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 [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 .
The Fenton and Wilkinson (FW) method  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 . Let us denote as and as . From , 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 , 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  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.