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A Semianalytical PDF of Downlink SINR for Femtocell Networks


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.

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 [68] 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 [1217]. 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 [1823]. 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.

Let be the transmission power of the BS in the CoI. It is attenuated by path loss and shadow fading. Let be the RV which models the shadow fading. It is generally assumed that follows a Gaussian distribution with zero mean and variance in dB. Thus the received signal power at the MS from the serving BS, , is denoted by


where and are antenna gains of the BS and the MS, respectively, is constant of path loss in the CoI, is path loss exponent of CoI, and . The denotes natural logarithm. can be rewritten as follows:


Note that an RV follows a log-normal distribution if is a Gaussian distributed RV. Thus, follows a log-normal distribution conditioned on the location of MS . The PDF of is given by


where and .

Let be the received interference power from the th interfering BS. By denoting as the transmission power from the th BS, results in


where and are the path loss constant and exponent, respectively, on the link between the th BS and MS , and is a Gaussian RV for shadow fading with zero mean and variance on the link between the th BS and MS . Note that the transmission power of each interfering BS can be different since an uncoordinated femtocell deployment is considered. Path loss parameters and standard deviation of shadow fading can also be different in each BS in practical systems. The distance between MS and the th interfering BS is , which is obtained from


In a similar fashion to , follows a log-normal distribution with PDF given by


where and .

Background noise can be regarded as a constant value by assuming the constant noise figure and the noise temperature. Let be the background noise power at MS , given by


where is the Boltzmann constant, is the ambient temperature in Kelvin, is the channel bandwidth, and is the noise figure of the MS. In order to make mathematically tractable, we introduce an auxiliary Gaussian RV with 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:


Let us consider a system with no interference arising from the serving cell such as an OFDMA or a TDMA system. The downlink SINR of MS is denoted by , which is given by


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 [1921].

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 .

The Fenton and Wilkinson (FW) method [19] is one of the most frequently adopted approximations in literature. It obtains and by assuming that the first and second moments of match the sum of the moments of . It should be noted that the FW method is the only approximate method that provides a closed-form expression of and [20]. Let us denote as and as . From [19], the PDF of conditioned on the location of MS is given as follows:


where and are given by


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 method proposed in [20], which is called MWMZ method in this paper after the initials of authors, exploits the property of the moment-generating function (MGF) that the product of MGFs of independent RVs equals to the MGF of the sum of RVs. The MGF of RV is defined as


By the property of MGF,


While the closed-form expression for the MGF of log-normal distribution is not available, a series expansion based on Gauss-Hermite integration was employed in [20] to approximate the MGF. For a real coefficient , the MGF of the log-normal RV is given by


where and are weights and abscissas of the Gauss-Hermite series which can be found in [24, Table ]. From (13), a system of two nonlinear equations can be set up with two real and positive coefficients and as follows:


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.

As shown in (3), the received signal power, , follows a log-normal distribution. The sum of the received interference and the background noise power, , was also approximated as a log-normal RV. Thus, the SINR of the MS , , is the ratio of two log-normal RVs, which also follows a log-normal distribution. From (3) and (10), the PDF of is shown as


where and .

2.2. The PDF of Downlink SINR in a Cell

Up to this point, the PDF of the downlink SINR has been derived conditionally on the location of the MS . Let us denote the location of MS by . Since it is assumed that MSs are uniformly distributed within a circular area, the PDF of , , is as follows:


From (16) and (17), the joint distribution of the SINR and the MS location is


Let be the RV of the downlink SINR of an MS in an arbitrary location within a circular cell area. The PDF of can be obtained by integrating over and . Thus, we get


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

3. Numerical Results

The PDF of downlink SINR derived in (19) is calculated numerically and compared with a Monte Carlo simulation result in order to validate the analysis. We consider the nonline of sight (NLOS) indoor environment at 5.25 GHz as specified in [25, page 19] to be the basic environment for the comparison. The path loss formula is given as follows:


where is a reference distance in the far field. The interfering BSs are assumed to be femto BSs located on the same floor of a building throughout the experiments. However, interference scenarios such as femto BSs in different floors or outdoor macro-BSs can be easily examined by employing appropriate path loss models. The basic parameters used for the comparison are summarised in Table 1.

Table 1 Simulation parameters.

We assume that all interfering BSs are located at the same distance from the serving BS as shown in Figure 1. Cells are assumed to overlap each other to consider a dense deployment of the femto BSs. Although it is unlikely that the interfering BSs are in regular shapes in practical deployments, it is useful to consider this topology for examining the effects of parameters such as standard deviation of shadow fading, the number of BSs, wall penetration loss, and transmission power of BSs. It should be emphasised that the PDF derived in Section 2 can effectively take into account irregular locations and transmission powers of BSs.

Figure 1
figure 1

Locations of the CoI and the interfering BSs.

The result of the comparison is illustrated in Figure 2 where the PDFs derived by FW and MWMZ methods are compared with the Monte Carlo simulation result in Figure 2(a) and the cumulative distribution functions (CDFs) of the PDFs are depicted in Figure 2(b). The standard deviation of shadow fading, and , is considered to be 3.5 dB since it represents a typical value in an indoor office environment according to the measurement results in [25]. It is observed that the numerically obtained PDFs from both of the methods are in good agreement with the Monte Carlo simulation.

Figure 2
figure 2

A comparison of the PDF and CDF obtained by the analysis with the result of Monte Carlo simulation ( = = 3. 5 dB).PDF of the downlink SINRCDF of the downlink SINR

The impact of the parameter on the performance of MWMZ method is shown in Figure 3 where the tail portion of the CDF (low SINR region) is depicted in Figure 3(a) and the head portion of the CDF (high SINR regime) is illustrated in Figure 3(b). Smaller tends to give more accurate match in low SINR region while resulting in larger error in high SINR region. is chosen in the experiments since it brings about relatively small difference from simulations throughout the whole SINR region.

Figure 3
figure 3

Impact of on the performance of MWMZ method: tail and head portions of CDF ( = = 3. 5 dB).Tail portion of the CDFHead portion of the CDF

Figure 4 shows the CDFs when the standard deviation of shadow fading is 8.0 dB. While the SINR obtained by MWMZ method is still in good agreement with the simulation result, the difference between the analysis and the simulation is apparent in case of FW method. It means that FW method cannot be used in an environment where high shadow fading is experienced by MSs. In order to quantify the effect of shadow fading standard deviation, we introduce 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


The KLD is a nonnegative entity which measures the difference of the 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.

Table 2 Kullback-Leibler Distance between the simulation and the analysis ().
Figure 4
figure 4

A comparison of the CDF obtained by the analysis with the result of Monte Carlo simulation ( = = 8. 0 dB).

The effect of the number of interfering BSs is examined in Figure 5. It is known that the sum of log-normal RVs is not accurately approximated by a log-normal distribution as the number of summands increases [22]. This means that the derived SINR may not be accurate for a large number of interfering BSs. Figure 5 shows the KLD of FW and MWMZ methods compared to simulation results when is between 2 and 60. An impairment in the accuracy is not observed as increases, which means that the derivation of SINR in this paper is useful for the practical range of interfering BSs in the downlink of cellular systems.

Figure 5
figure 5

Kullback-Leibler Distance between simulation and analysis ( = = 3. 5 dB).

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.

Clearly isolation of a cell by wall penetration loss is an inherent property of indoor femtocell networks which can be utilised as a means of interference mitigation. Let be the wall penetration loss between the CoI and the interfering BS . The effect of is examined in Figure 6 where is assumed to be identical for all interfering BSs. It is shown that has significant impact on the SINR of the femtocell. The outage probability drops to 3.7% when  dB and to 0.5% when  dB. This result implies that the implementation of the femtocell network is viable without complicated interference mitigation method if the wall isolation between BSs is provided.

Figure 6
figure 6

Effect of wall penetration loss on CDF of SINR (FW method, = = 3. 5 dB).

In Figure 7, different wall losses, , are considered. We examine the following scenarios:

  1. (i)

    scenario 1:  dB,

  2. (ii)

    scenario 2:  dB and  dB,

  3. (iii)

    scenario 3:  dB,

  4. (iv)

    scenario 4:  dB and  dB.

Figure 7
figure 7

Effect of different wall penetration losses on CDF of SINR (FW method, = = 3. 5 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.

Similar behaviours are observed in Figure 8 where different BS transmission powers are considered. The effect of the uncoordinated power is examined by considering the following scenarios where  dBm:

  1. (i)

    scenario 5:  dBm,

  2. (ii)

    (ii)scenario 6:  dBm and  dBm,

  3. (iii)

    scenario 7:  dBm and  dBm,

  4. (iv)

    scenario 8:  dBm and  dBm.

Figure 8
figure 8

CDF of SINR with uncoordinated BS transmission power (FW method, = = 3. 5 dB).

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.

Finally, the effects of the BSs transmission power and the background noise are shown in Figure 9. If the transmit power drops below a certain level, a change in the PDF can be observed. For 10 dBm transmit power, for example, a noticeable impairment of the SINR can be seen. This is because the noise power remains the same regardless of the transmission power. In the case of increased transmission power, however, little change in the SINR distribution is observed. This means that the SINR is already interference limited with a transmission power of 20 dBm. Thus, the increase in the transmit power of BSs does not result in an improvement as expected.

Figure 9
figure 9

Effect of BS transmission power and background noise on CDF of SINR (FW method, = = 3. 5 dB).

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.


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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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Sung, K., Haas, H. & McLaughlin, S. A Semianalytical PDF of Downlink SINR for Femtocell Networks. J Wireless Com Network 2010, 256370 (2010).

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