# A Semianalytical PDF of Downlink SINR for Femtocell Networks

- KiWon Sung
^{1}Email author, - Harald Haas
^{2}and - Stephen McLaughlin
^{2}

**2010**:256370

https://doi.org/10.1155/2010/256370

© KiWon Sung et al. 2010

**Received: **31 August 2009

**Accepted: **17 February 2010

**Published: **7 April 2010

## Abstract

This paper presents a derivation of the probability density function (PDF) of the signal-to-interference and noise ratio (SINR) for the downlink of a cell in multicellular networks. The mathematical model considers uncoordinated locations and transmission powers of base stations (BSs) which reflect accurately the deployment of randomly located femtocells in an indoor environment. The derivation is semianalytical, in that the PDF is obtained by analysis and can be easily calculated by employing standard numerical methods. Thus, it obviates the need for time-consuming simulation efforts. The derivation of the PDF takes into account practical propagation models including shadow fading. The effect of background noise is also considered. Numerical experiments are performed assuming various environments and deployment scenarios to examine the performance of femtocell networks. The results are compared with Monte Carlo simulations for verification purposes and show good agreement.

## Keywords

## 1. Introduction

Signal-to-interference and noise ratio (SINR) is one of the most important performance measures in cellular systems. Its probability distribution plays an important role for system performance evaluation, radio resource management, and radio network planning. With an accurate probability density function (PDF) of SINR, the capacity and coverage of a system can be easily predicted, which otherwise should rely on complicated and time-consuming simulations.

There have been various approaches to investigate the statistical characteristics of received signal and interference. The other-cell interference statistics for the uplink of code division multiple access (CDMA) system was investigated in [1], where the ratio of other-cell to own-cell interference was presented. The result was extended to both the uplink and the downlink of general cellular systems by [2]. In [3], the second-order statistics of SIR for a mobile station (MS) were investigated. In [4, 5], the prediction of coverage probability was addressed which is imperative in the radio network planning process. The probability that SINR goes below a certain threshold, which is termed outage probability, is another performance measure that has been extensively explored. The derivation of the outage probability can be found in [6–8] and references therein.

While most of the contributions have focused on a particular performance measure such as coverage probability or outage probability, an explicit derivation of the probability distribution for signal and interference has also been investigated [9, 10]. In [9], a PDF of adjacent channel interference (ACI) was derived in the uplink of cellular system. A PDF of SIR in an ad hoc system was studied in [10] assuming single transmitter and receiver pair.

In this paper, we derive the PDF of the SINR for the downlink of a cell area in a semianalytical fashion. A practical propagation loss model combined with shadow fading is considered in the derivation of the PDF. We also consider background noise in the derivation, which is often ignored in the references. Uncoordinated locations and transmission powers of interfering base stations (BSs) are considered in the model to take into account the deployment of femtocells (or home BSs) [11] in an indoor environment. It has been suggested that femto BSs can significantly improve system spectral efficiency by up to a factor of five [12]. It has also been found that in closed-access femtocell networks macrocell MSs in close vicinity to a femtocell greatly suffer from high interference and that such macrocell MSs cause destructive interference to femtocell BSs [13]. Thus, an accurate model for the probability distribution of the SINR assuming an uncoordinated placement of indoor BSs can be vital for further system improvements. In spite of the recent efforts for the performance evaluation of femtocells, most of the works relied on system simulation experiments [12–17]. To the best of our knowledge, the PDF of SINR for the outlined conditions and environment has not been derived before.

Since shadow fading is generally considered to follow a log-normal distribution, the PDF of the sum of log-normal RVs should be provided as a first step in the derivation of the SINR distribution. During the last few decades numerous approximations have been proposed to obtain the PDF of the sum of log-normal RVs since the exact closed-form expression is still unknown [18–23]. So far, no method offers significant advantages over another [18], and sometimes a tradeoff exists between the accuracy of the approximation and the computational complexity. We adopt two methods of approximation proposed by Fenton and Wilkinson [19] and Mehta et al. [20] which provide a good balance between accuracy and complexity. The performance of both methods is examined in various environments and a guideline is provided for choosing one of the methods.

The derivation of SINR distribution in this paper is semianalytical in the sense that the PDF can be easily calculated by applying standard numerical methods to equations obtained from analysis. Numerical experiments are performed to investigate the effects of standard deviation of shadow fading, the number of interfering BSs, wall penetration loss, and transmission powers of BSs. The results obtained are also validated by comparison with Monte Carlo simulations.

The paper is organised as follows. In Section 2, the PDF of the downlink SINR is derived. Numerical experiments are performed in various environments and the results are compared with Monte Carlo simulations in Section 3. Finally, the conclusions are provided in Section 4.

## 2. Derivation of the PDF of Downlink SINR

The derivation of the PDF of downlink SINR is divided into two parts. First, the SINR of an arbitrary MS is expressed depending on its location in Section 2.1. Methods of approximating the sum probability distribution of log-normal RVs are discussed and adopted in the SINR derivation. Second, the PDF of SINR unconditional on the location of the MS is derived in Section 2.2.

### 2.1. Location-Dependent SINR

Let us consider a femtocell which will be termed the cell of interest (CoI). The CoI is assumed to be circular with a cell radius . We assume the MSs in the CoI to be uniformly distributed in the cell area. An arbitrary MS is considered whose location is , where and . The MS receives interference from BSs that are a mixture of femto and macro-BSs. The network is modelled using polar coordinates where the BS of the CoI is located at the center and the location of the th interfering BS is denoted by . In a practical deployment of femtocell systems, the placement of BSs in a random and uncoordinated fashion is unavoidable and may generate high interference scenarios and dead spots particularly in an indoor environment.

*zero*mean and

*zero*variance so that can be treated as log-normal RV with parameters of and . Note that has a constant value, and this is accounted for by the fact that the defined RV has zero variance. This particular definition is useful for the determination of the final PDF. By introducing , can be rewritten as follows:

In (9), denotes the sum of the interference powers and the background noise power. Since all of and are log-normally distributed, is the sum of log-normal RVs. Note that the exact closed-form expression is not known for the PDF of the sum of log-normal RVs. The most widely accepted approximation approach is to assume that the sum of log-normal RVs follows a log-normal distribution. Various methods have been proposed to find out parameters of the distribution [19–21].

Let be independent but not necessarily identical log-normal RVs, where and is a Gaussian distributed RV with mean and variance . The sum of RVs is denoted by such that . Approximations assume that follows a log-normal distribution with parameters and .

In spite of its simplicity, the accuracy of the FW method suffers at high values of . This means that the method may break down when an MS experiences a large standard deviation of shadow fading from interfering BSs. Thus, we adopt another method of approximating the sum of log-normal RVs which gives a more accurate result at a cost of increased computational complexity.

The variables to be solved by (15) are and . The right-hand side of (15) is a constant value which can be calculated with known parameters.

By employing (15), and in (10) can be effectively obtained by standard numerical methods such as the function "fsolve" in Matlab. The coefficient adjusts weight of penalty for inaccuracy of the PDF. Increasing imposes more penalty for errors in the head portion of the PDF of , whereas smaller penalises errors in the tail portion. Thus, smaller is recommended if one is interested in the PDF of poor SINR region, while larger should be used to examine statistics of higher SINR.

### 2.2. The PDF of Downlink SINR in a Cell

Note that in (19) is a function of . We employ numerical integration methods to obtain the final PDF.

## 3. Numerical Results

Simulation parameters.

Parameter | Value |
---|---|

Cell radius | 50 m |

Path loss exponent | 3.68 |

Path loss constant | 43.8 dB |

Center frequency | 5.25 GHz |

Channel bandwidth | 10 MHz |

MS noise figure | 7 dB |

BS transmission power | 20 dBm |

BS antenna gain | 3 dBi |

MS antenna gain | 0 dBi |

Number of interfering cells | 6 |

Frequency reuse factor | 1 |

*Kullback-Leibler Distance*(KLD) which is a measure of divergence between two probability distributions [26]. For the two PDFs and the KLD is defined as

*estimated*distribution from the

*real*distribution in a statistical sense. It becomes zero if and only if = . Table 2 presents the KLD for various standard deviations of shadow fading by assuming that the simulation results represent the true PDFs of SINR. It is shown in the table that the KLD of FW method soars when the standard deviation of shadow fading is higher than 6 dB. This implies that the range of standard deviation in which FW method can be adopted is between 3 dB and 6 dB, which is a typical range of shadow fading in an in-building environment [14, 25]. On the contrary, the MWMZ method maintains an acceptable level of the KLD even for the high shadow fading standard deviation. FW method is preferred if both of the methods are applicable due to its simplicity.

The numerical results so far have focused on the verification of the derived PDF. Now we investigate the performance of femtocell network in various environments. An important observation in Figure 2 is that the probability of the SINR below 2.2 dB (a typical threshold for binary phase shift keying (BPSK) to achieve reasonable BER performance [27]) is about 0.38 for the parameters in Table 1. In other words, the outage probability is around 38%. This means that a dense deployment of femtocells in a building results in unacceptable outage, unless intelligent interference avoidance and interference mitigation techniques are put in place.

- (i)
- (ii)
- (iii)
- (iv)

It is shown that scenarios 1 and 2 give similar performance. This means that the isolation from one or few BSs does not result in the performance improvement when the CoI is not protected from the majority of interfering BSs. On the contrary, a considerable difference is observed between scenarios 3 and 4. Significant degradation in the SINR is caused by one BS which is not isolated by the wall.

- (i)
- (ii)
- (iii)
- (iv)

Figure 8 shows the CDFs of SINR by FW method with the assumption that dB . It is observed that scenario 6 results in the worst SINR. This means that the higher transmission powers of a few BSs result in significantly decreased SINR. However, reduced transmission power in only a subset of neighbouring BSs does not necessarily improve the SINR because the predominant interference largely depends on the BSs which use high transmission powers. A similar trend is shown when comparing scenario 7 and scenario 8. The SINR performance is worse in scenario 8 than in scenario 7 for the same reason.

## 4. Conclusion

In this paper, the PDF of the SINR for the downlink of a cell has been derived in a semianalytical fashion. It models an uncoordinated deployment of BSs which is particularly useful for the analysis of femtocells in an indoor environment. A practical propagation model including log-normal shadow fading is considered in the derivation of the PDF. The PDF presented in this paper has been obtained through analysis and calculated through standard numerical methods. The comparison with Monte Carlo simulation shows a good agreement, which indicates that the semianalytical PDF obviates the need for complicated and time-consuming simulations. The results also provide some insights into the performance of the indoor femtocells with universal frequency reuse. First, significant outage can be expected for a scenario where femto BSs are densely deployed in an in-building environment. This highlights that interference avoidance and mitigation techniques are needed. The isolation offered by wall penetration loss is an attractive solution to cope with the interference. Second, the SINR can be worsened by uncoordinated transmission powers of BSs. Thus, a coordination of BSs transmission power is needed to prevent a significant decrease in SINR.

## Declarations

### Acknowledgment

This work was supported by the National Research Foundation of Korea, Grant funded by the Korean Government (NRF-2007-357-D00165).

## Authors’ Affiliations

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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.