outage performance of dual branch diversity techniques in broadband fixed wireless access networks

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Published in IET CommunicationsReceived on 27th June 2013Revised on 6th February 2014Accepted on 10th February 2014doi: 10.1049/iet-com.2013.1176

648The Institution of Engineering and Technology 2014

ISSN 1751-8628

Outage performance of dual branch diversitytechniques in broadband fixed wireless accessnetworksCharilaos I. Kourogiorgas, Athanasios D. Panagopoulos

Mobile Radio Communications Laboratory, School of Electrical and Computer Engineering, National Technical University

of Athens, Athens, Greece

E-mail: [email protected]

Abstract: New physical–mathematical models for the calculation of the outage probability of the maximal ratio combining andselection combining dual branch spatial diversity schemes for broadband fixed wireless access networks operating above 10 GHzare presented. At these frequency bands and considering line-of-sight conditions, rain attenuation is the dominant fadingmechanism, which should be taken into account in the radio communications system design. The models are based onbivariate inverse Gaussian (IG) distribution and on the adoption of a spatial correlation coefficient for the convergentterrestrial wireless links. IG distribution has been shown that models accurately the rain attenuation fading for both temperateand tropical climatic regions. The models are validated with numerical results and some useful conclusions are drawn.

1 Introduction

Owing to the increased demand of high data rates formultimedia services at user-ends and the congestion ofconventional frequency bands, below 10 GHz, higherfrequencies are promising solutions for the next generationwireless terrestrial systems. Already in the fifth generation(5G) urban cellular communications, the millimetre wavebands, that is, frequency bands above 20 GHz, areconsidered to be adopted [1]. In these high-frequencybands, atmospheric phenomena affect the signal and causedegradation of the quality of the link. Rain attenuation isthe dominant fading mechanism, in case of line-of-sight(LOS) conditions, and therefore must be taken into accountin system modelling [2]. A static fade margin cannot beapplied because of the high values of rain attenuation forsmall time percentages. This becomes more severe forheavier climatic regions. Therefore mitigation techniquesmust be taken into account and be employed into thesystem [2]. An effective mitigation technique in terms ofgain of availability is the route diversity. In this technique,which may also be called cell-site diversity, consideringcellular broadband fixed wireless access (BFWA) networks,the receiver is communicating with two separated basestations, thus receiving the same signal from two different,in spatial domain, sources. Therefore applying differentcombination schemes such as the selection combining (SC)or maximal ratio combining (MRC), the outage probabilitywill be decreased. The latter result holds because of thespatial inhomogeneity of the rainfall medium. Moreparticularly, rain rate and hence, the rain attenuation exhibitspatial variations and a significant diversity gain can be

achieved in case that separated in spatial domain wirelessroutes exist. The performance of the models for theevaluation of the diversity techniques strongly depends onthe choice of the corresponding fading channel model. Thereader can observe this from the related literature [3–6]. Inthis paper because of the fact that we focus for radiosystems operating at frequencies above 10 GHz, where thetroposphere plays important role, for the accurate andreliable design of them, we have to investigate theirperformance by exploiting the rainfall inhomogeneity andevaluate the joint statistics of rain attenuation. For thisparticular reason, a physical–mathematical model in [6]has been very recently adopted by InternationalTelecommunication Union in ITU-R.P.1410-5 [7]. It isbased on the fundamental assumption that rain attenuationfollows the lognormal distribution and by the employmentof a spatial correlation coefficient the joint complementarycumulative distribution function (CCDF) of rain attenuationinduced on the two terrestrial links can be calculated. Veryrecently, in [8], it was firstly presented that the inverseGaussian (IG) distribution [9] can be used for the modellingof the first order statistics, that is, probability densityfunction (PDF) and CCDF, of wireless terrestrial links rainattenuation. Moreover, in [10], a physical–mathematicalmethod for the prediction of rain attenuation exceedanceprobability in terrestrial links given that rain rate and rainattenuation follow the IG distribution has been presented.Also, in [11], it is remarked that the IG distribution has thebest overall performance for modelling the exceedanceprobability of rain attenuation in terrestrial links for tropicalareas, using experimental data collected in Brazil.Considering joint statistics of rain attenuation, it was shown

IET Commun., 2014, Vol. 8, Iss. 9, pp. 1648–1653doi: 10.1049/iet-com.2013.1176

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in [12] that bivariate IG distribution may have also very goodresults for the modelling of joint CCDF of rain attenuation inthe Earth–Space systems.In this short contribution, the bivariate IG distribution is
used for the accurate evaluation of the performance of dualbranch terrestrial diversity schemes that employ SC andMRC diversity techniques in the reception. The use ofbivariate IG distribution is motivated by the observationsand the results of Kourogiorgas et al. [8, 10], that is, thesuitability of the IG distribution for the modelling offirst-order statistics of rain attenuation in both temperate andtropical climatic regions. New expressions are derived forthe calculation of the outage performance of dual branchcell-site diversity systems employing MRC and SCdiversity techniques, based on a model which strives tomodel the long-term joint statistics of rain attenuation onconvergent links. In Section 2, the analytical expressionsfor the evaluation of the system’s performance for SC andMRC techniques are given. In Section 3, numerical resultsare presented for validation and the system’s performance isevaluated. Finally, in Section 4 some useful conclusions aredrawn.

2 Outage performance prediction models

2.1 Outage performance prediction modelling

In this section, analytical expressions for the prediction ofoutage probability for SC and MRC diversity schemes arepresented. First of all, the geometry of the site diversitysystem in fixed wireless access networks is described. Insuch system, a receiver (R) with two highly directivecollocated antennas, is communicating with two basestations (BS1 and BS2), as this is shown in Fig. 1. L1 andL2 are the path lengths of the LOS links between R–BS1and R–BS2, respectively. The two links are converging andtheir separation angle is considered θ.In the SC scheme, the link with the highest SNR is selected

from the receiver in order to communicate, whereas in theMRC scheme, the receiver combines the two signalsreceived at the two antennas of the receiver, in order thatthe signal to noise ratio (SNR) of the received signal is thesum of the SNR of the two signals. The latter summationholds considering the SNR in linear terms. Another

Fig. 1 Geometry of a site diversity system


difference between the MRC and SC diversity techniques isthat for the former technique (MRC), the sum of the SNRof the two signals received at the antennas on the receiverside takes place after co-phasing these two signals at thereceiver side, whereas in the latter one, there is no need forco-phase of the two signals. The received SNRr in linearterms, that is, Watts, can be calculated, as

SNRr = max SNR1, SNR2

( ), SC

SNR1 + SNR2, MRC

{(1)

with SNR1 is the SNR of the signal transmitted from BS1 andreceived by R and SNR2 is the SNR of the signal transmittedfrom BS2 and received in R, both in linear terms. For bothSNR1 and SNR2 holds that

SNRi = 10 SNRcs,i−Ai( )/10, i = 1, 2( ) (2)

where SNRcs,i is the SNR in dB of the received signal in pathLi under clear sky conditions and Ai is the rain attenuation indB induced in link with path length Li. The SNR under clearsky conditions can be computed as

SNRcs,i = Pi + Gi,tr + Gi,rec − PLi − Ni, i = 1, 2( ) (3)

where Pi (dBW) is the transmitted power from transmitter Si,Gi,tr is the antenna gain of transmitter BSi and Gi,rec is theantenna gain of the receiver R in link Li, all in dBi. PLi isthe path loss in dB of link Li and is equal to PLi = 20 log(4πLi/λ), with λ the wavelength and Ni in dB is equal to Ni

= 10 log(kTiB)The constant k is k = 1.38 × 10−23(J/K ), Tithe antenna temperature of the receiver in link Li and B isthe bandwidth of the system.An outage occurs when thereceived SNR is below a certain threshold. Consequently,the outage probability of both SC and MRC systems isgiven by

Pout = P SNRr ≤ SNRth

[ ](4)

For the case of the SC scheme, from (1) the outage probabilityis

Pout,SC = P max SNR1, SNR2

( ) ≤ SNRth

[ ]= P SNR1 ≤ SNRth, SNR2 ≤ SNRth

[ ](5)

Using (5), the outage probability for a SC system afterstraightforward algebra becomes

Pout,SC = P A1 ≥ a1,th = SNRcs,1 − SNRth,dB, A2 ≥ a2,th[

= SNRcs,2 − SNRth,dB

](6)

where SNRth,dB is the SNRth in dB values, that is, 10 log(SNRth). For the MRC scheme the outage probability is given

Pout,MRC = P SNR1 + SNR2 ≤ SNRth

[ ]= P SNR1 ≤ SNRth, SNR2 ≤ SNRth − SNR1

[ ](7)

1649& The Institution of Engineering and Technology 2014

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Using the expression in (2), the outage probability for theMRC scheme is (see (8))
2.2 Outage performance prediction final formulae

For the calculation of the of the above outage probabilities,we employ the fundamental assumption that the raininduced attenuations on the terrestrial links A1, A2 bothfollowing the IG distribution with parameters λ1, μ1 and λ2,μ2, respectively, their joint PDF is given by [13]

fA1,A2 A1, A2

( ) = fA1 (A1) · fA2 (A2) · 1+ rd ·C(A1, A2)[ ]

(9)

where ρd is the correlation coefficient of the random variablesand fA1 (A1), fA2 (A2) are the PDF of the univariate IGdistribution [9] and

C A1, A2

( ) = 8��l1l2/m

31m

32

√· A1 − m1

( ) · A2 − m2

( )·exp − l1 A1 − m1

( )22m2

1A1

( )+ l2 A2 − m2

( )22m2

2A2

( )[ ](10)

The following spatial correlation coefficient is employed

rd =Dr��

D2r + d2

√ d =��L21 + L22 − 2L1L2 cos (u)

√( )(11)

with d the separation distance between the two transmittersand the coefficient Dr, also in km, is derived from [7] theequation

Dr = 0.644 ln Latitude| |( ) − 1.02, 50 ≤ Latitude| | ≤ 900

(12)

The final expression (6) for the outage probability of a SCdiversity system is given

Pout,SC = P A1 ≥ a1,th, A2 ≥ a2,th[ ] = 1− P A1 ≤ a1,th

[ ]− P A2 ≤ a2,th

[ ]+ P A1 ≤ a1,th, A2 ≤ a2,th[ ]

(13)

where the CDF of single links are given

P Ai ≤ ai,th[ ] = Q

��liai,th

√1− ai,th

mi

( )( )

+ e2 li/mi( )Q��liai,th

√1+ ai.th

mi

( )( )i = 1, 2( ) (14)

where Q is the Gaussian Q-function and the joint CDF iscalculated trough the following expressions [13]

P[A1 ≤ a1,th, A2 ≤ a2,th]

= P[A1 ≤ a1,th] · P[A2 ≤ a2,th]+ rd · H a1,th, a2,th( )

(15)

Pout,MRC = PA1 ≥ SNRcs,1 − SNRth,dB,

A2 ≥ SNRcs,2 − 10 log 10SNRth,dB/10 − 10 SNRcs,1(([


with

H a1,th, a2,th( ) = ∫a1,th

0

∫a2,th0

C(A1, A2)fA1 A1

( )fA2 A2

( )dA2dA1

= 16

��l1l2m1m2

√exp 4

l1m1

+ l2m2

( )[ ]Q C1

( )Q C2

( )(16)

Ci = 24limi

+ z2i

( )[ ]1/2

zi =��li

√(ai,th − mi)

mi��ai,th

√

⎫⎪⎪⎪⎬⎪⎪⎪⎭

i = 1, 2( ) (17)

For the MRC case (see (8)), the joint CCDF is more complex,considering that the threshold value of the one of two randomvariables depends on the value of the other, The computationof this joint CCDF using the bivariate IG distribution is notstraightforward and an analysis must be done. The novelmathematical framework is presented here.From (8), we have

Pout,MRC = P A1 ≥ a1,th, A2 ≥ g A1

( )[ ]

=∫+1

a1,th

∫+1

g A1( )fA1,A2 (A1, A2)dA2dA1 =

=∫+1

A1

∫+1

g A1( )fA1 (A1) · fA2 (A2)dA2dA1

+ rd

∫+1

A1

∫+1

g A1( )fA1 (A1) · fA2 (A2)C(A1, A2)dA2dA1

= I1 + I2 (18)

The calculation of I1 is straightforward

I1 =∫+1

a1,th

fA1 (A1) · P A2 ≥ g A1

( )[ ]dA1 (19)

where

P A2 ≥ g A1

( )[ ] = 1− P A2 ≤ g A1

( )[ ](20)

and P[A2≤ g(A1)] is given by (14).For the computation of I2 we denote the functionΨi(.), with

Ci(x) =��8

√��lim3i

√x− mi

( )exp − li x− mi

( )22m2

i x

( )[ ](21)

−A1)/10)]= P A1 ≥ SNRcs,1 − SNRth,dB, A2 ≥ g A1

( )[ ](8)


Fig. 2 Outage probability for MRC and SC diversity systems with60° and 90° separation angle and the corresponding single link, inAthens, Greece

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such as Ψ(A1, A2) =Ψ1(A1)·Ψ2(A2). Now it holds for I2 that
I2 = rd

∫+1

a1,th

fA1 (A1)C1 A1

( ) ∫+1

g A1( )fA2 (A2)C2 A2

( )dA2dA1

(22)

The inner integral of (22) is

∫+1

g A1( )fA2 (A2)C2 A2

( )dA2

=��8

√��l2m2

√1��2p

√∫+1

g A1( )1

x

��l2

√(x− m2)

m2

��x

√ e− l2(x−m2)2

( )/m2

2xdx

(23)

Making the substitution y = ��l2

√(x− m2)/m2

��x

√, (23)

becomes

I3 =��8

√��l2m2

√2��2p

√∫+1

yth

y��4(l2/m2)+ y2

√ e−y2dy (24)

with yth =��l2

√(g A1

( )− m2)/m2

��g A1

( )√. Finally, making a

last substitution w =��2 4 l2/m2

( )+ y2( )√

, I3, (A-15),

becomes

I3 = 4

��l2m2

√e4(l2/m2)Q(C2) (25)

with Q, the Gaussian Q-function and

C2 = 24l2m2

+ y22

( )[ ]1/2

y2 =��l2

√g A1

( )− m2

( )m2

��z2

√(26)

Consequently, from all the above (8) becomes

Pout,MRC = P A1 ≥ A1, A2 ≥ g A1

( )[ ]=

∫+1

a1,th

fA1 (A1) · P A2 ≥ g A1

( )[ ]dA1

+ 4rd

��l2m2

√e4(l2/m2)

∫+1

a1,th

fA1 (A1)C1 A1

( )Q(C2)dA1

(27)

The numerical calculation of the final expressions isstraightforward and the corresponding integral convergesquickly. Here, it must be noted that the complexity of thecalculation of MRC and SC techniques for dual diversitysystems based on the IG distribution or lognormaldistribution, that is, the model presented inITU-R. P. 1410-5 [7] is similar. However, as stated in theIntroductory section, the presentation and use of theseexpressions are motivated by the accurate modelling of rainattenuation long-term first-order statistics with the IGdistribution. Moreover, (15) and (27) refer only to the dualbranch site diversity systems and therefore cannot be used


for higher order diversity techniques. The whole analysisfor the calculation of the performance of dual diversitysystems can be extended in diversity systems with morebranches, considering the assumptions that the spatialcorrelation is considered only between the two links. Theconcept of n-dimensional IG distribution with correlatedlinks is considered as a future work and is beyond thescope of this contribution.

3 Numerical results and discussion

In this section, some brief numerical results are presentedregarding the outage probability of diversity systems inorder to validate the proposed models. For the computationof clear sky SNR we consider a transmitted power ofPi = −10 dBW and transmitter antenna gains of 19 dBi foreach transmitter. The antenna gains at the two receiverantennas are 34 dBi and the antenna temperatures areconsidered 300 °K. The bandwidth is set equal to 28 MHz,typical values from IEEE 802.16 Standard [2]. To proceedto a fair comparison of the performance of the diversitysystems with the single link, we consider that thetransmitted power at the case of single link is −7 dBW. Therain attenuation prediction is derived fromITU-R. P. 530-14 [14].Firstly, we consider the diversity scheme in Athens,

Greece. In Fig. 2, the outage probability for the SC andMRC systems are shown, as well as the outage probabilityfor the single link. The frequency of the link is considered25 GHz and the path lengths (L1, L2) are 4 km. Twoconfigurations of cell-site diversity were considered, inwhich the separation angle between the two links (θ) was60° and 90°.The same curves as shown in Fig. 2 are also shown in

Fig. 3; however, considering that the stations are located atSeoul with operating frequency of 27 GHz and path lengthsequal to 3 km. Again, the two configurations are consideredwith separation angles of 120° and 80°. It can be observedfrom Figs. 2 and 3 that the performance of MRC systems isbetter than this of SC and single link for all the SNRthresholds. However, it is noted again, as in Section 2.1,


Fig. 3 Outage probability for MRC and SC diversity systems with80° and 120° separation angle and the corresponding single link inSeoul, South Korea

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that MRC technique requires to co-phase the signals,whereas SC does not. Furthermore, SC has a higherperformance than single link for low probability levels. Thelatter is because of the fact that we proceed to a faircomparison, that is, the total transmitted power is equal forthe case of single links and the diversity systems.Moreover, the minimum SNR thresholds above which thesystem is in a permanent outage are the SNRcs values forthe SC technique (SNRcs,SC) and the case of single link(SNRcs,SL) which in linear terms the former is half of thisof SC technique. This occurs because of the fact that if theSNR threshold required for the realisation of thecommunications is larger than the obtained SNR in case ofclear sky conditions and considering that the losses in clearsky conditions are present for 100% of time, the systemwill be permanently in outage. Now, considering thethreshold for MRC case, this is twice of the SNRcs,SC inlinear terms because of the sum of the SNRs of the signalsreceived by the two different radiopaths, as given in (4).For higher order diversity techniques it is expected that theperformance will be further improved for both techniques.The relative diversity gain will decrease with the increase ofthe number of the branches. However, the costs will beincreased since more stations will be needed for therealisation of the multi-branch diversity, the receiver will bemore complex and the channel state information at thereceiver side required will be also increased. Moreover, itcan be observed that increasing the separation angle, theoutage probability is decreasing, since rain attenuation onthe two links is less and less correlated.Moreover, we may consider the diversity gain for the

evaluation of the improvement of the performance of adiversity system. The diversity gain for the MRC system isgreater comparing with the SC system for all the paths andthat for higher path lengths, the diversity gain of the systemis increasing. Moreover, from extended simulation thediversity gain for a given outage probability level of acell-site diversity system in a heavy rain climatic region isgreater comparing with a similar system locating in atemperate region, nevertheless the difference is very small.The difference is small because of the definition of thediversity gain that is defined in terms of single link rainattenuation (outage probability level), hence it absorbs the


differences. This is an important conclusion that can beused by the radio communication system designers.

4 Conclusions

In this paper, the IG distribution is used for modelling rainattenuation on terrestrial links operating above 10 GHz andtwo new models for the calculation of the outage performanceof MRC and SC diversity systems are presented. Moreover,their performance is examined for mid-latitude and heavy rainareas. It is found that the MRC has better performance thanthe SC system and that increasing the path length of the linkor the separation angle between the diversity terrestrial links,the availability time is also increasing. It is also remarked thatthe diversity gain for heavy rain climatic areas is higher thanthe gain in mid-latitude areas. However, the difference it issmall. Numerical evaluations of the system’s performance arepresented, which can then be used from system engineer. Afuture work to be addressed regarding BFWA networks anddiversity is the evaluation of the performance of N-branchesdiversity techniques.

5 Acknowledgment

This work has been co-financed by the European Union(European Social Fund – ESF) and Greek national fundsthrough the Operational Program ‘Education and LifelongLearning’ of the National Strategic Reference Framework(NSRF) – Research Funding Program: THALES:Reinforcement of the interdisciplinary and/orinter-institutional research and innovation and the BilateralCooperation Research Project between Greece and Czech.

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outage performance of dual branch diversity techniques in broadband fixed wireless access networks

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