On the Weibull Distribution with ArbitraryCorrelation

Zafiro G. Papadimitriou∗‡, P. Takis Mathiopoulos∗, Nikos C. Sagias†, Lazaros Merakos‡∗Institute for Space Applications and Remote Sensing, National Observatory of Athens, Metaxa & Vas. Pavlou Street,

15236 Athens, Greece, E-mail: {zpapadim;mathio}@space.noa.gr† Institute of Informatics and Telecommunications, National Centre for Scientific Research Demokritos, Agia Paraskevi,GR-15310 Athens, Greece, E-mail: nsagias@ieee.org‡ Department of Informatics and Telecommunications, University of Athens,

GR-15784 Athens, Greece, E-mail: merakos@di.uoa.gr

Abstract—This paper studies the performance of the quadri-variate Weibull distribution with arbitrary correlation assumingnon-identical fading parameters and average powers. Novelinfinite series representations for the joint probability densityfunction (PDF), the cumulative distribution function (CDF) andthe product moments are derived. Using these theoretical results,a performance analysis study in terms of outage probability, forreceivers employing selection combining (SC) diversity techniquesis presented. The proposed mathematical analysis is accompa-nied by various numerical results, with parameters of interestthe fading severity and the power decay factor. Furthermore,equivalent performance evaluation results, obtained by variouscomputer simulations have verified the accuracy of the proposedanalytical method.

Index Terms—Arbitrary correlation, selection combining (SC),diversity, outage probability, Weibull fading.

I. INTRODUCTION

One of the most efficient and widely used techniques toreduce the destructive effects of fading on the system’s per-formance is diversity reception. When space and polarizationdiversity is applied in practical systems (e.g. mobile terminal,indoor base-station etc.), the antennas placed at the receiverare not sufficiently separated and consequently the combinedsignals are correlated with each other. In order to analyze andmodel such realistic wireless channels with correlated fading,multivariate statistics are commonly used [1].

In the open technical literature there exist many papers thatconcern multivariate statistics for fading channel modeling.Most of them deal with the constant and exponential correla-tion model. For the first one, the correlation depends on thedistance among the combining antennas and as a consequenceit is more suitable for equidistant antennas [1]. For theexponential correlation model, widely used for performanceanalysis of space diversity techniques [2]–[4] or multiple-inputmultiple-output (MIMO) systems [5], the correlation betweenthe pairs of combined signals decays as the spacing betweenthe antennas increases [1]. In [6], the authors presented usefulclosed-form expressions for the multivariate, exponentiallycorrelated Nakagami-m probability density function (PDF)and the cumulative distribution function (CDF).

A more general approach would be to consider that thecorrelation between the pairs of combined signal can takea more general form. Thus, an arbitrary correlation model

will include the two previously mentioned models as specialcases. Past works concerning arbitrarily correlated multivariatedistributions can be found in [7]–[10]. In [7] infinite seriesrepresentations for the joint PDF and the CDF of three andfour arbitrarily correlated Rayleigh random variables werepresented. In [8] useful closed-form expressions for the jointNakagami-m multivariate PDF and CDF with arbitrary cor-relation were derived and the correlation matrix was approx-imated by a Green’s matrix so that the inverse matrix wouldbe tridiagonal. Similarly, in [9], the Green’s matrix was usedto approximate the correlation matrix of L - branch selectioncombining (SC) receivers and the outage probability for log-normal fading channels has been obtained. In [10] expressionsfor multivariate Rayleigh and exponential PDFs generatedfrom correlated Gaussian random variables were presented,while a general expression for the multivariate exponentialcharacteristic function (CF), in terms of determinants, was alsoderived.

The Weibull distribution although it was originally usedin reliability and failure data analysis, has recently receivedrenewed interest because it exhibits a very good fit to ex-perimental fading channel measurements for both indoor andoutdoor terrestrial radio propagation environments [11], [12].Moreover, in [13], it was argued that the Weibull distributioncould also been considered as a more generic channel modelfor land-mobile satellite systems.

For this distribution, in a recent paper [14], the joint PDF,CDF and the moment-generating function (MGF) for thebivariate Weibull distribution have been analytically presented.The multivariate Weibull distribution has also been studied forthe exponential and the constant correlation case with equalaverage fading powers. Finally in [15] the trivariate Weibulldistribution with arbitrary correlation is studied, the joint PDF,CDF and the product moments are obtained and the outageprobability of SC receivers is derived. It should be mentioned,however, that to the best of our knowledge, analyzing theperformance of an arbitrary Weibull fading channel with fourdiversity branches has not been presented in the open technicalliterature.

Motivated by this observation, in this paper we present novelinfinite series representations for the joint PDF, CDF and themoments for the quadrivariate Weibull distribution, assuming

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an arbitrary covariance matrix and non-identical fading param-eters and average powers. Furthermore, the outage probabilityof SC receivers is derived and various performance evaluationresults are presented.

The organization of this paper is as follows: After this in-troduction, in Section II the system model is presented and thequadrivariate Weibull distribution is analyzed. Additionally, inSection III the SNR statistics are derived while in Section IVthe performance of SC receivers in terms of outage probabilityis studied. Various numerical results are presented in SectionV showing the impact of correlation, the fading severity andthe power decay factor on the system’s performance. Finally,in Section VI several useful concluding remarks are given.

II. SYSTEM MODEL

Let G = {G1, G2, ..., GL} be the joint complex Gaus-sian random variables (RVs) with zero mean E 〈G�〉 = 0,� = 1, 2, ..., L and positive definite covariance matrix Ψ, withelements ψiκ = E 〈GiG

∗κ〉, where E 〈·〉 denotes expectation.

Let R = {R1, R2, ..., RL} be a set of Rayleigh RVs withR� = |G�| where | · | denotes absolute value. The Weibullfading channel h� can be expressed in terms of the Gaussianin-phase X� and quadrature Y� elements [14] as follows:

h� = (X� + jY�)2/β� = G

2/β�

�(1)

where β� > 0 represents the Weibull fading parameter and jthe imaginary operator. Since Z� = |h�|, Z� can be expressedas a power transformation of a Rayleigh distributed RV R� =|X� + jY�| as [16]

Z� = R2/β�

� . (2)

Quadrivariate Weibull distribution

For the quadrivariate Weibull distribution (L = 4) weconsider the inverse covariance matrix, Φ, given by

Φ = Ψ−1 =

⎡⎢⎢⎣φ11, φ12, φ13, 0φ∗12, φ22, φ23, φ24

φ∗13, φ∗23, φ33, φ34

0, φ∗24, φ∗34, φ44

⎤⎥⎥⎦ (3)

with φiκ = |φiκ| exp(jχiκ) where i, κ ∈ {1, 2, 3, 4}.Note here for the quadrivariate case, the obtained results

are more general than those presented by Blumenson andMiller in [17] for the multivariate Rayleigh distribution. Morespecifically, the statistical properties derived in [17] hold onlyunder the assumption that Ψ is tridiagonal so that φiκ = 0 for|i− κ| > 1.

The joint PDF of Z = {Z1, Z2, Z3, Z4} can be derivedusing [7, eq. (16)] and applying the power transformationdescribed in (2) as a product of the modified Bessel function

of the first kind as follows

fZ(z1, z2, z3, z4) = β1β2β3β4 det(Φ)

× exp[−

(zβ11 φ11 + zβ2

2 φ22 + zβ33 φ33 + zβ4

4 φ44

)]

× zβ1−11 zβ2−1

2 zβ3−13 zβ4−1

4

∞∑j=0

∞∑k=−∞

εj(−1)j+k cos(A)

× Ij(2zβ1/21 z

β2/22 |φ12|)Ij(2zβ1/2

1 zβ3/23 |φ13|)

× Ik(2zβ2/22 z

β4/24 |φ24|)Ik(2zβ3/2

3 zβ4/24 |φ34|)

× Ij+k(2zβ2/22 z

β3/23 |φ23|).

(4)

Using the infinite series expansion of the Bessel function[18, eq. (8.447/1)]

Iν(u) =∞∑

k=0

1(k + ν)!k!

(u

2)2k+ν (5)

and after some cumbersome but straightforward mathematicalsteps, (4) can be expressed as

fZ(z1, z2, z3, z4) = β1β2β3β4 det(Φ)

× exp[−

(zβ11 φ11 + zβ2

2 φ22 + zβ33 φ33 + zβ4

4 φ44

)]

× zβ1−11 zβ2−1

2 zβ3−13 zβ4−1

4

∞∑j=0

∞∑k=−∞

εj(−1)j+k cos(A)

×∞∑

�,m,n,p,q=0

C |φ12|2�+j |φ13|2m+j |φ24|2n+|k|

× |φ34|2p+|k| |φ23|2q+|j+k|z

β1(�+n+j)1

× zβ2(�+n+q+ j+|k|+|j+k|

2 )2 z

β3(m+p+q+ j+|k|+|j+k|2 )

3

× zβ4(n+p+ |k|

2 )4

(6)

where εj is the Neumann factor (ε0 = 1, εj = 2 for k =1, 2, · · · ), A = j(χ12 +χ23 +χ31) + k(χ23 +χ34 +χ42) and

C =1

�!(�+ j)!m!(m+ j)!n!(n+ |k|)!p!(p+ |k|)!q!(q + |k+j|2 )!

.

The corresponding CDF is obtained by integrating (6), as

ISCCSP 2008, Malta, 12-14 March 2008 589

follows

FZ(z1, z2, z3, z4) = det(Φ)∞∑j=0

∞∑k=−∞

εj(−1)j+kcos(A)

×∞∑

�,m,n,p=0

|φ12|2�+j |φ13|2m+j |φ24|2n+|k| |φ34|2p+|k|

�!(�+ j)!m!(m+ j)!n!(n+ |k|)!p!(p+ |k|)!

×γ

(�+m+ j + 1, zβ1

1 φ11

)

φ�+m+j+111

×γ

(n+ p+ |k| + 1, zβ4

4 φ44

)

φn+p+|k|+144

∞∑q=0

|φ23|2q+|j+k|

q!(q + |k+j|2 )!

×γ

(�+ n+ q + j+|k|+|j+k|

2 + 1, zβ22 φ22

)

φ�+n+q+

j+|k|+|j+k|2 +1

22

×γ

(m+ p+ q + j+|k|+|j+k|

2 + 1, zβ33 φ33

)

φm+p+q+ j+|k|+|j+k|

2 +133

(7)

with γ(·, ·) denoting the incomplete lower Gamma function[18, eq. (3.381/1)].

Moreover, the joint moments are defined as

E〈zα1 , z

η2 , z

ϑ3 , z

ζ4〉 Δ=

∫ ∞

0

∫ ∞

0

∫ ∞

0

∫ ∞

0

zα1 z

η2z

ϑ3 z

ζ4

× fZ(z1, z2, z3, z4)dz1dz2, dz3dz4(8)

and hence substituting (6) in (8) can be derived as follows

E〈zα1 , z

η2 , z

ϑ3 , z

ζ4〉 = det(Φ) ×

∞∑j=0

∞∑k=−∞

εj(−1)j+k cos(A)

×∞∑

�,m,n,p=0

C |φ12|2�+j |φ13|2m+j |φ24|2n+|k| |φ34|2p+|k|

× Γ (�+m+ j + α/β1 + 1)

φ(�+m+j+1)+α/β111

Γ (n+ p+ |k| + ζ/β4 + 1)

φ(n+p+|k|+1)+ζ/β444

×∞∑

q=0

|φ23|2q+|j+k| 1

φ(�+n+q+ j+|k|+|j+k|

2 + ηβ2

+1)

22

× 1

φ(m+p+q+

j+|k|+|j+k|2 + ϑ

β3+1)

33

× Γ(�+ n+ q +

j + |k| + |j + k|2

+η

β2+ 1

)

× Γ(m+ p+ q +

j + |k| + |j + k|2

+ϑ

β3+ 1

).

(9)

III. SNR STATISTICS

In order to obtain and study performance measures fordiversity receivers, the statistics of SNR are needed. Assuminga diversity receiver with L branches, the received at the �th

Fig. 1. Outage probability of a 4-branch SC receiver as a function of thenormalized outage threshold, γth/γ for different values of β.

branch baseband signal can be mathematically expressed asfollows

ζ� = wh� + n� (10)

where w is the complex transmitted symbol, with Es =E〈|w|2〉 being the transmitted average symbol’s energy andn� is the additive white Gaussian noise (AWGN) with single-sided power spectral densityN0. The instantaneous per symbolSNR of the �th diversity channel can be expressed as

γ� = Z2�Es/N0. (11)

Moreover, since the corresponding average SNR is expressedas

γ� = E〈Z2� 〉Es

N0= Γ(d2,�)Ω

2/β�

�

Es

N0(12)

where dτ,� = 1 + τ/β� with τ taking non-negative values,expressions for the statistics of γ� can be easily derivedby replacing β� with β�/2 and Ω� with (α�γ�)β�/2, in thecorresponding expressions for the fading envelope Z�.

Thus using (7), the CDF of the SNR for the quadri-variate Weibull distribution with arbitrary correlation,

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Fγ (γ1, γ2, γ3, γ4), can be obtained as

Fγ(γ1, γ2, γ3, γ4) = det(Φ′)∞∑j=0

∞∑k=−∞

εj(−1)j+kcos(A)

×∞∑

�,m,n,p=0

|φ′12|2�+j |φ′13|2m+j |φ′24|2n+|k| |φ′34|2p+|k|

�!(�+ j)!m!(m+ j)!n!(n+ |k|)!p!(p+ |k|)!

×γ

(�+m+ j + 1, γβ1/2

1 φ′11)

φ′�+m+j+111

×γ

(n+ p+ |k| + 1, γβ4/2

4 φ′44)

φ′44n+p+|k|+1

∞∑q=0

|φ′23|2q+|j+k|

q!(q + |k+j|2 )!

×γ

(�+ n+ q + j+|k|+|j+k|

2 + 1, γβ2/22 φ′22

)

φ′22�+n+q+ j+|k|+|j+k|

2 +1

×γ

(m+ p+ q + j+|k|+|j+k|

2 + 1, γβ3/23 φ′33

)

φ′33m+p+q+

j+|k|+|j+k|2 +1

(13)

where Φ′ with elements φ′iκ is the inverse covariance matrixobtained from (3) after the substitutions mentioned in (11) and(12).

IV. OUTAGE PROBABILITY OF SC RECEIVERS

Consider a SC receiver with four branches operating overan arbitrarily correlated Weibull fading environment and in thepresence of an AWGN channel. The instantaneous per symbolSNR at the output of a receiver of this type, will be the onewith the highest instantaneous value between the branches [19]

γsc = max{γ1, γ2, γ3, γ4}. (14)

The outage probability, Pout, is defined as the probabilitythat the instantaneous error probability exceeds a specifiedvalue or equivalently the probability that the receiver out-put SNR, γsc, falls below a certain specified threshold, γth

[1]. As a consequence and since the CDF of γsc equalsFγ (γsc, γsc, ..., γsc), Pout can be derived as follows

Pout(γth) = Fγsc(γth). (15)

Thus, with the aid of (13), Pout can be directly obtained.

V. NUMERICAL RESULTS

In this section, we use the previous mathematical analysisin order to present numerical results for the performance ofSC receivers, where without loss of generality, it is assumedthat β� = β ∀ �.

Consider an antenna array with normalized covariance ma-trix

Ψ =

⎡⎢⎢⎣

1, 0.4975, 0.2998, 0.11210.4975, 1, 0.1912, 0.15850.2998, 0.1912, 1, 0.18680.1121, 0.1585, 0.7868, 1

⎤⎥⎥⎦ (16)

Fig. 2. Outage probability of a 4-branch SC receiver as a function of the firstbranch normalized outage threshold, γth/γ1 for different values of ρ and δ.

The inverse covariance matrix now satisfies (3) and thus, using(15) the outage probability can be evaluated as shown in Fig.1. More specifically, assuming γ1 = γ2 = γ3 = γ4 = γ , theoutage probability is plotted as a function of the normalizedoutage threshold, γth/γ , for different values of β. It is evidentthat the system’s performance degrades with a decrease of βand/or an increase on γth/γ.

Moreover, Fig. 2 and Fig. 3 show the impact of an expo-nentially decaying power factor δ on the outage probabilityof SC receiver with exponentially correlated branches, as-suming non-identical distributed Weibull channels, i.e., γ� =γ1 exp[−(�− 1)δ] . In Fig. 2 the outage probability is plottedas a function of the first branch normalized outage thresholdγth/γ1 assuming β = 2.5 and for different values of δ and ρwhile in Fig. 3 the outage probability is plotted for ρ = 0.3 andfor different values of δ and β. In order to verify the proposedmathematical analysis some computer simulation results arealso included for comparison purposes. The obtained resultsshow clearly that the systems performance degrades with anincrease of δ and ρ.

In case the normalized covariance matrix is given as [8]

Ψ =

⎡⎢⎢⎣

1, 0.7, 0.538, 0.37770.7, 1, 0.769, 0.5380.538, 0.769, 1, 0.70.3777, 0.538, 0.7, 1

⎤⎥⎥⎦ (17)

following a similar procedure, the outage probability can bederived as shown in Fig. 4, for different values of β. Onceagain the obtained results show that the system’s performanceimproves with an increase of β. The theoretical results areagain in excellent agreement with the computer simulation

ISCCSP 2008, Malta, 12-14 March 2008 591

Fig. 3. Outage probability of a 4-branch SC receiver as a function of thefirst branch normalized outage threshold, γth/γ1 for different values of βand δ.

results.

VI. CONCLUSIONS

A performance analysis study for the quadrivariate Weibulldistribution with arbitrary correlation was presented. Initially,infinite series representations for the joint PDF, CDF andthe product moments were derived. These theoretical resultswere applied to analyze the performance of SC receivers withfour branches, operating in an arbitrarily correlated fadingenvironment. The outage probability, one of the most usefulperformance criteria, was studied. Various performance evalu-ation results were presented showing the effects of correlation,fading severity and the power decay factor on the system’sperformance.

ACKNOWLEDGMENTS

This paper is part of the 03ED910 research project, im-plemented within the framework of the ”Reinforcement Pro-gramme of Human Research Manpower” (PENED) and co-financed by National and Community Funds (25% from theGreek Ministry of Development-General Secretariat of Re-search and Technology and 75% from E.U.- European SocialFund).

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