A wide variety of wireless communication systems have been developed to provide access to the communications infrastructure for mobile or fixed users in a myriad of operating environments. Most of today’s wireless systems are based on the cellular radio concept. Cellular communication systems allow a large number of mobile users to seamlessly and simultaneously communicate to wireless modems at fixed base stations using a limited amount of radio frequency (RF) spectrum. The RF transmissions received at the base stations from each mobile are translated to baseband, or to a wideband microwave link, and relayed to mobile switching centers (MSCs), which connect the mobile transmissions with the Public Switched Telephone Network (PSTN). Similarly, communications from the PSTN are sent to the base station, where they are transmitted to the mobile. Cellular systems employ either frequency division multiple access (FDMA), time division multiple access (TDMA), code division multiple access (CDMA), or spatial division multiple access (SDMA) [1, 2].
Wireless communication links experience hostile physical channel characteristics, such as time-varying multipath and shadowing due to large objects in the propagation path. In addition, the performance of wireless cellular systems tends to be limited by interference from other users, and for that reason, it is important to have accurate techniques for modeling interference. These complex channel conditions are difficult to describe with a simple analytical model, although several models do provide analytical tractability with reasonable agreement to measured channel data [3, 4]. However, even when the channel is modeled in an analytically elegant manner, in the vast majority of situations it is still difficult or impossible to construct analytical solutions for link performance when error control coding, equalization, diversity, and network models are factored into the link model. Simulation approaches, therefore, are usually required when analyzing the performance of cellular communication links.
Like wireless links, the system performance of a cellular radio system is most effectively modeled using simulation, due to the difficulty in modeling a large number of random events over time and space. These random events, such as the location of users, the number of simultaneous users in the system, the propagation conditions, interference and power level settings of each user, and the traffic demands of each user, combine together to impact the overall performance seen by a typical user in the cellular system. The aforementioned variables are just a small sampling of the many key physical mechanisms that dictate the instantaneous performance of a particular user at any time within the system. The term cellular radio system, therefore, refers to the entire population of mobile users and base stations throughout the geographic service area, as opposed to a single link that connects a single mobile user to a single base station. To design for a particular system-level performance, such as the likelihood of a particular user having acceptable service throughout the system, it is necessary to consider the complexity of multiple users that are simultaneously using the system throughout the coverage area. Thus, simulation is needed to consider the multi-user effects upon any of the individual links between the mobile and the base station.
The link performance is a small-scale phenomenon, which deals with the instantaneous changes in the channel over a small local area, or small time duration, over which the average received power is assumed constant [1]. Such assumptions are sensible in the design of error control codes, equalizers, and other components that serve to mitigate the transient effects created by the channel. However, in order to determine the overall system performance of a large number of users spread over a wide geographic area, it is necessary to incorporate large-scale effects such as the statistical behavior of interference and signal levels experienced by individual users over large distances, while ignoring the transient channel characteristics. One may think of link-level simulation as being a vernier adjustment on the performance of a communication system, and the system-level simulation as being a coarse, yet important, approximation of the overall level of quality that any user could expect at any time.
Cellular systems achieve high capacity (e.g., serve a large number of users) by allowing the mobile stations to share, or reuse a communication channel in different regions of the geographic service area. Channel reuse leads to co-channel interference among users sharing the same channel, which is recognized as one of the major limiting factors of performance and capacity of a cellular system. An appropriate understanding of the effects of co-channel interference on the capacity and performance is therefore required when deploying cellular systems, or when analyzing and designing system methodologies that mitigate the undesired effects of co-channel interference. These effects are strongly dependent on system aspects of the communication system, such as the number of users sharing the channel and their locations. Other aspects, more related to the propagation channel, such as path loss, shadow fading (or shadowing), and antenna radiation patterns are also important in the context of system performance, since these effects also vary with the locations of particular users. In this chapter, we will discuss the application of system-level simulation in the analysis of the performance of a cellular communication system under the effects of co-channel interference. We will analyze a simple multiple-user cellular system, including the antenna and propagation effects of a typical system. Despite the simplicity of the example system considered in this chapter, the analysis presented can easily be extended to include other features of a cellular system.
Cellular systems provide wireless coverage over a geographic service area by dividing the geographic area into segments called cells as shown in Figure 17.1. The available frequency spectrum is also divided into a number of channels with a group of channels assigned to each cell. Base stations located in each cell are equipped with wireless modems that can communicate with mobile users. Radio frequency channels used in the transmission direction from the base station to the mobile are referred to as forward channels, while channels used in the direction from the mobile to the base station are referred to as reverse channels. The forward and reverse channels together identify a duplex cellular channel. When frequency division duplex (FDD) is used, the forward and reverse channels are split in frequency. Alternatively, when time division duplex (TDD) is used, the forward and reverse channels are on the same frequency, but use different time slots for transmission.
High-capacity cellular systems employ frequency reuse among cells. This requires that co-channel cells (cells sharing the same frequency) are sufficiently far apart from each other to mitigate co-channel interference. Channel reuse is implemented by covering the geographic service area with clusters of N cells, as shown in Figure 17.2, where N is known as the cluster size.
The RF spectrum available for the geographic service area is assigned to each cluster, such that cells within a cluster do not share any channel [1]. If M channels make up the entire spectrum available for the service area, and if the distribution of users is uniform over the service area, then each cell is assigned M/N channels. As the clusters are replicated over the service area, the reuse of channels leads to tiers of co-channel cells, and co-channel interference will result from the propagation of RF energy between co-channel base stations and mobile users. Co-channel interference in a cellular system occurs when, for example, a mobile simultaneously receives signals from the base station in its own cell, as well as from co-channel base stations in nearby cells from adjacent tiers. In this instance, one co-channel forward link (base station to mobile transmission) is the desired signal, and the other co-channel signals received by the mobile form the total co-channel interference at the receiver. The power level of the co-channel interference is closely related to the separation distances among co-channel cells. If we model the cells with a hexagonal shape, as in Figure 17.2, the minimum distance between the center of two co-channel cells, called the reuse distance DN, is
where R is the maximum radius of the cell (the hexagon is inscribed within the radius). Therefore, we can immediately see from Figure 17.2 that a small cluster size (small reuse distance DN), leads to high interference among co-channel cells.
The level of co-channel interference received within a given cell is also dependent on the number of active co-channel cells at any instant of time. As mentioned before, co-channel cells are grouped into tiers with respect to a particular cell of interest. The number of co-channel cells in a given tier depends on the tier order and the geometry adopted to represent the shape of a cell (e.g., the coverage area of an individual base station). For the classic hexagonal shape, the closest co-channel cells are located in the first tier and there are six co-channel cells. The second tier consists of 12 co-channel cells, the third, 18, and so on. The total co-channel interference is, therefore, the sum of the co-channel interference signals transmitted from all co-channel cells of all tiers. However, co-channel cells belonging to the first tier have a stronger influence on the total interference, since they are closer to the cell where the interference is measured.
Co-channel interference is recognized as one of the major factors that limits the capacity and link quality of a wireless communications system and plays an important role in the tradeoff between system capacity (large-scale system issue) and link quality (small-scale issue). For example, one approach for achieving high capacity (large number of users), without increasing the bandwidth of the RF spectrum allocated to the system, is to reduce the channel reuse distance by reducing the cluster size N of a cellular system [1]. However, reduction in the cluster size increases co-channel interference, which degrades the link quality.
The level of interference within a cellular system at any time is random and must be simulated by modeling both the RF propagation environment between cells and the position location of the mobile users. In addition, the traffic statistics of each user and the type of channel allocation scheme at the base stations determine the instantaneous interference level and the capacity of the system.
The effects of co-channel interference can be estimated by the signal-to-interference ratio (SIR) of the communication link, defined as the ratio of the power of the desired signal S, to the power of the total interference signal, I. Since both power levels S and I are random variables due to RF propagation effects, user mobility and traffic variation, the SIR is also a random variable. Consequently, the severity of the effects of co-channel interference on system performance is frequently analyzed in terms of the system outage probability, defined in this particular case as the probability that SIR is below a given threshold SIR0. This is
where pSIR(x) is the probability density function (pdf) of the SIR. Note the distinction between the definition of a link outage probability, that classifies an outage based on a particular bit error rate (BER) or Eb/N0 threshold for acceptable voice performance, and the system outage probability that considers a particular SIR threshold for acceptable mobile performance of a typical user.
Analytical approaches for estimating the outage probability in a cellular system, as discussed in Chapter 11, require tractable models for the RF propagation effects, user mobility, and traffic variation, in order to obtain an expression for pSIR(x). Unfortunately, it is very difficult to use analytical models for these effects, due to their complex relationship to the received signal level. Therefore, the estimation of the outage probability in a cellular system usually relies on simulation, which offers flexibility in the analysis. In this chapter, we present a simple example of a simulation of a cellular communication system, with the emphasis on the system aspects of the communication system, including multi-user performance, traffic engineering, and channel reuse. In order to conduct a system-level simulation, a number of aspects of the individual communication links must be considered. These include the channel model, the antenna radiation pattern, and the relationship between Eb/N0 (e.g., the SIR) and the acceptable performance.
In order to simulate a cellular system, we must mathematically model the individual components of the system. In the next section, the models of the system components to be simulated will be discussed.
This section treats several aspects of a cellular communication system that will be useful when developing a simulation model for a system to be investigated.
We begin by discussing some important issues related to the capability of a cellular radio system to provide service to a large number of users. Like fixed telephone systems, cellular radio systems rely on the trunking concept to provide communication service for a large number of users, employing a limited resource that, in the cellular communications case, is the available RF spectrum or the number of available channels. The use of trunking techniques is possible due to the statistical behavior of users, described mainly by the following two aspects:
A single user accesses the system, that is, requests a call, on a random basis during a period of time, and the interval τ between two consecutive call requests from the same user, follows an exponential distribution. Thus the underlying pdf is
where λu is the average number of call requests per unit time (calls per time) made by a single user. If we consider a population of U users, the distribution of the time interval between two consecutive call requests, made by any two users, is also exponential. The average number of call requests is λ = Uλu.
The call duration is also a random variable that follows an exponential distribution, such that short calls are more likely to occur than long calls. Denoting the duration of a call by s, the pdf of s is
where 1/µ = H is the average call duration (units of time).
Based on this statistical behavior, a large number of users can share a relatively small number of channels in a pool of channels. For each base station in a cellular system, a pool of C trunked channels is made available to all users that are within the coverage area of the base station. Since a single user does not require access to the cellular system at all times, channels can be allocated to users on a per call basis. Once the call is terminated, the channel returns to the pool of available channels. However, one can intuitively expect that a single user may not always be allowed to establish radio communication with its serving base station because of a lack of available radio channels at the base station. In this case, all channels would be busy serving calls placed by other users, and the call request is blocked. Based on the statistical behavior of the users, the number of channels available in the pool, and certain characteristics of the trunked system, we can determine the probability that a user will have its call request blocked due to a lack of idle channels. This probability, usually called blocking probability, is a measure of the “grade of service” of a trunked system. The statistical behavior of a single user can be summarized by the traffic Au generated by that user, given in Erlangs and defined as
For a system containing U users, the total offered traffic in Erlangs is
An important characteristic of the trunked system that dictates the quality of the service offered to the users concerns how the system handles blocked calls. There are two basic strategies. In the first strategy, call requests that do not find available free channels are blocked and cleared. In this case the trunked system is referred to as blocked calls cleared. In the second strategy, blocked calls are held in a queue and served as soon as a channel becomes available. Trunked systems using this strategy are called blocked calls delayed. We will focus our attention on the blocked calls cleared trunked system, since this type of system is more often found in practice.
We therefore assume that blocked calls are cleared. In addition, the following assumptions are made:
Under these conditions the probability PB that a call is blocked is given by the Erlang B formula. The Erlang B formula relates the number of trunked channels C, the blocking probability PB and traffic A, which can be either the offered traffic or the carried traffic. In the former case, PB is the blocking probability experienced by the population of users that generates the traffic A, offered to a trunked system with C channels. In the later case, A is the maximum carried traffic by a trunked system containing C channels at blocking probability PB. The carried traffic by a trunked system is also a measure of the capacity of the system. The Erlang B formula is
Given the offered traffic and the number of trunked channels, the blocking probability can be computed using the following MATLAB code
function erb = erlang_b(A,c)
% A = offered traffic in Erlangs.
% c = number of trunked channels.
num = A^c;
sum = 0;
for k=0:c
kfact = prod(1:k);
term = (A^k)/kfact;
sum = sum + term;
end
cfact = prod(1:c);
den = cfact*sum;
erb = num/den;Figure 17.3 shows the Erlang B chart with the number of trunked channels as a parameter. The number of trunked channels, C, is given across the top of the chart. The complete MATLAB program used to compute Figure 17.3 is given in Appendix A. The code listed in Appendix A makes repeated calls to the previously listed function erlang_b(A,c). Given the Erlang B formula, the effects of the cluster size of a cellular radio system on the capacity of the system, in terms of the number of users, can be evaluated.
Example 17.1.
Consider a cellular system where 400 pairs of forward and reverse channels are available for the entire system. Each cell has a radius of 5 km, and the base stations are equipped with omnidirectional antennas that are assumed to be located at the center of each cell. Assume that each user generates a traffic of 0.02 Erlang and that a cluster size N = 7 is used. Assuming that the distribution of users is uniform over the service area, each cell is allocated NC = 400/7 ≈ 57 channels. If we further assume that blocked calls are cleared and that a blocking probability PB = 0.02 is acceptable, the maximum traffic AC carried by each cell is given by Erlang B formula. The result is
AC = 46.8 | Erlangs per cell |
The number of users supported per cell is easily computed. Since Au = 0.02 Erlang per user, the number of users supported per cell is
Now reduce the cluster size to N = 3 without changing the coverage area of each cell. Each cell will now be allocated NC = 400/3 = 133 channels at PB = 0.02. This gives
AC = 120.1 | Erlangs per cell |
for the maximum carried traffic per cell. This gives
We see that we achieve higher capacity by reducing the cluster size due to increased reuse and trunking efficiency. However, co-channel cells are closer to each other in cluster size N = 3 than in cluster size N = 7. Using (17.1) we have
D3 = 15 | km |
for N = 3 and
D7 = 22.9 | km |
for N = 7. We intuitively see that the level of interference will be higher in cluster size N = 3 than in cluster size N = 7. Therefore, cluster size N = 3 will present a lower link quality.
When analyzing the performance of a cellular system, it is very important to accurately model the effects of the radio propagation on the received signal, as those effects are very often among the major sources of system performance degradation. As discussed in Chapter 14, the statistical characterization of the received signals (both desired and interference signals) involves mainly two propagation effects: small-scale fading, induced by multipath over a local area, and shadowing (large-scale fading), induced by random attenuators of the local mean signal, such as trees, buildings, and terrain [1]. Measurements have shown that the local mean signal level in a wireless communications system [5, 6] can be accurately modeled as a lognormal random variable. When expressed in decibel units, the local mean signal level follows a normal variation and is characterized by an area mean value and standard deviation, both in dB. The area mean value is a function of the transmitter to receiver separation (T-R) distance, transmitter power levels, and antenna gains, while the shadowing standard deviation depends on the physical environment. In the general case of system design or simulation, the effects of small-scale fading and shadowing must be taken into consideration, although in some cases, shadowing of the desired and interference signals is the main source of performance degradation. For example, spatial diversity, spread spectrum, and coding and interleaving techniques have been extensively employed to combat the effects of small-scale fading [1], such that received signals are mainly dependent on large-scale channel variations. In the analysis presented here, we assume, for simplicity, that the small-scale fading effects are averaged out and only shadowing and path loss are considered. However, the effects of small-scale fading can be easily incorporated in the analysis.
Assuming that the effects of small-scale fading are averaged out, the local mean power level of the desired or individual interference signal, denoted generically here by ρ, undergoes lognormal variation. In dBW, the local mean power level can be modeled as [1]
where mX is the area mean power level in dBW (or, alternatively, average large-scale propagation path loss in dB) and χ is a zero-mean normally distributed random variable in dB with standard deviation σX, also in dB, due to the shadowing caused by large obstacles [1]. The area mean power mX is usually modeled as a function of the T-R separation d, path loss exponent γ, transmitted power PT in dBW, and transmitter and receiver antenna gains GT and GR, both in dB. Specifically
The constant K in (17.9) comprises all terms that do not change in the model. The angles θT and φT are the elevation and azimuth angles of departure of the signal transmitted toward the receiver, while θR and φR are the elevation and azimuth angles of arrival of the signals impinging on the receiving antenna. Angles θT, φT, θR, and φR depend on the relative locations of the base station and the mobile antennas.
Example 17.2.
Consider that a mobile station is receiving a signal from a base station as depicted in Figure 17.4(a). The T-R separation is d = 1,200 meters. Let us assume that the communication link between the base station and mobile occurs through line of sight. Assuming the coordinate systems shown in the figure and using simple geometry, we can determine the azimuth angle of the signal transmitted by the base station [see Figure 17.4(b)]:
Likewise, the azimuth angle of the signal impinging on the mobile antenna, according to the coordinate system adopted, is
φR = 236.4°
Suppose that the base station antenna is 30 meters high, and that the user holds the mobile telephone 1.5 meters above the ground, as also shown in the figure. Using again simple geometry, the elevation angle of the signal transmitted by the base station toward the mobile is
Likewise for the elevation angle of the signal arriving at the mobile antenna, we have
θR ≈ 88.4°
We see that the azimuth and elevation angles depend on the relative locations of the transmitter and receiver antennas, and the coordinate system used, as well. In the simulation of cellular systems, the coordinate systems adopted to specify all angles and distances must be carefully defined [11].
Another important conclusion from the preceding example is that, for macro-cellular systems, where the cell radius is larger than 1 km and, consequently, T-R separations are much larger than the difference between the heights of the base station and mobile antennas, we can assume that
and omit both θT and θR in (17.9). This gives
Cellular communication systems often use several sectorized antennas at the base station, in order to reduce the co-channel interference. Each sectorized antenna radiates within a specified sector of the cell, and each sector is allocated a subset of the set of channels available for the cell. Therefore, due to the directivity of the base station antennas, the total co-channel interference impinging on the receiver antenna (at the base station or mobile) is reduced, as illustrated in the next example.
Example 17.3.
Consider a cellular system with the base stations equipped with sectorized antennas on both links. Assume a beamwidth BW = 120° and that the front-to-back ratio of the sectorized antennas is infinity, so that no power is radiated out of the beamwidth. Also assume that cluster size N = 4 is used. Considering first the forward link, the number of interfering base stations in the first tier reduces from six to only two (base stations at Cells 5 and 6), as shown in Figure 17.5, due to the directivity of the base station antennas. On the reverse link, the number of co-channel mobiles in the first tier interfering with the base station at the center cell also drops from six to two (mobiles at Cells 2 and 3). Therefore, sectoring is an effective way for reducing co-channel interference. It is clear that the amount of reduction in co-channel interference depends on the beamwidth of the sectorized antenna and the cluster size used.
Actual sectorized antennas used in practice have a finite front-to-back ratio. As a result, the number of co-channel interference signals from co-channel cells in the first tier would still be six, but some of them will be very weak, since they are attenuated by the front-to-back ratio. Also, sectorized antennas commonly found in practice have beamwidths of 120° (three sectors), or 60° (six sectors).
Based on the assumption that the received individual interference signals are affected by shadowing and path loss only, the total co-channel interference is modeled as a composition of interference signals, whose local mean power levels follow log-normal variations. It is usually assumed that the phase shift observed in each individual interference signal varies significantly due to scattering, such that we can assume that the phases are random and thus signals add incoherently (e.g., their powers add) when averaged over the local area. Therefore, the total co-channel interference I received at a given location is modeled as the sum of lognormally distributed signals. Thus
where Ii, when expressed in decibel units, is modeled as in (17.8).
It is well accepted that the distribution of the sum of lognormal random variables can be approximated by another lognormal distribution [7, 8, 9], and several methods have been proposed for computing the mean value and standard deviation, in dB, of the resulting lognormal distribution. The two most popular techniques, as discussed in Appendix C, are Wilkinson’s method [7] and Schwartz and Yeh’s method [8]. Once we know the distributions of the individual co-channel interference signals Ii, or in other words, the means mIi and standard deviations σIi of Ii, we can compute the mean mI and standard deviation σI of the total interference I, by using Wilkinson’s method or Schwartz and Yeh’s method.
Since we now know how to compute the total co-channel interference, let us analyze more closely the effects of sectoring on capacity and link quality of a cellular system. We have seen that cell sectoring reduces co-channel interference, but at the expense of a reduction in trunking efficiency which, in turn, reduces the total traffic carried by the cell. Each sector of the cell will be allocated a subset of the set of channels allocated to the cell. From traffic theory, we know that when a pool of channels is partitioned into subsets of channels, the sum of the maximum traffic carried by the subsets is always lower than the maximum traffic carried by the whole pool of channels.
Example 17.4.
Consider an AMPS (Advanced Mobile Phone System) cellular system using a cluster size of N = 4. In addition, assume that 395 pairs of forward and reverse traffic channels are available. In this example the maximum traffic carried by each cell and the co-channel interference will be estimated. Two different configurations will be considered: (1) omnidirectional antennas at the base station and (2) sectorized antennas at the base station. Only the forward link will be analyzed.
For onmidirectional antennas, base stations will radiate in all directions with equal strength. They will interfere with and will receive interference from all co-channel cells. Considering only the first tier of co-channel cells, the total co-channel interference received at a given mobile is the sum of all co-channel interference signals, as shown in Figure 17.6(a). Based on this figure, we can compute the area mean SIR at a mobile located at the boundary of the cell. This is a worst-case situation and is given by
where mS is the area mean power of the desired signal and mI is the area mean power of the total interference (note that both are expressed in dB which explains the substraction). In order to compute mS and mI, we will use (17.9) and (17.12), assuming for simplicity, that the mobile antenna is omnidirectical (GR(θR,φR) = 0 dB), K = 1, and all base stations transmit the same power level PT = 0 dBW. Therefore
and
where R is the cell radius, γ is the path loss exponent, and di, i = 1, ... , 6, are the T-R separations. Using the geometry of cluster size N = 4, assuming that R = 1,000 and γ = 4, we can show that
and
Therefore
and
which gives the result
For the traffic analysis, we again assume N = 4 and also assume that the distribution of users is uniform over the cell area. Each cell is allocated NC = 395/4 ≅ 98 channels. Assuming that blocked calls are cleared and that the blocking probability is PB = 0.02, the maximum traffic carried per cell is, from the Erlang B formula
If a single user generates 0.02 Erlang of traffic, the omnidirectional cell can support up to 86.0/0.02 ≅ 4,300 users at a blocking probability of PB = 0.02.
We now assume that 120° sectoring is used (three sectors per cell) as shown in Figure 17.6(b). In order to analyze the co-channel interference the same approach used in part I is applied. We assume an idealized antenna with an infinite front-to-back ratio so that only Cells 5 and 6 interfere with the center cell. Therefore, the number of interfering co-channel cells in this idealized example drops from six (all co-channel cells in the first tier) to only two, and
and
Finally
For the capacity traffic analysis, we recognize that each sector is allocated NS = 395/(4 × 3) ≅ 33 channels. The maximum traffic carried by each sector is, assuming a blocking probability of PB = 0.02, given by
For a three-sector cell this number is multiplied by three. Therefore:
Again assuming that each user generates 0.02 Erlang of traffic, the sectorized cell with three sectors can support up to 71.1/0.02 ≅ 3,550 users at a blocking probability of PB = 0.02.
We see from these results that a link quality improvement of approximately 10 dB (ΔSIR = 22.56 − 12.35 = 10.21 dB) is achieved by using sectoring, compared with the omnidirectional case. However, this link quality improvement is achieved at the expense of trunking efficiency, so that the total traffic carried per cell is reduced from 86.0 Erlangs per cell with the omnidirectional antenna to 71.1 Erlangs per cell with sectoring. After sectoring N can be reduced if desired.
In the preceding example, the signal-to-interference ratio was computed for the worst-case situation, since the mobile was assumed to be located at the cell boundary. It is obvious that at locations closer to the serving base station, the SIR will be higher. Another limitation of the simplified analysis presented in the preceding example is that the effects of shadowing were not considered. When we consider both the spatial distribution of mobiles and the effects of shadowing, SIR becomes a random variable, as we discussed before. The performance of the cellular system must then be measured through the outage probability, defined in (17.2) as the probability that SIR is below a minimum acceptable level SIR0.
In the next section, the methodology for simulating a simple cellular system is presented, taking into consideration both the user spatial distribution and shadowing effects on the received signals. The results of the simulation will be the SIR statistics, that will give us an estimate of the performance of the system. Despite the simplicity of the simulation, the methodology presented can be viewed as the core part of more complex simulations, such as those described by Cardieri and Rappaport [10, 11].
The simulation procedure consists of modeling a snapshot of the location of mobile stations. At each snapshot, the statistics (mean value and variance) of the SIR at a given base station (reverse link) and mobile station (forward link) are computed, taking into account user locations and propagation conditions. Several snapshots are simulated, in order to generate a sufficiently large sample set so that statistically valid results are achieved. Figure 17.7 shows the flowchart of the procedure. In the following sections, we describe some aspects of the simulation. Other aspects will be described subsequently, together with the steps required to simulate one snapshot of the system.
We will consider only the first tier of co-channel cells. The number of sectors in each cell is a simulation parameter and can be chosen among: (1) one, or omnidirectional antennas, (2) three, or 120° sectoring, and (3) six, or 60° sectoring. The gain of the sectorized antenna is assumed to be constant within the sector and equal to 0 dB, as shown in Figure 17.8. The sectorized antennas have a finite front-to-back ratio of B dB, which is a simulation parameter. For 120° sectoring, the radiation patterns of the sectorized antennas for Sector 1 are modeled as
The radiation pattern for Sectors 2 and 3 are obtained by rotating 
and 
120° and 240°, respectively. For 60° sectoring, we have
The radiation pattern for Sectors 2 through 6 are obtained by appropriate rotation of and by 60°. Both forward link and reverse link base station antennas are assumed to be identical. Mobile antennas are assumed to be omnidirectional antennas on both forward and reverse links.
The channel model includes both path loss and lognormal shadowing. The path loss exponent and the standard deviation of the lognormal shadowing are simulation parameters. The simulation consists of the following two major steps:
Definition of the target system to be simulated
Generation of snapshots of mobile locations and computation of the SIR
These are discussed in the following sections.
Here we define the propagation characteristics (channel parameters) and the location of the co-channel cells.
The parameters that define the channel characteristics consist of the following:
Cell radius R
Path loss exponent (γ)
Standard deviation in decibel units of lognormal shadowing (σ)
Base station transmission power level (PT,BS)
Mobile station transmission power level (PT,MS)
Number of sectors per cell
Front-to-back ratio of sectorized antennas
Number of snapshots to be simulated (M)
This part of the simulation can be implemented using the MATLAB code given in Appendix B. The complete simulation program presented in this section can be assembled by appending the code segments appearing later in this section to the MATLAB code given in Appendix B.
For convenience, we will adopt both the rectangular and polar coordinate systems to represent the locations of the base stations and mobiles in our simulation. The base stations will be located at the center of the corresponding cells. The base station at the center cell, where the co-channel interference will be measured, will be located at the center of the coordinate systems. The locations of the co-channel cells depend on the cluster size N of the cellular system and the cell radius R. For the first tier, all co-channel cells are located on a circumference of radius 
centered at the center cell, and equally distant from each other, as indicated in Figure 17.9. Also indicated in this figure is the angle θN (see Table 17.1), which determines the angular location of the first co-channel cell. Using simple geometry, we can show that, for cluster size N, the location of the ith co-channel cell, using vector notation, is
where 
and ŷ are the unit vectors in the direction of axes x and y, respectively. An example of the implementation of this part of the simulation is as follows:
% Location of base stations (center cell is located at x = 0, y = 0)
% Location (angular) of the center cell of each cluster in the
% first tier.
theta_N = [pi/6 0 pi/6 asin(1/(2*sqrt(7)))];
% Angular distance between the center cells of all 6 clusters in
% first tier.
theta = pi/3*[0:5]';
aux_1 = [1 0 2 3 0 0 4];
ind = aux_1(cluster_size);
% Location [x,y] of the center cells of all clusters in the
% first tier.
bs_position = [sqrt(3*cluster_size)*r_cell*cos(theta + ...
theta_N(ind)) sqrt(3*cluster_size)*r_cell*sin(theta + ...
theta_N(ind))];Note that, in this MATLAB program segment, bs_position(i,1) and bs_position(i,2) correspond to the components in the directions and ŷ, respectively, of the vector representing the location of the ith base station.
In this part of the simulation, the actual Monte Carlo evaluation is performed. Snapshots of the locations of the mobiles sharing the same channel are generated, and for each snapshot, the statistics of SIR are computed. This part of the simulation is performed M times.
The mobiles are assumed to be uniformly distributed over the cell area. As mentioned before, the sectors of a given cell are allocated different sets of channels. Therefore, co-channel interference occurs only among sectors allocated the same set of channels. In our simulation, the mobiles at the center cell and at the co-channel cells are assumed to be located within the same sector in their cells. Using a polar coordinate system, the location of the ith mobile within its cell can be described by the distance ri between the mobile and its serving base station, and the angle βi between a reference and the direction of propagation between the mobile and its base station (see Figure 17.10). Note that βi and ri are defined with respect to the coordinate system centered at the base station of the ith cell. Since the distribution of the mobile location is uniform over the cell area, βi is uniformly distributed over the range [0, 2π], and the distance r follows the pdf
Note that, for simplicity, the cell is assumed to be circular in the simulation.
The sector used in a particular snapshot is chosen randomly, with equal probabilities of choosing any sector:
120° sectoring | ⇒ | Pr{sector i is chosen} = 1/3, | i = 1, 2, 3. |
60° sectoring | ⇒ | Pr{sector i is chosen} = 1/6, | i = 1, ... ,6. |
Once the sector is chosen, the angles βi can be determined, noting that βi is uniformly distributed over the chosen sector:
120° sectoring | ⇒ | (2s - 3)π/3 < βi ≤ (2s - 1)π/3 |
60° sectoring | ⇒ | (s - 1)π/3 < βi ≤ sπ/3, |
where s is the sector selected (s = {1, 2, 3}, for 120° sectoring and s = {1, ... , 6}, for 60° sectoring). Figure 17.11 shows an example of a snapshot for 120° sectoring, where the mobiles are located in Sector 1. This part of the simulation can be implemented as shown in the following MATLAB code:
% Determination of the sector to simulated in this snapshot % % --- Select (randomly) a sector --- sector = unidrnd(num_sectors(sec)); % % --- Place the desired mobile within the select sector --- des_user_beta = rand(1)*phi_BW(sec) + phi_center(sector,sec); des_user_r = sqrt(rand(1).*(r_cell^2)); % % --- Place co-channel mobiles within the selected sector of % co-channel cells--- co_ch_user_beta = rand(6,1)*phi_BW(sec) + phi_center(sector,sec); co_ch_user_r = sqrt(rand(6,1))*(r_cell);
The (x,y) location of the the desired and co-channel mobiles are computed according to
des_user_position = des_user_r*[cos(des_user_beta) ...
sin(des_user_beta)];
co_ch_user_position = [co_ch_user_r.*cos(co_ch_user_beta) ...
co_ch_user_r.*sin(co_ch_user_beta)] + bs_position;The next step is to determine the distances between the co-channel mobiles to the base station at the center cell (to be used to compute the mean values of the interference signals on the reverse link) and the distances between the mobile at the center cell and the co-channel base stations (to be used to determine the mean values of the interference signals on the forward link).
Since we are interested in computing the co-channel interference at the base station and mobile in the center cell, we need to determine the following:
The location of the co-channel mobile stations with respect to the base station at the center cell: Using vector notation, the location of the ith mobile station with respect to the base station at the center cell is (see Figure 17.10)
The term
,i is the T-R separation between the base station at the center cell and the ith mobile, and αi is the angle of arrival of the signal from the ith mobile, impinging on the base station antenna. Therefore, ,i and αi are the length and direction of the vector given by the right hand side of (17.32).The location of the mobile station at the center cell with respect to the ith co-channel base station: Again using vector notation, we have (see Figure 17.10)
where
,i is the T-R separation between the ith base station and the mobile at the center cell, and θi is the angle of departure of the signal transmitted by the ith base station toward the mobile at the center cell.
In this step, three different approaches for computing the statistics of SIR will be presented. In all three approaches, the lognormal variation of the received signals will be taken into account. This requires the mean value of the desired and interference signals.
Mean value and standard deviation in dB of each signal
Desired signals: Using (17.9), the mean values in decibel units of the desired signals on the forward and reverse link are
and
respectively. Note that the base station antenna gains on both links in these expression are set to 0 dB, since the mobile is located within the sector of the base station. Also, the mobile antenna gains on both links are set to 0 dB, since we are assuming omnidirectional antennas. The standard deviations of the desired signals are equal to the shadowing standard deviation. Thus
Interference signals: The determination of the mean values of the total interference signals on both links is more involved. As we mentioned before, the total co-channel interference on both links is modeled as the sum of the individual co-channel interference signals. This gives
for the forward and reverse links, respectively. Figure 17.12 shows an example snapshot for 120° sectoring. Six co-channel cells are simulated but, for clarity, only signals from co-channel Cells 1 and 4 are shown. Since shadowing effects and path loss are taken into account in this model, the total interference signals If and Ir are the sum of the lognormally distributed signals
and 
, respectively. As discussed before, we assume that both If and Ir are lognormally distributed. The mean and standard deviation, in dB, of If and Ir are functions of the means and standard deviations of the individual interference signals and can be computed using Wilkinson’s or Schwartz and Yeh’s methods, as discussed in Appendix C. The mean values, in dBW, of and can be determined using (17.9). This givesand
The antennas gains GT,i(φT,i) and GR,0(φR,i) depend on the relative positions of the mobiles and follow the description given in (17.28) and (17.29). Note that we are assuming omnidirectional antennas at the mobiles on both links. The standard deviations of the interference signals are equal to the shadowing standard deviation σ
Once the means and standard deviations of all co-channel signals have been determined, we apply Wilkinson’s or Schwartz and Yeh’s method (see Appendix C) to compute the means
and 
, and standard deviations 
and 
, in dB, of the total co-channel interference on both links. The computation of the moments of the signals on the forward link is carried out using the following MATLAB code:% --- DESIRED USER --- m_S_fwd = P_BS - 10*K*n_path*log10(des_user_r); % % --- CO-CHANNEL USERS --- % --- Location of desired mobile with respect to % co-channel cells --- aux_01 = ((des_user_position(1) - bs_position(:,1))+ ... sqrt(-1)*(des_user_position(2) - bs_position(:,2))); beta_fwd = angle(aux_01); d_I_fwd = abs(aux_01); % --- Computation of antenna gain at co-channel cells clear gain_fwd for k = 1:n_co_ch_users if (beta_fwd(k) >= ... sector_min(sector,sec)) & (beta_fwd(k) < ... sector_max(sector,sec)) gain_fwd(k) = in_beam; else gain_fwd(k) = out_beam; end end % --- Computation of mean value and standard deviation --- m_I_fwd = P_BS - 10*K*n_path*log10(d_I_fwd) + gain_fwd.'; sigma_I_fwd = sigma_int*ones(length(m_I_fwd),1); [m_I_total_fwd, sigma_I_total_fwd] = ... wilkinson(m_I_fwd,sigma_I_fwd,corr_fwd);In a similar manner, the computation of the moments of the signals on the reverse link is carried out using the following MATLAB code:
% --- DESIRED USER --- m_S_rev = P_MS - 10*K*n_path*log10(des_user_r); % % --- CO-CHANNEL USERS --- % --- Location of co-channel users --- aux_02 = (co_ch_user_position(:,1) + ... sqrt(-1)*co_ch_user_position(:,2)); beta_rev = angle(aux_02); d_I_rev = abs(aux_02); % % --- Computation of antenna gain at center cell clear gain_rev for k = 1:n_co_ch_users if (beta_rev(k) >= ... sector_min(sector,sec)) & (beta_rev(k) < ... sector_max(sector,sec)) gain_rev(k) = in_beam; else gain_rev(k) = out_beam; end end % --- Computation of mean value and standard deviation --- m_I_rev = P_MS - 10*K*n_path*log10(d_I_rev) + gain_rev.'; sigma_I_rev = sigma_int*ones(length(m_I_rev),1); [m_I_total_rev, sigma_I_total_rev] = ... wilkinson(m_I_rev,sigma_I_rev,corr_rev);Note that in this example, we use Wilkinson’s method for computing the mean and standard deviation of the total interference [10]. The MATLAB code for implementing Wilkinson’s method is given in Appendix D.
Steps for computing the statistics of SIR
Step 1: Computation of mean and standard deviation of SIR: In this approach, we compute the means
and 
, and standard deviations 
and 
of the SIR, expressed in dB, on both the forward and reverse links. These moments will be used to compute the outage and reliability probabilities, as described later. Since the desired signal and the total co-channel interference are normal random variables, when expressed in dB, the signal to interference ratio SIR, in dB, is also a normal variable. For the forward link, SIRf is given bywith mean and standard deviation, in dB, given by
and
Equations (17.41), (17.42), and (17.43) are also used for the reverse link. The results of this step are
, , , and 
. The MATLAB code for this portion of the simulation follows:m_SIR_fwd(i) = m_S_fwd - m_I_total_fwd; sigma_SIR_fwd(i) = sqrt(sigma_S^2 + sigma_I_total_fwd^2 -... 2*corr_fwd*sigma_S*sigma_I_total_fwd); m_SIR_rev(i) = m_S_rev - m_I_total_rev; sigma_SIR_rev(i) = sqrt(sigma_S^2 + sigma_I_total_rev^2 -... 2*corr_rev*sigma_S*sigma_I_total_rev);Note that, at this point in the simulation we start collecting the simulation results. In Step 1 we save the means and standard deviations in arrays controlled by index
i.Step 2, Method A: Sampling the SIR using the mean and standard deviation of SIR: Since we know the means
and , and standard deviations 
and of SIR on both links from Step 1, we can sample the normal random processes SIRf and SIRr. Therefore, each snapshot will be associated with samples of SIR on each link, denoted by 
and 
. These samples are the result of sampling a normal random process with mean mSIR and standard deviation σSIR. The results of Step 2, Method A, are values for and as defined by the following MATLAB code.SIR_fwd_2(i) = normrnd(m_SIR_fwd(i),sigma_SIR_fwd(i)); SIR_rev_2(i) = normrnd(m_SIR_rev(i),sigma_SIR_rev(i));
Step 2, Method B: Sampling SIR using the mean and standard deviation of the desired and individual interference signals: To this point, we have assumed that the total co-channel interference, modeled as the sum of individual co-channel interference signals, is lognormally distributed, or normally distributed when expressed in dB. In Step 2, Method B, we do not make any assumption regarding the distribution of the total interference. A sample of each signal received at the mobile (forward link) and base station (reverse link) at the center cell is determined, assuming that the signals, expressed in dB, are normally distributed having moments Sf, Sr,
, and 
. The total interference signals on each link is computed asand
The samples of SIR on both links, denoted by
and 
, are given byand
These are computed using the following MATLAB code:
des_sig_spl_fwd = normrnd(m_S_fwd,sigma_S); int_sig_spl_fwd = normrnd(m_I_fwd,sigma_I_fwd); tot_int_sig_spl_fwd = 10*log10(sum(10.^... (int_sig_spl_fwd/10))); SIR_spl_fwd_2B(i) = des_sig_spl_fwd - tot_int_sig_spl_fwd; des_sig_spl_rev = normrnd(m_S_rev,sigma_S); int_sig_spl_rev = normrnd(m_I_rev,sigma_I_rev); tot_int_sig_spl_rev = 10*log10(sum(10.^... (int_sig_spl_rev/10))); SIR_spl_rev_2B(i) = des_sig_spl_rev - tot_int_sig_spl_rev;
At this point, we have completed the processing of one snapshot. The other M −1 snapshots needed to complete the overall simulation are processed generating collections of length M of the following:
From Step 1: mean values
and , and standard deviations and of SIR on each linkFrom Step 2, Method A: samples of SIR on both links,
and From Step 2, Method B: samples of SIR on both links,
and
By processing the results of the simulation, we can estimate the performance of the overall cellular system, in terms of outage probability and other performance indicators. In this section, we will present examples of simulation results comparing the performance of cellular systems operating under the following six different configurations:
The sectorized antennas at the base stations have a front-to-back ratio of 30 dB. The shadowing standard deviation is set to 8 dB, and the path loss exponent is assumed to be γ = 4. In order to obtain statistically valid results, 1,000 snapshots are simulated.
The mean and standard deviation of SIR, expressed in dB, were computed for each snapshot in the simulation using Step 1. These moments correspond to the SIR measured at the mobile and base station at the center cell for a specific snapshot of locations of the co-channel mobiles. Therefore, we can compute the outage probability of the cellular system, at the mobile and base station, for a given specific situation. In a previous section outage probability was defined. It is repeated here for convenience:
where Q(·) is the Gaussian Q-function. Therefore, by using the mean and standard deviation computed at each snapshot, from Step 1 in the simulation, we obtain a sample of outage probability. The average outage probability of the cellular system simulated, denoted as 
, is then computed by averaging the samples of outage probabilities Poutage(SIR0), computed for each snapshot. The average outage probability is often referred to as area-averaged outage probability, since each element in the averaging process corresponds to a location in the cell area. The average outage probabilities at different thresholds, computed using the results from Step 1 for cluster size N = 7 and 120° sectoring, are shown in Figure 17.13. The average outage probabilities for cluster size N = 7 and 120° sectoring, computed from the results of Step 2, Methods A and B, are also shown in Figure 17.13.
The average outage probability can also be computed using the results from Step 2, Methods A or B. This provides a collection of samples of SIR on both links. Based on these collections, we can estimate the probability density functions of the area-averaged SIR, computing first the histograms of the collections. For example, Figure 17.14 shows the histograms of the samples of area-averaged SIR obtained from Step 2, Method A. These histrograms approximate the actual pdf of the area-averaged SIR. Using the histogram, we can then estimate the average outage probabilities on both links.
We can now compare the performance of all six configurations simulated. Figure 17.15 shows the average outage probability of each configuration, using Step 1 on the forward link. As expected, sectoring improves the performance of a cellular system, that is, reduces the probability that the SIR drops below a given threshold. For example, a system using cluster size N = 7 and three sectors per cell performs better, in terms of link quality, than a system using cluster size N = 7, but with omnidirectional antennas. The outage probability at SIR0 = 18 dB (which is the threshold usually used in AMPS) in the former system is 15%, versus an outage probability of 35% in the later system.
However, sectoring degrades the capacity of the system, in terms of the maximum carried traffic. As an illustration, Table 17.2 shows the carried traffic per cell for each configuration used, together with the corresponding outage probability at SIR0 = 18 dB, obtained from Figure 17.15. The carried traffic was computed assuming that there are 395 channels available for the whole system and that a blocking probability of 0.02 is acceptable. We clearly see the tradeoff between capacity and link quality. As the capacity improves, the link quality degrades.
The average outage probability, as discussed and computed in the preceding section, tells us about the performance of the cellular system averaged over the cell area. As a consequence of the averaging operation, the existence of a high outage probability (undesirable) at a given location may be compensated by a low outage probability at another location. Sometimes, in performance analysis, it is desirable to have a measure of the percentage of the cell area where the performance of the system (outage probability) is acceptable or is not acceptable. This measure can be computed from the simulation results.
The reliability probability Prelia(SIR0) is defined as the probability that SIR is larger than a given threshold SIR0. Therefore, at a given location, we have
Let us now assume that the performance of the system is considered acceptable if the reliability probability Prelia(SIR0) at a given threshold SIR0 is larger than a threshold Pmin. Then, using the computed values of Prelia(SIR0), we can estimate the percentage of the cell area where the system performance is acceptable (i.e., where Prelia(SIR0) > Pmin holds) by computing
Note that the computation of Parea(SIR0) is only possible using the results of Step 1 in the simulation, since the mean and standard deviation of SIR are required. Figure 17.16 shows the percentage of the cell area where Prelia(SIR0) > 0.75 holds on the forward link for all configurations that were simulated.
This chapter provided a fundamental overview of the key factors involved in simulating the overall performance of a wireless communication system. Several design parameters, such as cellular frequency reuse (cluster size), blocking probability, average signal-to-interference ratio, and antenna beamwidth all impact the overall system performance of a wireless system. Furthermore, these design parameters are all interrelated, and often are used to make tradeoffs in system performance. For example, a smaller cellular cluster size increases the available channel capacity of a system, but increases the interference level of each user. Likewise, sectoring lowers the co-channel interference, yet decreases system capacity. Simulation is needed to explore the impact of various tradeoffs, and the diminishing returns offered by various system design strategies. The level of co-channel interference offered by the undesired users sets the operating noise floor as seen by a particular mobile user at a particular location, and this chapter illustrated how to properly model and simulate the effect of system design parameters across a mobile communication system. The simulation strategy is to create a Monte Carlo analysis of many snapshots of system performance seen by many mobile users operating in random locations, whereby the collection of snapshots are used to build statistics of operating performance over the desired coverage areas. System simulation requires careful modeling of the spatial effects of users, the specified antenna patterns, the assignment of radio channels over space, and the appropriate large-scale interference levels over that space. This chapter provided examples and methodologies to successfully account for these key issues for the design and simulation of wireless communication systems.
Over the past several years a number of textbooks have been developed that cover the basic concepts of cellular radio systems at varying levels of detail. A sampling of these includes the following:
T. S. Rappaport, Wireless Communications: Principles and Practice, 2nd ed., Upper Saddle River, NJ: Prentice Hall, 2002.
G. L. Stuber, Principles of Mobile Communication, Boston: Kluwer, 1996.
W. C. Y. Lee, Mobile Cellular Communication: Analog and Digital Systems, 2nd ed., New York: McGraw-Hill, 1995.
W. C. Y. Lee, Mobile Communications Engineering: Theory and Applications, 2nd ed., New York: McGraw-Hill, 1998.
V. K. Garg and J. E. Wilkes, Wireless and Personal Communications Systems, Upper Saddle River, NJ: Prentice Hall, 1996.
In addition to the above listed books, a number of other books are available that treat specific topics within the broader area of cellular and mobile communications. Although this list is growing rapidly, representative examples available at the current time include the following:
H. L. Bertoni, Radio Propagation for Modern Wireless Systems, Upper Saddle River, NJ: Prentice Hall, 2000.
V. K. Garg and J. E. Wilkes, Principles and Applications of GSM, Upper Saddle River, NJ: Prentice Hall, 1999.
J. C. Liberti, Jr. and T. S. Rappaport, Smart Antennas for Wireless Communications: IS-95 and Third Generation CDMA Applications, Upper Saddle River, NJ: Prentice Hall, 1999.
B. Pattan, Satellite-Based Cellular Communications, New York: McGraw-Hill, 1998.
B. Pattan, Robust Modulation Methods & Smart Antennas in Wireless Communications, Upper Saddle River, NJ: Prentice Hall, 2000.
17.1 | Determine the smallest allowable physical distance between the centers of co-channel cells if N = 4 reuse is used, and each cell has a radius of 2 km. |
17.2 | What would the proper cellular reuse factor be if the minimum distance in Problem 17.1 was designed for 6 km? |
17.3 | Using your answer in Problem 17.2, how many channels would be assigned to each cell assuming that the authorized spectrum was 20 MHz in bandwidth, and each duplex channel used 100 kHz forward link and 100 kHz reverse link? |
17.4 | Assume the signal to interference ratio SIR follows a Gaussian distribution with mean 30 dB and standard deviation 10 dB. Compute the outage probability at SIR0 = 17 dB. |
17.5 | Derive the Erlang B expression given in (17.7). |
17.6 | Execute the MATLAB script given in Appendix A to validate that it reproduces the Erlang B blocking probability shown in Figure 17.3. |
17.7 | Assume that the allocated spectrum for a cellular system is 25 MHz in bandwidth, for a single link, and that each channel uses 200 kHz. Also, assume that 50% of the users generate a traffic of 0.25 Erlang/user, while the other 50% generate 0.02 Erlang/user. For cluster sizes N = 3, 4, and 7, compute the maximum number of users per cell, at blocking probability 0.02. |
17.8 | Sectoring offers improvement in interference, but creates capacity loss due to trunking. This is shown in Example 17.4 in the text. Rework Example 17.4 for the following cellular architectures: (a) cluster size N = 3, with 3 sectors per cell; (b) cluster size N = 3, with 6 sectors per cell. |
17.9 | Using a Gaussian random number generator to represent independent lognormal shadowing about the distance-dependent mean path loss, estimate the mean and standard deviation of SIR at points A, B, and C indicated in Figure 17.17. Also, compute the outage probability at SIR0 = 17 dB at all three locations. Assume path loss exponent γ = 3.5 and shadowing with standard deviation 8 dB. Both base stations transmit at 10 W and are equipped with omnidirectional antennas. Assume that the T-R separation distances are much larger than the base station antenna height. Compare your simulation results with analytical results. |
17.10 | Using Wilkinson’s method compute the mean and standard deviation for the sum of two uncorrelated lognormal signals (see Figure 17.18). Assume the first signal has a mean of −50 dBm, and the second has a mean of −45 dBm. Assume both signals have the same standard deviation of 7 dB. |
% File: c17_erlangb.m
C_1 = [1:10];
A_1 = linspace(0.1,10,50);
C_2 = [12:2:20];
A_2 = linspace(3,20,50);
C_3 = [30:10:100];
A_3 = linspace(13,120,50);
for i = 1:length(C_1)
for j = 1:length(A_1)
p_1(j,i) = erlang_b(A_1(j),C_1(i));
end
end
for i = 1:length(C_2)
for j = 1:length(A_2)
p_2(j,i) = erlang_b(A_2(j),C_2(i));
end
end
for i = 1:length(C_3)
for j = 1:length(A_3)
p_3(j,i) = erlang_b(A_3(j),C_3(i));
end
end
%
% The following code plots the results.
x1 = [.1 .1 .2 .2 .3 .3 .4 .4 .5 .5 .6 .6 .7 .7 .8 .8 .9 .9 ];
y1 = 10.^[-4 1 1 -4 -4 1 1 -4 -4 1 1 -4 -4 1 1 -4 -4 -1];
y2 = [.1 .1 .2 .2 .3 .3 .4 .4 .5 .5 .6 .6 .7 .7 .8 .8 .9 .9];
x2 = 10.^[-1 3 3 -1 -1 3 3 -1 -1 3 3 -1 -1 3 3 -1 -1 3];
loglog(A_1,p_1,'k-',A_2,p_2,'k-',A_3,p_3,'k-',...
x1,y1,'k:',10*x1,y1,'k:',...
100*x1,y1,'k:',1000*x1,y1,'k:',...
x2,y2,'k:',x2,0.1*y2,'k:',x2,0.01*y2,'k:'),
xlabel('Offered traffic (Erlangs)')
ylabel('Blocking probability')
axis([0.1 120 0.001 0.1])
text(.115, .115,'C=1')
text(.6, .115,'C=2')
text(1.18, .115,'3')
text(2, .115,'4')
text(2.8, .115,'5')
text(7, .115,'10')
text(9, .115,'12')
text(17, .115,'20')
text(27, .115,'30')
text(45, .115,'50')
text(100, .115,'100')
% End of script file.The preceding MATLAB program makes the function call erlang_b(A,c). The code for erlang_b(A,c) is listed in Section 17.2.2.
The code that follows is the initialization code for the cellular simulation. The overall simulation can be developed by adding the code segments given in Section 17.3 to this initialization code.
% ===== Pre-defined simulation parameters ==========================
r_cell = 1000; % cell radius (in meters)
n_co_ch_users = 6; % number of co-channel users
P_BS = 0; % BS transmitter power (in dBW)
P_MS = 0; % MS transmitter power (in dBW)
corr_fwd = 0.0; % correlation coefficient - forward link
corr_rev = 0.0; % correlation coefficient - reverse link
K = 1; % constant in the link equation
in_beam = 0; % maximum gain of sectorized antennas (in dB)
%
% --- Limits (angles) of each sector ---
sector_min = zeros(6,3);
sector_max = zeros(6,3);
sector_min(:,1) = (pi/3)*[-3:2]';
sector_min([1:3],2) = pi/3*[-3 -1 1]';
sector_min(1,3) = -pi;
sector_max(:,1) = sector_min(:,1) + pi/3;
sector_max([1:3],2) = sector_min([1:3],2) + 2*pi/3;
sector_max(1,3) = pi;
%
% --- Center of each sector ----
phi_center = zeros(6,3);
phi_center(:,1) = (pi/3)*[-3:2]';
phi_center([1:3],2) = (pi/3)*[-3 -1 1]';
%
% --- Beamwidth of each sector ---
phi_BW = [1 2 6]*pi/3;
%
% --- Number of sectors -----
num_sectors = [6 3 1];
%
% ===== User Inputs =========================================
num_snapshots = input('Number of snapshots = '),
cluster_size = input('Cluster size (3, 4 or 7) = '),
n_path = input('Path loss exponent = '),
sigma_int = input('Shadowing std deviation - interference (dB) = '),
sigma_S = input('Shadowing std deviation - desired signal (dB) = '),
sec = input('Sectorization...
(1=>60 degree, 2=>120 degree, 3=>omni). enter: '),
ftb = input('Front-to-back ratio of the BS antennas (dB) = '),
out_beam = in_beam - ftb;
% End of script file.Consider the situation in which N interference signals arrive at a receiver from N co-channel transmitters. Assuming that the effects of small-scale fading are averaged out, the local mean power level Ii of the ith signal undergoes lognormal variation. Using decibel units, the local mean power level can be modeled as [1, 2]
where mXi is the area mean power (or, alternatively, average large-scale propagation path loss) and χi is a zero-mean normally distributed RV in dB with standard deviation σXi, also in dB, due to the shadowing caused by large obstacles [1, 2]. The area mean mXi is usually modeled as a function of the T-R separation distance di, the path loss exponent n, the transmitted power PT,i in dBm, and the transmitter and receiver antenna gains GT,i and GR,i, both in dB. This gives
Under the reasonable assumption that the individual signals Ii add incoherently, the total interference signal is modeled as the sum of N lognormally distributed signals
The distribution of I is an important consideration for modeling the impact of multiple interferers, and thus dictates the resulting total interference at a receiver. Multiple transmitters, displaced throughout a geographic area, may each provide different levels of interference, based on the particular physical distances to the receiver. Thus, the effects of shadowing upon each interfering signal and the sum of the interference at the receiver from all interfering signals must be considered to determine an accurate interference level at any particular location. Note that, depending on the strength of the particular individual interferers, the resulting composite interference level may vary widely, and if each interferer produces a random signal level, the composite signal level will also be random. It is well accepted that the distribution of I can be approximated by another lognormal distribution [7–10], or, equivalently, that
follows a normal distribution. Assuming that the sum I is lognormally distributed, Wilkinson’s method [9] and and Schwartz and Yeh’s method [8] allow for the computation of the mean mX and standard deviation σX of X.
In the derivation of these two methods, it is convenient to use the natural logarithm instead of the base 10 logarithm to define the normal RV that corresponds to a lognormal RV. Thus, we define the normal RV Yi as
with mean value mYi and standard deviation σYi given, respectively, by
where λ = ln(10)/10. Note that Yi = λXi.
Using (17.53) and (17.55) and recalling that we are approximating the distribution of I by a lognormal distribution, we have
where Z (in logarithmic units) and X (in dB) are normally distributed, and Z = λX. Either Wilkinson’s method or Schwartz and Yeh’s method can then be used to compute the mean value and standard deviation of Z(mZ and σZ) or X(mX and σX) from the mean values and standard deviations of the summands Yi, as shown subsequently.
For generality, it is useful to assume that the individual signals Ii may be correlated to each other. This correlation may be due to the fact that there is a common physical obstruction that induces shadowing loss for particular propagation paths, such as vegetation or building structures. Therefore, even signals coming from different directions may be attenuated by the same obstacles, leading to correlation among the received signals. To consider the correlated interference signals case, let us define the correlation coefficient rYi,Yj of Yi and Yj by
Since Yi is simply a scaled version of Xi, it follows that rYi,Yj is the correlation coefficient of Xi and Xj.
According to Wilkinson’s method, the mean value and standard deviation of Z in (17.57) are found by matching the first and second moments of I with those of I1 + I2 + ... + IN. For the first moment, we have
The moments in (17.59) are evaluated by observing that, for a normal RV u with mean value mu and variance 
, and any integer l, it turns out that [14]
where l denotes the moment order for the normal random variable u. Therefore, to evaluate the first moment of exp(Z) where the random variable Z is assumed to be Gaussian, we see that
and
Using (17.61) and (17.62) in (17.59), we have
The summation in (17.63) is a function of the mean values mYi and standard deviations σYi of the summands Yi, which are assumed to be known either through measurement or through the use of a propagation model.
Matching the second moments of I and I1 + I2 + ... + IN, we have
Using again the property (17.60) in both sides of (17.64), we obtain
which is
Equation (17.66) can be evaluated using the mean values mYi, standard deviations σYi and correlation coefficients rYi,Yj.
Equations (17.63) and (17.66) form a system of equations with unknowns mZ and σZ. By solving this system of equations, and using Z = λX, we finally obtain
Therefore, Wilkinson’s method consists of computing the terms u1 and u2 using (17.63) and (17.66), applying the means, standard deviations, and correlation coefficients of the summands, and solving the system of equations defined by (17.67) and (17.68). An important feature of Wilkinson’s method is that the assumption of ΣiIi being lognormally distributed is actually used in the computation of mX and σX. The MATLAB program for Wilkinson’s method is given in Appendix D.
Schwartz and Yeh proposed a method based on the exact computation of the mean value mX and standard deviation σX of the sum of N = 2 lognormal RVs. For N > 2, a recursive approach is used, approximating the sum of two lognormal RVs by another lognormal RV, and computing the mean and standard deviation of the sum. The details of this method are not included here. The interested student is referred to the literature [8, 10].
% File: wilkinson.m
function [m_out,std_out] = wilkinson(m_x,std_x,r)
% this function computes the mean and standard deviation
% of the sum of two lognormal RV's
% Input and ouput values are in dB.
lambda = 0.1*log(10);
m_x_cmp = m_x;
v_x_cmp = std_x.^2;
m_y = lambda*m_x_cmp;
v_y = (lambda^2)*v_x_cmp;
u_1 = 0;
for i = 1:length(m_y)
u_1 = u_1 + exp(m_y(i) + v_y(i)/2);
end
a = 0;
for i = 1:length(m_y)
a = a + exp(2*m_y(i) + 2*v_y(i));
end
b = 0;
for i = 1:length(m_y)-1
for j = i+1:length(m_y)
b = b + exp( m_y(i) + m_y(j) )*...
exp(0.5*(v_y(i) + v_y(j) + ...
2*r*sqrt(v_y(i))*sqrt(v_y(j))));
end
end
u_2 = a + 2*b;
% mean and variance or the variable Z, which is normal
% in natural units
m_z = 2*log(u_1) - 0.5*log(u_2);
std_z = sqrt(log(u_2) - 2*log(u_1));
% mean and variance of the variable X, which is normal in dB.
g = 10*log10(exp(1));
m_out = g*m_z;
std_out = g*std_z;
% End of function file.