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Correlation Phenomenon Versus Beam-Subset Scenario in MIMO-Femtocell System with Fading Environments

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Abstract

The channel correlation happens in a MIMO-femtocell (multiple-input multiple-output) system with beam-subset beamforming technique is investigated in this paper. MIMO-femtocell system constructed with MIMO signaling that is adapted to a tiered femtocellular network. The MIMO-femtocell system deployed with home base station (BS) in a single floor indoor environment is proposed. According to the analyses of both theoretical and simulation results, channel correlation definitely degrades the overall system performance of a MIMO-femtocell system with beam-subset, especially in indoor channels. Moreover, the coverage area of a MIMO-femtocell deployment is reduced because of channel correlation occurring in the propagation channel. Some analyzed of closed forms are utilized to discuss the phenomenon of channel correlation that exists in MIMO-femtocell systems. Specifically, many numerical results, which are applied to validate the performance of being derived throughput, and the curves match well with the simulation results obtained from the software package. It is noteworthy to remember that if the channel correlation phenomenon is neglected in the performance evaluation of a MIMO-femtocell system over the Rayleigh statistical model, there will be one to about four folds degradation existing in the BS coverage area. Moreover, the number of beam-subset is the other parameter dominates the system performance of a MIMO-femtocell system when the beamforming signaling is applied in the operation.

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Acknowledgments

The authors wish to express their appreciation for financial support from the National Science Council of the Republic of China under Contact NSC 102-2221-E-212 -002.

Author information

Correspondence to Joy Iong-Zong Chen.

Appendix 1

Appendix 1

The calculation of Eq. (20) with the help of binomial expansion, \( (1 - \alpha_{m} e^{{ - \beta_{m} r}} )^{Q} \), is discussed in the Appendix 1. It is well known that the equation can be expressed as

$$ \left( {1 - \alpha_{m} e^{{ - \beta_{m} r}} } \right)^{Q} = \sum\limits_{q = 0}^{Q} {\left( {_{q}^{Q} } \right)\left( { - \alpha_{m} } \right)^{q} \cdot e^{{ - \beta_{m} rq}} } $$
(21)

By substituting the pdf of SINR into aggregated throughput formula, shown in (15), which then can be obtained as

$$ \begin{aligned} Tp({\mathbf{Bs}}) & \cong {\mathbf{Bs}}\sum\limits_{m = 1}^{Card(M)} {\log_{2} \left( M \right)\int_{{\gamma = \gamma_{th,m} }}^{{\gamma_{th,m + 1} }} {\left( {1 - \alpha_{m} e^{{ - \beta_{m} \gamma }} } \right)^{Q} \cdot f_{SINR} (\gamma )d\gamma } } \\ & = {\mathbf{Bs}}\sum\limits_{m = 1}^{Card(M)} {\log_{2} \left( M \right)\int_{{\gamma = \gamma_{th,m} }}^{{\gamma_{th,m + 1} }} {\left[ {\sum\limits_{q = 0}^{Q} {\left( {_{q}^{Q} } \right)\left( { - \alpha_{m} } \right)^{q} \cdot e^{{ - \beta_{m} \gamma q}} } } \right]d\gamma } } \\ & \quad \times \frac{1}{{\Delta (\eta )}}\sum\limits_{n = 0}^{N - 1} {\sum\limits_{\ell = 0}^{L - 1} {\sum\limits_{k = 0}^{{T_{f} - U_{f} }} {\frac{{\Lambda_{n,\ell } (\eta )}}{(\ell - 1)!}\frac{{(\gamma Q_{f} \eta )^{k} }}{k!} \cdot \frac{\Gamma (\ell + k + 1)}{{\kappa_{2}^{(\ell + k + 1)} }}} } } \\ \end{aligned} $$
(22)

where \( \kappa_{2} = Q_{f} \eta + ({z \mathord{\left/ {\vphantom {z {C\eta \varphi_{n} }}} \right. \kern-0pt} {C\eta \varphi_{n} }}) \), then to separate the integral part in (22) and to list it as

$$ \begin{aligned} & \int_{{\gamma_{th,m} }}^{{\gamma_{th,m + 1} }} {\frac{{(c\eta \varphi_{n} )(Q_{f} \eta )^{k} \cdot \gamma^{k} \cdot e^{{ - \beta_{m} q\gamma }} }}{{(\gamma + c\varphi_{n} Q_{f} \eta^{2} )^{(\ell + k + 1)} }}d\gamma } , & = \int_{{\gamma_{th,m} }}^{{\gamma_{th,m + 1} }} {(c\eta \varphi_{n} )(Q_{f} \eta )^{k} \cdot \gamma^{k} \cdot (\gamma + c\varphi_{n} Q_{f} \eta^{2} )^{(\ell + k + 1)} \cdot e^{{ - \beta_{m} q\gamma }} d\gamma } \\ \end{aligned} $$
(23)

that is, the integral problem can be completed and reduced by the following steps. First, let \( u = - c\varphi_{n} Q_{f} \eta^{2} \) and \( \beta = \beta_{m} q \) in (23), then after applying the change of variables and resorting the identity [30, Eq. 3.583, p. 365], it can be re-written as

$$ \begin{aligned} & \int_{u}^{\infty } {x^{v - 1} \cdot (x - u)^{\mu - 1} } \cdot e^{ - \beta x} dx & = \beta^{{ - ({{\mu + u)} \mathord{\left/ {\vphantom {{\mu + u)} 2}} \right. \kern-0pt} 2}}} \cdot u^{{{{(u + v - 2)} \mathord{\left/ {\vphantom {{(u + v - 2)} 2}} \right. \kern-0pt} 2}}} \cdot \Gamma (u) \cdot \exp ({{\beta u} \mathord{\left/ {\vphantom {{\beta u} 2}} \right. \kern-0pt} 2}) \cdot W_{{{{(v - u)} \mathord{\left/ {\vphantom {{(v - u)} 2}} \right. \kern-0pt} 2}, \, {{(1 - u - v)} \mathord{\left/ {\vphantom {{(1 - u - v)} 2}} \right. \kern-0pt} 2} \, }} (\beta u),[\text{Re} (\mu ) > 0,\text{Re} (\beta u) > 0] \\ \end{aligned} $$
(24)

where the Whittaker’s function is adopted as

$$ W_{\lambda ,\mu } (z) = \frac{\Gamma ( - 2\mu )}{\Gamma (0.5 - \mu - \lambda )}M_{\lambda ,\mu } (z) + \frac{\Gamma (2\mu )}{\Gamma (0.5 + \mu - \lambda )}M_{\lambda , - \mu } (z), $$
(25)

where

$$ M_{\lambda ,\mu } (z) = z^{{\mu + \frac{1}{2}}} \cdot e^{ - z/2} \cdot \varPhi (\mu - \lambda + 0.5, \, 2\mu + 1;z), $$
(26)

and where \( \varPhi (\alpha ,\gamma ;z) =_{1} F_{1} (\alpha ;\gamma ;z) \) stands for confluent hyper geometric function, that is, it can be expanded as \( _{1} F_{1} (\alpha ;\gamma ;z) = \sum\nolimits_{k = 0}^{\infty } {{{(\alpha )_{k} \cdot z^{k} } \mathord{\left/ {\vphantom {{(\alpha )_{k} \cdot z^{k} } {(\gamma )_{k} k!}}} \right. \kern-0pt} {(\gamma )_{k} k!}}} \). Once the integral shown in (24) has been solved by plugging it back into (22) and treating the corresponding terms with linear algebra operation the Eq. (20) can be calculated finally as

$$ \begin{aligned} Tp({\mathbf{Bs}}) & = {\mathbf{Bs}}\sum\limits_{m = 1}^{Card(M)} {\frac{{\log_{2} \left( M \right)}}{\Delta (\eta )}} \sum\limits_{q = 0}^{Q} {\left[ {\left( {_{q}^{Q} } \right)\left( { - A_{m} } \right)^{q}} \right]} \beta^{{ - ({{\mu + u)} \mathord{\left/ {\vphantom {{\mu + u)} 2}} \right. \kern-0pt} 2}}} \cdot u^{{{{(u + v - 2)} \mathord{\left/ {\vphantom {{(u + v - 2)} 2}} \right. \kern-0pt} 2}}} \\ & \quad \times \Gamma (u) \cdot \exp ({{\beta u} \mathord{\left/ {\vphantom {{\beta u} 2}} \right. \kern-0pt} 2}) \cdot W_{{{{(v - u)} \mathord{\left/ {\vphantom {{(v - u)} 2}} \right. \kern-0pt} 2}, \, {{(1 - u - v)} \mathord{\left/ {\vphantom {{(1 - u - v)} 2}} \right. \kern-0pt} 2} \, }} (\beta u)\sum\limits_{n = 0}^{N - 1} {\sum\limits_{\ell = 0}^{L - 1} {\sum\limits_{k = 0}^{{T_{f} - U_{f} }} {\frac{{\Lambda_{n,\ell } (\eta ) \cdot \Gamma (\ell + k + 1)}}{\Delta (\eta ) \cdot k!(\ell - 1)!}} } } \\ \end{aligned} $$
(27)

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Chen, J.I., Huang, C. Correlation Phenomenon Versus Beam-Subset Scenario in MIMO-Femtocell System with Fading Environments. Wireless Pers Commun 86, 995–1011 (2016). https://doi.org/10.1007/s11277-015-2968-z

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Keywords

  • Beam-subset
  • Channel correlation
  • Coverage area
  • MIMO-femtocell