Performance analysis of system with selection combining over correlated Rician fading channels in the presence of cochannel interference

Int. J. Electron. Commun. (AEÜ) 63 (2009) 1061 – 1066 www.elsevier.de/aeue

LETTER

Performance analysis of system with selection combining over correlated Rician fading channels in the presence of cochannel interference ˇ Stefanovi´c, Dragan Lj. Draˇca Aleksandra S. Panajotovi´c∗ , Mihajlo C. Department of Telecommunications, University of Niš, Aleksandra Medvedeva 14, 18000 Niš, Serbia Received 6 March 2008; accepted 13 August 2008

Abstract Dual-diversity receiver employing selection combining (SC) is often used in wireless communication systems due to its simplicity. Ability of the Rician model to describe fading in wireless communications is the reason to derive infinity-series representations for both the probability density function (PDF) and the cumulative distribution function (CDF) of output signal-to-interference ratio (SIR) at the dual SC receiver over correlated Rician fading channels in the presence of correlated Rayleigh distributed cochannel interference (CCI). These expressions are used to study wireless system performance criteria, such as outage probability and average bit error probability (ABEP). 䉷 2008 Published by Elsevier GmbH Keywords: Selection combining; Correlated Rician fading; Cochannel interference; Outage probability; Average bit error probability

1. Introduction Several statistical models are used in communication systems analysis to describe fading in wireless environments. The most frequently used distributions are Rayleigh, Nakagami-m, Rice, and Weibull. The performance of the selection combining (SC) receiver has been extensively studied in the open technical literature for the abovementioned fading statistical models [1–6]. There are some problems in wireless communication theory that involve multivariate distributions, such as performance analysis of correlative fading applications with space or frequency diversity in multichannel reception. Particularly, in practice due to insufficient spacing between antennas, when the diversity system is applied on small terminals with multiple antennas, correlation arises between branches [7–11]. While multivariate distributions of other fading models such as Rayleigh [12–14], Nakagami [8,12,15–17], or ∗ Corresponding author.

E-mail address: [email protected] (A.S. Panajotovi´c). 1434-8411/$ - see front matter 䉷 2008 Published by Elsevier GmbH doi:10.1016/j.aeue.2008.08.001

Weibull [11,18] have been derived, the joint probability density function (PDF) and cumulative distribution function (CDF) of multivariate Rician distribution are not fully studied in the literature. Despite the usefulness of the Rice model, not many papers studied bivariate Rician PDF. The joint PDF of bivariate Rician distribution was first derived in [19]. The complicated form of bivariate Rician PDF was the reason for the use of numerical integration for calculation average bit error probability (ABEP) in [20]. Infinite-series representation of this PDF, presented in [10], converges rapidly, and thus it can be efficiently used for an analytical study of the performance criteria of a dual-diversity receiver. Another infinite-series of joint PDF and also infinite-series of joint CDF are developed in [21]. Very recently, one more infinite-series representation of joint PDF and joint CDF of bivariate Rician random variables is derived in [22]. The joint PDF of trivariate Rician distribution is also examined in [23], but it may not be used in many cases because of the involving assumption. In this paper, the dual SC system over correlated Rician fading channels in the presence of Rayleigh

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distributed cochannel interference (CCI) is considered. For instance, in a microcellular environment, an undesired signal from a distant cochannel cell may well be modeled by Rayleigh statistics, but Rayleigh fading may not be a good assumption for desired signal since a line-of-sight (LoS) path may exist within a microcell [24–26]. Therefore, in this situation, different fading statistics are needed to characterize the desired and undesired signals in a microcellular radio system. Exact closed-form expressions are derived for both the joint PDF of input signal-to-interference ratios (SIRs) and PDF of output SIR at the SC receiver. Capitalizing on these formulas, some important performance criteria, such as outage probability and ABEP, are obtained in this paper. Numerical results for outage probability and ABEP are graphically presented to show the effects of various system’s parameters, such as fading severity and level of correlation, to the system’s performance. A similar problem was studied in [27] for correlated Nakagami-m fading and in [28] for correlated Weibull fading. Signals from multiple antennas, or “spatial diversity”, can be used to reduce the effect of fading in wireless communication systems and to improve the received signal strength. The most popular linear diversity techniques are SC, equal-gain combining (EGC), and maximal-ratio combining (MRC). Among these types of diversity combining, SC, diversity technique considered in this paper, has the least implementation complexity. The two previous-mentioned combining techniques, MRC and EGC, require all or some of the channel state information (fading amplitude, phase, and delay) from all the received signals. In addition, a separate receiver chain is needed for each diversity branch, which adds to the overall receiver complexity. On the other hand, SC-type systems process only one of the diversity branches. Specifically, in its conventional form, the SC combiner chooses the branch with the highest SNR. In addition, since the output of the SC combiner is equal to the signal on only one of the branches, the coherent sum of the individual branch signals is not required. Therefore, the SC scheme can be used in conjunction with differentially coherent and noncoherent modulation techniques since it does not require knowledge of the signal phases on each branch as would be needed to implement MRC or EGC in a coherent system [1]. However, to the best of the authors’ knowledge, the performance of the SC receiver over correlated Rician fading channels in the presence of correlated Rayleigh distributed CCI has not been addressed yet.

2. PDF and CDF of output SIR at the SC receiver The Rician distribution is often used to model the propagation path consisting of one strong direct LoS signal and many randomly reflected and usually weaker signals. Such fading environments are typically encountered in some microcellular systems [24,29]. The Rician fading model has also been

used to model the mobile satellite channel [1,30]. Hence, the desired signal envelopes only two diversity branches are assumed to follow the correlated Rician distribution, whose probability density is given by pr1 r2 (r1 , r2 )=

r1r2 (K +1)2

2 (1−r 2 )   (r12 +r22 )(K +1)+4K (1−r ) × exp − 2(1−r 2 )    +∞  r1 2K (K +1) × k I k 1+r  k=0   r1r2 r (K +1) ×Ik (1−r 2 )    r2 2K (K +1) ×Ik 1+r 

(1)

where r is the correlation coefficient,  is the average power of r1 and r2 defined as  = r12 /2 = r22 /2, K is the Rice factor, defined as the ratio of the signal power in the dominant component over the scattered power, k = 1(k = 0), i.e. k = 2(k  0) and Ik (·) is the modified Bessel function of the first kind and kth order. If two correlated Rician variables, r1 and r2 , are expressed as r12 = (x1 + A)2 + y12 , r22 = (x2 + A)2 + y22

(2)

where A is the LoS component of Rician fading, Eq. (1) has been extracted using the joint PDF of Gaussian random variables x 1 , x2 , y1 , and y2 and converting px1 x2 y1 y2 (x1 , x2 , y1 , y2 ) into pr1 r2 1 2 (r1 , r2 , 1 , 2 ) where 1 = arctan(y1 /(x1 + A)), i.e. 2 = arctan(y2 /(x2 + A)). After many mathematical manipulations that involve using +∞ infinite-series representation of exp(z cos ) = k=0 k Ik (z) cos(k) and solving some basic integrals of trigonometric functions, the bivariate PDF of Rician distributed random variables obtains form (1). This form of bivariate Rician PDF is equal to PDF in [21]. In cellular mobile radio systems with fading, an exact performance analysis is usually quite complicated, and approximations are sometimes used to simplify the analysis. For example, the performance of the optimum combiner is studied by considering the effect of only the strongest interference signal while assuming that the remaining interference signals are uncorrelated between the diversity branches and can be combined and lumped with the noise [31]. Also, in the presence of large number of interference signals, which is typical in a wireless environment, the central limit theorem may be used to approximate the sum of the interference signals as Gaussian noise with power equal to the sum of the average interference power [32]. When the system is interference-limited, the effect of thermal noise may be ignored [33]. In this paper the case of a single CCI like is done in the paper [7,27,28] is considered.

A.S. Panajotovi´c et al. / Int. J. Electron. Commun. (AEÜ) 63 (2009) 1061 – 1066

The envelope of CCI on diversity branches is Rayleigh distributed because of its multipath propagation over a large distance [27]. Its correlative bivariate PDF, due to insufficient antenna spacing, is expressed by   A21 + A22 A1 A2 p A1 A2 (A1 , A2 ) = 4 exp − 2  A (1 − r A2 ) 2 A (1 − r A2 )   A1 A2 r A (3) × I0 2A (1 − r A2 )

be solved using [35, Eq. (3.194/1)], resulting in CDF:   2K F1 2 (1 ,2 ) = exp − 1+r +∞  k r 2n+k r A2m 2n+2l+2k+2 2 × (n+l+k+1)(m+1)

k, p,n,l,m=0 2n+2 p+2k+2  (K +1)2k+2n+ p+l+2 K p+l+k × 1 2 p+2l+2n+3k+1 (1+r ) ( p+k+1)n! p!m!l! 4n+2 p+2l+4k+4

×

branches of the SC receiver. PDF of these random variables can be obtained as +∞ +∞ p1 2 (1 , 2 ) = A1 A2 pr1 r2 (1 A1 , 2 A2 ) 0

× p A1 A2 ( A1 , A2 )d A1 d A2

(4)

Using the infinite-series representation of Ik (·) [34] Ik (x) =

+∞  n=0

x 2n+k + k + 1)

(5)

22n+k n!(n

and [35, Eq. (3.381/4)], p1 2 (1 , 2 ) can be expressed as p1 2 (1 , 2 )   2K = exp − 1+r ×

×

× × ×

4k

k, p,n,l,m=0

n! p!m!l!2k+2n+l+ p+2

(K + 1)2k+2n+l+ p+2 K p+l+k (1 + r )2 p+2l+3k+2n+1 (1 − r )2n+k+1 (n + l + m + k + 2)1 (n + k + 1)(l + k

+ 1)4m+4 ( p A

0

k, p,n,l,m=0

+ k + 1)

×

r A2m r 2n+k (1 − r A2 )2m+1 (m + 1) (n + p + m + k + 2) (1/2A (1 − r A2 ) + 22 (K + 1)/(1 − r 2 ))n+l+k+m+2 2n+2l+2k+1 2 (1/2A (1 − r A2 ) + 21 (K + 1)/(1 − r 2 ))n+ p+k+m+2 (6)

The corresponding bivariate CDF is, by definition, 1 2 F1 2 (1 , 2 ) = p1 2 (x1 ,x2 ) dx1 dx2

(8)

+ 1)/(1 − r 2 )

and 2 F1 (a, b, c, d) where is the hypergeometric function. Results of paper [27] showed the advantage of the SC diversity technique based on selection of signal to interference power ratio in relation to selection diversity based on the desired signal power algorithm. Let SC be the instantaneous SIR at the output of SC. The PDF of SC , pSC (), is defined as   p1 2 (,2 ) d2 + p1 2 (1 ,) d1 (9) pSC () = 0

+∞ 

0

= 2A (1 − r A2 )(K

Substituting (6) into the previous equation and using [35, Eq. (3.194/1)] finally results in pSC ()   2K = exp − 1+r +∞  2k K p+l+k r 2n+k r A2m × n! p!m!l!(l+k+1)

2n+2 p+2k+1

×

(1−r A2 )2n+2k+ p+l+3  A

2k+2n+ p+l+2 (l+k+1)(1−r )2n+k+1 (n+ p+m+k+2)(n+l+m+k+2) × (n+ p+k+1)(n+k+1) ×2 F1 [n+l+k+m+2, n+l+k+1, n+l+k+2, − 22 ] ×2 F1 [n+ p+k+m+2, n+ p+k+1, n+ p+k+2, − 21 ]

where r A is the correlation coefficient and 2A = A21 /2= A22 /2. Let 1 =r1 /A1 and 2 =r2 /A2 be SIR on the two diversity

0

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(n+ p+m+k+2)4n+4k+2 p+2l+3

2k+2n+ p+l+2 (1+r )2 p+2l+2n+3k+1 (1−r )2n+k+1 (K +1)2k+2n+ p+l+2 (n+l+m+k+2) × (m+1)(n+k+1)( p+k+1)  (1−r A2 )n+l+k+1−m × (1/2A (1−r A2 )+2 (K +1)/(1−r 2 ))n+ p+k+m+2 2n+2l+2k−2m A 2 F1 [n+l+k+m+2, n+l+k+1, (n+l+k+1) n+l+k+2, − 2 ]+2 F1 [n+ p+k+m+2, n+ p+k+1, n+ p+k+2, − 2 ]

×

2n+2 p+2k−2m

× (7)

0

Substituting (6) into (7) and changing the order of summation and integration, integrals in the same equation can

×

A (n+ p+k+1)

(1−r A2 )n+ p+k+1−m



(1/2A (1−r A2 )+2 (K +1)/(1−r 2 ))n+l+k+m+2 (10)

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CDF of SC , FSC (), is given by FSC () = F1 2 (, )

(11)

For binary noncoherent frequency shift keying (NCFSK), the ABEP for a given SIR is [36]   +∞ 1 2 1 exp −  pSC () d Pe () = (12) 2 2 0

3. Numerical results Outage probability is a measure of the system’s performance, helping the designers of wireless communication systems to readjust the system’s operating parameters in order to meet the quality-of-service (QoS) demands. In interference-limited environment, the outage probability is defined as the probability of failing to achieve sufficient SIR to give a radio reception over a level QoS, which is determined by a protection ratio (outage threshold). If qth is the protection ratio, defined as qth = 2th /2, the outage probability can be expressed as    qth Pout = FSC (13) 2 2 A q where q is the input average-signal to average-interference power ratio. It is plotted in Fig. 1 as a function of normalized protection ratio qth /q for several values of Rice factor and correlation coefficient. This figure shows that outage probability decreases, that is, the outage performance improves, as the Rice factor increases and correlation coefficient decreases. It is evident that, for qth /q > 5dB, the real effect of values of correlation coefficient and Rice factor on outage probability is significantly reduced.

Fig. 1. Outage probability versus the normalized protection ratio qth /q for a dual-diversity SC receiver for several values of Rice factor and correlation coefficients.

Fig. 2. Average bit error probability of binary NCFSK system versus input average-signal to average-interference power ratio for several values of Rice factor and correlation coefficients. Table 1. Number of terms of (13) required for four significant figure accuracies r \qth /q[dB]

0.2 0.4 0.6

K = 1 dB

K = 6.5 dB

−5

0

5

−5

0

5

6 6 12

8 9 16

8 10 17

14 14 22

14 15 25

15 15 28

The ABEP has been obtained using (12). In Fig. 2 the ABEP is plotted as a function of q. It considers the performance of the binary NCFSK system for different values of Rice factor and correlation coefficient. As expected, when the Rice factor increases and/or the correlation coefficient decreases, the ABEP improves. For example, Fig. 2 shows that the considered system has ABEP  10−4 only for K = 6.5dB and r  0.5 and it demonstrates very bad error performance (ABEP  10−3 ) for a very great value of correlation coefficient and a very small value of Rice factor, i.e. for r = 0.8 and K = 1dB. The proposed infinite-series representations (6), (8), and (10) have been efficiently used to study important performance criteria, such as outage probability and ABEP. The main problem in these infinite-series expressions can be their convergence. The obtained numerical results have showed that the number of required terms, in every of five sums, that need to be summed to attain a desired accuracy of outage probability and ABEP strongly depends on the fading severity and the correlation coefficient, as is shown in Tables 1 and 2 . Results in tables depict that outage probability converges less slowly than ABEP, i.e. it requires more number of terms to obtain four significant figure accuracies, especially for great values of correlation coefficient and Rice

A.S. Panajotovi´c et al. / Int. J. Electron. Commun. (AEÜ) 63 (2009) 1061 – 1066

Table 2. Number of terms of (12) required for four significant figure accuracies (input average-signal to average-interference power ratio equals 10 dB) r \K [dB]

1

3

5

7

0.2 0.4 0.6

5 5 10

6 6 10

9 6 11

11 10 13

factor. The number of terms increases as the correlation coefficient and the Rice factor increase in both of the tables.

4. Conclusions In cellular land mobile radio, the received signal suffers from CCI, which also arises in mobile satellite communication channels. With the increasing demand for wireless systems and services, microcell and picocell structures have been proposed to increase system capacity. Propagation measurements in such environments have shown that the received signal envelope has a Rician distribution, but the interference envelope has Rayleigh distribution. In this paper, the performance of a dual-diversity SC receiver, operating over correlated Rician fading channels in the presence of correlated Rayleigh distributed CCI, has been studied. First, the infinite-series representations of the PDF and CDF of SC output SIR are derived and they have been used for the analytical study of important performance criteria, such as outage probability and ABEP. Convergence of these functions enables great accuracy of the obtained outage probability and ABEP for binary NCFSK. The number of terms required for four significant figure accuracies of both performance criteria has been presented in tables. These results show slower convergence of the outage probability than the convergence of ABEP. Numerical results of the obtained performance criteria are presented, describing their dependence on correlation coefficient and fading severity. They show that system’s performance improves when the Rice factor increases (fading severity decreases) and/or correlation coefficient decreases.

Acknowledgements The authors would like to thank anonymous reviewers for the careful reading of the paper and for the constructive, focused, and detailed comments and suggestions. Also, the authors thank I. Kosti´c for his helpful suggestions.

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[35] Gradshteyn IS, Ryzhik IM. Table of integrals, series, and products, Academic Press, CD version 1.0, 5th ed., 1996. 473–79. [36] Aalo VA, Zhang J. On the effect of cochannel interference on average error rates in Nakgami-fading channels. IEEE Commun Lett series 1999;3(5):136–8. Aleksandra S. Panajotovi´c was born in Niš, Serbia, in 1974. She received her B.Sc., M.Sc. and Ph.D. degrees in electrical engineering from the Faculty of Electronic Engineering (Department of Telecommunications), University of Niš, Serbia, in 1999, 2003, and 2007, respectively. Her field of interest included telecommunications theory and optical communication systems. Her current research interests are in diversity systems, digital communications over fading channels and interference analysis. She works as Teaching Assistant at the Faculty of Electronic Engineering. ˇ Stefanovi´c was born in Mihajlo C. Niš, Serbia, in 1947. He received his B.Sc., M.Sc. and Ph.D. degrees in electrical engineering from the Faculty of Electronic Engineering (Department of Telecommunications), University of Niš, Serbia, in 1971, 1976, and 1979, respectively. His primary research interests are statistical communication theory, optical and wireless communications. He has authored or co-authored a great number of journal publications. He has written five monographs, too. Now, Dr. Stefanovi´c is a Professor at the faculty of Electronic Engineering in Niš. Dragan Lj. Draˇca was born in Niš, Serbia, in 1951. He received his B.Sc., M.Sc. and Ph.D. degrees in electrical engineering from the Faculty of Electronic Engineering (Department of Telecommunications), University of Niš, Serbia, in 1975, 1981, and 1995, respectively. His field of research is digital telecommunication systems and packet-based transport networking. Now, Dr. Draˇca is Professor at the University of Niš. He has authored or co-authored a great number of publications and two monographs.