- Petar Ć. Spalević
^{1}, - Stefan R. Panić
^{2}Email author, - Ćemal B. Dolićanin
^{3}, - Mihajlo Č. Stefanović
^{2}and - Aleksandar V. Mosić
^{2}

**2010**:142392

https://doi.org/10.1155/2010/142392

© Petar Ć. Spalević et al. 2010

**Received: **16 August 2009

**Accepted: **31 January 2010

**Published: **9 March 2010

## Abstract

This paper studies the performances of a dual-branch switched-and-stay combining (SSC) diversity receiver, operating over correlated fading in the presence of cochannel interference (CCI). Very useful, novel, infinite series expressions are obtained for the output signal to interference ratio's (SIR's) probability density function (PDF) and cumulative distribution function (CDF). The performance analysis is based on an outage probability (OP) and an average bit error probability (ASEP) criteria. ASEP is efficiently evaluated for modulation schemes such as noncoherent frequency-shift keying (NCFSK) and binary differentially phase-shift keying (BDPSK). The effects of various parameters, such as input SIR unbalance, the level of correlation between received desired signals and interferences, nonlinearity of the environment, and fading severity on systems performances are graphically presented and analyzed.

## 1. Introduction

Various techniques for reducing the fading effects and the influence of the cochannel interference (CCI) are used in wireless communication systems [1]. The two main goals of the diversity techniques are upgrading the transmission reliability without increasing transmission power and bandwidth and increasing the channel capacity. Space diversity is an efficient method for amelioration of system's quality of service (QoS) when multiple receiver antennas are used. There are several principal types of combining techniques and division, that can be generally performed depending on the complexity restriction put on the communication system and the amount of channel state information available at the receiver.

However, switch and stay combining (SSC) is the least complex and can be used in conjunction with coherent, noncoherent, and differentially coherent modulation schemes. In general, with 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 [1–3]. In fading environments as in cellular systems, where the level of CCI is sufficiently high as compared to the thermal noise, SSC selects a particular branch until its signal-to-interference ratio (SIR) drops below a predetermined threshold (SIR-based switched diversity). Then the combiner switches to another branch and stays there regardless of SIR of that branch. This type of diversity, in which switching is based on the SIR value, can be measured in real time both in base stations (uplink) and in mobile stations (downlink) using specific SIR estimators.

The multipath fading in wireless communications is modelled by several distributions including Weibull, Nakagami-m Hoyt, Rayleigh, and Rice. By considering two important phenomena inherent to radio propagation, namely, nonlinearity and clustering, the - fading model was proposed and discussed in [4–7]. The model provides a very good fit to measured data over a wide range of fading conditions. The - distribution is written in terms of physicallybased fading parameters, namely, and . Roughly speaking, is related to the nonlinearity of the environment, whereas is associated with the number of multipath clusters. The diversity reception over - fading channels was previously discussed in [8–11].

There is a number of papers concerning performance analysis of SSC receivers, for example, [12–15]. The performance analysis of SSC diversity receivers, operating over correlated Ricean fading satellite channels, can be found in [13], where the performance is evaluated based on a bunch of novel analytical formulae for the outage probability (OP), average symbol error probability (ASEP), channel capacity (CC), the amount of fading (AoF), and the average output SNR obtained in infinite series form. The similar performance analysis of the switched diversity receivers operating over correlated Weibull fading channels in terms of OP, ASEP, moments, and moment generating function (MGF) can be found in [14]. Reference [15] studies the performance of a dual-branch SSC diversity receiver with the switching decision based on SIR, operating over correlated Ricean fading channels in the presence of correlated Nakagami-m distributed CCI. Moreover to the best of author's knowledge, no analytical study of switch and stay combining involving assumed correlated - fading for both desired signal and cochannel interference has been reported in the literature.

In this paper, desired signal and CCI are considered to be affected by - fading distribution, which is adequate for multipath waves, propagating in a nonhomogenous environment. An approach to the performance analyses of given SSC diversity receiver is presented. Effectiveness of any modulation scheme and the type of diversity used is studied thought evaluating the system's performance over the channel conditions. Infinite series expressions for probability distribution function (PDF) and cumulative distribution function (CDF) of the output SIR for SSC diversity are derived. Furthermore, infinite series expression for important performance measure, such as OP, is obtained. An OP is shown graphically for different system parameters. Using infinite series formulae, ASEP is efficiently evaluated for several modulation schemes such as noncoherent frequency-shift keying (NCFSK) and binary differentially phase-shift keying (BDPSK). This analysis has a high level of generality because - fading distribution model includes as special cases other important distributions such as Weibull and Nakagami-m (therefore the one-sided Gaussian and Rayleigh are also special cases of it).

## 2. System Model and Received Statistics

Independent fading assumes antenna elements in diversity systems to be placed sufficiently apart, which is not general case in practice due to insufficient spacing between antennas. When diversity system is applied on small terminals with multiple antennas, correlation arises between branches. Now, due to insufficient antennae spacing, desired signal envelopes and experience correlative - fading, with joint distribution [5]:

Similarly, due to insufficient antennae spacing, envelopes of interference and are correlated with joint - distribution [5]:

It is important to quote that and denote the desired and interfering signal power correlation coefficients defined as , and , respectively. For the desired signal with correlated envelopes and the arbitrary power parameters , connected with the nonlinearity of the environment, , explained in [5], denote the -root mean values of desired signal. Similarly denote the -root mean values of interferer signal. Parameters and are associated with the number of multipath clusters through which desired and interference signals propagate, and they are defined in the terms of normalized variances as in [5, 11]. and are, respectively, the covariance and variance operators, and stands for Gamma function [16]. defines generalized Laguerre polynomial with the property of [17]

Let and represent the instantaneous SIR on the diversity branches, respectively [18]. The joint PDF of and and can be expressed by [18, 19]

Substituting (1), (2), and (3) in (4), we get

where denotes , denotes denotes , with denoting the Pochhammer symbol [16]. Derivation of this expression is presented in the appendix. Let represent the instantaneous SIR at the SSC output, and the predetermined switching threshold for the both input branches. Following [20], the PDF of SIR is given by

where , according to [20], can be expressed as . Moreover, can be expressed as infinite series

with , being the Gaussian hypergeometric function [16], and

In the same manner, the can be expressed as

Similarly, the CDF of the SSC output SIR, that is, the is given by [11, 20]

By substituting (5) in and (10) in and can be expressed as the following infinite series, respectively,

with

## 3. Outage Probability

## 4. Average Symbol Error Probability

The ASEP ( ) can be evaluated by averaging the conditional symbol error probability for a given SIR, that is, , over the PDF of , that is, [8]:

where depends on applied modulation scheme. For BDPSK and NCFSK modulation schemes the conditional symbol error probability for a given SIR threshold can be expressed by , where for BDPSK and for NCFSK [14]. Hence, substituting (7) into (17) gives the following ASEP expression for the considered dual-branch SSC receiver:

## 5. Conclusion

In this paper, the system performances of a dual-branch SSC diversity receiver over correlated - channels are analyzed based on SIR. Fading between the diversity branches and between interferers is correlated and - distributed. The complete statistics for the SSC output SIR is given in the infinite series expressions form, that is, PDF, CDF, and OP. Using these new formulae, ASEP was efficiently evaluated for some modulation schemes such as DPSK and NCFSK. The main contribution of this analysis for dualbranch signal combiner is that it has been done for a general case of - (Generalized Gamma) distribution, which includes as special cases important other distributions such as Weibull and Nakagami-m (therefore, the One-Sided Gaussian and Rayleigh are also special cases of it), so our analysis has a high level of generality.

## Declarations

## Authors’ Affiliations

## References

- Stuber GL:
*Mobile Communication*. 2nd edition. Kluwer Academic Publishers, Boston, Mass, USA; 2001.Google Scholar - Lee WCY:
*Mobile Communications Engineering*. Mc-Graw-Hill, New York, NY, USA; 2001.Google Scholar - Simon MK, Alouini MS:
*Digital Communication over Fading Channels*. John Wiley & Sons, New York, NY, USA; 2005.Google Scholar - Yacoub MD:The
distribution: a physical fading model for the Stacy distribution.
*IEEE Transactions on Vehicular Technology*2007, 56(1):27-34.View ArticleGoogle Scholar - De Souza RAA, Fraidenraich G, Yacoub MD:On the multivariate
distribution with arbitrary correlation.
*Proceedings of International Telecommunications Symposium (ITS '06), September 2006, Fortaleza, Brazil*38-41.Google Scholar - Coulson AJ, Williamson AG, Vaughan RG: Improved fading distribution for mobile radio.
*IEE Proceedings: Communications*1998, 145(3):197-202. 10.1049/ip-com:19981991View ArticleGoogle Scholar - Piboongungon T, Aalo VA, Iskander CD, Efthymoglou GP: Biyariate generalised gamma distribution with arbitrary fading parameters.
*Electronics Letters*2005, 41(12):709-710. 10.1049/el:20050435View ArticleGoogle Scholar - Sagias NC, Mathiopoulos PT: Switched diversity receivers over generalized gamma fading channels.
*IEEE Communications Letters*2005, 9(10):871-873. 10.1109/LCOMM.2005.10026View ArticleGoogle Scholar - Sagias NC, Karagiannidis GK, Mathiopoulos PT, Tsiftsis TA: On the performance analysis of equal-gain diversity receivers over generalized Gamma fading channels.
*IEEE Transactions on Wireless Communications*2006, 5(10):2967-2975.View ArticleGoogle Scholar - Panić SR, Stefanović MČ, Mosić AV:Performance analyses of selection combining diversity receiver over
Fading channels in the presence of co-channel interference.
*IET Communications*2009, 3(11):1769-1777. 10.1049/iet-com.2009.0023View ArticleGoogle Scholar - Bithas PS, Sagias NC, Tsiftsis TA: Performance analysis of dual-diversity receivers over correlated generalised Gamma fading channels.
*IET Communications*2008, 2(1):174-178. 10.1049/iet-com:20070207View ArticleGoogle Scholar - Abu-Dayya AA, Beaulieu NC: Switched diversity on microcellular Ricean channels.
*IEEE Transactions on Vehicular Technology*1994, 43(4):970-976. 10.1109/25.330159View ArticleGoogle Scholar - Bithas PS, Mathiopoulos PT: Performance analysis of SSC diversity receivers over correlated ricean fading satellite channels.
*EURASIP Journal on Wireless Communications and Networking*2007, 2007:-9.Google Scholar - Bithas PS, Mathiopoulos PT, Karagiannidis GK: Switched diversity receivers over correlated weibull fading channels.
*Proceedings of International Workshop on Satellite and Space Communications (IWSSC '06), September 2006, Madrid, Spain*143-147.Google Scholar - Bandjur DV, Stefanovic MC, Bandjur MV: Performance analysis of SSC diversity receiver over correlated Ricean fading channels in presence of co-channel interference.
*Electronics Letters*2008, 44(9):587-588. 10.1049/el:20080270View ArticleGoogle Scholar - Gradshteyn I, Ryzhik I:
*Tables of Integrals, Series, and Products*. Academic Press, London, UK; 1980.Google Scholar - , January 2009 http://mathworld.wolfram.com/
- Karagiannidis GK: Performance analysis of SIR-based dual selection diversity over correlated Nakagami-m fading channels.
*IEEE Transactions on Vehicular Technology*2003, 52(5):1207-1216. 10.1109/TVT.2003.816612View ArticleGoogle Scholar - Stefanović MČ, Milović DM, Mitić AM, Jakovljević MM: Performance analysis of system with selection combining over correlated Weibull fading channels in the presence of cochannel interference.
*International Journal of Electronics and Communications*2008, 62(9):695-700. 10.1016/j.aeue.2007.09.006View ArticleGoogle Scholar - Ko Y-C: Analysis and optimization of switched diversity systems.
*IEEE Transactions on Vehicular Technology*2000, 49(5):1813-1831. 10.1109/25.892586View ArticleGoogle Scholar

## Copyright

This article is published under license to BioMed Central Ltd. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.