Second-order statistics of selection macro-diversity system operating over Gamma shadowed κ-μ fading channels
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Abstract
In this article, infinite-series expressions for the second-order statistical measures of a macro-diversity structure operating over the Gamma shadowed κ-μ fading channels are provided. We have focused on MRC (maximal ratio combining) combining at each base station (micro-diversity), and selection combining (SC), based on output signal power values, between base stations (macro-diversity). Some numerical results of the system's level crossing rate and average fading duration are presented, in order to examine the influence of various parameters such as shadowing and fading severity and number of the diversity branches at the micro-combiners on concerned quantities.
Keywords
Fading Channel Channel State Information Maximal Ratio Combine Selection Combine Diversity Branch1 Introduction
Wireless channels are simultaneously affected by short-term fading and long-term fading (shadowing) [1]. Shadowing is the result of the topographical elements and other structures in the transmission path such as trees, tall buildings, etc. Short-term fading (multipath) is a propagation phenomenon caused by atmospheric ducting, ionospheric reflection and refraction, and reflection from water bodies and terrestrial objects such as mountains and buildings. By considering important phenomena inherent to radio propagation, κ-μ short-term fading model was recently proposed in [2], as a model which describes the short-term signal variation in the presence of line-of-sight (LoS) components, and includes Rayleigh, Rician, and Nakagami-m fading models as special cases [3]. An efficient method for reducing short-term fading effect at micro-level (single base station) with the usage of multiple receiver antennas is called space diversity. Upgrading transmission reliability without increasing transmission power and bandwidth while increasing channel capacity is the main goal of space diversity techniques. There are several principal types of space combining techniques that can be generally performed by considering the amount of channel state information available at the receiver [4, 5, 6].
While short-term fading is mitigated through the use of diversity techniques typically at a single base station (micro-diversity), the use of such micro-diversity approaches alone will not be sufficient to mitigate the overall channel degradation when shadowing is also concurrently present. Since they coexist in wireless systems, short- and long-term fading conditions must be simultaneously taken into account. Macro-diversity reception is used to alleviate the effects of shadowing, where multiple signals are received at widely located base stations, ensuring that different long-term fading is experienced by these signals [7, 8]. At the macro-level, selection combining (SC) is used as a basically fast response handoff mechanism that instantaneously or, with minimal delay chooses the best base station to serve mobile based on the signal power received [9].
The performance analysis of diversity systems operating over κ-μ fading channels is rather scarce in the literature [10, 11]. In [10], standard performance measures of maximal ratio combining (MRC) in the presence of κ-μ fading were discussed. Analytical expressions for the switching rate of a dual branch SC in κ-μ fading were derived in [11]. Macro-diversity over the shadowed fading channels was discussed by several researches [7, 8, 9, 12]. Discussions about the second-order statistics of various diversity systems can be easily found in the literature [13, 14, 15]. Second-order statistics analysis of macro-diversity system operating over Gamma shadowed Nakagami-m fading channels was recently proposed in [16]. Moreover, to the best knowledge of the authors, no analytical study investigating the second-order statistics of macro-diversity system operating over Gamma shadowed κ-μ fading channel has been reported in the literature.
This article delivers infinite-series expressions for level crossing rate (LCR) and average fading duration (AFD) at the output of SC macro-diversity operating over the Gamma shadowed κ-μ fading channels. Macro-diversity system of SC type consists of two micro-diversity systems and the selection (switching) is based on their output signal power values. Each micro-diversity system is of MRC type with an arbitrary number of branches in the presence of κ-μ fading. Received signal powers of the micro-diversity output signals are modelled by statistically independent Gamma distributions. Numerical results for these second-order statistical measures are also presented in order to show the influence of various parameters such as shadowing and fading severity and the number of the diversity branches at the micro-combiners on the system's statistics.
2 System model
The κ-μ distribution fading model corresponds to a signal composed of clusters of multipath waves, propagating in a nonhomogeneous environment. The phases of the scattered waves are random and have similar delay times, within a single cluster, while delay-time spreads of different clusters are relatively large. It is assumed that the clusters of multipath waves have scattered waves with identical powers, and that each cluster has a dominant component with arbitrary power. This distribution is well suited for LoS applications, since every cluster of multipath waves has a dominant component (with arbitrary power). The κ-μ distribution is a general physical fading model which includes Rician and Nakagami-m fading models as special cases (as the one-sided Gaussian and the Rayleigh distributions) since they also constitute special cases of Nakagami-m. κ parameter represents the ratio between the total power of dominant components and the total power of scattered components. Parameter μ is related to multipath clustering. As μ decreases, fading severity increases. For the case of κ = 0, the κ-μ distribution is equivalent to the Nakagami-m distribution. When μ = 1, the κ-μ distribution becomes the Rician distribution with κ as the Rice factor. Moreover, the κ-μ distribution fully describes the characteristics of the fading signal in terms of measurable physical parameters [2].
In the previous equation, I_{ r } (·)denotes the r th-order modified Bessel function of first kind [[17], eq. 8.445], μ_{ i } and κ_{ i } are well-known κ-μ fading parameters of each micro-diversity system, while L_{ i } denotes the number of channels at each micro-level.
where f_{ d } is a Doppler shift frequency.
In the previous equation, c_{1} and c_{2} denote the order of Gamma distribution, the measure of the shadowing present in the channels. Ω_{01} and Ω_{02} are related to the average powers of the Gamma long-term fading distributions.
3 Second-order statistics
Second-order statistical quantities complement the static probabilistic description of the fading signal (the first-order statistics), and have found several applications in the modelling and design of wireless communication systems. Two most important second-order statistical measures are the LCR and the AFD. They are related to the criterion that can be used to determine parameters of equivalent channel, modelled by the Markov chain with the defined number of states and according to the criterion used to assess error probability of packets of distinct length [19].
Terms need to be summed in each sum of (14) to achieve accuracy at the 5th significant digit
c_{1} = c_{2} = 1 | Ω_{01} = Ω_{02} = 1 | μ_{1} = μ_{2} = 2 | μ_{1} = μ_{2} = 3 |
---|---|---|---|
z = -10 dB | |||
L = 2 | κ_{1} = κ_{2} = 0: 5 | 9 | 12 |
L = 2 | κ_{1} = κ_{2} = 1 | 12 | 15 |
L = 3 | κ_{1} = κ_{2} = 0: 5 | 12 | 14 |
L = 3 | κ_{1} = κ_{2} = 1 | 16 | 21 |
z = 0 dB | |||
L = 2 | κ_{1} = κ_{2} = 0: 5 | 10 | 12 |
L = 2 | κ_{1} = κ_{2} = 1 | 13 | 17 |
L = 3 | κ_{1} = κ_{2} = 0: 5 | 11 | 17 |
L = 3 | κ_{1} = κ_{2} = 1 | 17 | 21 |
z = 10 dB | |||
L = 2 | κ_{1} = κ_{2} = 0: 5 | 12 | 13 |
L = 2 | κ_{1} = κ_{2} = 1 | 15 | 18 |
L = 3 | κ_{1} = κ_{2} = 0: 5 | 12 | 15 |
L = 3 | κ_{1} = κ_{2} = 1 | 16 | 23 |
4 Numerical results
5 Conclusion
In this article, the second-order statistic measures of SC macro-diversity system operating over Gamma shadowed κ-μ fading channels with arbitrary parameters were analyzed. Useful incite-series expressions for LCR and AFD at the output of this system were derived. The effects of the various parameters such as shadowing and fading severity and the number of the diversity branches at the micro-combiners on the system's statistics were also presented.
Notes
Supplementary material
References
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