Detection Performance of Cooperative Spectrum ... - Semantic Scholar

November 24, 2014

International Journal of Electronics

Paper

To appear in the International Journal of Electronics Vol. 00, No. 00, November 2014, 1–24

Detection Performance of Cooperative Spectrum Sensing with Hard Decision Fusion in Fading Channels S. Nallagonda1 , A. Chandra2,∗ , S. D. Roy1 , S. Kundu1 , P. Kukolev2 , and A. Prokes2 1

Electronics & Communication Engineering Department, National Institute of Technology, Durgapur 713209, WB, India. 2 Department of Radio Electronics, Brno University of Technology, Brno 61600, Czech Republic. (November 2014)

In this paper, we investigate the detection performance of cooperative spectrum sensing (CSS) using energy detector (ED) in several fading scenarios. The fading environments comprises of relatively less-studied Hoyt and Weibull channels in addition to the conventional Rayleigh, Rician, Nakagami-m, and log-normal shadowing channels. We have presented an analytical framework for evaluating different probabilities related to spectrum sensing, i.e. missed detection, false alarm, and total error due to both of them, for all the fading/ shadowing models mentioned. The major theoretical contribution is, however, the derivation of closed-form expressions for probability of detection. Based on our developed framework, we present performance results of CSS under various hard decision fusion strategies such as OR rule, AND rule, and Majority rule. Effects of sensing channel signal to noise ratio (SNR), detection threshold, fusion rules, number of cooperating CRs, and fading/ shadowing parameters on the sensing performance have been illustrated. The performance improvement achieved with CSS over a single CR based sensing is depicted in terms of total error probability. Further, an optimal threshold that minimizes total error probability has been indicated for all the fading/ shadowing channels. Keywords: Cognitive radio, energy detection, cooperative spectrum sensing, fusion rules, detection probability, Hoyt fading, Weibull fading.

1.

Introduction

1.1

Motivation

Radio spectrum crowding has occurred over last few decades as a result of a series of regulations imposed by various international and national administrative bodies. Allocation of huge spectrum pools to services that are intermittent and less demanding created an artificial shortage of spectrum to the ever-increasing wireless applications. The prospect of spectrum re-allocation is infeasible from economic viewpoint, and future allocations require negotiations on a global scale, a task almost impossible due to misalignment of national interests. One interesting solution is offered by the cognitive radio (CR) technology, where secondary users, ∗ Corresponding

author. Email: [email protected]

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hereafter termed as cognitive radio nodes or simply as CRs, are allowed to share the spectrum with the primary users (PUs) holding license to use the frequency band. Considering the fact that in a typical developed urban area the average spectrum utilization rate is under 10% [Valenta et al., 2010], the idea of sharing seems quite promising to solve the conflict between spectrum under utilization and spectrum scarcity. A CR node should not, however, interrupt the PU activity. Thus it is important to sense the PU signal, accurately and quickly [Haykin, 2005], [Wang and Ray Liu, 2011], in order to find the spectrum holes or white spaces. Energy detection based sensing schemes, which measures the energy in the received waveform over an observation time window [Urkowitz, 1967], [Akyildiz et al., 2011], are the simplest to implement as they do not need a priori knowledge of the PU signal. Quite naturally, the spectrum sensing decision is affected by fading, shadowing, and time-varying nature of wireless channels [Cabric et al., 2004] connecting PU and CR, and quite often, due to severe multipath fading/ shadowing, a single CR may fail to notice the presence of the PU. The detection performance may be improved with cooperative spectrum sensing (CSS), i.e. multiple CR nodes may try to sense the PU individually and send their sensing information in the form of single bit binary decisions to a fusion center (FC). Next, the local decisions may be combined following a hard decision/ counting (k-out-of-N ) rule at the FC to make the final decision regarding the presence of the PU [Kyperountas et al., 2008], [Duan and Li, 2010]. It may be noted that there exists alternative approaches too. For example, a soft decision fusion [Chaudhari et al., 2012], [Mustonen et al., 2009] may be realized by asking the CRs to send the entire test statistics to the FC instead of sending of binary values obtained by local decisions. The FC finally takes a global decision by combining the test statistics from all cooperating CRs.

1.2

Literature Survey

Existing works mostly examined the collaborative spectrum sensing, using energy detection in Rayleigh and Nakagami fading channels [Ghasemi and Sousa, 2005], [Nallagonda et al., 2011]. Our present study not only includes these popular fading models but also focuses on Rician, Hoyt (or Nakagami-q), Weibull, and log-normal fading environments. For the single CR case (non-collaborative), the average probability of detection (P¯d ) expression has already been derived for Rayleigh, Rician, Nakagami-m, and log-normal fading channels [Digham et al., 2003, 2007, Sun, 2011]. However, the expression of P¯d for Rician fading channel [Digham et al., 2003] is only valid when time-bandwidth product (u) is equal to unity. So, it is a challenging task to derive the expression of P¯d for Rician fading channel for any value of u. It is also equally important to analyze the spectrum sensing performance for more generalized fading channels. Recently, Weibull and Hoyt fading are used to model wireless channel in several scenarios. Moreover, they can be used to depict generalized fading as they encompass a wide range of fading cases such as Rayleigh, Nakagami-m, and Rician etc. for various values of the fading parameter. Weibull fading has been proved to exhibit an excellent fitting for indoor [Hashemi, 1993] and outdoor [Adawi et al., 1988] environments. The Weibull distribution encompasses Rayleigh fading for a certain value of the fading parameter (v = 2) [Ismail and Matalgah, 2006]. On the other hand, Hoyt or Nakagami-q distribution has been found useful in describing the random attenuation in some satellite as well as terrestrial wireless links [Simon and Alouini, 2004]. Thus, it is also necessary to derive the expression of probability of detection in Hoyt and Weibull

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fading channels. Although there had been some recent publications in this regard, namely [Atapattu et al., 2009, Fathi and Tawfik, 2012], where generalized η-µ and α-µ fading channels were dealt with (Hoyt and Weibull are special cases of η-µ and α-µ respectively), none of the authors considered cooperation among CRs. Further, the results in [Atapattu et al., 2009] were expressed in terms of contour integrals which decompose to closed-form expressions only for certain values of the fading parameters.

1.3

Contributions of the Paper

In this paper, we investigate the performance of a single CR spectrum sensing and cooperative spectrum sensing using energy detection over Additive white Gaussian noise (AWGN) channel and several fading environments (e.g. Rayleigh, Nakagamim, Hoyt, Rician, Weibull, and log-normal). We develop appropriate analytical formulations as well as a simulation test bed using MATLAB to validate our formulations. Specifically our contributions in this paper are as follows: • Development of new analytical expressions for probability of detection (P¯d ) in Hoyt and Weibull fading channels. Apart from that, generalized expression of P¯d for Rician channel and alternate expressions of P¯d for Rayleigh, Nakagamim, and log-normal channels are provided. • Evaluation of the performance of a single CR in terms of missed detection and false detection probabilities over AWGN and fading/ shadowing channels. Results obtained via analytical approach for the case of Rayleigh, Nakagami (m = 3), Hoyt (q = 0.25), Rician (K = 3), Weibull (v = 6), and lognormal (σ = 5 dB) fading match with the simulation results (based on our developed test bed) under same conditions, which validates the accuracy of our analytical expressions. • Impact of fading/ shadowing parameters and the number of cooperating CRs (N ) on sensing performance over several channels (Rayleigh, Nakagami-m, Hoyt, Rician, Weibull, and log-normal) have been assessed and portrayed through overall probability of missed detection (Qm ) versus overall probability of false alarm (Qf ) plots and total error probability (Qm + Qf ) versus threshold (λ) plots. • Impact of average SNR on overall sensing performance is indicated. • Quantitative comparison of the performances of spectrum sensing schemes with and without cooperation among the CRs have been studied. • Several hard decision fusion rules such as AND (N -out-of-N ), OR (1-out-ofN ) and Majority logic are investigated and performance of these fusion rules are compared. • We indicate optimal sensing threshold to minimize total error probability for various fading channels.

1.4

Organization of the Paper

The rest of the paper is organized as follows. In Section 2, the system model under consideration is described and important notations are listed. Further, the average probability of detection (P¯d ) expressions for Rayleigh, Nakagami-m, Hoyt, Rician, Weibull, and log-normal fading channels are derived. Results and discussions are presented in Section 3. Finally we conclude in Section 4.

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2. 2.1

Paper

Derivation of Probability of Detection Notations • s(t) : Signal transmitted from PU. • n(t) : Noise waveform which is modeled as a zero-mean white Gaussian random process. • N01 : One-sided noise power density. ∫ T spectral 2 • Es : Signal energy; Es = 0 s (t)dt. • γ = Es /N01 : Signal-to-noise ratio (SNR). • γ¯ : Average SNR. • W : One-sided bandwidth (Hz), i.e., positive bandwidth of low-pass (LP) signal. • u = T W : Time-bandwidth product. • fc : Carrier frequency. • Pd : Probability of detection at CR. • P¯d : Average probability of detection at CR in fading channels. • Pf : Probability of false alarm at CR. • Pm = 1 − Pd : Probability of missed detection at CR. • Qd : Overall probability of detection at FC. • Qf : Overall probability of false alarm at FC. • Qm = 1 − Qd : Overall probability of missed detection at FC. • H0 : Hypothesis 0 that corresponds to presence of PU. • H1 : Hypothesis 1 that corresponds to absence of PU. • N (µ, σ 2 ) : A Gaussian variate with mean µ and variance σ 2 . • χ2α : A central chi-square variate with α degrees of freedom. • χ2α (β) : A non-central chi-square variate with α degrees of freedom and noncentrality parameter β.

2.2

System Model

We consider a network of N CRs, one PU and one FC. Each CR is equipped with one ED, whose details are shown in Fig. 1, having identical thresholds (λi = λ; i ∈ {1, 2, · · · , N }). CRs make hard binary decisions (either binary bit ‘1’ or binary bit ‘0’) and transmit their decisions to FC for data fusion (shown in Fig. 2). It is assumed that the distance between any two CRs is less than the distance between the PU and a CR or the distance between a CR and the FC. Further, the channels between CRs and FC have been considered as ideal channels (noiseless) in this paper. ˜

x t

H0

T

³ ˜ dt

2

or

0

BPF

Signal squarer

Integrator

Decision device

H1

Figure 1. Block diagram of an energy detector [Urkowitz, 1967].

An energy detector receives a signal x(t) as defined below at input and gives a binary decision regarding the presence of the PU { x(t) =

n(t)

: H0

(1)

h(t)s(t) + n(t) : H1

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Cognitive radio users CR 1

PU

FC

CR 2

Primary user

Fusion center Fading channels

CR N

Ideal channels

Figure 2. Cooperative spectrum sensing scenario.

According to the sampling theorem, the noise process can be expressed as [Shannon, 1949]

n(t) =

∞ ∑

ni sinc(2W t − i)

(2)

i=−∞

where sinc(x) = sin(πx)/(πx) and ni = n(i/(2W )). One can easily check that ni is Gaussian with zero mean and variance N01 W , i.e. ni ∼ N (0, N01 W ); ∀i. The noise energy can be approximated over the time interval (0, T ), as [Urkowitz, 1967], [Digham et al., 2003] ∫

T

n2 (t)dt = 0

2u 1 ∑ 2 ni 2W

(3)

i=1

√ If we define n′i = ni / N01 W , the decision statistic Y can be written as

Y =

2u ∑

n′2 i

(4)

i=1

Clearly, Y is a sum of squares of 2u standard Gaussian variates with zero mean and unit variance. Therefore, Y follows a central chi-square (χ2 ) distribution with 2u degrees of freedom. The same approach can be extended to find the statistics when signal s(t) is present by replacing each of the ni terms with ni + si , where si = s(i/(2W )). The decision statistic Y in this case will have a non-central χ2 distribution with 2u degrees of freedom and a non centrality parameter 2γ [Urkowitz, 1967], [Digham et al., 2003]. We can describe the decision statistic in short-hand notations as { Y ∼

2.3

χ22u

: H0 (5)

χ22u (2γ) : H1

Non-fading Environment (AWGN Channel)

In non-fading environment i.e. when the sensing channels are corrupted by AWGN only, the probabilities of detection and false alarm at i-th CR are given by the

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following formulas [Urkowitz, 1967], [Digham et al., 2003] √ ) λ

(6)

Pf = Pr[Y > λ|H0 ] = Γ(u, λ/2)/Γ(u)

(7)

Pd = Pr[Y > λ|H1 ] = Qu

(√

2γ,

where Qu (·, ·) is the generalized Marcum Q-function [Nuttall, 1975] of order u, Γ(·) is the gamma function [Gradshteyn and Ryzhik, 2007, (8.310.1)], and Γ(·, ·) is the complementary incomplete gamma function [Gradshteyn and Ryzhik, 2007, (8.350.2)]. For simplicity, identical average SNRs (¯ γi = γ¯ ; ∀i) has been assumed for all CRs. The assumption stems from the fact that the PU is located far away from all CRs, i.e. the distance between any two CRs is negligible compared to the distance from the PU to a CR, and all the CRs receive the PU signal with same local mean signal power. As seen from (7), Pf is independent of γ¯ since under H0 there is no primary signal present. If the signal power is unknown, we can first set the false alarm probability at CR (Pf ) to a specific desired level. For a given desired level of Pf and for a fixed number of samples (2u), the threshold (λ) of energy detector is set which determines the detection probability at CR (Pd ) along with instantaneous SNR (γ) of sensing channel as governed by (6). Under a fading scenario, (6) gives the probability of detection as a function of the instantaneous SNR (γ). In this case, the average probability of detection (P¯d ) may be derived by averaging (6) over fading statistics [Ghasemi and Sousa, 2005], i.e. ∫ P¯d =

∞

Qu

(√ √ ) 2x, λ fγ (x)dx

(8)

0

where fγ (x) is the probability density function (PDF) of γ under fading.

2.4

Rayleigh Fading Channel

If the signal amplitude follows a Rayleigh distribution, then the SNR (γ) follows an exponential PDF and is given by [Simon and Alouini, 2004] ( ) γ 1 fγ (γ) = exp − γ¯ γ¯

;γ ≥ 0

(9)

The average Pd in this case, P¯d,Ray can now be evaluated by substituting (9) in (8), i.e. 1 P¯d,Ray = γ¯

∫

∞

Qu 0

(√

( ) √ ) γ 2γ, λ exp − dγ γ¯

(10)

Let, γ/¯ γ = z 2 . Then equation (10) can be rewritten as ∫ P¯d,Ray = 2

∞

( √ √ ) ( ) Qu z 2¯ γ , λ z exp −z 2 dz

0

6

(11)

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Let us now consider the following result ∫ 0

∞

( 2 2) p x Qu (ax, b)x exp − dx 2 [ ] u−1 ( )n ∑ b (ab)n a2 b2 = F 1; n + 1; 1 1 a 2n (a2 + p2 )Γ(n + 1) 2(a2 + p2 ) n=1 [ ] 1 b 2 p2 + 2 exp − p 2(a2 + p2 )

(12)

where 1 F1 (·; ·; ·) denotes confluent hypergeometric function [Gradshteyn and Ryzhik, 2007, (9.210.1)]. A formal proof of (12) is provided in Appendix A. Now using the result in (12), the integral in (11) can be solved and the final expression for average probability of detection becomes

P¯d,Ray =

[ ] u−1 ( ) 1 ∑ λ n exp(−λ/2) γ¯ λ 1 F1 1; n + 1; 1 + γ¯ 2 n! 2(1 + γ¯ ) n=1 [ ] λ + exp − 2(1 + γ¯ )

(13)

It may be noted that an alternate expression of P¯d,Ray was also derived earlier by Digham et al. (see [Digham et al., 2003, (16)] and [Digham et al., 2007, (9)]). However, their expression contained two different series while the expression derived above contains only one.

2.5

Nakagami-m Fading Channel

If the signal envelope follows a Nakagami-m distribution, then the random variable γ follows a Gamma PDF given by [Simon and Alouini, 2004] ( fγ (γ) =

m γ¯

)m

( ) γ m−1 mγ exp − Γ(m) γ¯

;γ ≥ 0

(14)

where m(0.5 ≤ m ≤ ∞) is the fading severity parameter. The average Pd in this case,P¯d,N ak , can now be evaluated by substituting equation (14) in (8) ( P¯d,N ak =

m γ¯

)m

1 Γ(m)

∫

∞

Qu

(√

2γ,

0

( ) √ ) m−1 mγ λ γ exp − dγ γ¯

(15)

With a simple substitution of variables, mγ/¯ γ = z 2 , (15) can be rewritten as

P¯d,N ak

2 = (m − 1)!

∫

( √

∞

Qu 0

z

) 2¯ γ √ , λ z 2m−1 exp(−z 2 )dz m

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(16)

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Let us now consider the following result ∫ 0

∞

( 2 2) p x Qu (ax, b)x exp − dx 2 ( 2) u−1 ( )n ∑ b b (ab)n Γ(q) = exp − a 2 2n+1 [(a2 + p2 )/2]q Γ(n + 1) n=1 [ ] a2 b2 × 1 F1 q; n + 1; 2(a2 + p2 ) [ ] b2 p2 2q−1 (q − 1)! a2 exp − + p2q p2 + a2 2(p2 + a2 ) { q−2 ( )k [ ] ∑ p2 b2 a2 × Lk − p2 + a 2 2(p2 + a2 ) k=0 ( )( )q−1 [ ]} p2 p2 b 2 a2 + 1+ 2 Lq−1 − a p2 + a 2 2(p2 + a2 ) 2q−1

(17)

where Ln (·) is the Laguerre polynomial of degree n [Gradshteyn and Ryzhik, 2007, Section 8.970]. A formal proof of (17) is provided in Appendix B. Now using the result in (17), the integral in (16) can be solved and the final expression for average probability of detection becomes (

P¯d,N ak =

)m ∑ [ ] u−1 ( )n λ γ¯ λ exp(−λ/2) 1 F1 m; n + 1; 2 n! 2(m + γ¯ ) n=1 [ ] γ¯ λm + exp − m + γ¯ 2(m + γ¯ ) {m−2 ( )k [ ] ∑ m λ¯ γ × Lk − m + γ¯ 2(m + γ¯ ) k=0 ( )( )m−1 ]} [ m + γ¯ m λ¯ γ + Lm−1 − γ¯ m + γ¯ 2(m + γ¯ ) m m + γ¯

(18)

As a double check, we can verify using [Gradshteyn and Ryzhik, 2007, (8.970.3)] that (18) reduces to (13) for m = 1. Further, (18) also matches with the expressions derived by Digham et al. (see [Digham et al., 2003, (20)] and [Digham et al., 2007, (7)])

2.6

Hoyt/ Nakagami-q Fading Channel

Hoyt or Nakagami-q, q (0 ≤ q ≤ 1) being the fading severity parameter, distribution is generally used to characterize the fading environments that are more severe than Rayleigh fading. The PDF of γ in Hoyt fading channel i.e., fγ (γ), may be defined as [Chandra et al., 2013], [Chandra et al., 2012] ( ) ( √ ) γ γ 1−p 1 I0 fγ (γ) = √ exp − p¯ γ p¯ γ p¯ γ

8

;γ ≥ 0

(19)

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where p = 4q 2 /(1 + q 2 )2 ; 0 ≤ p ≤ 1. For q = p = 1 the distribution reduces to Rayleigh fading PDF and for q = p = 0 it represents the one-sided Gaussian PDF. The average Pd in this case, P¯d,Hoy , can now be evaluated by substituting (19) in (8) 1 P¯d,Hoy = √ p¯ γ

∫

∞

Qu 0

(√

( ) ( √ ) √ ) γ γ 1−p 2γ, λ exp − I0 dγ p¯ γ p¯ γ

(20)

Using an infinite series form of the Marcum Q function [Andras et al., 2011, (2.4)] Qu

∞ (√ ∑ √ ) Γ(u + n, λ/2) γ n 2γ, λ = exp(−γ) Γ(u + n) n!

(21)

n=0

the expression in (20) may be written in the following form ∞ 1 ∑ Γ(u + n, λ/2) P¯d,Hoy = √ p¯ γ Γ(u + n)n! n=0 )] ( √ ) [ ( ∫ ∞ γ 1−p 1 n × I0 dγ γ exp −γ 1 + p¯ γ p¯ γ 0

(22)

Next, utilizing the following result [Chandra and Bose, 2012, (2.3)] ∫

∞

xµ−1 exp(−αx)Iν (βx)dx

0

β ν Γ(µ + ν) = ν µ+ν 2 F1 2 α Γ(ν + 1)

(

µ+ν+1 µ+ν β2 , ; ν + 1; 2 2 2 α

)

(23)

where 2 F1 (·, ·; ·; ·) is the Gaussian hypergeometric function [Gradshteyn and Ryzhik, 2007, (9.100)], the integration in (22) can be solved in closed-form and the final expression for P¯d,Hoy becomes ( )n+1 ∞ 1 ∑ Γ(u + n, λ/2) p¯ γ ¯ Pd,Hoy = √ p¯ γ Γ(u + n) 1 + p¯ γ n=0 [ ] n+2 n+1 1−p × 2 F1 , ; 1; 2 2 (1 + p¯ γ )2

(24)

To the best of our knowledge, this is a new result. As Rayleigh fading is a special case of Hoyt fading (when q = p = 1), we may obtain an alternative expression for P¯d,Ray by setting p = 1 (i.e., q = 1) in (24). 2.7

Rician/ Nakagami-n Fading Channel

If the signal strength follows a Rician distribution, the PDF of γ will be [Simon and Alouini, 2004] ) [ ] ( √ (1 + K)γ 1+K K(1 + K)γ exp −K − I0 2 fγ (γ) = γ¯ γ¯ γ¯

9

; γ ≥ 0 (25)

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where K is the Rician factor and In (·) is nth order modified Bessel function of first kind. The average Pd in the case of a Rician fading channel, P¯d,Ric , is then obtained by substituting (25) in (8), i.e. 1+K P¯d,Ric = exp(−K) γ¯ ∫

∞

× 0

] [ (√ √ ) (1 + K)γ I0 Qu 2γ, λ exp − γ¯

( √ 2

K(1 + K)γ γ¯

)

(26) dγ

To solve the integral in (26) we proceed by first expressing the Marcum Q function in the infinite series form as described in (21) ∞

∑ Γ(u + n, λ/2) 1+K exp(−K) P¯d,Ric = γ¯ Γ(u + n)n! n=0 ) [ ( )] ( √ ∫ ∞ 1 + K + γ ¯ K(1 + K)γ × γ n exp −γ I0 2 dγ γ¯ γ¯ 0

(27)

and then solving the integral containing power, exponential, and Bessel function through the following result [Chandra, 2011, (B.16)] ∫

∞

√ xµ−1/2 exp(−αx)Iν (β x)dx

0

2Γ[µ + (ν + 1)/2] = βαµ Γ(ν + 1)

(

β2 4α

) ν+1 2

( ) ν+1 β2 ; ν + 1; 1 F1 µ + 2 4α

(28)

to obtain ( )n+1 ∞ ∑ 1+K Γ(u + n, λ/2) γ¯ ¯ Pd,Ric = exp(−K) γ¯ Γ(u + n) 1 + K + γ¯ n=0 [ ] K(1 + K) × 1 F1 n + 1; 1; 1 + K + γ¯

(29)

which is valid for any time-bandwidth product (u) in contrast to the expression derived by Digham et al. [Digham et al., 2003, (25)] which was valid only for u = 1. Also, just like the Hoyt case, we can obtain an alternative expression for P¯d,Ray by setting K = 0 in (29).

2.8

Weibull Fading Channel

The PDF of SNR (γ) in Weibull fading channel is given by [Sagias et al., 2004, (1)] [

Γ(p) fγ (γ) = c γ¯

]c γ

c−1

[ { }] γΓ(p) c exp − γ¯

;γ ≥ 0

(30)

where c = v/2 and p = 1 + 1/c. The parameter v (v >)0 is the Weibull fading parameter expressing how severe the fading can be and for the special case of

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v = 2, (30) reduces to the Rayleigh PDF in (9). Also, for v = 1 the Weibull PDF reduces to the well known negative exponential PDF. The average probability of detection for Weibull channel (P¯d,W ei ) can be obtained analytically by substituting equation (30) in (8) [ ] ∫ [ { }] (√ √ ) c−1 Γ(p) c ∞ γΓ(p) c ¯ Pd,W ei = c Qu 2γ, λ γ exp − dγ γ¯ γ¯ 0

(31)

Using the infinite series form of Marcum Q function in (21), we first express (31) as [ ] ∞ Γ(p) c ∑ Γ(u + n, λ/2) ¯ Pd,W ei = c γ¯ Γ(u + n)n! n=0 }] [ { ∫ ∞ γΓ(p) c n+c−1 × γ exp(−γ) exp − dγ γ¯ 0

(32)

Next, the integration (32) may be solved through making use of the following result [Cheng et al., 2004, (4)], i.e. ∫

∞

xp−1 exp(−βx − αxr )dx

0

[ = (2π)

1−r 2

r

p−

1 2

β

−p

Gr,1 1,r

] 1 (β/r)r p+r−1 p α ,··· , r r

(33)

( ) a ,··· ,a where Gm,n x b11,··· ,bqp denotes Meijer’s G function [Gradshteyn and Ryzhik, p,q 2007, (9.301)]. Using (33), the final expression for P¯d,W ei can be evaluated as [ ] ∞ 1−c Γ(p) c ∑ n+c+ 1 Γ(u + n, λ/2) ¯ 2 c Pd,W ei = (2π) 2 γ¯ Γ(u + n)n! n=0 [ ] 1 1 c,1 n+c n + 2c − 1 ×G1,c ,··· , {cΓ(p)/¯ γ }c c c

(34)

which is also an entirely new result.

2.9

Log-normal Shadowing Channel

The PDF of SNR in log-normal shadowing channel is given by [Simon and Alouini, 2004] [ ] (10 log10 γ − µ)2 10 √ exp − fγ (γ) = 2σ 2 ln(10) 2πσγ

;γ ≥ 0

(35)

where µ and σ, both expressed in dB scale, denote the mean and standard deviation of the SNR in logarithmic scale (10log10 γ) respectively. Unfortunately, an exact closed-form expression for the average probability of detection can not be found for the log-normal case by simply substituting (35) in (8). Nevertheless, a tractable

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expression can be found if the log-normal distribution is approximated by the inverse Gaussian distribution [Sun, 2011] √ fγ (γ) =

[ ] α 1 α(γ − β)2 exp − 2πγ γ 2β 2 γ

;γ ≥ 0

(36)

where β denotes the expectation of γ (β = E{γ}) and α is the shape parameter. In order to approximate the log-normal distribution, [Sun, 2011] obtained a relation between the parameter sets, {α, β} and {µ, σ}, as ( β = exp

µ σ2 + ψ 2ψ 2

(

α= exp

) (37)

β ) σ2 −1 ψ2

(38)

where ψ=10/ln(10). Once the approximation is done, we continue to follow our regular approach for finding the average probability of detection, i.e. we express the Marcum’s Q function in (8) with an infinite series with the help of (21), and then we insert the approximated PDF for log-normal shadowing channel given by (36) to obtain P¯d,ln as P¯d,ln =

√

( ) α Γ(u + n)n! β n=0 [ ( )] ∫ ∞ α α n−3/2 × γ exp − −γ 1+ 2 dγ 2γ 2β 0 ∞ ∑ Γ (u + n, λ/2)

α exp 2π

(39)

Next, the integration in (39) may be solved through making use of the following result [Gradshteyn and Ryzhik, 2007, (3.471.9)], i.e. ∫

∞

x 0

v−1

( ) ( )v/2 ( √ ) θ θ exp − − ηx = 2 Kv 2 θη x η

(40)

where Kv (·) denotes modified Bessel function of the second kind [Gradshteyn and Ryzhik, 2007, (9.301)] with order v. Using (40), the final expression for P¯d,ln can be evaluated as P¯d,ln

(√ )n−1/2 √ ( )∑ ∞ α α Γ (u + n, λ/2) αβ 2 =2 exp 2π β Γ(u + n)n! α + 2β 2 n=0 √ ) ( α(α + 2β 2 ) ×Kn−1/2 β2

(41)

An alternate expression of P¯d,ln was also derived earlier by Sun [Sun, 2011, (3.31)]. However, their expression contained three different series while the expression derived above contains only one. The readers may also note that the expression

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in (41), as well as those in (24), (29), and (34), contain infinite series which should be truncated after a finite number of terms for numerical evaluation purpose. A detailed discussion regarding the truncation guideline is made available in Section 3. 2.10

Hard Decision Combining

In this section, we describe how the detection performance cane be improved with cooperation among different CRs. Each CR makes its own local decision regarding the presence or absence of PU (i.e. H1 or H0 ), and forwards the binary decision (‘1’ or ‘0’) to FC for data fusion as shown in Fig. 2. As described earlier, we have assumed that the noise, fading statistics, and average SNR in sensing channel are the same for each CR to simplify our modelling. The overall probability of detection (Qd ) for the FC following a k-out-of-N rule may be found in the following manner [Kyperountas et al., 2008], [Duan and Li, 2010]. We begin with P¯d , the average probability of detection for each individual CR, as defined in (8). Now, noting that for all the cases where at least k (k or more than k < N ) CRs detect the presence of PU, the FC decides that PU is present, we have N ( ) ∑ N ¯l Qd = Pd (1 − P¯d )N −l l

(42)

l=k

The overall probability of detection under OR-fusion rule (i.e., 1-out-of-N rule) can be evaluated by setting k = 1 in (42) Qd,OR

N ( ) ∑ N ¯l = Pd (1 − P¯d )N −l = 1 − (1 − P¯d )N l

(43)

l=1

Similarly, the performance with AND-fusion rule (i.e., N -out-of-N rule) can be evaluated by setting k = N in (42) Qd,AN D

N ( ) ∑ N ¯l = Pd (1 − P¯d )N −l = P¯dN l

(44)

l=N

Finally, for the case of Majority-fusion rule, or simply for (N/2 + 1)-out-of-N rule, the Qd,M aj is evaluated by setting k = ⌊N/2⌋ in (42). The overall probability of false alarm (Qf ) for all these three cases (OR, AND, and Majority fusion rules) can be evaluated by replacing P¯d with Pf . It is interesting to note that the probability of false alarm (Pf ) expression is independent of SNR (γ) and remains same for all fading models.

3.

Results and Discussions

In this section, numerical results for different channel and network parameters (e.g. fading parameter, average SNR, number of CRs) as well as for different hard decision fusion rules are discussed. In addition, simulation results are also provided in order to prove the accuracy of our analytical derivations introduced in Section 2. For all the subsequent plots, the time-bandwidth product and average SNR are

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assumed to be u = 5 and γ¯ = 10 dB respectively, except in the cases where the SNR is itself an independent variable. Also the AWGN curve is shown for comparison purpose and OR logic at FC is considered unless specified otherwise. While deriving the average probability of detection (P¯d ) for various fading channels, we ended up with closed-form expressions for Rayleigh and Nakagami-m channels. However, the end expressions for Hoyt (or Nakagami-q), Rician, Weibull, and log-normal channels contained infinite series. The series converge fast, and higher order terms can be neglected without affecting accuracy if the series is truncated reasonably. The rationale behind choosing the series solution over the basic integral form is, a trade-off may be easily realised between precision and complexity by choosing the number of terms to calculate P¯d which was not possible in integral solutions. For practical purposes, it is important to investigate how many terms are necessary to obtain an accurate approximation of the exact result. The number of terms (TN ) needed in (24), (29), (34), and (41) to achieve five significant figure accuracy was empirically determined using Mathematica/ MATLAB and presented in Table 1. The reference value for the exact result was calculated assuming a very high value of TN (> 250) so that a precision of 20 digits can be maintained. It is clear from the table that the increase in either average SNR (¯ γ ) or fading severity parameter (lower q or higher {K, v, σ}) increases the number of terms to be summed for a given accuracy. Table 1. The number of terms that are required to achieve five significant figure accuracy Average SNR (¯ γ) 5 dB 7 dB 9 dB

Hoyt parameter q = 0.3 q=1 62 96 >100

45 80 97

Rician parameter K=0 K=3 45 80 99

Weibull parameter v=2 v=4

29 37 54

47 64 99

Log-normal parameter σ = 3 dB σ = 4 dB

20 56 72

34 50 88

75 93 >100

0

10

Probability of Missed detection (Pm)

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−1

10

−2

10

AWGN (analytical) Weibull (analytical) Hoyt (analytical) Rician (analytical) Nakagami (analytical) Rayleigh (analytical) Log−normal (analytical) Simulation

−3

10

−4

10

−4

10

−3

10

−2

10

−1

10

0

10

Probability of False alarm (P ) f

Figure 3. Performance of single CR spectrum sensing in different fading/ shadowing channels.

Fig. 3 shows the probability of missed detection at CR (Pm ) vs. probability of false alarm at CR (Pf ) curves under AWGN and several fading scenarios. Simulation results are superimposed on theoretical curves obtained from analytical expressions derived in Section 2, and they exhibit a perfect match. Rician, Nakagami-m,

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Hoyt, Weibull, and log-normal fading/ shadowing parameters are assumed to be K = 3, m = 3, q = 0.25, v = 6, and σ = 5 dB respectively. As expected, spectrum sensing is difficult when both fading/ shadowing and noise is present (compared to the noise only case) as fading/ shadowing introduces additional missed detections. Further, we also observe that under Rayleigh or more-severe-than Rayleigh (Hoyt) fading scenarios, the performance of ED degrades significantly compared to less-severe-than Rayleigh (Rician, Nakagami with m > 1 and Weibull) fading. In a Hoyt fading channel, to achieve a Pm of 1%, resultant Pf goes beyond 0.9. On the contrary, in a Weibull fading channel, to achieve the same value of Pm , Pf is equal to 0.625. A lower Pf ensures better spectrum utilization. 0

10

Probability of Missed detection (Pm)

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−1

10

AWGN Rayleigh, Rician (K=0), Nakagami (m=1), Weibull (v=2), Hoyt (q=1) Rician (K=2) Rician (K=5) Nakagami (m=2) Nakagami (m=5) Hoyt (q=0.25) Hoyt (q=0.5) Weibull (v=6) Log−normal (σ = 5 dB) Log−normal (σ = 8 dB)

−2

10

−3

10

−3

10

−2

−1

10

0

10

10

Probability of False alarm (P ) f

Figure 4. Impact of fading/ shadowing parameters on performance of single CR spectrum sensing.

Fig. 4 shows the effect of different fading/ shadowing parameters on performance of single CR spectrum sensing. The parameters for Rician fading, K = {0, 2, 5}, for Nakagami-m fading, m = {1, 2, 5}, for Nakagami-q fading, q = {0.25, 0.5, 1}, for Weibull fading, v = {2, 6}, and for log-normal shadowing, σ = {5 dB, 8 dB} are considered. Rayleigh fading channel characteristics would be achieved in Rician, Nakagami-m, Hoyt (/Nakagami-q), and Weibull fading channel if K, m, q, and v are set to 0, 1, 1, and 2 respectively. Increase in fading/ shadowing severity (characterized by a lower q or a higher {K, m, v, σ} value) significantly decreases the probability of missed detection at CR. Fig. 5, Fig. 6, and Fig. 7 show the overall probability of missed detection at FC (Qm ) vs. overall probability of false alarm at FC (Qf ) plot for different number of CRs (N ) in Rayleigh and Rician (K = 5), in Nakagami (m = 3) and Weibull (v = 6), and in Hoyt (/Nakagami-q) (q = 0.25) fading respectively. While Fig. 5 demonstrates how the presence of line-of-sight (LoS) in a Rician fading environment improves the detection performance over non-LoS Rayleigh scenarios, the other two better-than-Rayleigh fading scenarios (e.g. Nakagami-m and Weibull) are grouped in Fig. 6. As shown in Fig. 5, when Qf is equal to 0.3 and N increases from 1 to 6, Qm decreases from 0.6 to 0.003 in Rayleigh fading while it decreases from 0.25 to 0.0001 in presence of Rician fading. Again, from Fig. 6 one can find that, for Qf = 0.3 as N goes up from 1 to 6, Qm decreases from 0.1 to 0.0001 in presence of Nakagami fading and decreases from 0.075 to 0.00001 in presence of Weibull fading. On the other hand, the performance in worse-than-Rayleigh scenario (e.g. Hoyt) is depicted in Fig. 7. For Qf = 0.1 as N grows from 1 to 10, Qm decreases from

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0

Probability of Missed detection (Qm)

10

−1

10

−2

10

Rayleigh, N=1 (no cooperation) Rayleigh, N=3 Rayleigh, N=6 Rician, N=1 (no cooperation) Rician, N=3 Rician, N=6

−3

10

−4

10

−4

10

−3

10

−2

10

−1

10

0

10

Probability of False alarm (Q ) f

Figure 5. Qm vs. Qf in Rayleigh and Rician fading (K = 5) channels for different cooperative CRs.

0

10

Probability of Missed detection (Qm)

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−1

10

−2

10

Nakagami, N=1 (no cooperation) Nakagami, N=3 Nakagami, N=6

−3

10

Weibull, N=1 (no cooperation) Weibull, N=3 Weibull, N=6

−4

10

−4

10

−3

10

−2

10

−1

10

0

10

Probability of False alarm (Q ) f

Figure 6. Qm vs. Qf in Nakagami (m = 3) and Weibull fading (v = 6) channels for different cooperative CRs.

0.75 to 0.0001. We can observe from all the three figures that fusing the received decisions that are taken by N CRs, the effect of fading on the detection performance may be canceled effectively. Moreover, with an increase in number of cooperating CRs (N ), the performance gap between CSS and single CR spectrum sensing widens. 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-cooperation case. In Fig. 8, the performance of single CR and cooperative spectrum sensing is evaluated via total error probability (Qm + Qf ) versus threshold (λ) in various fading/ shadowing channels. At lower values of λ, although the Qm value is less, we experience a high value of Qf . At the other extreme, when λ value is high, the situation simply reverses, i.e. we have large Qm and small Qf values. Thus, it is possible to find an optimum value of λ, λ∗ , where the total error probability attains its minimum value. This optimum value is, however, heavily dependent on the propagation environment as well as on the number of CRs (N ). For example, as seen from Fig. 8, for channels experiencing Weibull (v = 6) fading, one have λ∗ = 17 for single CR (N = 1) case and λ∗ = 23 for N = 3.

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0

Probability of Missed detection (Qm)

10

−1

10

−2

10

N=1 N=2 N=3 N=5 N=7 N=10 AWGN, N=1

−3

10

−4

10

−4

10

−3

10

−2

10

−1

0

10

10

Probability of False alarm (Q ) f

Figure 7. Qm vs. Qf in Hoyt fading (q = 0.25) channel for different cooperative CRs.

0

10

Total error probability (Qm + Qf)

N=1

−1

10

10

AWGN Rayleigh Rician (K=2) Nakagami (m=3) Weibull (v=6) Hoyt (q=0.25) Log−normal (σ = 5 dB) 15

N=3

20

25

30

Detection threshold

Figure 8. Total error probability (Qm + Qf ) vs. threshold (λ) in different fading/ shadowing channels.

1 0.9 0.8

Probability of detection (Qd)

November 24, 2014

0.7 0.6 0.5

N=1 (no cooperation)

0.4

N=2

0.3

N=3 0.2

N=5 N=7

0.1

N=10 0 −5

0

5

10

15

SNR (dB)

Figure 9. Qd vs. γ ¯ in Rician fading channel (K = 5) for different number of cooperative CRs, Qf = 0.05.

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1 0.9

d

Probability of detection (Q )

0.8 0.7 0.6 0.5

N=1 (no cooperation)

0.4

N=2 N=3 N=5 N=7 N=10

0.3 0.2 0.1 0 −5

0

5

10

15

SNR (dB)

Figure 10. Qd vs. γ ¯ in Weibull fading channel (v = 6) for different number of cooperative CRs, Qf = 0.05.

Fig. 9 and Fig. 10 show the variation of overall probability of detection (Qd ) with average SNR (¯ γ ) under Rician (K = 5) and Weibull (v = 6) fading scenarios for various values of N . No-cooperation (N = 1) curves are also shown for comparison purpose. We observe that there is an excellent improvement in detection performance of CSS with increase in either N or γ¯ . In Rician fading (see Fig. 9), for Qd = 0.9, spectrum sensing without cooperation requires γ¯ = 12.35 dB while cooperative sensing with N = 10 only needs γ¯ = 6.42 dB for individual CRs, i.e. an SNR gain of 5.93 dB is achieved. Under similar conditions (Qd = 0.9), in Weibull fading (see Fig. 10), the required SNR for spectrum sensing without cooperation is γ¯ = 11.15 dB, which is 4.6 dB more than the cooperative sensing case with N = 10. Note that, the detection performance with respect to average SNR for Rayleigh, Nakagami-m [Nallagonda et al., 2012b], and Hoyt [Nallagonda et al., 2012a] fading channels are reported earlier, and hence are not repeated here. 0

10

m

Probability of Missed detection (Q )

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−1

10

−2

10

1 out of 5 (OR−rule) 2 out of 5 3 out of 5 (Majority−rule) 4 out of 5 5 out of 5 (AND−rule) AWGN

−3

10

−4

10

−4

10

−3

10

−2

10

−1

10

0

10

Probability of False alarm (Qf)

Figure 11. Performance of hard decision fusion rules (k-out-of-N ; 1 ≤ k ≤ N, N = 5) in Rician fading channel (K = 5).

Fig. 11 depicts the performance of hard decision fusion rules, i.e. k-out-of-N (1 ≤ k ≤ N, N = 5) fusion rules (or voting/ counting rule). We assumed Rician environment as a test case. As expected, fusing the decisions of individual CRs at fusion center cancels the effect of fading to a great extent. For a particular value of Qf = 0.1, the overall probability of missed detection (Qm ) is 0.002, 0.2, and above

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0.8 for the 1-out-of-5 rule (OR-rule), 3-out-of-5 rule (Majority-rule), and 5-out-of-5 rule (AND-rule) respectively. Interestingly, performance with 1-out-of-5 and 2-outof-5 fusion rules under fading is better than performance of single CR spectrum sensing in non-fading AWGN channel. Finally, the OR-logic performs better than Majority-logic while the AND-logic performs worst among all fusion rules.

1 0.9 0.8

Probability of detection (Qd)

November 24, 2014

0.7 0.6 0.5

1 out of 5 (OR−rule) 2 out of 5 rule 3 out of 5 (Majority−rule) 4 out of 5 rule 5 out of 5 (AND−rule) AWGN

0.4 0.3 0.2 0.1 0 −5

0

5

10

15

SNR (dB)

Figure 12. Performance of hard decision fusion rules (k-out-of-N ; 1 ≤ k ≤ N, N = 5) in Weibull fading channel (v = 6), Qf = 0.05.

Fig. 12 shows the performance of hard decision fusion rules and their comparison based on Qd vs. γ¯ plots. In this case, however, we have chosen a Weibull fading channel as our sample propagation environment. The overall probability of detection (Qd ) increases when more number of CRs (k) are allowed to participate in the decision making process. Again, Qd value is more for higher sensing channel SNR. This is obvious as, at a higher SNR, the noise contribution is less in the received signal at each CR and a high probability of detection is achieved. In order to compare between the fusion rules, it is sufficient to observe that for a particular value of average SNR of 6 dB, Qd is above 0.95, 0.35, and 0 for the 1-out-of-5 rule (OR-rule), 3-out-of-5 rule (Majority-rule), and 5-out-of-5 rule (AND-rule) respectively. Thus, just like the previous figure, we can say that OR-rule performs better than Majority and AND-rules. It may be noted that the curve for non-fading case (N = 1) in Fig. 12 serves as a reference level.

4.

Conclusions

We have presented an analytical framework to evaluate the performance of energy detector based spectrum sensing with coooperative CRs as well as single CR in various small scale fading as well as large scale shadowing channels. Based on our developed framework we presented sensing performance in terms of overall missed detection probability and total error probability under several hard decision fusion rules. The performance in fading channels has been compared with that in the AWGN channel. The spectrum sensing performance in Weibull channel is seen to outperform that in other channels for the fading parameters assumed in our study. Higher sensing channel SNR and increased number of CRs are found to improve the overall performance. Optimal threshold at individual CR, which minimizes total error, is relatively lower in case of Weibull as compared to other fading cases.

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Results obtained on the basis of Monte Carlo simulations show excellent match with analytical results derived in the paper.

Acknowledgment This work was done under SoMoPro II grant, Project No. 3SGA5720 Localization via UWB, financed by the People Programme (Marie Curie action) of the Seventh Framework Programme of EU according to the REA Grant Agreement No. 291782 and by the South-Moravian Region. The research is co-financed by the Czech Science Foundation, Project No. 13-38735S Research into wireless channels for intra-vehicle communication and positioning.

Appendix A. Proof of (12) Consider an integral ∫

∞

I1 = 0

( 2 2) p x Qm (ax, b)x exp − dx 2

(A.1)

From the definition of Marcum’s Q function [Nuttall, 1975] 1 Qm (ax, b) = (ax)m−1

∫

∞

b

( 2 ) y + a2 x2 y exp − y m−1 Im−1 (axy)dy 2

(A.2)

∫ ∫ and with repetitive use of integration by parts, i.e. udv = uv − vdu where u = y m−1 Im−1 (axy) and v = exp[−(y 2 + a2 x2 )/2], the following recursion formula can be obtained

Qm (ax, b) =

m−1 ∑( n=1

b ax

)n

) ( 2 b + a2 x2 In (abx) + Q1 (ax, b) exp − 2

(A.3)

Using the above recursion formula, the integral in (A.1) can be rewritten as ( 2)∫ ∞ [ ] b (p2 + a2 )x2 2−n−1 I1 = exp − x exp − In (abx)dx 2 2 0 n=1 ∫ (A.4) ( ) ∞ p2 x2 + Q1 (ax, b)x exp − dx 2 0 m−1 ∑(

b a

)n

Next, using [Chandra, 2011, (B.16)] ∫ 0

∞

xµ−1 exp(−d2 x2 )Iν (cx)dx ( 2 ) ( ) c ν−µ c2 cν Γ[(µ + ν)/2] exp F + 1; ν + 1; − = ν+1 µ+ν 1 1 2 d Γ(ν + 1) 4d2 2 4d2

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(A.5)

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the first integral in equation (A.4) is evaluated as ∫

[ ] (p2 + a2 )x2 x exp − In (abx)dx 2 [ ] [ ] (A.6) (ab)n a2 b2 a2 b2 = n 2 exp 1 F1 n; n + 1; − 2 (a + p2 )Γ(n + 1) 2(a2 + p2 ) 2(a2 + p2 )

∞

2−n−1

0

Making use of the Kummer’s transformation, 1 F1 (a; b; z) = exp(z) 1 F1 (b − a; b; −z) [Gradshteyn and Ryzhik, 2007, (9.212.1)], this integral may be further simplified to ∫

] (p2 + a2 )x2 In (abx)dx x exp − 2 [ ] (ab)n a2 b2 = n 2 1 F1 1; n + 1; 2 (a + p2 )Γ(n + 1) 2(a2 + p2 ) [

∞

2−n−1

0

(A.7)

On the other hand, the solution to the second integral in (A.4) is readily available [Nuttall, 1975] ∫ 0

∞

( 2 2) ] [ p x 1 b 2 p2 Q1 (ax, b)x exp − dx = 2 exp − 2 p 2(a2 + p2 )

(A.8)

Finally, substituting (A.7) and (A.8) in (A.4), we have m−1 ∑(

] [ (ab)n a2 b2 I1 = 1 F1 1; n + 1; 2n (a2 + p2 )Γ(n + 1) 2(a2 + p2 ) n=1 ] [ 1 b2 p2 + 2 exp − p 2(a2 + p2 ) b a

)n

(A.9)

Appendix B. Proof of (17) Let us consider another integral ∫

∞

I2 =

2q−1

Qm (ax, b)x 0

( 2 2) p x exp − dx 2

(B.1)

Using the recursion formula given in (A.3), the integral in (B.1) can be rewritten as m−1 ∑(

( 2)∫ ∞ [ ] b (p2 + a2 )x2 2q−n−1 I2 = exp − x exp − In (abx)dx 2 2 0 n=1 (B.2) ( 2 2) ∫ ∞ p x 2q−1 + Q1 (ax, b)x exp − dx 2 0 b a

)n

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Using (A.5), the first integral in (B.2) may be solved as ∫

∞

0

[ ] (p2 + a2 )x2 x exp − In (abx)dx 2 [ ] a2 b2 (ab)n Γ(q) = n+1 2 exp 2 [(a + p2 )/2]q Γ(n + 1) 2(a2 + p2 ) [ ] a2 b2 × 1 F1 n − q + 1; n + 1; − 2(a2 + p2 ) [ ] (ab)n Γ(q) a2 b2 = n+1 2 1 F1 q; n + 1; 2 [(a + p2 )/2]q Γ(n + 1) 2(a2 + p2 ) 2q−n−1

(B.3)

where the simplification in the final step was achieved using the Kummer’s transformation 1 F1 (a; b; z) = exp(z) 1 F1 (b − a; b; −z) [Gradshteyn and Ryzhik, 2007, (9.212.1)]. Next, making use of [Nuttall, 1975, (9)] ∫

∞

Q1 (ax, b)x2q−1 exp(−p2 x2 /2)dx

0

=

[ ] b 2 p2 2q−1 (q − 1)! a2 exp − p2q p2 + a2 2(p2 + a2 ) { q−2 ( [ ] )k ∑ p2 b2 a2 × Lk − p2 + a2 2(p2 + a2 ) k=0 ( )( )q−1 [ ]} p2 p2 b2 a2 + 1+ 2 Lq−1 − a p2 + a 2 2(p2 + a2 )

(B.4)

the solution to the second integral in (B.2) can be determined. Finally, substituting (B.3) and (B.4) in (B.2), we have ( 2) (ab)n Γ(q) b I2 = exp − 2 2n+1 [(a2 + p2 )/2]q Γ(n + 1) n=1 ] [ a2 b2 × 1 F1 q; n + 1; 2(a2 + p2 ) [ ] 2q−1 (q − 1)! a2 b2 p2 + exp − p2q p2 + a2 2(p2 + a2 ) { q−2 ( )k ] [ ∑ p2 b2 a2 × Lk − p2 + a 2 2(p2 + a2 ) k=0 ( )( )q−1 ]} [ p2 p2 b 2 a2 + 1+ 2 Lq−1 − a p2 + a 2 2(p2 + a2 ) m−1 ∑(

b a

)n

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(B.5)

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References N. S. Adawi et al. Coverage prediction for mobile radio systems operating in the 800/900 MHz frequency range. IEEE Trans. Veh. Tech., 37(1):3–72, Feb. 1988. I. F. Akyildiz, B. F. Lo, and R. Balakrishnan. Cooperative spectrum sensing in cognitive radio networks: A survey. Physical Commun., 4(1):40–62, Mar. 2011. S. Andras, A. Baricz, and Y. Sun. The generalized Marcum Q-function: An orthogonal polynomial approach. Acta Univ. Sapientiae Math., 3(1):60–76, 2011. S. Atapattu, C. Tellambura, and H. Jiang. Energy detection of primary signals over η-µ fading channels. In Proc. IEEE ICIIS, pages 118–122, Dec. 2009. S. D. Cabric, S. M. Mishra, and R. W. Brodersen. Implementation issues in spectrum sensing for cognitive radios. In Proc. IEEE ACSSC, volume 1, pages 772– 776, Nov. 2004. A. Chandra. Performance analysis of diversity combining techniques for digital signals in wireless fading channels. PhD Thesis, Jadavpur University, Kolkata, India, 2011. A. Chandra and C. Bose. Series solutions for π/4-DQPSK BER with MRC. Intl. J. Electron., 99(3):391–416, Mar. 2012. A. Chandra, C. Bose, and M. K. Bose. Symbol error probability of non-coherent M ary frequency shift keying with postdetection selection and switched combining over Hoyt fading channel. IET Commun., 6(12):1692–1701, Aug. 2012. A. Chandra, C. Bose, and M. K. Bose. Performance of non-coherent MFSK with selection and switched diversity over Hoyt fading channel. Wireless Pers. Commun., 68(2):379–399, Jan. 2013. S. Chaudhari, J. Lunden, V. Koivunen, and H. V. Poor. Cooperative sensing with imperfect reporting channels: Hard decisions or soft decisions? IEEE Trans. Sig. Proc., 60(1):18–28, Jan. 2012. J. Cheng, C. Tellambura, and N. C. Beaulieu. Performance of digital linear modulations on Weibull slow-fading channels. IEEE Trans. Commun., 52(8):1265–1268, Aug. 2004. F. F. Digham, M. -S. Alouini, and M. K. Simon. On the energy detection of unknown signals over fading channels. In Proc. IEEE ICC, pages 3575–3579, May 2003. F. F. Digham, M. -S. Alouini, and M. K. Simon. On the energy detection of unknown signals over fading channels. IEEE Tran. Commun., 55(1):21–24, Jan. 2007. J. Duan and Y. Li. Performance analysis of cooperative spectrum sensing in different fading channels. In Proc. IEEE ICCET, volume 3, pages 64–68, Jun. 2010. Y. Fathi and M. H. Tawfik. Versatile performance expression for energy detector over α-µ generalised fading channels. Electron. Lett., 48(17):1081–1082, Aug. 2012. A. Ghasemi and E. S. Sousa. Collaborative spectrum sensing for opportunistic access in fading environments. In Proc. IEEE DySPAN, pages 131–136, Nov. 2005. I. S. Gradshteyn and I. M. Ryzhik. Table of Integrals, Series and Products. Academic Press/ Elsevier, San Diego, CA, USA, 7th edition, Mar. 2007. H. Hashemi. The indoor radio propagation channel. Proc. IEEE, 81(7):943–968, Jul. 1993. S. Haykin. Cognitive radio: Brain-empowered wireless communications. IEEE J. Select. Areas Commun., 23(2):201–220, Feb. 2005. M. H. Ismail and M. M. Matalgah. BER analysis of BPSK modulation over the

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International Journal of Electronics

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Weibull fading channel with CCI. In Proc. IEEE VTC, pages 1–5, Sep. 2006. S. Kyperountas, N. Correal, Q. Shi, and Z. Ye. Performance analysis of cooperative spectrum sensing in Suzuki fading channels. In Proc. IEEE CROWNCOM, pages 428–432, Jun. 2008. M. Mustonen, M. Matinmikko, and A. Mammela. Cooperative spectrum sensing using quantized soft decision combining. In Proc. IEEE CROWNCOM, pages 1–5, Jun. 2009. S. Nallagonda, S. Suraparaju, S. D. Roy, and S. Kundu. Performance of energy detection based spectrum sensing in fading channels. In Proc. IEEE ICCCT, pages 575–580, Sep. 2011. S. Nallagonda, A. Chandra, S. D. Roy, and S. Kundu. Performance of cooperative spectrum sensing in Hoyt fading channel under hard decision fusion rules. In Proc. IEEE CODEC, pages 1–4, Dec. 2012a. S. Nallagonda, S. D. Roy, and S. Kundu. Performance of cooperative spectrum sensing in fading channels. In Proc. IEEE RAIT, pages 202–207, Jan. 2012b. A. H. Nuttall. Some integrals involving the QM function. IEEE Trans. Inf. Th., 21(1):95–96, Jan. 1975. N. C. Sagias, G. K. Karagiannidis, and G. S. Tombras. Error-rate analysis of switched diversity receivers in Weibull fading. Electron. Lett., 40(11):681–682, May 2004. C. E. Shannon. Communication in the presence of noise. Proc. IRE, 37(1):10–21, Jan. 1949. M. K. Simon and M. -S. Alouini. Digital Communication over Fading Channels. John Wiley and Sons, NJ, USA, 2nd edition, Dec. 2004. H. Sun. Collaborative spectrum sensing in cognitive radio networks. PhD thesis, The University of Edinburgh, Jan. 2011. H. Urkowitz. Energy detection of unknown deterministic signals. Proc. IEEE, 55 (4):523–531, Apr. 1967. V. Valenta, R. Marˇs´alek, G. Baudoin, M. Villegas, M. Suarez, and F. Robert. Survey on spectrum utilization in Europe: measurements, analyses and observations. In Proc. IEEE CROWNCOM, pages 1–5, Jun. 2010. B. Wang and K. J. Ray Liu. Advances in cognitive radio networks: A survey. IEEE J. Select. Topics Signal Process., 5(1):5–23, Feb. 2011.

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