Outage probability of SSTS for distributed antenna systems over Rayleigh fading channels in multi-cell environment

Abstract

In this paper, downlink outage probability for distributed antenna systems (DAS) in multicell environment is proposed, while based on maximum desired signal criterion, a single selection transmission scheme (SSTS) is proposed. Usually, adopting central limit theorem (CLT) method, the component of interference plus noise is considered as a fixed-variance Gaussian random variable in most papers. However, the aforementioned method does not reflect the effect of short-term fading on interference and its usage is in the constraints of restrictive conditions. To relax the constraints, non-central limit theorem (NCLT) is introduced, which treats the variance of interference plus noise as a changeable-variance random variable influenced by short-term fading. It is assumed that channels are independent identical Rayleigh fading with propagation path-loss, and the closed-form expression of outage probability for DAS is derived. Finally, simulation results demonstrate the validity of theoretical analysis.

Appendices

Appendix A: derivation of Eq. (10)

$$\begin{aligned}&\Pr \left( {\gamma>\gamma _{th} } \right) \nonumber \\&\quad =\Delta \Pr \left( {\frac{z_{\max } }{L_{q,2} P_{q,2} \left| {h_{q,2} } \right| ^{2}+L_{l,3} P_{l,3} \left| {h_{l,3} } \right| ^{2}+\sigma _n^2 /S_s }>\gamma _{th} } \right) \nonumber \\&\quad =\int _{t_2 =0}^\infty {\int _{t_3 =0}^\infty {\Pr \left( {z_{\max } >\gamma _{th} \left( {t_2 +t_3 +\sigma _n^2 /S_s } \right) } \right) } }\nonumber \\&\times \exp \left( {\frac{-t_2 }{L_{q,2} P_{q,2} }} \right) \exp \left( {\frac{-t_3 }{L_{l,3} P_{l,3} }} \right) dt_2 dt_3 \nonumber \\&\quad =\Delta \sum _{u=1}^{\left| D \right| } {\left( {-1} \right) ^{u-1}}\sum _{u=1}^{\left( \left( {\mathop {\left| D \right| }\limits _u } \right) \right) }\int _{t_2 =0}^\infty \int _{t_3 =0}^\infty \exp \left( -\lambda _{uv} \gamma _{th} \right. \nonumber \\&\quad \left. \left( {t_2 +t_3 +\sigma _n^2 /S_s } \right) \right) \nonumber \\&\times \exp \left( {\frac{-t_2 }{L_{q,2} P_{q,2} }} \right) \exp \left( {\frac{-t_3 }{L_{l,3} P_{l,3} }} \right) dt_2 dt_3 \nonumber \\&\quad =\Delta \sum _{u=1}^{\left| D \right| } {\left( {-1} \right) ^{u-1}} \nonumber \\&\times \sum _{u=1}^{\left( \left( {\mathop {\left| D \right| }\limits _u } \right) \right) } \exp \left( {-\lambda _{uv} \gamma _{th} \sigma _n^2 /S_s } \right) \nonumber \\&\quad \int _{t_2 =0}^\infty {\int _{t_3 =0}^\infty {\hbox {!`}\exp \left( {\frac{-\lambda _{uv} \gamma _{th} L_{q,2} P_{q,2} -1}{L_{q,2} P_{q,2} }} \right) } } \nonumber \\&\quad \times \exp \left( {\frac{-\lambda _{uv} \gamma _{th} L_{l,3} P_{l,3} -1}{L_{l,3} P_{l,3} }t_3 } \right) dt_2 dt_3 \nonumber \\&\quad =\Delta \sum _{u=1}^{\left| D \right| } {\left( {-1} \right) ^{u-1}} \sum _{u=1}^{\left( \left( {\mathop {\left| D \right| }\limits _u } \right) \right) } {\exp \left( {-\lambda _{uv} \gamma _{th} \sigma _n^2 /S_s } \right) } \nonumber \\&\times \left[ {\frac{L_{q,2} P_{q,2} }{\lambda _{uv} \gamma _{th} L_{q,2} P_{q,2} +1}\frac{L_{l,3} P_{l,3} }{\lambda _{uv} \gamma _{th} L_{l,3} P_{l,3} +1}} \right] \nonumber \\&\quad =\Delta \sum _{u=1}^{\left| D \right| } {\left( {-1} \right) ^{u-1}} \sum _{u=1}^{\left( \left( {\mathop {\left| D \right| }\limits _u } \right) \right) } {\exp \left( {-\lambda _{uv} \gamma _{th} \sigma _n^2 /S_s } \right) } \nonumber \\&\quad \left( {\frac{1}{\lambda _{uv} \gamma _{th} L_{q,2} P_{q,2} +1}\frac{1}{\lambda _{uv} \gamma _{th} L_{l,3} P_{l,3} +1}} \right) \end{aligned}$$

(17)

$$\begin{aligned} where{\begin{array}{ll} &{} \Delta \\ \end{array} }=\frac{1}{L_{q,2} P_{q,2} }\frac{1}{L_{l,3} P_{l,3} } \end{aligned}$$

Appendix B: Gauss-Hermite quadrature

The Gauss-Hermite formula is expressed as [29]:

$$\begin{aligned} \int _{-\infty }^{+\infty } {\exp \left( {-x^{2}} \right) } f\left( x \right) dx= & {} \sum _{i=1}^n {H_i } f\left( {x_i } \right) +R_n\nonumber \\\simeq & {} \sum _{i=1}^n {H_i } f\left( {x_i } \right) \end{aligned}$$

(18)

where n is the order of Hermite polynomial (the number of sample points utilized for the approximation). The value $x_{i}$ is the roots of the Hermite polynomial$H_n \left( x \right) {\begin{array}{ll} &{} {\left( {i=1,\cdots ,n} \right) } \\ \end{array} }$ and the associated weights $H_{i}$ are given as:

$$\begin{aligned} H_i =\frac{2^{n-1}n!\sqrt{\pi }}{n^{2}\left[ {H_{n-1} \left( {x_i } \right) } \right] ^{2}} \end{aligned}$$

(19)

The Hermite polynomial $H_n \left( x \right)$is written as [28]:

$$\begin{aligned} \begin{array}{l} H_n \left( x \right) =\left( {-1} \right) ^{n}\exp \left( {x^{2}} \right) \frac{d^{n}}{dx^{n}}\left( {\exp \left( {-x^{2}} \right) } \right) {\begin{array}{ll} &{} {or} \\ \end{array} } \\ H_n \left( x \right) =2^{n}x^{n}-2^{n-1}\left( {{\begin{array}{l} n \\ 2 \\ \end{array} }} \right) x^{n-1}+ \\ 2^{n-2}\cdot 1\cdot 3\cdot \left( {{\begin{array}{l} n \\ 4 \\ \end{array} }} \right) x^{n-4}-2^{n-3}\cdot 1\cdot 3\cdot 5\cdot \left( {{\begin{array}{l} n \\ 6 \\ \end{array} }} \right) x^{n-6}+\cdots \\ H_0 \left( x \right) =1 \\ \end{array} \end{aligned}$$

(20)

The remainder in Eq. (15) is given as:

$$\begin{aligned} R_n =\frac{n!\sqrt{\pi }}{2^{n}\left( {2n} \right) !}f^{\left( {2n} \right) }\left( \xi \right) {\begin{array}{ll} &{} {\left( {-\infty<\xi <\infty } \right) } \\ \end{array} } \end{aligned}$$

(21)

where $\xi$ is arbitrarily chosen and $f^{\left( n \right) }\left( x \right)$ is the $n^{\mathrm{th}}$ derivative of f(x).

The precision of the Gauss-Hermite approximation is dominated by n. If the number of sample points (n) is insufficient, the approximate and exact curves do not match exactly. On the contrary, the larger n, the more accurate the approximation obtained.