Spectrum Sensing with Spatial Diversity Over KG

Composite FadingHaroon Rasheed∗, Farah Haroon†, Charles T. Spracklen‡ and Fumiyuki Adachi§

Department of Electrical Engineering, Bahria University, Karachi Pakistan†

Institute of Industrial Electronics, PCSIR, Karachi Pakistan∗

Habib University Foundation, Karachi Pakistan‡

Wireless Signal Processing Lab, Tohoku University, Sendai Japan§

Email: ∗haroon.rasheed@bimcs.edu.pk, †farah@iiee.edu.pk, ‡tim.spracklen@habib.edu.pk, §adachi@ecei.tohoku.ac.jp

Abstract—Cognitive ratio (CR) is aimed to be the answer ofinadequate spectrum utilization problem. Spectrum sensing is oneof the most demanding aspects in CR design and implementation.Low signal to noise ratio (SNR) and fading effects posed limita-tions to deploying CR in realistic propagation scenario with fastand fine sensing features. Energy detection (ED) is the simplestnon coherent technique for sensing and extensively referred asCR standard technique for primary user detection hypothesis.In this paper, we come back on the fundamentals of fadingeffects mitigation and to achieve inherent multipath propagationgain in wireless environment for ED performance improvement.An effective rectification approach to varying channel effects isto employ space diversity. We consider the performance of EDover composite Generalized-K (KG) fading to deal with bothsmall and large scale fading. Diversity combining using Maximalratio Combining (MRC) and Selective Combining (SC) over KG

fading channel are investigated. A novel tractable expression forEnergy detection based average detection probability for optimalMRC diversity combining scheme is derived and closed formpossibility is analyzed for SC. Both numerical and simulationmodels are examined for practical low to moderate shadowfading conditions. The results highlight the notable impact ofshadowing spread and fading severity on detection performanceand meliorating effects of employing combining techniques. Tothe best of knowledge, such analytical framework and resultingobservations were unexplored and have never been reportedbefore.

I. INTRODUCTION

Wireless spectrum is a confined asset which demands ef-ficient utilization of present spectrum allotment as the avail-able and accessible bandwidth often relinquishes the actualrequirement. Cognitive radio (CR) is an innovative and mostcompelling solution having tremendous possibility to improvespectrum efficiency and quality of services through sharedutilization [1]. The system is being explored to increase presentband engagement occupied by licensed users by allocation ofunused spectrum pool to secondary users (Cognitive users).For that purpose, this intelligent system will observe first anyparticular spectrum which is idle for a long duration beforeaccessing it. With this intentions, that detection mechanismmust be an artifact scheme, which senses the primary signalpresence with high detection possibility, while maintaining theerroneous exploration extremely small.

Several schemes are employed to expeditiously detect thepresence of primary transmitter. Some well-known spectrum

sensing techniques are Energy Detection, Matched Filter De-tection and Cyclostationary Feature Detection. Each methodpossesses distinctive and diverse features which are suitablein various environment conditions. Among these, ED is ofparticular focus because of its implementation simplicity andpotential to detect any shape of waveforms with intrinsicprivacy. Energy detector first proposed in classic work ofUrkowitz [2] for detection of unknown deterministic signalscorrupted by Gaussian noise. ED is the simplest efficienttechnique that essentially computes a running average of thesignal power over a window of pre-specified spectrum lengthand also requires no a priori information of transmitted signals.Concurrently, when knowledge of primary user signal is notpossible, energy detector for spectrum sensing in cognitiveradio is the optimal choice. However, the ability of an energydetector to classify between noise and signal is very compro-mising [3]. Furthermore, the spatial and transient variations ofwireless signal are also major contributing factors in sensingdeterioration using energy detection technique.

ED based Spectrum sensing in cognitive radio networksuffers performance loss during varying propagation envi-ronment and shadowing and frequency selective channelscharacteristics. However, as a matter of fact an energy detectorcan collect the multipath energy very easily by means of asimple integrator without any channel estimation. The timingaccuracy is also acceptable and the Bit Error Rate(BER) isnot largely affected due to synchronization errors. For fadingmitigation, the authors for eg., in [4]–[7] thoroughly discussedand analyzed ED performance with diversity reception. Thework is devoted to various receive diversity techniques overadditive white Gaussian noise (AWGN), Rayleigh, Rician andNakagami-m small scale fading. Exact closed-form expres-sions has been evaluated for probability of detection overabove fading channels assuming both single-channel and mul-tichannel schemes.

As both short and long-term fading conditions coex-ist simultaneously in wireless systems, observations andanalysis suggest that composite or mixed distribution ismore desirable to realistically model the shadow-fadingconditions. The performance evaluation of energy detec-tors with composite fading channels and various shadowfading environments has been of focus in literature [9]–

[12] and the references therein. The studies are quanti-fied ED based spectrum sensing for versatile compounddistributions like Suzuki, Loo, Rayleigh-Lognormal, Rician-Lognormal, Nakagami-Lognormal, Generalized-K (KG) torepresent shadow-fading environment. Specifically, in [11]closed-form expression is derived for average detection prob-ability over (KG) fading. Numerous respective approacheslike user cooperation and relay based networks for ED basedsensing has been devoted which convey vital significance EDas a candidate for sensing within researchers. Recently, coop-erative spectrum sensing with relay based user collaboration isreported in [8] and the references provide in it. Amplify andforward (AF) and Decode and Forward (DF) protocols areconsidered and expressions are derived for different diversityreception schemes.

In this paper, we mainly focus our work to KG fadingchannel. Since in KG fading model, shadowing is modeledby Gamma distribution and Nakagami-m distribution is usedto represent the small-scale random variations of the receivedsignal envelope. We extended the work showed in [11] byanalyzing the impact of diversity reception. We obtain anew closed-form expression for spatial diversity in compositeKG fading channel. By using the same probability densityfunction (PDF) approach and alternative series representa-tion of Marcum-Q function we derive expressions for EDbased average detection probability considering MRC and SCcombining techniques. Our analytical results are confirmedthrough comparison with Monte Carlo simulations. Differentdiversity strategies are presented over KG distribution alongwith explicitly new derived expressions for energy detectionbased MRC and SC schemes. For multiple antenna branches atthe receiver, detector performances are numerically quantifiedfollowed by their simulation counterparts.

The rest of this paper is outlined as follows. Section II givesthe description of the system model comprised of ED sensingstatistics used in the analysis followed. Diversity receptionanalysis is carried out in Section III. The numerical andsimulation results are presented in Section IV, followed bythe Section V which concludes the whole paper.

II. SYSTEM MODEL

A simplified diagram of a energy detector is shown inFig. 1, the basic operative structure requires a filter, squaringdevice, an integrator and a comparator. Over a deterministicand stationary signal model with white Gaussian noise havingnoise spectral density N0, the primary user signal x(t) withchannel gain h(t) is applied to band pass filter (BPF) usingcarrier frequency fc and signal bandwidth W . To opt for thebandwidth of interest and reduce the noise at this point, thenoise has a band-limited, flat spectral density at the output ofthe filter. Subsequently, to evaluate the power of the receivedsignal, the output of BPF is squared and integrated overobservation time duration T to measure the energy of receivedsignal at the detector. The number of samples u for eachcomponent of received signal is to select as an integer i.e.,u=TW . The output signal Y from the integrator for AWGN

for specified T is given as

Y =1

N0

∫ t

t−T|x(τ)|2 dτ (1)

Lastly, Y is match up to a given specific threshold λ todetermine the presence of a primary signal H1 or H0. Thethreshold λ value is consistent with the statistical propertiesof the output Y when ambient noise is present appearing fromthe receiver itself or from environing RF interference.

Fig. 1. Block diagram of energy detector.

The exact closed-form equations for probabilities of detec-tion (Pd), false alarm (Pf ), and missed detection (Pm) overAWGN channel are given by [4]

Pd = P̃{Y > λ|H1} = Qu(√

2γ,√λ) (2)

where P̃ is the probability of an expectation and QM (., .)is generalized M th order Marcum-Q function defined in itsintegral form as [13]

QM (α, β) =

∫ β

α

tM

αM−1e−

t2+α2

2 IM−1(αt)dt (3)

where IM−1(.) is the modified Bessel function of (M − 1)th

order. The probability of false alarm is expressed as

Pf = P̃{Y > λ|H0}

=Γ

(u,λ2

)Γ(u) (4)

here Γ(., .) is an upper incomplete gamma function whichis defined as Γ(m,n) =

∫∞ntm−1e−t dt . Similarly, the

probability of missed detection can be evaluated as

Pm = 1− P̃{Y > λ|H1}= 1−Qu(

√2γ,√λ)

(5)

Threshold λ for detector is calculated for a specified Pfusing (4). Whereas conventional optimality principle, Neyman-Pearson criterion maximizes Pd for a given Pf and equivalentto the likelihood ratio test (LRT) of Y indicated as LR(Y ). Ingeneral, there is no LOS path present between the secondaryuser and the primary transmitter. Hence, the received primarysignal is a superposition of many non LOS signals and iswell approximated to Gaussian random variables accordingto central limit theorem. In our consideration, when both theprimary signal and noise are treated as Gaussian processes, en-ergy detector can meet any desired Pd and Pf simultaneously,hence the threshold λ is optimal.

Another essential distinctive series form of M th orderMarcum-Q is shown in(6) as given by [13]

QM (α, β) =∑∞n=0 exp

(−α

2

2

)(α2/2)n

n!∑n+M−1k=0 exp

(−β

2

2

)(β2/2)k

k! (6)

by equating the variables (6) with (2) we can get the modifiedform as

Qu(√

2γ,√λ) =

∞∑n=0

γne−γ

n!

n+u−1∑k=0

e−λ2

k!

(λ

2

)k(7)

Subsequently, after utilizing the series expression in (6), γwill have just exponentials and powers (possibly with specialfunctions), which can be analyzed from (7). Besides theprevention of Bessel function present in (6), this form alsopermit us to deal with Marcum-Q function averaging withPDF’s of γ, that may involve special functions.

III. DIVERSITY COMBINING

To recuperate the sensing information and achieve the diver-sity gain, signals at multiple antenna branches are combinedat the cost of intricacy of the receiver. Hence, the transmittedsignal is conveyed through various channels which are entailedto combine at the receiver. As far as these channels arebelieved independent, there is a very high probability that thepresence of the primary signal is sensed correctly.

However, our goal is to present the evaluation of P d,c at themaximal ratio and selective combiners over i.i.d. Generalized-K (KG) composite fading branches using Energy detection.For this, the PDF approach is applied in which PDF of SNRat the combiner output fγ,c(γ) is first determined. Later,averaging (2) over the obtained PDF furnishes the final result.

P d,c =

∫γ

Qu(√

2γ,√λi)fγ,c(γ)dγ (8)

A. Maximal Ratio Combining

Irrespective of the nature of fading, in an interferencefree environment, MRC is an optimal combining methodwhich involves perfect knowledge of amplitudes and phases ofchannel fading. In this generic technique, the stronger signal isstrengthen while the weaker one is abated. Firstly, the multiplesignal are co phased and weighted accordingly to the SNR ofindividual channel. Finally, all the branch signals are linearlycombined and the MRC output signal for L diversity branchesis given by

ymrc(t) =

L∑l=1

h∗l rl(t) (9)

where hl is the impulse response and rl(t) is the receivedsignal at the lth branch. Hence, the aggregated SNR persymbol is expressed as

γmrc =

L∑l=1

γl (10)

In KG composite fading, the small scale variations areNakagami-m distributed and the closed-form exact expressionfor the PDF of the sum of such L i.i.d. random variables isnot available. However, it can precisely be approximated byanother Nakagami-m distribution with fading parameter Lm[13, pp.340]. Therefore, the PDF of MRC output fγ,mrc(γ)for L i.i.d. KG faded diversity branches is expressed as

fγ,mrc,KG(γ) =

2Γ(Lm)Γ(m0)

(c02

)m0+Lm

γ(m0+Lm

2 )−1KLm−m0(c0√γ)

γ > 0,m > 0,m0 > 0, L ≥ 1 (11)

where c0

= 2√

mm0

γ0is scaling parameter related to γ and

Km0−m(.) is the modified Bessel function of order (m0−m).Thus, on substitution of (11) and (7) in (8), detection

probability Pd,mrc,KG is obtained in (12).

Pd,mrc,KG =

2 e

c208

c0Γ(Lm)Γ(m0)

(c0

2

)m0+Lm

×∑∞n=0

Γ(n+m0)Γ(n+Lm)n!

×W 1−2n−m0−Lm2 ,

Lm0−m2

(c20

4 )∑n+u−1j=0

e−λ2

j! (λ2 )j (12)

where W−µ,ν is defied as Whittaker function. Using se-ries representation of incomplete Gamma function from [14,eq. (8.352.4), pp.900], the second summation in (12) can bereplaced. Now, the expression (12) can be written as

Pd,mrc,KG =

2 e

c208

c0Γ(Lm)Γ(m0)

(c02

)m0+Lm

×∑∞n=0

Γ(n+m0)Γ(n+Lm)n!

×W 1−2n−m0−Lm2 ,

Lm0−m2

(c20

4 )× Γ(n+u,λ2 )

Γ(n+u) (13)

B. Selection Combining

In contrast to MRC in which all diversity branches arehandled, SC only treats the branch with the highest SNR.Hence, the fading amplitude, phase and delay of the selectedbranch is processed as

γsc = arg maxl∈(1,..,L)

γl (14)

According to [15], the PDF of the output SNR for L diversitybranches at SC is given by

fγ,sc(γ) = L[Fγ(γ)]L−1fγ(γ) (15)

In case of KG composite fading, the CDF of output SNR isachieved by integrating [11, eq (14)] which equals to

Fγ,KG(γ) =

Γ(m−m0)(γc2

04 )m0

γ(m)Γ(m0+1) 1F2(m0; 1−m+m0, 1 +m0;γc204 )

+Γ(m0−m)(

γc20

4 )m

γ(m0)Γ(m+1) 1F2(m; 1−m0 +m, 1 +m;γc204 ) (16)

Fig. 2. MRC combining for L diversity branches, m=1, m0= 1.64

Therefore,

fγ,sc,KG(γ) =

L[Fγ,KG(γ)]L−1 2Γ(m)Γ(m0)

(c02

)m0+m

γ(m0+m

2 )−1 Km0−m(c0√γ) (17)

Thus, on substitution of (17) in (8) and following the similarsteps, we get Pd,sc,KG

Pd,sc,KG =

2LΓ(m)Γ(m0)

(c02

)m0+m∑∞n=0

∫∞0

[Fγ,KG(γ)]L−1

n!

×γ2n+m0+m−2

2 e−γ Km0−m(c0√γ)

Γ(n+u,λ2 )

Γ(n+u) (18)

The above equation does not lead to a closed-form solutionand hence the results are found numerically and throughsimulations.

IV. RESULTS AND DISCUSSIONS

On the basis of closed form expression for Energy detectionbased MRC combiner over KG fading, the analytical resultsare obtained exploiting (13) over varying SNR. The resultsare presented for L = 2 , 3 diversity branches in Fig. 2.Performance melioration is evident when compared with singleterminal i.e. L = 1. Maximum diversity gain is achieved fromL = 1 to L = 2 which drops between the change over fromL = 2 to L = 3. In support, the corresponding simulationresults are also provided which overlap the analytical coun-terparts. While in Fig. 3 for no diversity,L = 1 and L = 2.Further, the performance is evaluated in comparison to MRCfor L diversity branches in Fig. 4. It is evident that MRC isan optimal technique and always ameliorates SC and providesthe diversity gain.

It has already been revealed that the underlying fadingand shadowing severely degrade the detection performance.

Fig. 3. SC combining for L diversity branches, selecting m=1, m0=1.64.

Fig. 4. MRC combining for L diversity branches, m=1, m0= 1.64

Although diversity combining techniques at the receiver helpto combat against such issues, but an energy detector is unableto attain the required performance limits particularly in lowSNR regions.

V. CONCLUSION

In this paper, the performance analysis of ED based spec-trum sensing with diversity reception techniques is evaluatedfor KG shadow-fading condition. First, the diversity com-bining using multiple antenna at the receiver is consideredand novel expressions for average detection probability at theoutput of MRC and SC for KG fading channel are obtained.Later, numerical results are demonstrated with Monte Carlosimulation support. In either case, MRC proves to renderbetter detection capability as compared to SC, which is moreapparent at larger value of L branches. Hence, improvedenergy detection results are arrived for KG distributed channel.

Theses spatial diversity framework can be effectively deployedto overwhelm the low SNR region problems as in otherconventional fading and shadowing scenarios.

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