Relay Placement for Coverage Extension in Cellular Wireless Networks Under Composite Fading Model

Abstract

In relay-assisted cellular networks, mobile stations are connected to base station through two or more single-hop communication links, where the intermediate nodes act as relay stations (RSs). The focus of this paper is on two-hop relay assisted cellular networks, where optimal relay placement is a crucial issue for achieving maximum extension of the cell coverage. However, the location of RS has significant impact on signal-to-interference plus noise ratio (i.e., SINR) and outage probability experienced on the access and backhaul links. Moreover, the frequency re-use factor also has significant influence on the SINR. In this paper, we develop analytical models for computing the SINR and outage probability performance of a two-hop relay assisted cellular network for both downlink (DL) as well as uplink (UL) transmission scenarios, considering the impact of path loss, shadowing, Nakagami fading and co-channel interference. We then investigate optimal placement of RS while satisfying the required criterion on probability of correct decoding, initially by considering the DL scenario alone and then by considering both DL and UL scenarios jointly. Through extensive evaluations, we report the impact of realistic propagation models on outage probability, optimal relay position and the cell coverage radius. Further, the model can be used to find the impact of co-channel re-use factor on optimal relay positioning in two-hop cellular networks.

Appendix: Extended Fenton–Wilkinson’s Method

Let $({I_i: i= 1, 2,...N})$ be the log-normal random variables and $(X_i; i=1, 2,.. N)$ be the corresponding Gaussian random variable in decibels.

$$\begin{aligned} X_i = 10 \log _{10} I_i = \mu _{X_i}+\chi _{i} \end{aligned}$$

(44)

where $\mu _{X_i}$ is the mean and $\chi _{i}$ is a zero-mean normally distributed RV in dB with standard deviation $\sigma _{X_i}$, also in dB. Further, let $Y_i = \ln I_i$ then $Y_i$ is normally distributed random variable with mean $\mu _{Y_i}$ and standard deviation $\sigma _{Y_i}$ in logarithmic units, given respectively, by

$$\begin{aligned} \mu _{Y_i} = a \mu _{X_i}, \quad \sigma _{Y_i} = a \sigma _{X_i} \end{aligned}$$

(45)

Consider the sum of the log-normal random variables $I = \sum _{i=1}^{N} I_i$, It is well accepted that the distribution of I can be approximated by another lognormal distribution or equivalently, that $X = 10 \log _{10} I$ follows a normal distribution. Following the derivation detailed in [30] for the extended Fenton-Wilkinson method, the simple closed-form expressions for mean and variance of approximated log-normal random variables follows as

$$\begin{aligned} \mu _{X}&= \frac{1}{a_0} \left( 2 \ln (v_1)-\frac{1}{2}\ln (v_2)\right) \end{aligned}$$

(46)

$$\begin{aligned} \sigma ^2_{X}&= \frac{1}{a_0^2} \left( \ln (v_2)-2\ln (v_1)\right) \end{aligned}$$

(47)

where $v_1$ and $v_2$ are given as

$$\begin{aligned} v_1&= \sum _{i=1}^{N} \exp \left( \mu _{Y_i}+ \frac{\sigma ^2_{Y_i}}{2} \right) \end{aligned}$$

(48)

$$\begin{aligned} v_2&= \sum _{i=1}^{N} \exp \left( 2 \mu _{Y_i}+ 2{\sigma ^2_{Y_i}} \right) + 2 \sum _{i=1}^{N-1}\sum _{j=i+1}^{N} \exp \left( \mu _{Y_i}+\mu _{Y_j} \right) \nonumber \\&\times \exp \left( \frac{1}{2}(\sigma ^2_{Y_i}+\sigma ^2_{Y_j}+ 2 r_{i,j} \sigma _{Y_i}\sigma _{Y_j} ) \right) \end{aligned}$$

(49)

Here $r_{i,j}$ is the correlaton coefficient and it is defined as follows

$$\begin{aligned} r_{i,j} = \frac{E[(Y_i -\mu _{Y_i}) (Y_j -\mu _{Y_j})]}{\sigma _{Y_i} \sigma _{Y_j}} \end{aligned}$$

(50)

Since $Y_i$ is a scaled version of $X_i$, $r_{i,j}$ is also the correlation coefficient of $X_i$ and $X_j$.