- Research Article
- Open Access
A Semianalytical PDF of Downlink SINR for Femtocell Networks
© KiWon Sung et al. 2010
- Received: 31 August 2009
- Accepted: 17 February 2010
- Published: 7 April 2010
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.
- Probability Density Function
- Transmission Power
- Mobile Station
- Outage Probability
- Path Loss
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 , 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 . In , 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 , 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  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)  in an indoor environment. It has been suggested that femto BSs can significantly improve system spectral efficiency by up to a factor of five . 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 . 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 , 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  and Mehta et al.  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.
Path loss exponent
Path loss constant
MS noise figure
BS transmission power
BS antenna gain
MS antenna gain
Number of interfering cells
Frequency reuse factor
Kullback-Leibler Distance between the simulation and the analysis ( ).
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 ) 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.
scenario 1: dB,
scenario 2: dB and dB,
scenario 3: dB,
scenario 4: dB and dB.
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.
scenario 5: dBm,
(ii)scenario 6: dBm and dBm,
scenario 7: dBm and dBm,
scenario 8: dBm and dBm.
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.
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.
This work was supported by the National Research Foundation of Korea, Grant funded by the Korean Government (NRF-2007-357-D00165).
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