Joint Mobility and Co-channel Interference Characterization at System Level for SDMA

Cellular Systems Carmen B. Rodríguez-Estrello and Felipe A. Cruz-Pérez

Communication Section, Electrical Engineering Department, CINVESTAV Av. IPN 2508, Col. San Pedro Zacatenco, CP 07360, Mexico City, MEXICO

Tel.: (52) 55 57473800 ext. 6352 and 6351, Fax: (52) 55 5747 3799 e-mail: crodriguez@cinvestav.mx and facruz@cinvestav.mx

Abstract. Even though degraded link condition due to the excessive intra-cell co-channel interference is the major cause of dropping calls in SDMA mobile cellular systems, most of the published system level studies have not considered co-channel interference as a cause of call forced termination. To include the effect of co-channel interference at system level models, “beam overlapping time” has been previously introduced. Beam overlapping time is the period of time in which users sharing the same radio resource degrade the link condition of each other. This time depends on the current co-channel interference conditions which are the result of the real beam pattern and users mobility. Due to the necessity to understand the behavior of this time, some published works have characterized it based on geometrical considerations. However, if only geometrical considerations are contemplated current co-channel interference conditions are ignored. Thus, to fill this void in this paper beam overlapping time is statistically characterized considering real beam patterns as well as realistic users’ movement instead of geometrical considerations. Beam overlapping time is statistically characterized via simulation by means of its cumulative distribution function (CDF), mean, variance, and coefficient of variation. Numerical results show that beam overlapping time strongly depends on the antenna array parameters as well as on the users’ mobility.

Keywords: SDMA, cellular networks, system level model, mobility, beam overlapping time, cell dwell time, co-channel interference.

I. INTRODUCTION

Spatial Division Multiple Access (SDMA) has been proposed in 3G and 4G cellular systems (i.e. WiMAX, LTE, WCDMA) [1]-[2] as one of the leading technologies for achieving high spectral efficiency by reusing basic radio resources within cells1. SDMA technology is based on the beamforming capability of smart antennas to create a maximum in the direction of the signal of interest while nulling interference. However, if the beamforming process is not ideal, interferers are not eliminated at all. Then, more interference is carried both within the cell and into the co-channel cells. Moreover, users’ mobility produces a constantly changing radio environment. Therefore, mobile users share a basic radio resource within a cell may degrade each other link condition. As a consequence, calls can be dropped due to the excessive co-channel interference. Thus, it is necessary to include co-channel interference as a cause of forced termination at system level models2.

1 Basic radio resources are frequency in FDMA systems, time slots in TDMA systems, codes in CDMA systems or chunks in OFDMA systems. This research was performed under the support of CONACyT project 50434

Although several published papers have studied SDMA systems at system level [3]-[9], most of them have not considered the joint effect of mobility and non-ideal beamforming on co-channel interference [3]-[9]. Only some efforts have been made in order to include current co-channel interference conditions at system level through the probability of failure in the attempt to reuse a resource [4]-[6]. This probability reflects radio environment at the beginning of the call. However, it does not reflect mobility conditions. To fill this void, in [10] a system level analytical model that includes users’ mobility and link unreliability due to the co-channel interference has been proposed. In [10] the effect of co-channel interference is captured at system level model by introducing a Poisson call interferential process and a potential associated time which is called “beam overlapping time”. Beam overlapping time is the period of time in which users sharing the same radio resource would affect the link condition of each other under the assumption that both cell dwell time and unencumbered service time are of infinite duration. Beam overlapping time reflects not only the users’ movement but also the link condition. Thus, beam overlapping time reflects the intracell dynamic. Moreover, beam overlapping time has a great impact on the dropping / handoff rate [10].

Previous published papers have attempted to characterize beam overlapping time [7]-[9]. However they have considered ideal beamforming as well as straight line movement. Contrary to these papers, in the present paper beam overlapping time and cell dwell time are characterized in a macroscopic way3 by considering realistic beam patterns as well as a realistic users’ mobility model [12]. The characterization of beam overlapping time is conducted via simulation and it is made in terms of their CDF, mean value, standard deviation and coefficient of variation.

The rest of the paper is organized as follows: In Section II the previous published work is described. Section III shows the system model. Then the simulation procedure is explained in Section IV. Finally numerical results are presented.

2 Link level analysis is not very useful for the system capacity analysis because it is related to the individual physical channel characterization and does not reflect the network’s dynamic. 3 The macroscopic users’ moving modeling considers general statistics of users’ movement such as CDF, mean, and variance of the involved times [11]

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II. PREVIOUS PUBLISHED WORK

As it was established, many published papers have studied the performance of SDMA cellular systems at system level [3]-[9]. However, most of them have ignored the joint effect of mobility and non-ideal beamforming on intracell co-channel interference in multicell environments.

In [3] the performance evaluation of a cellular WiMAX IEEE 802.16 with SDMA is presented. The effect of co-channel interference is analyzed through the statistical characterization of SIR. In addition, the total cell throughput is analyzed in [3]. On the other hand, the goal of [4] is to assess the impact of different propagation conditions on the probability of failure in the attempt to reuse a resource in a Wireless Local Loop (WLL) system. In [5] a teletraffic analysis of a WLL system which uses SDMA is conducted and the probability of failure in the attempt to reuse a resource is calculated via simulation considering real beam patterns of ULA arrays. Although non-ideal beam-forming is considered in [3]-[5], only fixed users are contemplated. In [6] an analytical model for evaluating the probability of failure to reuse a resource in the general case of k-fold resource is treated. However, [6] considers only ideal beamforming and non mobile users.

Only few published works have dealt with users’ mobility [7]-[9]. However, they are based only on geometrical considerations (i.e., hexagonal/circular shaped cells, users’ movement in straight line, ideal beam patterns); thus, current co-channel interference conditions experienced by the users and non-ideal beamforming are ignored at all. The influence of users’ mobility on a switched beam SDMA system is evaluated in [7]. The mobility model in [7] considers an idealized circular coverage area serving a linear section of a motorway; that is, only straight line movement is considered and no interference conditions are taken into account.

In [8]-[9] flow fluid model is used to characterize users’ mobility. Cell dwell time and beam overlapping time are calculated based on the average outgoing cell rate and the average beam overlapping rate, respectively. Cell dwell time is found through the flow fluid model which implies high randomness of mobility, while beam overlapping time is calculated by using the relative moving method which considers only straight line movement. So, there is no agreement between two methods. In [8]-[9], ideal beam patterns are used, thus the SIR is considered to be degraded when the main beams of both users physically overlap. The previous works [3]-[9] has neither considered realistic beam patterns nor a realistic users’ mobility model. Moreover, surprisingly the beam pattern’s symmetry in the uniform linear arrays4, as well as grating lobes5, has been largely ignored.

4 Notice that the beam pattern of a linear array is symmetric to the array’s axis.

III. SYSTEM MODEL

A homogeneous mobile multi-cellular system with smart antennas located at the center of cells is assumed. It is considered that SDMA is used as multiple access scheme in conjunction with a basic multiple access scheme (TDMA, CDMA, OFDMA). Due to SDMA, two or more users could share a basic resource within a cell only if they accomplish the SIR requirements. Thus, there is a probability that the basic resource could not be replicated in the admission process. In addition, due to users’ mobility, beam patterns of users served by the same radio resource could interfere each other causing severe co-channel interference during a call. As a consequence, users’ SIR would be severely degraded. Thus, beam overlapping time reflects the link condition and it is measured as the period of time since the call initiates until the user’s SIR drops below a given threshold due to excessive co-channel interference [10].

IV. SIMULATION PROCEDURE

The objective of the simulations carried out in this paper, is to statistically characterize beam overlapping time (To) [10] and the cell dwell time (Td) of users in a mobile cellular system with SDMA capability in terms of their CDF, mean value, standard deviation and coefficient of variation. In this section the simulation scenario and methodology are described.

A. Simulation scenario

Monte Carlo simulations were conducted by considering many of the parameters involved in the design of the smart antenna, such as:

• Array’s geometry, • Number of elements in the antenna array, • Separation between elements in the antenna array, • Excitation (amplitude and phase) of the elements in

the antenna array, • Beamforming algorithm and • Number of cancelled interferers6.

The evaluated scenario consists of one observed cell and

two tiers of interferer co-channel cells. The cell radius is normalized to one and the propagation model is the single slope model with a path loss exponent of 4, considering also the shadowing losses.

5 A grating lobe is a lobe of the same gain as the main lobe but in different direction. When the separation between elements is more than 0.5λ, grating lobes appear. 6 The number of cancelled interferers is an important parameter in the design of a smart antenna. Previous works have considered that the smart antenna [3]-[9] only deals with the intracell interferers (i.e., the number of users sharing the same radio resource within cell); however, in multicell environments, physical conditions should also include interference due to the co-channel cells.

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B. Shadowing model

In order to obtain a correlated shadowing both in space and time, the shadowing model used here is a combination of the ones proposed in [13] and [14]. In the former, cross correlation function of the shadowing components between one mobile user and two base stations is considered, while in the latter, spatial, angular and temporal correlation of shadowing are considered.

First, following the procedure proposed by [13] a two-dimensional map of spatial correlated shadowing for each base station involved in the analysis is generated. Then, in order to complete the shadowing model, the procedure proposed in [14] is used: the two-dimensional maps obtained with procedure described in [13] are convoluted with the proposed correlation function in [14] in order to get spatially correlated shadowing. Finally, in order to achieve temporal correlation of the shadowing fading, values of the two-dimensional map are smoothly interpolated every Tmea, as proposed in [14]

C. Mobility model

Contrary to the previously published papers [3]-[9], in the present paper, users’ mobility is characterized by considering correlated directions of movement by means of a smooth random mobility model [12]. Not only does the mobility characterization here presented considers a linear movement but also a smoothed random movement. The mobility model proposed in [12] is characterized by the parameter α: the probability of the variation of the mobile direction along its path is a uniform distribution limited in the range of ±α relative to the current direction.

D. Simulation methodology

Simulation was conducted in two general steps. The first one simulates the admission process and the second one, the mobility process.

1. Admission process

a. The number of users determined by the intracell reuse factor is generated in each cell7: each user is uniformly generated in the total area of the cell.

b. Beam pattern of each user is calculated according to the physical parameters of the array, beamforming algorithm and user’s position.

c. The shadowing matrix associated to each link is generated following the general steps in [13].

d. The SIR of each user is calculated. If the SIR of any of the users is below a given threshold (SIRmin), another user is generated and the process is repeated until all users accomplish with the minimum required threshold.

7 A fully loaded system is evaluated. This case is the worst one in a cellular network in terms of interference.

2. Users’ mobility process

The second step consists in analyzing how users’ mobility impacts on the beam overlapping time based on the users’ SIR.

a. Position of each user as well as shadowing matrix is updated according to the mobility and shadowing models every Tmea.

b. The users’ SIR within the central cell is calculated. If the SIR of any of the users in the central cell decays below that of a given threshold (SIRmin) simulator counts the time in which this condition is met and it is registered as the beam overlapping time (to).

c. When a user leaves the cell (td) (i.e., its position is outside the cell) simulator registers the period of time as the cell dwell time.

d. If any of these events occurs, the time when it happens is reported and a new user is randomly generated. Simulation stops after Tsim..

Leaving a cell and beam overlapping are mutually exclusive events.

V. NUMERICAL RESULTS

In this section, numerical results of CDF, mean, variance, and coefficient of variation obtained from the conducted simulations are presented. In order to make an extensive analysis of beam overlapping time (To) and cell dwell time (Td) simulations were carried out in several different scenarios. There is a set of parameter that remain fixed for all the scenarios: the number of antenna elements is 6, the separation between elements of the array is 0.5λ, the excitation is uniform and the beamforming algorithm is MMSE. Each scenario differs from others in one parameter, while the rest are fixed. Table I shows the value of the different parameters and the scenario in which they vary. For instance, in scenario A the geometry of the array is changed. Thus, for scenario A the following parameters are fixed: randomness of mobility is α = 0°, minimum SIR requirement is 0 dB, inter-cell reuse pattern is 7, intracell reuse is 2, and only the intracell interferers are cancelled, while the geometry of the array varies taking Uniform Circular Array (UCA), Uniform Linear Array (ULA) and Ideal. In Scenario B the minimum SIR requirement varies and so on.

TABLE I. SCENARIOS OF EVALUATION

Parameter Value Varies in Scenario Geometry UCA A (UCA, ULA, Ideal) Minimum SIR requirement 0 dB B(0 dB, 17 dB) Intercell reuse pattern 7 C (3,4,7) Intracell reuse 2 D (2,3) Number of cancelled interferes Intracell E (intra and intercell) Randomness of mobility (α) 0° F (0°, 20°, 40°)

Figures 1-6 shows the CDF of the beam overlapping time

for the different cases specified in Table I. Tables II to VII shows mean (Xo, Xd), standard deviation ( σo , dσ ) and the

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coefficient of variation (CVo, CVd) of beam overlapping time and cell dwell time. In Figure 1 ideal, UCA and ULA arrays are selected to have the same Half Power Beamwidth (HPBW).

From Figure 1 it could be observed that the scenario in which UCA arrays are used mean beam overlapping time is larger compared with the scenario in which ULA arrays and ideal beams are considered. This is because the beam pattern formed with UCA arrays is not symmetric and it only has one main lobe [20]-[21]. It is observed that the mean beam overlapping time is greater when ideal patterns are considered than when real patterns are considered. This is because ideal patterns do not take into account secondary lobes or symmetry. As a consequence, considering ideal patterns will result in an underestimating of intra-cell rate at system level.

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Circular

LinealIdeal

Figure 1. CDF of Beam overlapping time for scenario A

Table II. Statistical Characterization of beam overlapping time and cell dwell time for scenario A

Geometry oX σo CVo dX σd

CVd

Ideal 573.80 671.53 1.17 191.62 243.76 1.27

UCA 301.80 354.04 1.17 191.53 245.02 1.27

ULA 236.94 275.25 1.16 191.79 239.88 1.25

Figure 2 and table III show how the minimum required SIR influences the statistical characterization of beam overlapping time. Observe that if the minimum SIR requirement is stricter, the mean beam overlapping time is shorter. In this way, it is not recommended to use SDMA as an access scheme in cellular systems with high minimum SIR requirements (i.e., 17dB as in analog systems). Contrary, SDMA is more recommendable as an access scheme in third and fourth generation of mobile wireless systems in which SIR requirements are lower (i.e., LTE, WIMAX, E-UTRA, UMTS and so on).

Figure 3 and table IV show how inter-cell reuse patterns impact on beam overlapping time. If the inter-cell reuse pattern is shorter, mean beam overlapping time is also shorter. This is because inter-cell interference becomes greater due the proximity of inter-cell interferers. In that sense, neither of the previous published works [3]-[9] had taken into account the impact of inter-cell reuse pattern in characterizing neither

probability of failure to reuse a resource nor the beam overlapping time.

Figure 4 and table VI show the case when two and three users are sharing the same radio resource within a cell. As it was expected, mean beam overlapping time is greater when only two users are considered because less intra-cell co-channel interference is brought into the cell. Thus, system model should evaluate the compromise between system capacity in terms of number of resources together with the probability of dropping due to the excessive co-channel interference.

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Time

CD

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SIRmin = 17

SIRmin = 0

Figure 2. CDF of Beam overlapping time for scenario B

Table III. Statistical Characterization of beam overlapping time and cell dwell time for scenario B

SIRmin oX σo CVo dX σd

CVd

17 5.14 9.68 1.88 210.64 303.92 1.44

0 301.80 354.04 1.17 191.53 245.02 1.27

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Intercell pattern reuse = 3

Intercell pattern reuse = 4Intercell pattern reuse = 7

Figure 3. CDF of Beam overlapping time for scenario C

Figure 5 and table VII show how the number of

considered interferers affect the characterized times. Notice that when all of the inter-cell co-channel interferers are considered to be eliminated in the beamforming, mean beam overlapping time is shorter, and its coefficient of variation is

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greater. This is because of the performance of the array. It is well known that an array of N-elements [17]-[18] is able to eliminate only N-1 interferers, however when considering all of the intercell co-channel interferers, beamforming algorithms are not able to eliminate all of the interferers. Although, in multi-cellular systems is very important to consider this fact in the design, previous papers [3]-[9] have not considered intercell interference in evaluating SDMA cellular systems at system level. Table IV. Statistical Characterization of beam overlapping time and cell

dwell time for scenario C PR oX σo

CVo dX σd CVd

3 96.99 128.73 1.32 209.77 285.49 1.36

4 138.24 176.48 1.27 199.58 258.04 1.29

7 301.80 354.04 1.17 191.53 245.02 1.279

From the previous numerical results it is observed that the CDF of the beam overlapping time may follow many different forms. Then, to adequately model beam overlapping time should be preferable modeled as a generally distributed random variable. Moreover, numerical values of coeficient of variation suggest that it could be modeled through a hiper-exponential distribution (i.e., CV > 1).

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Time

CD

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2 users sharing the same channel

3 users sharing the same channel

Figure 4. CDF of Beam overlapping time for scenario D

Table V. Statistical Characterization of beam overlapping time and cell

dwell time for scenario D Intracell

Reuse oX σo CVo dX σd

CVd

2 301.80 354.04 1.17 191.53 245.02 1.27

3 156.34 192.85 1.23 194.48 247.47 1.27

The random walk mobility model [12] can represent several different scenarios [15]-[16]: randomness of mobility could be interpreted as different mobility environments. α = 0° corresponds to the case of straight line movement (i.e., highways scenarios), α = 20° corresponds to the case in which user changes its direction more constantly (i.e., suburban scenarios) and α = 40° corresponds to the case of urban scenarios where there are rapid changes in users direction [18]-[19]. Thus, Figure 6 and Table VII show the

CDF of the beam overlapping time for different mobility scenarios. Notice that more randomness entails larger beam overlapping time. This is because users’ movement is reduced when randomness increases and the radio environment changes slowly. That is, SDMA systems have larger beam overlapping times in more randomness environment. On the other hand, notice that the coefficient of variation increases as randomness increases. So that it is necessary to consider that the beam overlapping time should be described as a generally distributed random variable.

Respect to the cell dwell time, in Tables II-VII, it is observed that different considered scenarios do not affect the statistical characterization of cell dwell time. This is because cell dwell time is only affected by the cell size and users mobility characteristics (i.e., velocity and randomness of mobility) as it is shown in Figure 7 and Table VII. Figure 7 shows the CDF of the cell dwell time while randomness of mobility changes. From Table VII and Figures 6 and 7. More randomness implies larger mean cell dwell times. This is because users tend to stay around the same place because they are constantly changing their direction.

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Figure 5. CDF of Beam overlapping time for scenario E

Table VI. Statistical Characterization of beam overlapping time and cell

dwell time for scenario E Interferers oX σo

CVo dX σd CVd

Only Intracell 301.80 354.04 1.17 191.53 245.02 1.27

Inter and intracell 122.56 164.21 1.33 196.02 248.70 1.26

All of the reported results in this paper could be used directly in the system model proposed in [10] in order to include the effect of co-channel interference in the performance of cellular systems with SDMA capability.

VI. CONCLUSIONS

In this paper the characterization of beam overlapping time in cellular systems with SDMA capability has been conducted. From the system level point of view, beam

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overlapping is an important parameter because it reflects users’ mobility and non-ideal beamforming together in the co-channel interference. Characterization of beam overlapping time was based on the user’s SIR. More important, the characterization of beam overlapping time was made by considering not only realistic user’s movement but also, realistic beam patterns. From numerical results it has been demonstrated that previous published works that only consider ideal beam patterns through geometrical considerations have overestimated beam overlapping time.

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α = 0°

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α = 40°

Figure 6. CDF of Beam overlapping time for scenario F

Table VII. Statistical Characterization of beam overlapping time and cell

dwell time for scenario F α oX σo

CVo dX σd CVd

0 301.80 354.04 1.17 191.53 245.02 1.27

20 358.23 447.69 1.2497 309.00 459.02 1.48

40 482.64 671.29 1.39 595.93 887.02 1.48

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Figure 7. CDF of cell dwell time for scenario F

Numerical results show that all of the array parameters as well as network parameters involved in the system design have a great impact on the beam overlapping time. Moreover, it was shown that considering ideal beam patterns conduct to an underestimated beam overlapping time.

Future work includes the proposal of an analytical approximation for a beam pattern that reflects most of the involved characteristics on the beam patterns. Finally, the statistical characterization presented here is a very useful tool for teletraffic models because it directly reflects the intracell dynamics.

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