } } dsdt \\ \end{aligned} $$
(21)
(21) can be written using [21, 2.57] as
$$ \begin{aligned} I_{2} & = \frac{{\alpha \mu^{{\left( {\mu + {1 \mathord{\left/ {\vphantom {1 2}} \right. \kern-0pt} 2}} \right)}} h^{\mu } }}{{\varGamma \left( \mu \right)H^{{\left( {\mu - {1 \mathord{\left/ {\vphantom {1 2}} \right. \kern-0pt} 2}} \right)}} \bar{\gamma }^{{\left( {{\alpha \mathord{\left/ {\vphantom {\alpha 2}} \right. \kern-0pt} 2}} \right)\left( {\mu + {1 \mathord{\left/ {\vphantom {1 2}} \right. \kern-0pt} 2}} \right)}} }}\sum\limits_{n = 0}^{u - 1} {\frac{1}{{2^{n} (n!)}}} \frac{1}{{0.5^{{^{{\frac{\alpha }{2}\left( {\mu + {\raise0.7ex\hbox{$1$} \!\mathord{\left/ {\vphantom {1 2}}\right.\kern-0pt} \!\lower0.7ex\hbox{$2$}}} \right) + n}} }} }} \\ & \quad \times \,H_{1,0:0,1:1,2}^{0,1:1,0:1,1} \,\,\left[ {\left. {\begin{array}{*{20}c} {\frac{{2\mu \left( {h - H} \right)}}{{\left( {0.5\bar{\gamma }} \right)^{{\left( {{\alpha \mathord{\left/ {\vphantom {\alpha 2}} \right. \kern-0pt} 2}} \right)}} }}} \\ {\frac{4\mu H}{{\left( {0.5\bar{\gamma }} \right)^{{\left( {{\alpha \mathord{\left/ {\vphantom {\alpha 2}} \right. \kern-0pt} 2}} \right)}} }}} \\ \end{array} } \right|\,\begin{array}{*{20}c} {\left( {1 - \frac{\alpha }{2}\left( {\mu + \frac{1}{2}} \right) - n;\frac{\alpha }{2},\frac{\alpha }{2}} \right): - ;\left( {\frac{1}{2},1} \right)} \\ { - :\left( {\mu - \frac{1}{2},1} \right);\left( {\frac{1}{2} - \mu ,1} \right)} \\ \end{array} } \right] \\ \end{aligned} $$
(22)
Using [2, 27] for the expansion of \({}_{1}\tilde{F}_{1} \left( {.;.;.} \right)\) and using [19, 8.4.3.1], [19, 8.4.22.3], [19, 8.3.2.21] the last term of (19) can be written as
$$\begin{aligned} I_{3} & = \frac{{\alpha \mu^{{\left( {\mu + {1 \mathord{\left/ {\vphantom {1 2}} \right. \kern-0pt} 2}} \right)}} h^{\mu } }}{{\varGamma \left( \mu \right)H^{{\left( {\mu - {1 \mathord{\left/ {\vphantom {1 2}} \right. \kern-0pt} 2}} \right)}} \bar{\gamma }^{{\left( {{\alpha \mathord{\left/ {\vphantom {\alpha 2}} \right. \kern-0pt} 2}} \right)\left( {\mu + {1 \mathord{\left/ {\vphantom {1 2}} \right. \kern-0pt} 2}} \right)}} }}\sum\limits_{n = 1 - u}^{u - 1} {\frac{{\varGamma \left( {u + n} \right)}}{{2^{u + n} \varGamma \left( u \right)}}} \sum\limits_{k = 0}^{\infty } {\frac{{\left( {u + n} \right)_{k} }}{{\left( {1 + n} \right)_{k} \varGamma \left( {1 + n} \right)\left( {k!} \right)}}} \\ & \quad \int\limits_{0}^{\infty } {\gamma^{{\frac{\alpha }{2}\left( {\mu + {\raise0.7ex\hbox{$1$} \!\mathord{\left/ {\vphantom {1 2}}\right.\kern-0pt} \!\lower0.7ex\hbox{$2$}}} \right) + k - 1}} } H_{0}^{1} \,_{1}^{0} \,\left[ {\left. {\frac{{2\mu \left( {h - H} \right)\gamma^{{\left( {{\alpha \mathord{\left/ {\vphantom {\alpha 2}} \right. \kern-0pt} 2}} \right)}} }}{{\bar{\gamma }^{{\left( {{\alpha \mathord{\left/ {\vphantom {\alpha 2}} \right. \kern-0pt} 2}} \right)}} }}} \right|\,_{(0\,,1)\,} } \right]H_{1}^{1} \,_{2}^{1} \,\left[ {\left. {\frac{{4\mu H\gamma^{{\left( {{\alpha \mathord{\left/ {\vphantom {\alpha 2}} \right. \kern-0pt} 2}} \right)}} }}{{\bar{\gamma }^{{\left( {{\alpha \mathord{\left/ {\vphantom {\alpha 2}} \right. \kern-0pt} 2}} \right)}} }}} \right|\,\begin{array}{*{20}c} {\left( {\frac{1}{2},1} \right)} & {} \\ {\left( {\mu - \frac{1}{2},1} \right)} & {\left( {\frac{1}{2} - \mu ,1} \right)} \\ \end{array} } \right]\exp \left( { - \,\gamma } \right)\,d\gamma \\ \end{aligned}$$
(23)
Using [21, 2.57], (23) can be simplified as
$$\begin{aligned} I_{3} & = \frac{{\alpha \mu^{{\left( {\mu + {1 \mathord{\left/ {\vphantom {1 2}} \right. \kern-0pt} 2}} \right)}} h^{\mu } }}{{\varGamma \left( \mu \right)H^{{\left( {\mu - {1 \mathord{\left/ {\vphantom {1 2}} \right. \kern-0pt} 2}} \right)}} \bar{\gamma }^{{\left( {{\alpha \mathord{\left/ {\vphantom {\alpha 2}} \right. \kern-0pt} 2}} \right)\left( {\mu + {1 \mathord{\left/ {\vphantom {1 2}} \right. \kern-0pt} 2}} \right)}} }}\sum\limits_{n = 1 - u}^{u - 1}\sum\limits_{k = 0}^{\infty } {\frac{{\left( u \right)_{n} }}{{2^{u + n + k} }}} {\frac{{\left( {u + n} \right)_{k} }}{{\left( {n + k} \right)!\left( {k!} \right)}}} \\ & \quad \times \,H_{1,0:0,1:1,2}^{0,1:1,0:1,1} \,\,\left[ {\left. {\begin{array}{*{20}c} {\frac{{2\mu \left( {h - H} \right)}}{{\bar{\gamma }^{{\left( {{\alpha \mathord{\left/ {\vphantom {\alpha 2}} \right. \kern-0pt} 2}} \right)}} }}} \\ {\frac{4\mu H}{{\bar{\gamma }^{{\left( {{\alpha \mathord{\left/ {\vphantom {\alpha 2}} \right. \kern-0pt} 2}} \right)}} }}} \\ \end{array} } \right|\,\begin{array}{*{20}c} {\left( {1 - \frac{\alpha }{2}\left( {\mu + \frac{1}{2}} \right) - k;\frac{\alpha }{2},\frac{\alpha }{2}} \right): - ;\left( {\frac{1}{2},1} \right)} \\ { - :\left( {\mu - \frac{1}{2},1} \right);\left( {\frac{1}{2} - \mu ,1} \right)} \\ \end{array} } \right] \\ \end{aligned}$$
(24)
Putting the values of \(I_{1} ,I_{2}\) and \(I_{3}\) in (19), we get the final expression for AUC.

5 Results and Discussion

In this section, the effect of non-linearity and the inhomogeneity on the system performance will be demonstrated. For validating the accuracy of the proposed expressions, Monte-Carlo simulations are also included with 107 numbers of samples for generating α–η–μ fading distribution. All the computations and simulations were carried out in MATLAB (version R2014a). It is noted that the simulation results are in close match with the numerical results obtained by keeping enough number of terms in the infinite series. For the implementation of bivariate Fox-H function, the contours \(L_{1} = \sigma_{1} \pm j\ell\) and \(L_{2} = \sigma_{2} \pm j\ell\). The values of \(\sigma_{1} = \sigma_{2} = 0.2\) and \(\ell = 10\) are taken for all the calculations performed here. We have considered format 2 for the values of η in all the computations.

Figure 1 shows the CROC curve [average probability of missed detection (\(\bar{P}_{m} (\lambda ) = 1 - \bar{P}_{d} (\lambda )\)) versus \(P_{f} (\lambda )\)] for several values of fading parameters α, η and μ. The performance of the system increases with increase in μ, this is because of the fact that number of multipath clusters increases with increase in μ. Similar the ED performance increases with increase in α, which represents the non-linearity of the fading channel. It is also observed that when η increase, the system performance improves. The CROC curve for η–μ, which is derived as a special case by putting α = 2 in α–η–μ distribution is shown in Fig. 2. The analytical results produced here are in close agreement with the simulated results, verifying the accuracy of the proposed results. The effect of the number of samples used for decision statistic (\(n\)) and average received SNR on CROC is shown in Fig. 3. It is noted that the value of \(\bar{P}_{m}\) increases with increase in number of samples used. The performance improvement in the system can be clearly seen with increase in average received SNR.
Fig. 1

CROC curve for several values of α, η and μ

Fig. 2

CROC curve for η–μ fading channel (derived as a special case of α–η–μ by putting α = 2)

Fig. 3

CROC curve for various values of number of samples and average received SNR

Figure 4 illustrates the detection performance for SS with up to six cooperative users. It is shown that the performance of ED improves substantially as the number of cooperative users increases. More specifically, for \(Q_{f}\) = 0.2, the probability of detection for 6 cooperative users is 50% larger than for a single user SS. ROC curve for SLS diversity scheme with up to five branches are presented in Fig. 5. The average SNR of each branch is taken as \(\bar{\gamma }_{1}\) = 0 dB, \(\bar{\gamma }_{2}\) = 1 dB, \(\bar{\gamma }_{3}\) = 2 dB, \(\bar{\gamma }_{4}\) = 3 dB and \(\bar{\gamma }_{5}\) = 4 dB. It is noted that as the number of diversity branch increases the performance of ED increases. Particularly, for \(P_{f}\) = 0.1, the probability of detection for L = 5 is 80% larger than the corresponding value for L = 1. The highest diversity gain is observed between the no diversity case to the dual branch (L = 2) scheme, where an increase of 20% is observed on the average probability of detection.
Fig. 4

ROC curve for cooperative SS for different number of cooperative users

Fig. 5

ROC curve with SLS diversity for different number of diversity branches

Figure 6 shows the variation of CAUC (\(1 - \bar{A}\)) versus the average received SNR for several values of the system parameters. This figure shows that the plot shifts downwards for increase in α, η and μ. It is clear from the results that system performance improves with every small increment in the values of α, η and μ. It is noted that the numerical results produced here are highly accurate and in closed agreement with their Monte-Carlo simulation counterparts.
Fig. 6

CAUC against average received SNR for various values of α, η and μ

6 Conclusion

In this paper, novel and closed-form analytical expressions for \(\bar{P}_{d}\) and \(\bar{A}\) over α–η–μ generalized fading channel has been presented. The expressions are derived in terms of Fox-H function. The performance of the ED in terms of CROC and CAUC is evaluated for the said channel. It is shown that the performance of the ED increases with the increase in α, η and μ values. In the end, it was indicated that diversity reception and cooperative SS can significantly improve the detection performance over extreme fading conditions. Furthermore, it was shown that ED-based SS requires large time-bandwidth product values for robust sensing in low SNR regions over α–η–μ generalized fading channel. The derived results are generic and can be directly used for the performance analysis of η–μ and α–μ fading channels and their special cases. The analytical results produced here have been validated with the Monte-Carlo simulations.

Notes

Acknowledgements

The authors would like to thank the anonymous reviewers for their useful suggestions for improving the presentation of the material in this paper.

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Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Central Research LaboratoryBharat Electronics LimitedGhaziabadIndia

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