Energy detection of unknown signals in Gamma-shadowed Rician fading environments with diversity reception

8/10/2019 Energy detection of unknown signals in Gamma-shadowed Rician fading environments with diversity reception

1/15

Published in IET Communications

Received on 15th August 2013

Revised on 4th September 2014

Accepted on 29th September 2014

doi: 10.1049/iet-com.2014.0170

ISSN 1751-8628

Energy detection of unknown signals inGamma-shadowed Rician fading environmentswith diversity receptionKostas P. Peppas1, George Efthymoglou2, Valentine A. Aalo3, Mohammed Alwakeel4,Sami Alwakeel4,5

1National Centre for Scientic Research Demokritos, Patriarhou Grigoriou and Neapoleos, Institute of Informatics and

Telecommunications, 15310 Agia Paraskevi, Athens, Greece2Department of Digital Systems, University of Piraeus, 80 Karaoli and Dimitriou Street, Piraeus 18534, Greece3Department of Computer and Electrical Engineering and Computer Science Florida, Atlantic University, Boca Raton, FL

33431, USA4Sensor Networks and Cellular Systems, (SNCS) Research Center, University of Tabuk, Tabuk, Saudi Arabia5Department of Computer Engineering, College of Computer and Information Sciences, King Saud University, Riyadh,

11543, Saudi Arabia

E-mail: [email protected]

Abstract:This study presents a comprehensive performance analysis of an energy detector over Gamma-shadowed Rician fadingchannels, namely Rician fading channels with the uctuating line-of-sight components following the Gamma distribution. Thiscomposite multi-path/shadowing model has been shown to provide a remarkably accurate fading characterisation while leading toclosed-form expressions for important channel statistics. Rapidly convergent innite series representations are rstly derived forthe average probability of detection and the area under the receiver operating characteristic curve for the no-diversity receptioncase. These results are then extended to the case of maximal ratio, equal gain and selection diversity. To this end, novel analyticalexpressions for the statistics of the end-to-end signal-to-noise ratio of equal gain and selection diversity receivers, operating overGamma-shadowed Rician fading channels are derived. Analytical results are substantiated by Monte Carlo simulation, as well as

by extensive numerically evaluated results.

1 Introduction

Energy detection is a popular method of spectrum sensing incognitive radio systems, because of its implementationsimplicity, fast sensing, low computational cost and thecapability to detect any shape of signal. The energy detectoris a non-coherent device which samples the received signalenergy over an observation time window and compares witha predened threshold, in order to determine the presenceor absence of an unknown signal. In the seminal work ofUrkowitz [1], where the problem of energy detection ofunknown deterministic signals over at bandlimitedGaussian noise channel was rst addressed, analyticalexpressions for the probability of detection, Pd and the

probability of false alarm, Pf were derived. Theseperformance metrics are based on the assumption that thedecision problem is a binary hypothesis test in a non-fadingenvironment where the decision statistics follows the centralchi-square and the non-central chi-square distribution,

respectively.The performance assessment of energy detectors with

single and multi-channel reception in the presence of fadinghas been extensively studied in the wireless literature.

Representative past examples can be found in [213] andreferences therein. For example, in [2], energy detectionover Nakagami-m fading with square-law combining andsquare-law selection was addressed. Using a contourintegral representation for the Marcum Q-function, in [3], amoment generating function-based approach was utilised toderive analytical expressions for the average probability ofdetection over Nakagami-m and Rician fading channelswith maximal ratio combining (MRC), square lawcombining, selection combining (SC) and equal gaincombining (EGC). In [7], the area under the receiveroperating characteristic (ROC) curve, denoted (AUC), wasutilised to address the performance of energy detectors withdiversity reception over Nakagami-m fading channels. In[11], the performance of energy detectors over andextreme channels were discussed. Finally, cooperativespectrum sensing was addressed in [1214].

On the other hand, apart from the multi-path fading,shadowing is also present in practical wireless systems and

can signicantly degrade the performance of the energydetector. Thus, analytical results concerning the impact of

both multi-path fading and shadowing on energy detectionare important to the system design engineer for performance

www.ietdl.org

IET Commun., pp. 115

doi: 10.1049/iet-com.2014.0170

1

& The Institution of Engineering and Technology 2014


2/15

evaluation purposes. Moreover, in many practical casesmulti-path fading and shadowing occur simultaneously. Thestatistics of this composite propagation environment maywell be described by the so-called log-normal-based fadingmodels, such as the Loos model, which assumes that thesignal is affected with log-normal shadowing on the directcomponent only [15], the Suzuki model, in which the signalamplitude variations are Rayleigh distributed [16], the

Rician/log-normal model, in which shadowing on both thedirect and diffuse component is introduced, and the

Nakagami-m/log-normal model [17]. However, these fadingmodels are not widely used in the context of performanceanalysis for energy detection because of their rathercomplicated mathematical expressions. To deal with this

problem, accurate approximations, such as power sumcorrelated log-normal and shifted log-normal are presentedin [18, 19], respectively. The performance of spectrumsensing in composite Rician/log-normal channels isdiscussed in [20]. By approximating log-normal shadowingwith a Gamma distribution, in [21, 22], the performance ofenergy detectors in K and generalised-K fading channelswas analysed in detail. In [23], using the inverse-Gaussiandistribution to approximate log-normal shadowing, athorough performance analysis of energy detection withdiversity reception was presented. In [24], a mixtureGamma distribution was proposed to model compositefading and the performance of energy detection with asingle antenna reception in a composite propagationenvironment was addressed.

Recently, in order to model channels with compositemulti-path fading and shadowing, an analytically tractable,yet accurate shadowed Rician model was proposed in [25].This model is obtained by approximating the statistics ofthe direct component of the Loos model with a Gammadistribution and can accurately model composite fading in

land mobile satellite (LMS) systems [15]. Such systemsplay a key role in the third and fourth generation wirelesscommunications systems because of their ability to provideservices over a wide area with low cost, which are notfeasible via conventional land mobile terrestrial systems.Representative applications of such systems includenavigation, communications and broadcasting.

An important advantage of this approximate model incomparison with the exact Loos model is that it leads toclosed-form expressions for the fundamental rst- andsecond-order statistics of the wireless communicationchannel. Therefore this model is very convenient for the

performance analysis of complicated LMS systems, with orwithout diversity. In [25], the exibility of this model incharacterising a variety of channel conditions and

propagation mechanisms was demonstrated by comparingits rst- and the second-order statistics with different sets of

published channel data. In the same work, it was alsodemonstrated that this model provides a similar t to theexperimental data as the Loos model, with signicantlyless computational complexity.

The performance of MRC diversity receivers overshadowed Rician channels has been addressed in [26, 27],where analytical expressions for important metrics such asthe outage probability and the average channel capacityhave been derived. However, to the best of our knowledge,the performance of an energy detector in such channels

with diversity reception is still not available in the literatureand thus is the topic of our contribution.

Motivated by these considerations, in this work acomprehensive performance analysis of the energy detector

over Gamma-shadowed Rician fading channels withantenna diversity reception is presented. The maincontributions of this paper are summarised as follows:

For the MRC diversity case, expressions for the averageprobability of detection are derived by using either theprobability density function (PDF) method or the momentsgenerating function (MGF) method when independent andidentically distributed (i.i.d.) branches. The former methodis valid for arbitrary-valued fading parameters of theGamma component, and is given in terms of rapidlyconvergent innite series representation. The latter methodis valid for integer-valued fading parameters of the Gammacomponent and is based on a contour integral representationfor the Marcum Q-function [28]. High SNR (signal-to-noiseratio) approximations and the detection diversity gain arealso derived. When independent but not necessarilyidentically distributed (i.n.i.d) branches are assumed, aninnite series representation for the average probability ofdetection is also derived. For the EGC diversity case, a simple, yet highly accurate

closed form approximation to the PDF of the sum of i.i.d.shadowed Rician envelopes is presented. Based on thisformula, innite series representations for the average

probability of detection are deduced. The averageprobability of detection when i.n.i.d. branches areconsidered is studied by means of the Pad approximantstechnique. For the SC case, assuming integer-valued fadingparameters of the Gamma component and i.i.d. branches,similar results are also derived. Finally, for all considered diversity schemes wecomprehensively analyse the AUC of the energy detectorunder consideration.

Extensive numerically evaluated results accompanied withMonte Carlo simulations are presented to validate the

proposed analysis. The remainder of this paper is structuredas follows: in Section 2, the system and channel models aredescribed in detail. In Section 3, the detector performancewith MRC, EGC and SC diversity reception is addressed.

Numerically evaluated and computer simulation results arepresented in Section 4, whereas Section 5 concludes thepaper.

Mathematical notations: Throughout this paper, E{} denotesthe expectation operator andPr[] denotes probability. The PDFof a random variable X is denoted as fX(), its cumulativedistribution function (CDF) as F

X() and its MGF as

MX().

In terms of mathematical functions used in this paper, Ia() isthe modied Bessel function of the rst kind and ordera [29,eq. (8.431)], () is the Gamma function [29, eq. (8.310/1)],G(,) is the lower incomplete Gamma function [29, eq.(8.350/1)] and (,) is the upper incomplete Gamma function[29, eq. (8.350/2)]. In addition, pFq() denotes the generalisedhypergeometric function [29, eq. (9.14/1)], p

Fq() denotesthe regularised hypergeometric function [30, eq.(07.32.02.0001.01)], Ln(x) is nth order Laguerre polynomial[29, eq. (8.970)], U(n) is the discrete unit step function andQm(a,b) = a1m

1

b xm exp(x2 + a2)/2 Im1(ax), m 1 is

the generalised Marcum Q-function [31]. Moreover,(x)kW G(x

+k)/G(x) designates the Pochhammer symbol,

= 1 and Res(f:z0;k) denotes the residue of the pole atz=z0 of order k 1 for complex valued function f(z). Finally,the notation f(x) = o[g(x)] as xx0 stands forlimxx0 (f(x)/g(x)) = 0.

www.ietdl.org

2



doi: 10.1049/iet-com.2014.0170


3/15

2 System and channel model

2.1 Energy detector

Assuming narrow band signal detection, the received signalr(t), which contains either an unknown deterministic signaland noise or noise only, can be expressed as [1]

r(t) = n(t) :H0hs(t) + n(t) :H1

(1)where s(t) is an unknown deterministic signal, h denotes thecomplex channel gain and n(t) is an additive whiteGaussian noise (AWGN) process. Moreover, the hypotheses

H0 and H1 refer to signal absence and signal presence,respectively. The energy detector lters, squares andintegrates r(t) over the time interval T. The output of theintegrator, Y acts as decision statistic that determineswhether the received energy corresponds only to the energyof n(t) or to the energy of both s(t) and n(t). Finally, theenergy detector compares the decision statistic Y with a

pre-de

ned threshold, l, and determines that the signal ispresent ifY> l, or absent otherwise.The decision statistic follows a central chi-square

distribution under H0, and a non-central chi-squaredistribution underH1, namely

fY(y|g) =

1

2uG(u)yu1 exp y

2

:H0

1

2

y

2g

(u1/2)exp 2g+y

2

Iu1(

2gy

) :H1

(2)

whereu= TW is the time-bandwidth product andis the SNR

dened by g= |h|2E2s /N0withEsbeing the signal energy andN0 is the noise-power spectral density. The probability ofdetection, Pd(l) and the probability of false alarm, Pf(l),can be expressed as [2]

Pd(l) = Pr Y. l|H1 = Qu( 2g , l ) (3)

and

Pf(l) = Pr Y . l|H0 = G u, l2

G(u) (4)

respectively.

2.2 Channel model

The PDF of the instantaneous SNR in a Gamma-shadowedRician fading channel, , is obtained using [25, eqn. (6)] as

fg(g) = m

m +K

m1 +Kg

exp (1 +K)gg

1

1F1 m; 1; K(1 +K)gg(m +K)

, g. 0 (5)

where m> 0 is the shadowing severity index [It is noted that

the range of m is different from that of the parameter ofconventional Gamma distribution for which m 0.5.], K isthe ratio of the average power of the LOS component tothat of the scattered component andg is the average SNR.

The parametersKandgcan also be expressed as K= /2b0andg= V+ 2b0 where is the average power of the LOScomponent and 2b0 is the average power of the scatteredcomponent.

The corresponding MGF of , dened asMg(s) = E{exp (sg)} can be obtained using [25, eq. (7)] as

Mg(s) =(sg

+K

+1)m1(1

+K)

sg 1 + Km

+ 1 +K m (6)3 Energy detection with diversity reception

Diversity reception is a well-known technique to mitigate thedeleterious effects of multi-path fading in wireless mobilechannels at relatively low cost. In the following analysis wederive analytical expressions for the average probability ofdetection and the average AUC under MRC, EGC and SCdiversity receptions. The proposed schemes incorporate Hindependent diversity branches, operating overGamma-shadowed Rician channels.

3.1 Maximal ratio combining

As it was pointed out in [3,7], the use of MRC with energydetection is not desirable because of the fact that MRCrequires accurate channel estimation for optimal

performance. Nevertheless, the performance of this setup isstill of interest since it serves as a benchmark (tight upper

bound) on the achievable performance of energy detectionschemes employing diversity reception. The instantaneousSNR at the output of the combiner is given bygMRC=

H=1 g, where is the instantaneous SNR of the

th branch.

3.1.1 Average probability of detection: PDF-basedapproach: For communication scenarios over fadingchannels, the average probability of detection is obtained as

Pd(l) = 1 FY l|H1 = 1 l

0

fY y|H1,l

dy (7)

where

fY(y|H1,l) =1

2

1

0

y

2g

u1/2 exp 2g+y

2

Iu1(

2gy

)fg(g) dg

(8)

and the decision thresholdl is obtained from (4), for a givenvalue of the false alarm probabilityPf. The following cases ofinterest are considered:

i.i.d. branches:Assuming i.i.d. branches, the PDF ofMRCcan be expressed as [26, eq. (7)]

fgMRC (g) = m

m +K mH

1 +Kg

Hg

H1

(H 1)!

exp (1 +K)gg

1F1 mH,H,

K(1 +K)gg(m +K)

(9)

www.ietdl.org


doi: 10.1049/iet-com.2014.0170

3



4/15

Substituting (9) in (8), one obtains

fY y|H1, l = (1 +K)HmmH

2 gH (m +K)mH (H 1)!y

2

u1/2exp y

2

1

0

g(1u/2)+H1

exp g 1 + 1 +Kg

1F1 mH;H; K(K+ 1)g(m +K)g

Iu1 2yg dg(10)

As it can be observed, in order to evaluate fY(y|H1, l), anintegral involving exponentials, hypergeometric functionsand modied Bessel functions needs to be solved. We areunaware of a closed-form solution for this integral;nevertheless, using an innite series representation of themodied Bessel function [29, eq. 8.445], as well as [29, eq.7.621/4], fY(y|H1, l) can be obtained as

fY y|H1, l = (1 +K)HmmH2 (m +K)mH (H 1)! exp

y2

1i=0

(y2)

u1+iG(i +H)

i!G(u + i)g

i

(g+ 1 +K)i+H

2F1 Hm,H+ i;H; K(K+ 1)

(m +K)(g+ 1 +K)

(11)

Finally, using (7) and the denition of the lower incompleteGamma function, the following innite series representationfor Pd(l) is deduced

Pd(l) = 1 (1 +K)HmmH

(m +K)mH (H 1)!

1i=0

G u + i, l2

G(i +H)

G(u + i)i!g

i

(g+ 1 +K)i+H

2F1 Hm,H+ i;H; K(K+ 1)

(m +K)(g+ 1 +K)

(12)

Although Pd(l) can be implemented easily in common

mathematical software such as Mathematica, seriestruncation is required to achieve a given numericalaccuracy. As far as the corresponding truncation error isconcerned, the following result holds.

Proposition 1: The error result in truncating the innite seriesin (12) by M terms is given by (see (13))

where k0 > 1 is a parameter that depends on channel

parameters, m andK, as well as on l and can be selected ina suitable manner to guarantee the tightness of the bound.

Proof: See Appendix 1.

In the following and assuming high values ofg, approximateexpressions for Pd(l) will be derived that provide usefulinsight regarding the factors affecting system performance.

It can be observed that for high values of g the Gausshypergeometric function in (12) tends to unity. Also, by

observing that (g+ 1 +K)H gH, (12) can beapproximated as

Pd(l) 1 (1 +K)HmmH

gH (m +K)mH (H 1)!

1i=0

G u + i, l2

G(i +H)

G(u + i)i! (14)

from where it is deduced that the detection diversity gain,dened in [7], is equal to the number of the receiveantennas, H, and is independent of the fading parameters. Itshould be noted that this is not the case when othercommonly used fading models, such as the Nakagami-mareconsidered. Using the identity [32, eq. (4.4)]

1F1(a,b,x) =xa G(b)

G(b a)G(a)1i=0

(a + b 1)ii!xi

G(i + a,x)

(15)

Equation (14) is deduced in closed form as

Pd(l) 1 (1

+K)

Hm

mH

gH (m +K)mH (H 1)!G(H)G(1 + u) l

2

u1F1 H+ u, 1 + u,

l

2

(16)

Interestingly, (16) can also be obtained by using amethodology similar to the one developed in [33]. Thismethodology enables the evaluation of Pd(l) for sufciently large SNR values by employing a Taylor seriesapproximation for the PDF of the SNR around zero.Specically, fgMRC (g) can be approximated as

fgMRC (g) = (1

+K)HmmH

gH (m +K)mH (H 1)!gH1 + o(gH) (17)

Using [29, eq. (6.643.2)] and [29, eq. (9.220.2)], fY(y|H1, l)can be obtained as

fY y|H1, l = 2uyu1 G(H)

G(u) 1F1 u H,u,

y

2

(18)

|E| (1 +K)H

mmH

gM+1

(l/2)u+M+1

G(M+H+ 1)(m +K)mH (H 1)!(g+ 1 +K)M+H+1G(M+ u + 2)G(M+ 2) 2F1 Hm,H+M+ k0;H;

K(K+ 1)(m +K)(g+ 1 +K)

1F1(u +M+ 1,u +M+ 2,l/2) 2F2 1,M+H+ 1;M+ 2,u +M+ 2; lg2(g+ 1 +K)

(13)

www.ietdl.org

4



doi: 10.1049/iet-com.2014.0170


5/15

By substituting (18) into (7) and employing [34,eq. (1.14.1.7)], (16) is readily deduced.

i.n.i.d distributed branches: Assuming independent butnon identically distributed branches, an innite seriesrepresentation for the PDF of MRC is given by [27,eq. (17)] and [35, eq. (19)]

fgMRC (g) = B1i=0

cigH+i1 exp(g/b)bH+i(H+ i 1)! (19)

where

B = bHHi=1

(2bi)mi1

(2bi +Vi/mi)mi(20)

is an arbitrary positive real parameter whose value isselected to ensure the uniform convergence of (19) and the

coefcients of the series, ci, are recursively calculated as[35, eq. (16)]

c0= 1 andci=1

i

i1=0

dic, i . 0 (21)

with

dj=

H

i=1mk 1

b

2bi + Vi/mi

j

H

i=1(mi 1) 1

b

2bi

j(22)

The average probability of detection is obtained as

Pd(l) =1

0

fgMRC (g)Qu(2g

,

l

) dg (23)

Since the series in (19) is uniformly convergent forappropriate selection of , integration and summation can

be interchanged. Therefore Pd(l) is obtained as

Pd(l) = B1i=0

ci

bH+i(H+ i 1)!

1

0

gH+i1 exp (g/b)Qu(

2g

,

l

)dg

(24)

Assuming integer values ofu and employing [31, eq. (29)],Pd(l) is deduced as

Pd(l) = B1i=0

ci

bH+i(H+ i 1)! J H+ i, 1/b ,u, 1, l

(25)

where

J N,p,M,x,y) = G(N,p,M,x,y)+M1i=1

Di(N,p,M,x,y)Fi+1(N,p,M,x,y)

(26a)

G(N,p,M,x,y)=2N1(N1)!x2p2N(p2 +x2) exp

y2p2

2(p2 +x2)

N2k=0

p2

p2 +x2 k

Lk y2x2

2(p2 +x2)

+ p2

p2 +x2 u1

1+p2

x2

LN1

y2x2

2(p2 +x2)

(26b)

Fi(N,p,M,x,y) = 1F1 N,i, x2y2

2(p2

+x2)

(26c)

Di(N,p,M,x,y) = G(N)y2i2Ni

2i!(p2 +x2)Nexp y2

2

(26d)

3.1.2 Average probability of detection: MGF-basedapproach: Using a contour integral representation of theMarcum Q-function, in [3] an MGF-based approach was

proposed to evaluate the average probability of detection.Specically, Pd(l) can be expressed in terms of a contourintegral as

Pd(l) =exp(l/2)

2p

D

Mg 1 1

z

exp (l/2)z

zu(1 z) dz (27)

where is a circular contour of radius r [0, 1). Assuminginteger values of u as well as integer-valued fading

parameters of the Gamma component, m, the integral in(27) can be efciently evaluated by means of the residuetheorem. For i.i.d. diversity branches, the MGF of theoutput SNR can be obtained as

MgMRC

(s)

=

(sg+K+ 1)H(m1)(1 +K)H

sg 1 + (K/m) + 1 +K mH (28)Consequently, the probability of detection can be expressedas

Pd(l) = exp l

2

(1 +K)HmmH

2p

D

g(z) dz (29)

where

g(z) = exp (l/2)z

(g+K+ 1)z g H(m1)zuH(1 z) gm + gK+ m +Km

z g(m +K)

mH

(30)

In order to evaluate Pd(l) by using the MGF-based approach,two cases are considered.

www.ietdl.org


doi: 10.1049/iet-com.2014.0170

5



6/15

u >H: As it can be observed, g(z) has amHth order pole atz= g(m +K)/g(m +K) + m(1 +K) and a pole at z= 0 oforder uH. When H(m 1) and Hm are integers,application of the residue theorem yields the followingexpression forPd(l):

Pd(l) = exp l

2

(1 +K)HmmH

Res g: g(m +K)g(m +K) + m(1 +K) ;mH

+ Res( g:0; u H)

(31)

uH: In this case, no pole at the origin exists, andtherefore only the pole at z= g(m +K)/g(m +K) + m(1+

K) needs to be considered. Assuming integer values ofmHand applying the residue theorem, the following expressionforPd(l) is obtained

Pd(l) = exp l2 (1 +K)HmmH Res g: g(m +K)

g(m +K) + m(1 +K) ;mH

(32)

3.1.3 Average AUC: The area under the ROC curve is asingle gure of merit that provides better insights as to whatfactors affect the performance of the energy detector. Thisis because of the fact that AUC varies from 0.5 to 1 as theenergy threshold, l, varies from 0 to [7, 36]. In [37], it

was pointed out that AUC represents the probability thatchoosing the correct decision at the detector is more likelythan choosing the incorrect decision. The average AUC ofthe system under consideration can be evaluated as [7]

A = 1

0

Pd(l)Pf(l)

ldl (33)

where

Pf(l)

l= l

u1el/2

2uG(u) (34)

i.i.d. distributed branches: By substituting (12) and (34) to(33) and using [29, eq. (6.455/2)], the following innite seriesrepresentation for the average AUC is deduced

A = 1 (1 +K)HmHm

(m +K)mH (H 1)!1i=0

22ui

G(2u + i)G(i +H)i!G(u)G(u + i + 1)

gi

(g+ 1 +K)i+H

2F1 1, 2u

+i;u

+i

+1;

1

2 2F1 Hm,H+ i; 1;

K(K+ 1)(m +K)(g+ 1 +K)

(35)

As far as the corresponding truncation error is concerned,the following result holds.

Proposition 2: The error result in truncating the innite seriesin (35) by M terms can be bounded as

|E| gM+1

(1 +K)HmHm(m

+K)mH (H

1)!

2F1 1, 2u +M+ 1; u +M+ 2;1

2 2F1 Hm,H+M+ k0; 1;

K(K+ 1)(m +K)(g+ 1 +K)

2

2uM1G(2u +M+ 1)G(H+M+ 1)

(1 +K+g)H+M+1G(M+ 2)G(u)G(u +M+ 2)

3F1

1, 2u +M+ 1,H+M+ 1;M+ 2,

u +M+ 2; g2(g+ 1 +K)

(36)

where k0 > 1 is a parameter that depends on channel

parameters, m andK, as well as on l and can be selected ina suitable manner to guarantee the tightness of the bound.

Proof: See Appendix 2.

i.n.i.d distributed branches: When i.n.i.d diversitybranches are considered the average AUC can be obtained as

A =1

0

A(g)fgMRC (g) dg (37)

where A() is the unfaded AUC given by [7, eq. (9)]

A(g) = 1 u1k=0

1

2kk!g

ke(g/2)

+u1

k=1u

G(u + k)2u+kG(u)

eg1F1 u + k; 1; g

2

(38)

and fgMRC is given by (19). By employing [7, eq. (30)], ananalytical expression for A can be deduced as

A = 1 Bu1k=0

1

2kk!

1i=0

ciG(H+ k+ i)bH+i(H+ i 1)!

1

b+ 1

2

Hki

+ AG(u)

u1k=1u

G(u + k)2u+k

1i=0

ci

bH+i(H+ i 1)!

K H+ i, 1b

+ 1,u + k, 1 + k, 12

(39)

where

K(a,p,b,d,c) = paG(a) 2F1 a,b;d, c

p

(40)

3.2 Selection combining

In this scheme, the combiner selects the branch with thestrongest SNR among all diversity branches.Mathematically speaking, the instantaneous SNR at the

www.ietdl.org

6



doi: 10.1049/iet-com.2014.0170


7/15

combiner output can be expressed as SC= max{1, , H}.In order to derive analytical expressions for the average

probability of detection and the average AUC, the PDF ofSC is required. To the best of our knowledge, an analyticalexpression for the PDF of SC is not available in thetechnical literature. In Appendix 3, assuming that m isrestricted to integer values and i.i.d. diversity branches, ananalytical expression forSC is derived as in (66).

3.2.1 Average probability of detection: To evaluate theaverage probability of detection, a similar procedure as in thederivation of (12) is adopted. To this end, using an inniteseries representation of the modied Bessel function andthe denition of the Gamma function, an innite seriesrepresentation for the average probability of detection can

be obtained as

Pd(l) = 1 1i=0

1p=0

H=0

1=0

12=0

. . .

m2m1=0

(1)

m

m +K (m1) H

1 12 m2m1 G u + i,

l2

i!G(u + i) 1 +

(1 +K)mg(1 +K)

pm1k=1 ki1

Gm1k=1

k+ i + 1 p

cp(;1, . . . , m1)b10 b121 , . . . ,bm1m1

U

m1

k=1k+ i + 1 p

(41)

3.2.2 Average AUC: The average AUC under SC, can beevaluated using (33) where Pd(l) is given by (41). Followinga similar procedure as in the derivation of the average AUC insingle antenna and MRC diversity cases, Acan be obtained as

A = 1 1i=0

1p=0

H=0

1=0

12=0

. . .

m2m1=0

(1) mm +K (m1)

H

1 1

2 . . .

m2m

1

1 + (1 +K)mg(1 +K)

pm1k=1 ki1

G

m1k=1

k + i + 1 p

Um1k=1

k + i + 1 p

cp(;1, . . . ,m1)b10 b121 , . . . ,bm1m1 22ui

G(2u + i)G(i +H)i!G(u)G(u + i + 1)

2F1 1, 2u

+i;u

+i+

1;1

2 (42)Finally, it is noted that the series in (41) and (42) mainlyinvolve elementary and common Gamma functions. Hence,

they are both easy to be numerically evaluated and theiraccuracy is corroborated by the simulation results in Section 4.

3.3 Equal gain combining

EGC provides a performance comparable to that of MRC, butwith a simplied receiver structure. Hence, analysis of energydetectors employing EGC is of considerable interest. The

instantaneous SNR at the output of the EGC combineremploying Hdiversity branches is dened as

gEGC= Es

N0H

Hk=0

|hk| 2

(43)

where hkis the signal envelope of the kth branch. In order toevaluate the average detection probability and the averageAUC, the PDF of EGC is required. It is well known,however, that the analytical determination of the statistical

properties of the sums of fading signals envelopes in termsof tabulated functions is a rather difcult task [17]. This

intricacy of the exact sum statistics can be circumvented byusing accurate closed-form approximations to the PDF ofEGC or the Pad approximants method [14,38].

i.i.d. distributed branches:When i.i.d. distributed branchesare considered, the so-called small argument approximation(SAA) [39, pp. 453457] is employed to provide anaccurate closed-form approximation to the PDF of EGC.SAA is based on the observation that at small values ofSNRs, the statistics of EGC can be accurately approximated

by the statistics of MRC with appropriately scaledarguments. The validity of this approximation, even forlarge SNR values, has been demonstrated in several recentworks, including [4042]. In the following analysis, SAA is

applied to yield a highly accurate closed-formapproximation to the PDF ofEGC.

Let Z= Hk=0 |hk|. We propose to approximate the PDFofZwith the PDF ofZW W with W= Hk=1 |hk|2. As itis evident, W describes the power of the signal componentat the combiner output of an MRC diversity receiver. ThePDF ofWcan be expressed as

fW(w) = m

m + K

mH1 + KV

HwH1

(H 1)!

exp (1

+ K)w

V 1F1 mH,H, K(1 + K)wV(m + K) (44)

and its corresponding MGF as

MW(s) =(sV+ K+ 1)H(m1)(1 + K)H

s V 1 + Km

+ 1 +K

mH (45)The unknown parameters V, m and K can be estimatedmatching the rst, second and third moments of Z2 and W,namely

E{Z2} = E{W}, E{Z4} = E{W2},

E{Z6} = E{W3}

(46)

www.ietdl.org


doi: 10.1049/iet-com.2014.0170

7



8/15

The rst three moments of W can be straightforwardlyevaluated by the application of the moment theorem, namely

E{W} = dMW(s)ds

s=0

= HV (47a)

E{W2

} = d2MW(s)ds2

s=0

= HV2

(1 + K)2m

Hm + m + 2HmK+HmK2 + 2mK+ K2

(47b)

E{W3} = d

3MW(s)ds3

s=0

= HV

3

(1 + K)3m2 3Hm2 + 2K3

+ 6m2 K+ 2m2 +H2m2

+ 3H2

m2

K+ 3H2

m2

K2

+ 6Hm2

K2

+H2

m2

K3

+ 3HmK3 + 9Hm2K+ 3HmK2 + 6mK2

(47c)

The nth moment of |hk| can be expressed in closed form byperforming a transformation of random variables in (9) andapplying [29, eq. 7.621/4] as

m|hk|(n) = m

K+ m

mg

1 +K

n/2G 1 + n

2

2F1 m, 1 + n2 ; 1;

KK+ m (48)

Finally, the required moments E{Z2}, E{Z

4} and E{Z6} are

determined using (48) and the multinomial identity as

E{Zn} =

nj1=0

j1j2=0

jH2

jH1=0

n

j1

j1

j2

jH2

jH1

m|hk|(n j1)m|hk|(j1 j2), . . . , m|hk|(jH1) (49)

wherenis a positive integer. The resulting system of equationthat determines V, mand Kcan be easily solved using any ofthe most popular mathematical software packages, forexample, Matlab, Maple or Mathematica. As it will becomeevident, the proposed approximation yields highly accurateresults for both the average probability of detection and theaverage AUC which are practically indistinguishable fromthe exact solutions obtained using Monte Carlo simulations.

By performing a random variable transformation, anaccurate closed-form approximation to the PDF ofEGC can

be obtained as

fgEGC (g) H EsN0

1

fWHg

Es/N0 (50)Consequently, the average probability of detection and the

average AUC can be approximated as

Pd(l) 1 (1 + K)HmmH

(m +K)mH(H 1)!

1i=0

G u + i, (l/2) G(i +H)G(u + i)i!

vi

(v+ 1 + K)i+H

2F1 Hm,H+ i;H; K( K+ 1)(m + K)(v+ 1 + K)

(51)

and

A 1 (1 + K)H mH

m

(m + K)mH (H 1)!1i=0

22ui

G(2u + i)G(i +H)i!G(u)G(u + i + 1)

vi

(v+ 1 + K)i+H

2F1 1, 2u + i;u + i + 1; 1

2 2F1 Hm,H+ i; 1;

K(K+ 1)(m + K)(v+ 1 + K)

(52)

respectively, where

v= VH

EsN0

1(53)

i.n.i.d distributed branches: When non-identicallydistributed branches are considered, it is convenient to

employ the Pad approximants method to accuratelyapproximate the MGF of EGC,MgEGC (s) and eventuallyobtain Pd(l). A Pad approximant to the MGF is a rationalfunction of a specied order B for the denominator and Afor the nominator, whose power series expansion agreeswith the (A +B)-order power expansion of the MGF

MgEGC (s) R[A/B](s) =A

i=0cisi

1 +Bi=0bisi=Bj=0

rj

s +pjA+Bn=0

mgEGC (n)sn

n!

(54)

where mgEGC (n) is the nth moment of EGC, bi, ci are realnumbers and pj, rj are the poles and the residues of thePad approximant, respectively. In order to obtain anaccurate approximation of the MGF, sub-diagonal Padapproximants (B=A+ 1) are used [38]. Given the nthmoment of EGC, the coefcients bi, ci can be readilyevaluated using any of the most popular softwaremathematical packages, such as Maple, Matlab orMathematica. The nth moment of EGC can be obtained byemploying the multinomial identity as

mgEGC (n) = 1

Hn 2n

j1=0j1

j2=0

jH2

jH1=0

n

j1 j1

j2 jH2

jH

1 m|h1|(n j1)m|h2|(j1 j2), . . . , m|hH|(jH1)

(55)

www.ietdl.org

8



doi: 10.1049/iet-com.2014.0170


9/15

Finally, Pd(l) can be readily evaluated by employing (29)and applying the residue theorem.

4 Numerical and computer simulationresults

In this section, various performance evaluation results

obtained using the detection probability and AUCexpressions presented in Section 3 will be presented. In

particular, the following performance evaluation resultshave been obtained: (i) complementary ROC curves fordiversity reception (obtained using (12), (31), (32), (25),(26a), (26), (41) and (51) with (48) and (49) see Figs. 1and 2); (ii) probability of a missed detection as a functionof SNR (obtained using (12) and (16), see Fig. 3) (iii)complementary AUC curves for diversity reception(obtained using (35), (42) and (52) with (48) and (49) seeFigs.4 and5); and (iv) truncation error bounds of the seriesin (12) and (35), respectively, as a function of the numberof terms (obtained using (13) and (36), respectively, seeFigs.6 and7).

To validate the accuracy of the previously mentionedexpressions, comparisons with complementary Monte Carlosimulated performance results are also included in thesegures by generating 105 random samples from theGamma-shadowed Rice distribution. This specic samplesize guarantees statistical convergence for values of averagedetection probability of the order of 104. In order togenerate random samples from a Gamma-shadowed Ricedistribution, random samples from the non-centralchi-square distribution with two degrees of freedom andnon-centrality parameter modulated by a Gammadistribution with parameters m and/m are generated. Forthe MRC and EGC cases, it is assumed that perfect channel

estimates are available for diversity reception in a similarfashion as in [2].

In order to numerically quantify the impact of shadowingon the probability of detection and AUC when i.i.d.

branches are considered, the corresponding distribution

parameters (b0, m, ) are selected in accordance with theentries of [25, Table 3], to account for four differentshadowing scenarios: (i) frequent heavy shadowing (b0,m, ) = (0.063, 0.739, 8.97 104); (ii) overall results (b0, m, ) = (0.251, 5.21, 0.278); (iii) infrequent lightshadowing (b0, m, ) = (0.158, 19.4, 1.29); and (iv)average shadowing (b0, m, ) = (0.126, 10.1, 0.835).

For all the above considered scenarios, the parameters (b0,m, ) have been obtained by employing a moment matching

technique that associates the parameters of the exact Loosmodel with those of (5). It is noted that the parameters ofthe Loos model for the considered scenarios, have beenused in several studies such as [4347], for system

Fig. 1 Complementary ROC curves of dual branch diversity

receivers over i.i.d. Gamma-shadowed Rician fading channels

Fig. 2 Complementary ROC curves of triple branch diversity

receivers over i.i.d. Gamma-shadowed Rician fading channels

Fig. 3 Probability of missed detection of dual branch diversity

receivers over i.i.d. Gamma-shadowed Rician fading channels with

Pf= 100.5

www.ietdl.org


doi: 10.1049/iet-com.2014.0170

9



10/15

simulation, analysis and performance prediction purposes.Moreover, as it was shown in [25], for all consideredscenarios, the correspondingrst and second order statisticsof the exact Loos model and the considered Gamma-shadowed Rician model are almost indistinguishable, and both are close enough toexperimental data obtained through a measurement campaign.

In the i.n.i.d. case, a triple-branch diversity receiver isconsidered with {b}

3=1=

{0.1, 0.2, 0.3}, {m}3=1={5.21, 3.2, 1.5} and {V}3=1= {2, 3, 4}. Finally, in all test

cases u = TW = 4 is assumed.In Figs.1and2, complementary ROC curves for dual- and

triple-branch diversity are portrayed, assuming MRC, SC and

EGC receivers. To avoid curve entanglement, only twoshadowing scenarios are considered, namely overall resultsand average shadowing. Two different values of SNR,namely Es/N0 = 0 dB and Es/N0= 10 dB are assumed. Forcomparison purposes, the single antenna case (single-inputsingle-output) is also illustrated. For Es/N0 = 10 dB, it can

be observed that there exists an obvious diversity gainwhen MRC, EGC or SC is employed compared to thesingle antenna case. However, for low values of Es/N0,namely when Es/N0 = 0 dB diversity reception alone cannotguarantee a satisfactory operation of the energy detector. Asit was pointed out in [7], a possible solution in suchconditions is to employ both diversity reception and

Fig. 4 Complementary AUC against Es/N0 curves of dual branch

diversity receivers over i.i.d. Gamma-shadowed Rician fading

channels

Fig. 5 Complementary AUC against Es/N0curves of triple branch

diversity receivers over i.i.d. Gamma-shadowed Rician fading

channels

Fig. 6 Truncation error for the series in (12), assuming dualbranch MRC diversity receivers k0 = 2 and various shadowing

scenarios

Fig. 7 Truncation error for the series in (35), assuming dual

branch MRC diversity receivers and various shadowing scenarios

www.ietdl.org

10



doi: 10.1049/iet-com.2014.0170


11/15

cooperative sensing. Another interesting observation is thatthe performance difference between EGC and MRC is verysmall, especially under less severe shadowing conditions,that is, when average shadowing is considered. This is dueto the fact that the concept behind EGC diversity receptionis to provide performance close to the optimal MRCschemes while minimising hardware complexity. It is alsonoted that this nding is in agreement with similar ones

reported in [7]. Therefore since MRC requires high-qualitychannel estimates for reasonable performance and suchestimations are unlikely to be present for energydetection-type applications, EGC is a very attractive choicefor practical energy detectors, because of its considerablylower implementation complexity. Moreover, it can beobserved that the performance degrades as the amount ofshadowing increases (b0, m or ), whereas the impact ofshadowing increases with the increase of the average SNR.This is because an increase in b0 results in diminishing theimpact of shadowing. Moreover, as m increases, a higherdetection probability with a lower false alarm probability isobserved because the channel fading conditions improve,that is, the uctuations of the signal strength reduce.Furthermore, as shown from (16), channel parameters b0, mor do not affect the detection diversity gain. Thereforethe impact of the aforementioned parameters on system

performance will show up at high SNR values only.In Fig.3, the probability of a missed detection when dual

branch MRC diversity receivers are considered, is depictedas a function of SNR assuming Pf= 10

0.5. The propagationscenarios under consideration are frequent heaveshadowing, infrequent shadowing and overall results. In thesame plot, approximate results obtained using (16) are alsoincluded. As it is evident, the asymptotic expressions forthe probability of detection correctly predict the detectiondiversity gain for all scenarios under consideration. It can

also be observed that the detection diversity gain is indeednot affected by the channel parameters b0, m and.

As far as the validation of the proposed analysis isconcerned, it is noted that for the MRC case, both PDF-and MGF-based approaches are used to evaluate the

probability of missed detection. In order to use theMGF-based approach for the evaluation of the probabilityof detection when MRC diversity is employed, thecorresponding values of the m parameter are rounded to thenearest integer ones. A similar approach is applied to theevaluation of the probability of detection and AUC in thecase of SC diversity reception. Interestingly enough, theMGF-based approach yields almost identical results withthe PDF-based one, for all shadowing scenarios, despite thefact that the value ofm parameter is rounded to the nearestinteger. This observation, also implies that performancemetrics such as the probability of detection and AUC, arerelatively insensitive to small variations of m. This is

because of the fact that small changes in m do notseemingly cause signicant changes in the moments ofMRC, obtained by the differentiation of (6) as shown inSection 3.3. Moreover, as the detection diversity gain doesnot depend on the channel parameters but only on thenumber of diversity branches, it is expected that thedetection probability is not signicantly affected by smallchanges in m. Therefore, the MGF-based method obtainedusing (31) and (32), provides a simple and efcient means

to quantify the performance of energy detection overGamma-shadowed Rician fading channels. The onlyrestriction is the limitation of u to integer values. For theSC case, analytical results have been obtained by rounding

m to the nearest integer. Nevertheless, one can observe thatthe corresponding analytical results match very well withthe exact ones, obtained using Monte Carlo simulations,thus verifying that (41) can be efciently used to predictthe energy detection capabilities of practical systemsemploying SC diversity. As far as the EGC case isconcerned, a comparison of the approximate resultsobtained using (51) with (48) and (49) with the exact ones,

obtained using Monte Carlo simulation, indicates that theproposed approximation is highly accurate for allconsidered shadowing conditions, yielding results which are

practically indistinguishable to the exact solution.In Figs. 4 and 5, complementary AUC curves against

Es/N0, for dual- and triple-branch diversity are provided,assuming MRC, SC and EGC receivers and i.i.d. diversity

branches. The shadowing scenarios under consideration arefrequent heavy shadowing, overall results and averageshadowing. As it is evident, diversity reception can beefciently used to improve the detectors capabilities.Regarding the accuracy of (42) in the case of SC diversityreception, as well as of (52), in the case of EGC reception,similarndings to the ones reported from the observation ofFigs. 1 and2 can be veried. Also, it can be observed thatalthough the performance gap between MRC and EGC issmall, it slightly increases at high SNR values.

It is noted that the numerical evaluation of the analyticalexpressions containing innite series, namely (12) and (35),has been carried out by truncating the appropriate seriesexpressions. Figs. 6 and 7 depict the truncation error of (12)and (35), respectively, as a function of the number of terms,assuming dual branch MRC diversity receivers. Theshadowing scenarios under consideration are frequent heavyshadowing and infrequent light shadowing. In both cases, theexact truncation errors were obtained by evaluating theinnite series in (12) and (35) using 10

3terms and the

corresponding bounds by employing (13) and (36),respectively. In Fig. 6, the arbitrary parameter k0 is set to beequal to 2 whereas in Fig. 7, k0 is selected in an appropriatemanner to provide a tight truncation error bound. By theobservation of both gures, it is evident that the truncationerrors of both series under consideration decrease to zeromonotonically and very quickly as the number of termsincreases. Moreover, it can be observed that by selecting k0 ina suitable manner, the closed forms error bounds are verytight and, in some cases, almost indistinguishable to the exacttruncation error. The optimal selection of k0 is, however, anopen issue and left to a future research work. Tables1and2depict the number of terms, M, required to achieve a relativeerror er< 10

6 when MRC diversity reception is consideredfor different shadowing scenarios. For the considered cases,our results have shown that all innite series rapidlyconverged with the speed of convergence depending on thefading/shadowing parameters, the probability of false alarm

Pf, the number of samples u and the SNR. As shown,although M increases as shadowing conditions improve or uincreases, a relatively small M (with a maximum of M= 36for the evaluation of (12) and M= 28 for the evaluation of(35) for all cases considered in this paper) is required.

Finally, in order to demonstrate the validity of (25) and thePad approximants framework when non-identicallydistributed diversity branches are considered, in Fig. 8complementary ROC curves are depicted when triple branch

MRC or EGC diversity reception is employed, assumingSNR = 0 dB, 10 dB and 15 dB. For the MRC case, in orderto apply (25), the value of the arbitrary parameter should

be selected in an appropriate manner to ensure the uniform

www.ietdl.org


doi: 10.1049/iet-com.2014.0170

11



12/15

convergence of the corresponding innite series. In ourconducted numerical experiments, the series in (25) weretruncated to 20 terms by selecting = 0.39 for SNR = 0 dB,= 3.8 for SNR = 10 dB and= 11.2 for SNR = 20 dB. Forthe EGC case, the Pad approximants of order (3, 4) forSNR= 0 dB and (4, 5) for SNR = 10 dB and SNR = 20 dBalong with the residue theorem were employed. In all casesof interest, numerically evaluated results are compared toequivalent Monte Carlo simulation ones. These comparisonsclearly show that the ROC curves coincide with square

pattern signs obtained via simulations, verifying thecorrectness of the proposed analysis.

5 Conclusion

In this work, the performance of an energy detector underboth multi-path fading and shadowing effects in a

Gamma-shadowed Rician fading environment was studiedin detail. Novel analytical expressions were derived for theaverage probability of detection, as well the area under theROC curves, for single antenna and diversity receptioncases. It is shown that the detection diversity gain does notdepend on the channel parameters but on the number ofreceive antennas only. This nding provides useful insightas to the factors that affect the performance of the energydetector. Closed-form bounds for the truncation error ofsuch series have been derived that enable the estimation ofthe terms required to achieve a given numerical accuracy. Itis also found that the Pad approximants theory is useful

for the numerical approximation of the average probabilityof detection when EGC diversity reception is employed.Our conducted numerical experiments have alsodemonstrated the impact of composite fading andshadowing on the detectors performance. It is shown thatthe probability of detection increases with the fading

parameter m and shadowing parameter b. Furthermore,when diversity reception is employed to enhance the

performance of the energy detector, the ROC reveals thatdiversity gains are obvious for medium and high SNRvalues. Moreover, MRC provides better performanceenhancements than SC but comparable to EGC. Acomparison of the analytical results with extensive Monte

Carlo simulations validated the correctness of the proposedanalysis. Our newly derived results are useful to systemdesign engineers in quantifying the impact of fading andshadowing in energy detection spectrum sensing, a fact thatcan lead to the design of more efcient cognitive radiocommunication systems.

6 Acknowledgments

This work was supported by the Sensor Networks andCellular Systems Research Center of the University of Tabuk.

7 References

1 Urkowitz, H.: Energy detection of unknown deterministic signals.Proc. IEEE, April 1967, vol. 55, no. 4, pp. 523531

Table 1 Number of required terms, Nq, for convergence of the Pd(l) given by (12) to achieve a relative error er < 106 for various

values ofPf,uand for different shadowing scenarios

Pf u= 4 u= 10

Frequent heavyshadowing

Overallresults

Averageshadowing


Overallresults

Averageshadowing

103 22 27 29 27 35 36102 19 22 22 23 30 30101 15 17 16 19 23 23100.1 8 8 5 11 12 10

Table 2 Number of required terms,Nq, for convergence of the Agiven by (35) to achieve a relative error er < 106 for various values of

SNR, uand for different shadowing scenarios

SNR, dB u= 4 u= 10


Overallresults

Averageshadowing


Overallresults

Averageshadowing

0 6 13 14 7 14 165 10 18 20 11 22 2410 15 20 21 18 27 28

Fig. 8 Complementary ROC curves of triple branch diversity

receivers over i.n.i.d. Gamma-shadowed Rician fading channels

www.ietdl.org

12



doi: 10.1049/iet-com.2014.0170


13/15

2 Digham, F.F., Alouini, M.S., Simon, M.K.:On the energy detection ofunknown signals over fading channels, IEEE Trans. Commun., 2007,55, (1), pp. 2124

3 Herath, S.P., Rajatheva, N., Tellambura, C.: Energy detection ofunknown signals in fading and diversity reception, IEEE Trans.Commun., 2011, 59, (9), pp. 24432453

4 Hemachandra, K.T., Beaulieu, N.C.: Novel analysis for performanceevaluation of energy detection of unknown deterministic signals usingdual diversity. Proc. IEEE Vehicular Tech. Conf. (VTC-fall 11),September 2011, pp. 15

5 Herath, S.P., Rajatheva, N.:Analysis of equal gain combining in energydetection for cognitive radio over Nakagami channels. Proc. IEEEGlobal Commun. Conf. (GLOBECOM), 2008

6 Atapattu, S., Tellambura, C., Jiang, H.: Spectrum sensing via energydetector in low SNR. Proc. IEEE Int. Conf. Commun. (ICC 11), June2011, pp. 15

7 Atapattu, S., Tellambura, C., Jiang, H.:Analysis of area under the ROCcurve of energy detection, IEEE Trans. Wirel. Commun., 2010, 9, (3),pp. 12161225

8 Atapattu, S., Tellambura, C., Jiang, H.: Energy detection of primarysignals over -- fading channels. Proc. Fourth Ind. Inf. Systems(ICIIS 09), December 2009, pp. 15

9 Ruttik, K.K.K., Jantti, R.: Detection of unknown signals in a fadingenvironment, IEEE Commun. Lett., 2009, 13, (7), pp. 498500

10 Banjade, V.R.S., Rajatheva, N., Tellambura, C.: Performance analysisof energy detection with multiple correlated antenna cognitive radio in

Nakagami-m fading, IEEE Commun. Lett., 2010, 16, (4), pp. 50250511 Sofotasios, P., Rebeiz, E., Zhang, L., Tsiftsis, T., Cabric, D., Freear, S.:

Energy detection based spectrum sensing over - and - extremefading channels, IEEE Trans. Veh. Technol., 2013, 62, (3),pp. 10311040

12 Atapattu, S., Tellambura, C., Jiang, H.: Relay based cooperativespectrum sensing in cognitive radio networks. Proc. IEEE GlobalCommun. Conf. (GLOBECOM), 2009, pp. 15

13 Rao, A., Alouini, M.-S.: Performance of cooperative spectrum sensingover non-identical fading environments, IEEE Trans. Commun., 2011,59, (12), pp. 32493253

14 Atapattu, S., Tellambura, C., Jiang, H.: Energy detection basedcooperative spectrum sensing in cognitive radio networks, IEEETrans. Wirel. Commun., 2011, 10, (4), pp. 12321241

15 Loo, C.: A statistical model for a land mobile satellite link, IEEETrans. Veh. Technol., 1985, 34, pp. 122125

16 Suzuki, H.: A statistical model for urban radio propagation, IEEETrans. Commun., 1977, COM-25, (7), pp. 673680

17 Simon, M.K., Alouini, M.S.: Digital communication over fadingchannels (Wiley, New York, 2005, 2nd edn.)

18 Renzo, M.D., Graziosi, F., Santucci, F.: Cooperative spectrum sensingin cognitive radio networks over correlated log-normal shadowing.IEEE Vehicular Technology Conf., 2009

19 Ghasemi, A., Sousa, E.S.:Interference aggregation in spectrum-sensingcognitive wireless networks,IEEE J. Sel. Top. Signal Process., 2008, 2,(1), pp. 4156

20 Rasheed, H., Haroon, F., Rajatheva, N.: Performance analysis ofRice-lognormal channel model for spectrum sensing. IEEE Int. Conf.on Electrical Engineering/Electronics Computer Telecommunicationsand Information Technology, May 2010, pp. 420424

21 Atapattu, S., Tellambura, C., Jiang, H.: Performance of an energydetector over channels with both multipath fading and shadowing,IEEE Trans. Wirel. Commun., 2010, 9, (12), pp. 36623670

22 Rasheed, H., Haroon, F., Adachi, F.: On the energy detection overgeneralized-K (KG) fading. IEEE 14th International MultitopicConference (INMIC), December 2011, no. 4, pp. 367371

23 Olabiyi, O., Alam, S., Odejide, O., Annamalai, A.: Further results onthe energy detection of unknown deterministic signals overgeneralized fading channels. IEEE Int. Workshop on RecentAdvances in Cognitive Communications and Networking, 2011

24 Atapattu, S., Tellambura, C., Jiang, H.: A mixture gamma distributionto model the SNR of wireless channels, IEEE Trans. Wirel. Commun.,2011, 10, (12), pp. 41934203

25 Abdi, A., Alouini, W.C.L.M.-S., Kaveh, M.: A new simple model forland mobile satellite channels: First and second order statistics, IEEETrans. Wirel. Commun., 2003, 2, (3), pp. 519528

26 Alfano, G., Maio, A.D.: Sum of squared shadowed-Rice randomvariables and its application to communication systems performanceprediction,IEEE Trans. Wirel. Commun., 2007, 6, (10), pp. 35403545

27 Ropokis, G.A., Rontogiannis, A.A., Mathiopoulos, P.T., Berberidis, K.:An exact performance analysis of MRC/OSTBC over generalizedfading channels,IEEE Trans. Commun., 2010, 9, (58), pp. 24862492

28 Tellambura, C., Annamalai, A., Bhargava, V.:Closed form and inniteseries solutions for the MGF of a dual-diversity selection combiner

output in bivariate Nakagami fading, IEEE Trans. Commun., 2003,51, (4), pp. 539542

29 Gradshteyn, I., Ryzhik, I.M.: Tables of integrals, series, and products(Academic Press, New York, 2000, 6th edn.)

30 Wolfram, T.: function site, available at http://www.functions.wolfram.com

31 Nuttall, A.H.: Some integrals involving the QM-function. TechnicalReport, Naval Underwater Systems Center (NUSC), 1974

32 Muller, K.: Computing the conuent hypergeometric function, M(a,b, x), Numer. Math., 2001, 90, (1), pp. 179196

33 Wang, Z., Giannakis, G.B.: A simple and general parameterizationquantifying performance in fading channels, IEEE Trans. Commun.,2003, 51, (8), pp. 13891398

34 Prudnikov, A.P., Brychkov, Y.A., Marichev, O.I.: Integrals and SeriesVolume 3: More Special Functions (Gordon and Breach SciencePublishers, 1986, 1st edn.)

35 Ropokis, G.A., Rontogiannis, A.A., Berberidis, K., Mathiopoulos, P.T.:Performance analysis of maximal ratio combining over shadowed-ricefading channels. Int. Workshop on Satelli te and SpaceCommunications (IWSSC), 2009

36 Alam, S., Annamalai, A.: Energy detectors performance analysis overthe wireless channels with composite multipath fading and shadowingeffects using the AUC approach. IEEE Nineth Annual ConsumerCommunications and Networking Conf. Wireless ConsumerCommunication and Networking, 2012

37 Hanley, J.A., McNeil, B.J.: The meaning and use of the area under a

receiver operating characteristic (ROC) curve, Radiology, 1982,

143,(1), pp. 2936

38 Karagiannidis, G.K.: Moments-based approach to the performanceanalysis of equal-gain diversity in Nakagami-m fading, IEEE Trans.Commun., 2004, 52, (5), pp. 685690

39 Schwartz, M., Bennett, W.R., Stein, S.: Communication systems andtechniques (McGraw-Hill, New York, 1966)

40 Hu, J., Beaulieu, N.C.:Accurate closed-form approximations to Riciansum distributions and densities, IEEE Commun. Lett., 2005, 9, (2),pp. 133135

41 Zlatanov, N., Hadzi-Velkov, Z., Karagiannidis, G.K.: An efcientapproximation to the correlated Nakagami-m sums and its applicationin equal gain diversity receivers, IEEE Trans. Wirel. Commun., 2010,9, (1), pp. 302310

42 Peppas, K.:Accurate closed-form approximations to generalised-K sumdistributions and applications in the performance analysis of equal-gaincombining receivers, IET Commun., 2011, 5, (7), pp. 982989

43 Jamali, S.H., Ngoc, T.L.: Coded-modulation techniques for fadingchannels (Kluwer, Boston, 1994)

44 Loo, C.: Digital transmission through a land mobile satellite channel,IEEE Trans. Commun., 1990, 38, pp. 693697

45 Loo, C., Secord, N.: Computer models for fading channels withapplications to digital transmission, IEEE Trans. Veh. Technol., 1991,40, pp. 700707

46 Chau, Y.A., Sun, J.T.: Diversity with distributed decisions combiningfor direct-sequence CDMA in a shadowed Rician-fading land-mobilesatellite channel, IEEE Trans. Veh. Technol., 1996, 45, pp. 243247

47 van Nee, R.D.J., Prasad, R.: Spread spectrum path diversity in ashadowed Rician fading land-mobile satellite channel, IEEE Trans.Veh. Technol., 1993, 42, pp. 131136

48 Abramovitz, M., Stegun, I.:Handbook of mathematical functions withformulas, graphs, and mathematical tables (Dover, New York, 1964),ISBN 0-486-61272-4

8 Appendix

8.1 Appendix 1: Proof of Proposition 1

The error result in truncating the innite series in (12) by Mterms is given by

|E| = (1 +K)HmmH

(m +K)mH (H 1)!1

i=M+1

G u + i, l2

G(i +H)

G(u + i)i!

gi

(g+ 1 +K)i+

H

2F1 Hm,H+ i;H; K(K+ 1)

(m +K)(g+ 1 +K)

(56)

www.ietdl.org


doi: 10.1049/iet-com.2014.0170

13



14/15

By employing the identity G u + i, l2

= (u + i)1(l/2)

u+i1F1(u + i,u + i + 1,l/2) [29, eq. (8.351.2)],

(56) can be written as

|E| = (1 +K)HmmH

(m +K)mH (H 1)!1k=0

(l/2)u+k+M+1

G(u + k+M+ 2)(k+M+ 1)!

gk

+M

+1

(g+ 1 +K)k+M+H+1

2F1 Hm,H+ k+M+ 1;H; K(K+ 1)

(m +K)(g+ 1 +K)

1F1(u + k+M+ 1,u + k+M+ 2,l/2)G(k+M+H+ 1)

(57)

The conuent hypergeometric function 1F1(u + k+M+ 1,u + k+M+ 2, l/2) is a monotonically decreasing functionwith respect to k. On the other hand, the Gauss

hypergeometric function

2F1 Hm,H+ k+M+ 1;H; (K(K+ 1)/(m +K)(g+ 1 +K))

is monotonically increasing with respect tok. However, thereexist a suitable integerk0 such that fork> k0 the product of

both hypergeometric functions decreases monotonically,that is the conuent hypergeometric function decreases in afaster rate than the one that the Gauss hypergeometricfunction increases. Therefore it can be observed that |E| can

be bounded as

|E| (1

+K)HmmH

(m +K)mH (H 1)!2F1

Hm,H+M+ k0;H; K(K+ 1)

(m +K)(g+ 1 +K)

1F1(u +M+ 1,u +M+ 2,l/2)

1k=0

(l/2)u+k+M+1

G(u + k+M+ 2)(k+M+ 1)!

G(k+M+H+ 1)gk+M+1

(g+ 1 +K)k+M+H+1

=(1

+K)HmmHgM+1 (l/2)u+M+1G(M

+H

+1)

(m +K)mH (H 1)!(g+ 1 +K)M+H+1G(M+ u + 2)G(M+ 2)

2F1 Hm,H+M+ k0;H; K(K+ 1)

(m +K)(g+ 1 +K)

1F1(u +M+ 1,u +M+ 2,l/2)

1k=0

lg

2(g+ 1 +K) k

(1)k(M+H+ 1)kk!(M+ 2)k(M+ u + 2)k

(58)

The innite series in (58) is recognised as the seriesrepresentation of the 2F2 generalised hypergeometricfunction. Hence, a bound for |E| can be obtained in closedform as in (13) and this concludes the proof.

8.2 Appendix 2: Proof of Proposition 2

The truncation error is given by

|E| = (1 +K)HmHm

(m +K)mH (H 1)!1

i=M+122ui

G(2u

+i)G(i

+H)

i!G(u)G(u + i + 1)gi

(g+ 1 +K)i+H

2F1 1, 2u + i; u + i + 1; 1

2

2F1 Hm,H+ i; 1;

K(K+ 1)(m +K)(g+ 1 +K)

(59)

Following a similar line of arguments as in the proof ofProposition 1, it can be observed that (35) can be bounded as

|E| (1 +K)HmHm

(m +K

)mH (H

1)!

2F1 1, 2u +M+ 1;u +M+ 2; 1

2

2F1 Hm,H+M+ k0; 1;

K(K+ 1)(m +K)(g+ 1 +K)

1

i=M+122ui

G(2u + i)G(i +H)i!G(u)G(u + i + 1)

gi

(g+ 1 +K)i+H

= gM+1

(1 +K)HmHm(m +K)mH (H 1)!

2F1(1, 2u +M+ 1;u +M+ 2; 1

2 2F1 Hm,H+M+ k0; 1;

K(K+ 1)(m +K)(g+ 1 +K)

22uM1

G(2u +M+ 1)G(H+M+ 1)(1 +K+ g)H+M+1G(M+ 2)G(u)G(u +M+ 2)

1i=0

(1)i(2u +M+ 1)i(H+M+ 1)ii!(M+ 2)i(u +M+ 2)i

g

2(g+ 1 +K)

i(60)

The innite series in (60) is recognised as the seriesrepresentation of 3F2 generalised hypergeometric function.Consequently, a closed-form bound for |E| can be deducedas in (36) thus completing the proof.

8.3 Appendix 3: Derivation of fgSC (g)

Assuming i.i.d. diversity branches, the CDF ofSC, FgSC (g),

is mathematically expressed as FgSC (g) = [Fg(g)]H, where

Fg(g) =g

0 fg(t)dt is the CDF of the individual SNR per

branch, , andf(t) is given by (9). Assuming integer valuesof m, by applying the Kummer transformation for theconuent hypergeometric function in (9) [48], that is

1F1 m;n;z( ) = exp(z)mnk=0

(m n)k( z)kk!(n)k

, n m (61)

www.ietdl.org

14



doi: 10.1049/iet-com.2014.0170


15/15

Equation (9) can be written as

fg(g) = m

m +K

m1 +Kg

m1k=0

exp m(1 +K)g(m + k)g

(1)k(1 m)kk!(1)k

m(1 +K)g(m + k)g

k (62)

Using the denition of the lower incomplete Gamma functionas well as [29, eq. (8.353/6)], F() can be obtained as

Fg(g) = 1 m

m +K m1m1

k=0

kj=0


(1)kKkmjk(1 m)k

j!(1)k

(1 +K)g(m + k)g j

= 1 mm +K m1m1

j=0

m1k=j


(1)kKkmjk(1 m)kj!(1)k

(1 +K)g(m + k)g j

(63)

By applying the multinomial identity, [F()]Hand henceforth

FgSC (g) can be deduced as

FgSC (g)=H=0

1=0

12=0

m2

m1=0(1) m

m+K (m1)

H

1 1

2 m2

m

1 gm1

k=1 k exp m(1+K)g(m+k)g

b10 b

121 , . .. , b

m1m1

(64)

where

bj=m1k=j

(1)kKkmjk(1 m)kj!(1)k

(1 +K)(m + k)g j

(65)

Finally, fgSC

(g) is obtained by taking the derivative of (64)

with respect to yielding

fgSC (g) =1p=0

H=0

1=0

12=0

. . .

m2m1=0

(1) mm +K

(m1)

H

1

1

2

m2

m1

g

m1k=1 kp exp m(1 +K)g

(m + k)g

b10 b

121 , . . . ,b

m1m

1 cp(;1, . . . , m

1)

(66)

where

c0(;1, . . . ,m1) = m(1 +K)(m +K)g (67)

and

c1(;1, . . . ,m1) = m1

k=1 k (68)

www.ietdl.org

IET Commun., pp. 115 15

Documents

Energy detection of unknown signals in Gamma-shadowed Rician fading environments with diversity reception