doi:10.1155/2007/25361 Research Article Performance Analysis of SSC Diversity Receivers over Correlated Ricean Fading Satellite Channels

This paper studies the performance of switch and stay combining (SSC) diversity receivers operating over correlated Ricean fading satellite channels. Using an infinite series representation for the bivariate Ricean probability density function (PDF), the PDF of the SSC output signal-to-noise ratio (SNR) is derived. Capitalizing on this PDF, analytical expressions for the corresponding cumulative distribution function (CDF), the moments of the output SNR, the moments generating function (MGF), and the average channel capacity (CC) are derived. Furthermore, by considering several families of modulated signals, analytical expressions for the average symbol error probability (ASEP) for the diversity receivers under consideration are obtained. The theoretical analysis is accompanied by representative performance evaluation results, including average output SNR (ASNR), amount of fading (AoF), outage probability, average bit error probability (ABEP), and average CC, which have been obtained by numerical techniques. The validity of some of these performance evaluation results has been verified by comparing them with previously known results obtained for uncorrelated Ricean fading channels.


INTRODUCTION
The mobile terrestrial and satellite communication channel is particularly dynamic due to multipath fading propagation, having a strong negative impact on the average bit error probability (ABEP) of any modulation scheme [1]. Diversity is a powerful communication receiver technique used to compensate for fading channel impairments. The most important and widely used diversity reception methods employed in digital communication receivers are maximal-ratio combining (MRC), equal-gain combining (EGC), selection combining (SC), and switch and stay combining (SSC) [2]. For SSC diversity considered in this paper, the receiver selects a particular branch until its signal-to-noise ratio (SNR) drops below a predetermined threshold. When this happens, the combiner switches to another branch and stays there regardless of whether the SNR of that branch is above or below the predetermined threshold. Hence, among the abovementioned diversity schemes, SSC is the least complex and can be used in conjunction with coherent, noncoherent, and differentially coherent modulation schemes. It is also well known that in many real life communication scenarios the combined signals are correlated [2,3]. A typical example for such signal correlation exists in relatively small-size mobile terminals where typically the distance between the diversity antennas is short. Due to this correlation between the signals received at the diversity branches there is a degradation in the achievable diversity gain.
The Ricean fading distribution is often used to model propagation paths consisting of one strong direct line-ofsight (LoS) signal and many randomly reflected and usually weaker signals. Such fading environments are typically encountered in microcellular and mobile satellite radio links [2]. In particular for mobile satellite communications the Ricean distribution is used to accurately model the mobile satellite channel for single- [4] and clear-state [5] channel conditions. Furthermore, in [6] it was depicted that the Ricean K-factor characterizes the land mobile satellite channel during unshadowed periods.
The technical literature concerning diversity receivers operating over correlated fading channels is quite rich, for example, see [7][8][9][10][11][12][13]. In [7] expressions for the outage probability (P out ) and the ABEP of dual SC with correlated Rayleigh fading were derived either in closed form or in terms of 2 EURASIP Journal on Wireless Communications and Networking single integrals. In [8] the cumulative distribution functions (CDF) of SC, in correlated Rayleigh, Ricean, and Nakagamim fading channels were derived in terms of single-fold integrals and infinite series expressions. In [9] the ABEP of dual-branch EGC and MRC receivers operating over correlated Weibull fading channels was obtained. In [10] the performance of MRC in nonidentical correlated Weibull fading channels with arbitrary parameters was evaluated. In [11] an analysis for the Shannon channel capacity (CC) of dual-branch SC diversity receivers operating over correlated Weibull fading was presented. In [12], infinite series expressions for the capacity of dual-branch MRC, EGC, SC, and SSC diversity receivers over Nakagami-m fading channels have been derived.
Past work concerning the performance of SSC operating over correlated fading channels can be found in [14][15][16][17]. One of the first attempts to investigate the performance of SSC diversity receivers operating over independent and correlated identical distributed Ricean fading channels was made in [14]. However, in this reference only noncoherent frequency shift keying (NCFSK) modulation was considered and its ABEP has been derived in an integral representation form. In [15] the performance of SSC diversity receivers was investigated for different fading channels, including Rayleigh, Nakagami-m and Ricean, and under different channel conditions but dealt mainly with uncorrelated fading. For correlated fading in this reference only the Nakagami-m distribution was studied. In [16] the moments generating function (MGF) of SSC was presented in terms of a finite integral representation for the correlated Nakagamim fading channel. In [17] expressions for the average output SNR (ASNR), amount of fading (AoF) and P out for the correlated log-normal fading channels have been derived.
All in all, the problem of theoretically analyzing the performance of SSC over correlated Ricean fading channels has not yet been thoroughly addressed in the open technical literature. The main difficulty in analyzing the performance of diversity receivers in correlated Ricean fading channels is the complicated form of the received signal bivariate probability density function (PDF), see [14,Equation (17)], and the absence of an alternative and more convenient expression for the multivariate distribution. An efficient solution to these difficulties is to employ an infinite series representation for the bivariate PDF, such as those that were proposed in [18] or [19]. Such an approach was used in [20] to analyze the performance of MRC, EGC, and SC in the presence of correlated Ricean fading. Similarly here the most important statistical metrics and the capacity of SSC diversity receivers operating over correlated Ricean fading channels will be studied. In particular, we derive the PDF, CDF, MGF, moments and the average CC of such receivers operating over correlated Ricean fading channels. Furthermore, analytical expressions for the average symbol error probability (ASEP) of several modulation schemes will be obtained. Capitalizing on these expressions, a detailed performance analysis for the P out , ASNR, AoF, and ASEP/ABEP will be presented.
The remainder of this paper is organized as follows. After this introduction, in Section 2 the system model is intro-duced. In Section 3, the SSC received signal statistics are presented, while in Section 4 the capacity is obtained. Section 5 contains the derivation of the most important performance metrics of the SSC output SNR. In Section 6, various numerical evaluation results are presented and discussed, while the conclusions of the paper can be found in Section 7.

SYSTEM MODEL
By considering a dual-branch SSC diversity receiver operating over a correlated Ricean fading channel, the baseband received signal at the th ( = 1 and 2) input branch can be mathematically expressed as In the above equation, s is the transmitted complex symbol, h is the Ricean fading channel complex envelope with magnitude R = |h |, and n is the additive white Gaussian noise (AWGN) having single-sided power spectral density of N 0 . The usual assumption for ideal fading phase estimation is made, and hence, only the distributed fading envelope and the AWGN affect the received signal. Moreover, the AWGN is assumed to be uncorrelated between the two diversity branches. The instantaneous SNR per symbol at the th input branch is where K is the Ricean K-factor defined as the power ratio of the specular signal to the scattered signals and I 0 (·) is the zeroth-order modified Bessel function of the first kind [21,Equation (8.406)]. The CDF of γ is given by [14,Equation (8)] where Q 1 (·) is the first-order Marcum-Q function [2, Equation (4.33)].
The joint PDF of γ 1 and γ 2 , presented in [14,Equation (17)], can be expressed in terms of infinite series by following a similar procedure as for deriving [18,Equation (9)]. Hence, substituting I 0 (·) with its infinite series representation [ where Γ(·) is the Gamma function [21, Equation (8.310/1)] and ρ is the correlation coefficient between γ 1 and γ 2 . It can be proved that the above infinite series expression always converges [18].

RECEIVED SIGNAL STATISTICS
In this section, the most important statistical metrics, namely, the PDF, CDF, MGF, and moments of dual branch SSC output SNR diversity receivers operating over correlated Ricean fading channels will be presented.

Cumulative distribution function (CDF)
Similar to [23,Equation (20)], the CDF of γ ssc , F γssc (γ), is given by which after some manipulations can be expressed in terms of CDFs as Hence, by substituting (4) In order to verify the validity of the above derivations, (10) and (11) have been numerically evaluated for the special case of uncorrelated, that is, ρ = 0, Ricean fading channels. The resulting CDF was found to be identical to the same CDF presented in [2,Equation 9.273], which was derived using a different mathematical approach as a closed-form expression.

CHANNEL CAPACITY (CC)
CC is a well-known performance metric providing an upper bound for maximum errorless transmission rate in a Gaussian environment. The average CC, C, is defined as [26] where BW is transmission bandwidth of the signal in Hz. Hence, substituting (6) in (18), C becomes  [28], I 5 can be solved as Due to the very complicated nature of I 6 , it is very difficult, if not impossible, to derive a closed-form solution for this integral. However, I 6 can be evaluated via numerical integration using any of the well-known mathematical software packages, such as MATHEMATICA or MATLAB.

PERFORMANCE ANALYSIS
In this section a detailed performance analysis, in terms of P out , ASEP, ASNR and AoF, for SSC diversity receivers operating over correlated Ricean fading channels will be presented.

Outage probability (P out )
P out is the probability that the output SNR falls below a predetermined threshold γ th , P out (γ th ), and can be obtained by replacing γ with γ th in (10) as

Average output SNR (ASNR) and amount of fading (AoF)
The ASNR, γ ssc , is a useful performance measure serving as an excellent indicator for the overall system fidelity and can be obtained from the first-order moment of γ ssc as γ ssc = μ γssc (1). Ricean K-Factor The AoF, defined as AoF Δ = var(γ ssc )/γ 2 ssc , is a unified measure of the severity of the fading channel [2] and gives an insight to the performance of the entire system. It can be expressed in terms of first-and second-order moments of γ ssc as

PERFORMANCE EVALUATION RESULTS
Using the previous mathematical analysis, various performance evaluation results have been obtained by means of numerical techniques and will be presented in this section. Such results include performances for the ASNR, AoF, P out , ABEP 1 , and C and will be presented for different Ricean channel conditions, that is, different values for K and ρ, as well as for various modulation schemes. In Figures 1 and 2 the normalized ASNR (γ ssc /γ) and AoF are plotted as functions of the Ricean K-factor for several values of the correlation coefficient ρ. These performance evaluation results have been obtained by numerically evaluating (15)- (17), (28), and (29). The results presented in Figure 1 1 For the consistency of the presentation from now on instead of the ASEP the ABEP performance will be used. As it is well known [2] for M-ary (M > 2) modulation schemes, assuming Gray encoding, the ABEP can be simply obtained from the ASEP as P be ∼ = P se / log 2 M, since E s = E b log 2 M, where E b denotes the transmitted average bit energy. show that as K increases, that is, the severity of the fading decreases, and/or ρ increases, the normalized ASNR decreases, resulting in a reduced diversity gain. We note that similar observations have been made in [3,30]. Furthermore, the results presented in Figure 2 indicate that as K increases and/or ρ decreases, AoF is degraded. Next the ABEP has been obtained using (23)- (27). In Figures 3 and 4 the ABEP is plotted as a function of the average input SNR per bit, that is, γ b = γ/ log 2 M, for several values of K. Figure 3 considers the performance of DBPSK, BPSK, and M-ary PSK signaling formats and ρ = 0.5. As expected, when K increases, the ABEP improves and BPSK exhibits the best performance. Figure 4 presents the ABEP of 16-QAM for different values of ρ and K. For comparison purposes, the performance of an equivalent single receiver, that is, without diversity, is also included. Similar to the previous cases, it is observed that the ABEP improves as K increases and/or ρ decreases, while significant overall performance improvement, as compared to the no-diversity case, is also noted.
In Figure 5, P out is plotted as a function of the normalized outage threshold per bit, γ th /γ b , for several values of K and ρ. These performance results have been obtained by numerically evaluating (10), (11), and (21) and for ρ = 0 they are identical to the ones obtained by using [2,Equation 9.273]. It is observed that P out decreases, that is, the outage performance improves, as K increases and/or ρ decreases.
Finally, the normalized average CC can be obtained as C = C/BW (in b/s/Hz) by employing (19) and (20). In Figure 6, C is plotted as a function of γ b for several values P. S. Bithas and P. T. Mathiopoulos      8 EURASIP Journal on Wireless Communications and Networking of ρ and for K = 1. These results illustrate that as ρ increases, C decreases, as expected [12], and the receiver without diversity has always the worst performance.

CONCLUSIONS
In this paper, the performance of dual branch SSC diversity receivers operating over correlated Ricean fading channels has been studied. By deriving a convenient expression for the bivariate Ricean PDF, analytical formulae for the most important statistical metrics of the received signals and the capacity of such receivers have been obtained. Capitalizing on these formulas, useful expressions for a number of performance criteria have been obtained, such as ABEP, P out , ASNR, AoF, and average CC. Various performance evaluation results for different fading channel conditions have been also presented and compared with equivalent performance results of receivers without diversity.