Institutionen för systemteknik

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1 Institutionen för systemteknik Department of Electrical Engineering Examensarbete Estimation of Inter-cell Interference in 3G Communication Systems Examensarbete utfört i Reglerteknik vid Tekniska högskolan vid Linköpings universitet av Dan Gunning & Pontus Jernberg LiTH-ISY-EX--/456--SE Linköping 2 Department of Electrical Engineering Linköpings universitet SE Linköping, Sweden Linköpings tekniska högskola Linköpings universitet Linköping

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3 Estimation of Inter-cell Interference in 3G Communication Systems Examensarbete utfört i Reglerteknik vid Tekniska högskolan i Linköping av Dan Gunning & Pontus Jernberg LiTH-ISY-EX--/456--SE Handledare: Examinator: Ylva Jung isy, Linköpings universitet Graham Goodwin cdsc, University of Newcastle Katrina Lau cdsc, University of Newcastle Thomas Schön isy, Linköpings universitet Linköping, September, 2

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5 Avdelning, Institution Division, Department Division of Automatic Control Department of Electrical Engineering Linköpings universitet SE Linköping, Sweden Datum Date 2-9- Språk Language Svenska/Swedish Engelska/English Rapporttyp Report category Licentiatavhandling Examensarbete C-uppsats D-uppsats Övrig rapport ISBN ISRN LiTH-ISY-EX--/456--SE Serietitel och serienummer Title of series, numbering ISSN URL för elektronisk version Titel Title Estimation of Inter-cell Interference in 3G Communication Systems Författare Author Dan Gunning & Pontus Jernberg Sammanfattning Abstract In this thesis the telecommunication problem known as inter-cell interference is examined. Inter-cell interference originates from users in neighboring cells and affects the users in the own cell. The reason that inter-cell interference is interesting to study is that it affects the maximum data-rates achievable in the 3G network. By knowing the inter-cell interference, higher data-rates can be scheduled without risking cell-instability. An expression for the coupling between cells is derived using basic physical principles. Using the expression for the coupling factors a nonlinear model describing the inter-cell interference is developed from the model of the powercontrol loop commonly used in the base stations. The expression describing the coupling factors depends on the positions of users which are unknown. A quasi decentralized method for estimating the coupling factors using measurements of the total interference power is presented. The estimation results presented in this thesis could probably be improved by using a more advanced nonlinear filter, such as a particle filter or an Extended Kalman filter, for the estimation. Different expressions describing the coupling factors could also be considered to improve the result. Nyckelord Keywords inter-cell interference, thermal noise, Decision Feedback Equalizer, estimation, 3G, Kalman filter, nonlinear model, quasi-decentralized estimator

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7 Abstract In this thesis the telecommunication problem known as inter-cell interference is examined. Inter-cell interference originates from users in neighboring cells and affects the users in the own cell. The reason that inter-cell interference is interesting to study is that it affects the maximum data-rates achievable in the 3G network. By knowing the inter-cell interference, higher data-rates can be scheduled without risking cell-instability. An expression for the coupling between cells is derived using basic physical principles. Using the expression for the coupling factors a nonlinear model describing the inter-cell interference is developed from the model of the power-control loop commonly used in the base stations. The expression describing the coupling factors depends on the positions of users which are unknown. A quasi decentralized method for estimating the coupling factors using measurements of the total interference power is presented. The estimation results presented in this thesis could probably be improved by using a more advanced nonlinear filter, such as a particle filter or an Extended Kalman filter, for the estimation. Different expressions describing the coupling factors could also be considered to improve the result. v

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9 Acknowledgments This thesis has been performed at the University of Newcastle Australia at the Centre for Complex Dynamic Systems and Control. We would like to thank everyone at the department, especially Professor Graham Goodwin and Dr. Katrina Lau without whom this thesis could not have been successfully completed. We would also like to acknowledge Ericsson for providing the motivation for this thesis. We especially thank Professor Torbjörn Wigren and Dr. Erik Geijer Lundin for their assistance. Thank you for all your help, patience and enthusiasm for the project. It has been a very interesting and challenging project and we have learned a lot. We would also like to thank our examiner Dr. Thomas Schön and our supervisor Ylva Jung for all their support and encouragement. Dan Gunning & Pontus Jernberg Newcastle, July 2 vii

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11 Contents Introduction. Purpose & Goals Background The Evolution of 3G communicators Rate constraints Power Control and Interference Model Limitations Related research Theory 7 2. Power control loop description and analysis Single cell, n equal users Multiple cells, n equal users Inter-cell model Loads Stability - Multi-cell constraints cell case cell case n-cell case Stability - conservatism versus failure-rate A linear model for estimating the coupling Interference model Matrix inversion model Iterative model Fast model Model with fixed λ State simulation Kalman filter Decision Feedback Equalizer Results Stability experiments Experiment Experiment ix

12 x Contents 3..3 Experiment Experiment Experiment Estimation experiments Experiment A Experiment B Experiment C Experiment D Experiment E Concluding remarks 6 4. Conclusions Future work Bibliography 63 A Abbreviations 65 B Notation 66 C Matlab code for the estimation experiments 68

13 Chapter Introduction This chapter contains the purpose, goals and background of this thesis. It also includes the limitations and a short description of similar research that has been done before.. Purpose & Goals The purpose of this thesis is to better understand single cell and multi-cell interference in 3G mobile communication systems. The motivation for this is that the entire operation of 3G communication depends critically on interference. The sources of interference are thermal noise on the receiver antennas. interference from other users in the same cell. self interference (auto interference). interference from users in other cells (part of this is the neighboring cell responding to the own cell). Some of these sources are known, some are partially known, some are assumed known and some are completely unknown and therefore models are needed. These models are nonlinear and contain unknown parameters and random variables. There are also noisy measurements which are nonlinear functions of the states of the model. The goals of this thesis are to. Derive an expression for the coupling between cells. 2. Develop a nonlinear model for inter-cell interference. 3. Estimate coupling factors using measurements generated by the model.

14 2 Introduction.2 Background This section contains a brief history of the evolution of 3G communicators. It also contains some details concerning rate constraints, power control and interference models..2. The Evolution of 3G communicators Mobile telephony started to become international in the early 98s with the introduction of the analog NMT system (Nordic Mobile Telephony). NMT made it possible for users traveling outside the area of their home operator to still receive service (roaming), which gave mobile phones a larger market. At this point mobile telephones were still bulky and could only support standard phone calls. During the mid 98s digital communication made it possible to develop a second generation communication system (2G) that could not only make phone calls but also send and receive small amounts of data. This introduced the possibility to send and receive and also added support for Short Message Services (SMS). The highest data rates in early 2G were 9.6 kbps. In the second half of the 99s new technology made it possible to send packet data over the cellular system. This is usually called 2.5G. An international standardization of cellular systems and a larger market for products were two of the main driving forces behind the development of the third generation communication system (3G). This system provided a higher-bandwidth radio interface which made new services, which were only implied in 2G and 2.5G, possible. Since the introduction of 3G, mobile devices have become multi-purpose devices, not only used for phone calls. As the demand on data and Internet services for mobile devices has increased so has the need for higher data rates. Therefore a main goal for the evolution of 3G is to achieve higher end-user data rates compared to what was possible in the earlier releases of 3G. This includes a higher overall data rate for the whole cell area and also a higher peak data rate. A few of the limiting factors for data rates are the available bandwidth, the signal power and the noise interfering with transmissions..2.2 Rate constraints According to Shannon s capacity theorem [3] the maximum rate that can be sent over a channel can be determined by T = BW log 2 ( + P ). (.) N In (.), T is the channel capacity, BW is the available bandwidth, P is the received signal power and N is the noise power. The noise is assumed to be white

15 .2 Background 3 and additive. The signal power can be written as P = E b R, where E b is the energy received per bit (J/bit) and R is the rate of data communication (bit/s). Also, the noise power can be expressed as N = φ BW where φ is the constant noise power spectral density (W/Hz). This gives the inequality R T = BW log 2 (+ E b R φ BW ). Defining the bandwidth utilization β = R/BW gives the expression R T = BW log 2 (+ E b β) β log φ 2 (+ E b β) 2β E b. φ β φ This expresses the signal-to-noise ratio (SNR) needed to send data with a certain rate over a limited bandwidth. Thus, for bandwidth utilization larger than one the minimum required E b /φ increases quickly, see Figure.. [2] 2 Minimum E b /φ required (db) 5 5 Power limited region Band limited region 5 β Figure.. Minimum E b /φ required at the receiver as a function of β.

16 4 Introduction.2.3 Power Control and Interference Model The channel is affected by time variations in the channel gains (fading). Users also interfere with each other due to the fact that the codes used to separate the users are not completely orthogonal. Another problem is that a user close to a base station may overpower a user who is further away (near-far effect). To solve these problems the signal-to-interference ratio (SIR) for each user is monitored by a power control loop which adjusts the users transmission power to compensate for variations in the channel conditions. The goal is to keep the received SIR at an approximately constant level to successfully transmit data. The control loop increases the power for a user who experiences poor channel quality and decreases it for a user who experiences good channel quality. At time k the user transmits data at a power of γ(k) p (k), where γ(k) is a scaling factor called the power grant and p (k) is the transmitted power. A users data-rate is determined by the power grant. The received power on the control channel at the base station is given by p(k) = p (k) g(k), where g(k) is the fading gain. [9] Note that, in the rest of this thesis, both linear and logarithmic scales (i.e. db) are used. A bar is used to denote a linear quantity. The SIR S for user i at time k is given by S i (k) = p i(k) I i (k), where I i (k) is the interference to the user, and is given by I i (k) = j n, j i (+γ j (k)) p j (k)+α (+γ i (k)) p i (k)+n +I other (k). (.2) In equation (.2), (+γ j (k)) p j (k) is the interference from user j, α (+γ i (k)) p i (k) is the interference from the user to itself (α is a constant) and I other (k) is the interference from users in other cells. The parameter N is unknown and consists of thermal noise and other sources of interference. [9] By rewriting (.2) as I i (k) = C ( α) (+γ i (k)) p i (k), where C = j n (+ γ j (k)) p j (k)+n +I other (k) it can be shown that N +I other (k) is a scaling factor for the total interference. Therefore, by estimating N + I other (k) it is possible to get a better estimation of the interference I i (k). This will give a better model to use for power control, which in turn makes higher data rates possible.

17 .3 Limitations 5.3 Limitations The experiments on inter-cell interference are limited to the interaction between two cells only. The reason for this is that at least two cells are needed to examine the problem, but using more than two cells would just increase the difficulty of interpreting the results. However, the models and all the formulas have been developed with an arbitrary number of cells in mind. Since there is no way to measure the coupling factors without geographical information about every user in a cell, the quality of the estimates can only be measured against simulations. Variations in the channel gain (fading) are not considered..4 Related research Both [2] and [8] are good choices for better understanding mobile communications, especially the evolution of the 3G network and the techniques used. They also give a description of the background and the technical details surrounding data transfers in the 3G network. [2] also contains details concerning the upcoming LTE system. Relevant aspects and discussions on Universal Mobile Telecommunication System (UMTS) power control using an automatic control framework are described in [7]. The problem of estimating inter-cell interference is closely linked to the problem of estimating the thermal noise level. One approach to the latter problem is described in [4] where a nonlinear three stage algorithm is used. The only measurement required is the Received Total Wideband Power (RTWP). In the first stage a Kalman filter is used to estimate a Gaussian Probability Density Function (PDF) of the estimated RTWP. In the next stage the PDF is further processed to produce an estimate of the thermal noise power floor. The last stage of the algorithm uses the estimated RTWP and the thermal noise power floor to compute the Wideband Code Division Multiple Access (WCDMA) load in a single cell. The algorithm is a workaround for the problem that the thermal noise power floor is not observable due to neighbor cell interference. This algorithm is further developed in [2] to include inter-cell interference in a single Radio Base Station (RBS). To reduce memory usage of the algorithm without compromising performance a recursive scheme to estimate the thermal noise power floor has been developed. This gives the possibility to run several instances of the algorithm in parallel in the RBS. [3] In [9] a nonlinear decoupling algorithm for the uplink of the WCDMA 3G cellular system is described. This is a way to combat the interference from users within the same cell. The paper shows that decoupling strategies lead to significant performance gains relative to the decentralized strategies used today. The basic theory for constructing mathematical and physical models for a process as well as model validation and simulation can be found in []. A basic framework for stability analysis of Multiple-Input Multiple-Output (MIMO) systems and nonlinear models is treated in [4]. A popular method for estimating

18 6 Introduction unobservable (latent) states of a nonlinear model is to use a Particle filter (PF). The fundamental PF theory as well as its use in tracking applications is described in [5]. The nonlinear filtering problem is also presented. In addition the article contains an overview of the Marginalized (often called Rao-Blackwellized) Particle filter (MPF or RBPF) and a general framework of how PF can be applied to complex systems.

19 Chapter 2 Theory This chapter describes the theoretical expressions and calculations that are later used in the experiments in chapter 3. It also contains theory taken from literature as well as models and mathematical expressions developed in this thesis. 2. Power control loop description and analysis To control the SIR of a User Equipment (UE) within a cell in a mobile network, power control is used. The power control is commonly handled in a decentralized manner with one Single-Input Single-Output loop (SISO) for each UE. A simplified block diagram of the power control loop for a single user is shown in Figure 2., where S is the desired SIR for the user (in db), p (k) is the transmitted power, p(k) is the received power, g(k) is the channel gain, I(k) is interference power and S(k) is the SIR at time k. A typical choice for the controller is K(q) =. An alternative which has been considered in some detail in the literature is K(q) = /( + q q d+ ) where d is the delay and the mobile station G(q) = q d /( q ), (see [9] and the references therein). g(k) I(k) Desired SIR Controller Mobile station S + + e(k) u(k) p (k) p(k) K(q) G(q) S(k) + + Measured SIR Figure 2.. Power control loop for a single user. 7

20 8 Theory 2.. Single cell, n equal users By assuming d = and g(k) = the following can be derived p(k + ) = p(k) + u(k), u(k) = e(k), e(k) = S S(k), S(k) = p(k) I(k). For the case of n equal users I(k) = n (+γ(k)) p(k)+(α (+γ(k)) p(k)+n ), where α ( + γ(k)) p(k) is the auto-interference or self-interference and N is the thermal noise at the antenna. As noted in section.2.3 a bar denotes a linear quantity. This gives the expression p(k + ) = p(k) + S S(k) = p(k) + S p(k) + I(k) = S + I(k) which in linear scale becomes p(k+) = S I(k) = S (α (+γ(k)) p(k)+n ). (2.) Equation (2.) is a linear state-space representation of the system and can be rewritten as p(k + ) = A p(k) + b, where A = S α ( + γ(k)) and b = S N. The poles of a system, which determine its stability properties, are given by the eigen-values of the matrix A [4]. By examining the eigenvalues for increasing values of the power-grant γ it is possible to find the maximum value of γ before the system reaches instability. It is assumed that the data-rate is proportional to the power grant and hence, maximizing the cell throughput is equivalent to maximizing the sum of the users grants. In the case of n equal users (i.e. γ i = γ), this is equivalent to maximizing γ. The upper limit on the data-rate can also be estimated using the expression for steady state power, described by p = S N (n + α) ( + γ) S. (2.2)

21 2. Power control loop description and analysis 9 For the case of a single user (n = ) Figure 2.2 shows the value of the power grant γ that gives the highest achievable data-rate with these two approaches. As expected the two methods yield the same result and in this case instability is reached at γ = 22. Real component of pole x 9 Power [mw] γ Figure 2.2. Poles (top) and Power (bottom) for increasing power grant (γ) where S = /64, α =.3 and n =. Figure 2.4 shows the value of the power grant γ that gives the highest achievable data-rate for n = 2. It also assumes that S = S 2 = /64 and α = α 2 =.3. Just as before the two approaches give the same result. As seen in the figure instability is reached at γ = γ 2 = 49 which is earlier compared to the single-user case Multiple cells, n equal users For the case of users in neighboring cells I i (k) = n ( + γ i (k)) p i (k) + α i ( + γ i (k)) p i (k) + N + I otherji (k), where I otherji (k) is the interference from user j to user i. The resulting SISO-loop, for the case of two users, is shown in Figure 2.3, where IC l is a gain factor for how much user l interferes with the other user and H l is the interference transfer function for user l, in this case l =, 2.

22 Theory IC Iother2(k) g(k) H N Desired SIR S + Controller Mobile station e(k) + u(k) p (k) K(q) G(q) p(k) + + I(k) S(k) Measured SIR IC2 Iother2(k) g2(k) H2 N Desired SIR S2 + Controller Mobile station e2(k) + u2(k) p 2(k) K2(q) G2(q) p2(k) + + I2(k) S2(k) Measured SIR Figure 2.3. The common SISO-loop for two users in neighboring cells. Using the same assumptions as in the single-user case the following equations can be derived p (k + ) = p (k) + u (k), u (k) = e (k), e (k) = S S (k), S (k) = p (k) I (k), p 2 (k + ) = p 2 (k) + u 2 (k), u 2 (k) = e 2 (k), e 2 (k) = S2 S 2 (k), S 2 (k) = p 2 (k) I 2 (k).

23 2. Power control loop description and analysis The interference for the users, expressed in linear scale, are described by where I (k) = α ( + γ (k)) p (k) + N + I other2 (k), I 2 (k) = α 2 ( + γ 2 (k)) p 2 (k) + N + I other2 (k), I other2 (k) = IC 2 ( + γ 2 (k)) p 2 (k), I other2 (k) = IC ( + γ (k)) p (k). In the same way as in the single-user case above, the linear recursive power equations are given by p (k + ) = S (α ( + γ (k)) p (k) + N + IC 2 ( + γ 2 (k)) p 2 (k)), (2.3) p 2 (k + ) = S 2 (α 2 ( + γ 2 (k)) p 2 (k) + N + IC ( + γ (k)) p (k)). (2.4) Rewriting (2.3) and (2.4) using vector notation gives the state-space equation p k+ = A p k + b. (2.5) To find the maximum achievable data-rate before reaching instability the eigenvalues of the matrix A in (2.5) are examined for equal increase of the power grants γ (k) and γ 2 (k) as well as IC = IC 2 =. In other words the users interfere fully with one another. The result can then be compared to the steady-state power given by (2.2) with n = 2.

24 2 Theory Real component of pole 2 Pole Pole x 9 Power [mw] γ Figure 2.4. Poles (top) and Power (bottom) for increasing power grant (γ) and n = 2.

25 2.2 Inter-cell model Inter-cell model To get a basic understanding of the effects of inter-cell interference, consider the simple example in Figure Cell Cell 2 Distance y [m] UE 2 UE UE Distance x [m] Figure 2.5. Two cells with a total of three users. Base stations are illustrated with a circle. By writing the power equations on state-space form as in (2.5) we get the following A-matrix [ S α ( + γ (k)) S ( + γ 2 (k)) S IC 2 ( + γ 2 (k)) S 2 ( + γ (k)) S 2 α 2 ( + γ 2 (k)) S 2 IC 2 ( + γ 2 (k)) S 2 IC 2 ( + γ (k)) S 2 IC 2 2 ( + γ 2 (k)) S 2 α 2 ( + γ 2 (k)) ] Let the positions of the UEs and the base stations as well as the radius of the cells be known. To determine the value of IC ij l, which is the interference gain from user j in cell i to cell l, the power reaching the own base station, P, and the power reaching the other base station, P 2, are needed. If it is assumed that there is no fading these two quantities can be expressed as in equation (2.6) and equation (2.7) using the inverse square law. P = P ij d 2, ij i (2.6) P 2 = P ij d 2, ij l (2.7) where P ij is the transmission power from user j in cell i, d ij i is the distance from user j in cell i to the base station in cell i and d ij l is the distance from user j in cell i to the base station in cell l.

26 4 Theory The coupling factors can then be expressed as the ratio P 2 /P as in IC ij l = ( d ij i d ij l ) 2. (2.8) Figure 2.6 shows a map of the resulting IC-values when two cells with the same radius are placed next to each other with no overlap (as in Figure 2.5). The Figure shows the IC-values as a function of position for the cell on the left hand side. Figure 2.6. The value of the coupling factors depending on the UE position in the cell. By assuming that the positions of the users and base stations are known it is possible to calculate the A-matrix and examine the eigenvalues for different values of γ ij. Furthermore the results in Figure 2.7 assume that S ij are equal for all [i, j], no fading, α ij are equal for all [i, j], γ ij increases equally for all users in all cells and that the setup is the same as in Figure 2.5. Instability is reached for γ = 45 which is almost the same as in Figure 2.4 where there are two users and the interference gain IC =. If full coupling would be assumed instability is reached at γ = 27. The simple case described above can be extended to a more general case with an arbitrary number of users in each cell. The structure of the resulting A-matrix is shown in Figure 2.8.

27 2.3 Loads Real component of pole γ Figure 2.7. Pole placement for increasing γ values where S = S 2 = S 2 = /64, α = α 2 = α 2 =.3 and n = 3. Self-interference elements Elements for interference from users in the same cell Elements for interference from users in other cells Figure 2.8. The structure of the general A-matrix. 2.3 Loads In section 2. the eigenvalues of the A-matrix were calculated in order to get the stability properties of the system. An alternative to this is to calculate the load L i (k) for i =... n where n is the number of users in the cell, as described in (2.9). It is known that, for a single cell, if n i= L i(k) <, the system is stable [8]. This stability criterion is valid under the assumption that the total interference power is arbitrarily large, which is shown at the end of this section. ( + γ L i (k) = i (k)) + ( α S i(k) i) ( + γ i (k)) (2.9)

28 6 Theory The stability criterion n i= L i(k) < can be derived by looking at the SIRequation for user i in cell l S li (k) = P li(k) I li (k). (2.) Let C l be the total interference power for cell l. Then C l is given by n C l (k) = (+γ i (k)) P li (k)+i otherml (k)+n l. (2.) i= It follows that I li (k) = C l (k) ( α li ) ( + γ li (k)) P li (k). Equation (2.2) is then given by rearranging (2.) as described below S li (k) = P li (k) C l (k) ( α li ) ( + γ li (k)) P li (k) C l (k) ( α li ) ( + γ li (k)) P li (k) = P li(k) S li (k) ( ) C l (k) = S li (k) + ( α li) ( + γ li (k) P li (k) C l (k) P li (k) = + ( α S li (k) li) ( + γ li (k)). (2.2) Let I otherml (k) be the inter-cell interference from cell m to cell l described by I otherml (k) = m, n m j= m l IC mj l ( + γ mj (k)) P mj (k). (2.3)

29 2.3 Loads 7 Inserting (2.3) into equation (2.) gives n C l (k) = ( + γ li (k)) P li (k) i= n m + m, j= m l IC mj l ( + γ mj (k)) P mj (k) + N l (2.4) Equation (2.5) is given by inserting (2.2) into (2.4). C l (k) = n l i= + m, C l (k) = n l j= m l n l ( + γ li (k)) C l (k) S li (k) + ( α li) ( + γ li (k)) n m i= L li (k) + m, ( + γ mj (k)) C m (k) IC mj l + ( α S mj(k) mj) ( + γ mj (k)) + N l i= L li (k) C l (k) + m, n m j= m l n m j= m l IC mj l L mj (k) C m (k) + N l IC mj l L mj (k) Cm(k) C l (k) + N l C l (k) = (2.5) Normally the contribution from the other cells, I otherml (k), is not taken into account. It is then easy to see that the stability criterion n i= L i(k) < is only valid if C l (k) is arbitrarily large. However, if I otherml (k) is taken into account and C l (k) is assumed to be arbitrarily large the stability of cell m will be affected. The reason for this is that the factor C m (k)/c l (k) in (2.5) will be inverted. Clearly it is not possible to have n i= L i(k) arbitrarily close to when the contribution from the other cells is considered.

30 8 Theory 2.4 Stability - Multi-cell constraints This section describes different approaches to derive necessary and sufficient conditions for stability that consider inter-cell interference cell case In this section the 2-cell case, which is a special case of (2.5) is examined. The total interference power for cell and cell 2 are given by (2.6) and (2.7) respectively. a {}}{{}}{ n n 2 C (k) L i (k) C (k) IC 2j L 2j (k) C 2 (k) = N (2.6) i= j= b n 2 n C 2 (k) L 2j (k) C 2 (k) IC i 2 L i (k) C (k) = N 2 (2.7) j= } {{ } a 2 i= }{{} By writing (2.6) and (2.7) on matrix-form we get [ ] a b b 2 a 2 }{{} A [ C C 2 }{{} C ] [ ] N = N 2 }{{} N Solving this expression for C gives: b 2 [ ] C = det(a) a2 b N (2.8) b 2 a }{{} A For a feasible solution to (2.8), which is necessary for stability, C > must hold as it is not possible to have a negative total interference power. Since N, a, a 2, b, b 2 > the solution is only positive if a, a 2 < and det(a) >. The determinant is given by det(a) = ( a ) ( a 2 ) b b 2. The condition that det(a) > gives the inequality b b 2 < ( a ) ( a 2 ). (2.9) Together with a, a 2 <, (2.9) forms a necessary and sufficient condition for stability.

31 2.4 Stability - Multi-cell constraints 9 If full coupling from the other cell is assumed, b = a 2 and b 2 = a hold. Using this, the following sufficient condition for stability can be derived a a 2 < ( a ) ( a 2 ) a a 2 < + a a 2 a a 2 a + a 2 <. (2.2) The sufficient stability condition above also holds in the case of coupling less than one, since b b 2 a a 2 < ( a ) ( a 2 ). If the coupling is greater than one a necessary condition for stability is that either b or b 2 has to be less than one. They can not both be greater than one at the same time. As mentioned above, C has to be greater than zero for a feasible solution. This means that A has to be elementwise positive. Using Perron-Frobenius theorem, see [], it can be shown that (I B) iff ρ(b) = λ max (B) <, (2.2) where B is a nonnegative matrix. It is not hard to see that the A-matrix can be written in the form [ ] A = I a b, where B is a nonnegative matrix. b 2 a 2 }{{} B The necessary and sufficient condition given by (2.2) states that if the spectral radius of B, ρ(b) which is equal to the absolute of the largest eigenvalue of B, is less than one, then A is positive and we have a feasible solution for C.

32 2 Theory The eigenvalues of B are calculated as below det(b λ I) = a λ b a 2 λ = b 2 (a λ) (a 2 λ) b b 2 = a a 2 a λ a 2 λ + λ 2 b b 2 = (λ a + a 2 2 λ = a + a 2 2 ) 2 ( a + a 2 ) 2 + a a 2 b b 2 = 2 ± ( a + a 2 ) 2 2 a a 2 + b b 2. It can now be seen that the largest absolute eigenvalue is given for the case of full coupling. In this case, λ = a + a 2, λ 2 =. Hence, λ max (B) a + a 2. This together with Perron-Frobenius theorem gives the sufficient condition a + a 2 <, which is the same as in (2.2) cell case In this section the 3-cell case is analyzed and stability conditions are derived. The calculations are similar to the ones in 2.4. but since there are 3 cells the calculations are a bit more tedious. C is calculated as in (2.8) but in this case the A-matrix is given by a b 2 b 3 A = b 2 a 2 b 23 b 3 b 32 a 3 By rewriting A on the following form, (2.2) can once again be used. a b 2 b 3 A = I b 2 a 2 b 23, where B is a nonnegative matrix. b 3 b 32 a 3 }{{} B

33 2.4 Stability - Multi-cell constraints 2 The eigenvalues of B are calculated as below det(b λ I) = λ 3 + (a + a 2 + a 3 ) λ 2 + ( a a 2 a a 3 a 2 a 3 + b 2 b 2 + b 3 b 3 + b 23 b 32 ) λ + a a 2 a 3 + b 2 b 32 b 3 + b 3 b 2 b 23 a b 23 b 32 a 2 b 3 b 3 a 3 b 2 b 2 =. If full coupling is assumed the eigenvalues are λ = a + a 2 + a 3, λ 2 = λ 3 =. But what happens to the eigenvalues when the coupling is less than one? Is it possible to get a matrix with an absolute eigenvalue greater than in the case of full coupling? The short answer is no. For a matrix where the elements in each column are a scaling (between and ) of that columns diagonal element it can be shown, using Gersgorin discs, that the largest absolute eigenvalue occurs when all the scaling factors are equal to one. Details on Gersgorin discs can be found in []. In other words, it can be concluded that a + a 2 + a 3 < is a sufficient condition for stability in the case of full coupling (or less) n-cell case Since the A-matrix always has the same structure, the necessary and sufficient condition given by (2.2) as well as the sufficient condition a + a a n < are valid also in the n-cell case. One might be tempted to make things more simple by only considering two cells at a time (pairwise stability). However, it can be shown that even if a cell has pairwise stability with all of its neighboring cells the macro-cell can still be unstable. In other words pairwise stability is not a sufficient condition for stability.

34 22 Theory 2.5 Stability - conservatism versus failure-rate The sufficient stability condition a + a a n < is quite conservative given the fact that it is not possible to have full coupling with more than one neighbor. The purpose of the experiments in this section is to find out whether it would be a good idea to estimate the coupling factor from one cell to another. This section assumes a total of two cells. The modified stability condition used in this experiment is given by a + µ a 2 < ɛ, (2.22) where µ is a scaling factor between and and ɛ is a threshold. Let µ be defined as µ i = n i j= IC ij l n i, (2.23) where IC ij l is the coupling factor user j in cell i affects cell l with. Let us now simulate the case where three UEs with grants between 5 and 3 are placed randomly in each cell. Stability is then checked by analyzing the poles of the system. The left hand side of equation (2.22) is also calculated using equation (2.23). This simulation is then repeated for a total of times. Using the simulation data an ɛ, as well as a false alarm probability, that gives a missed detection probability of.% can be calculated. The result of the simulation is ɛ =.97 and a false alarm probability of.45%. From this we may conclude that knowing the µ-parameter of a cell is very useful when scheduling the loads in each cell.

35 2.6 A linear model for estimating the coupling A linear model for estimating the coupling In this section the linear model that will be used for estimating the coupling factors is described. Equation (2.5) shows that the total interference power for a cell can be written as C l (k) = n l i= L li (k) C l (k) + m, n m j= m l IC mj l L mj (k) C m (k) + N l, where n l and n m are the number of UEs in the cells l and m. Let us now look at the two-cell case and let us assume that the coupled load from the other cell can be expressed using only one coupling factor instead of one for each UE. The equation above can then be written as a (k) {}}{{}}{ n n 2 C (k) = L i (k) C (k) + µ 2 (k) L 2j (k) C 2 (k) + N, i= j= a 2(k) n 2 n C 2 (k) = L 2j (k) C 2 (k) + µ (k) L i (k) C (k) + N 2. (2.24) j= Since C (k) and C 2 (k) can be measured and a (k) and a 2 (k) are assigned by the scheduler and therefore known, let y(k) = x(k) = i= [ ] C (k) be the measurements and (2.25) C 2 (k) µ (k) µ 2 (k) N (k) N 2 (k) be the states. (2.26)

36 24 Theory Let us assume that the time update of the first two states can be described as a semi-random walk and that the time update of the last two states can be described by a random walk as shown below x (k + ) = ρ x (k) + ω (k), x 2 (k + ) = ρ 2 x 2 (k) + ω 2 (k), x 3 (k + ) = x 3 (k) + ω 3 (k), x 4 (k + ) = x 4 (k) + ω 4 (k), where ρ and ρ 2 are constants between and and ω i (k), i =,..., 4, denotes the process noise. This gives the linear state-space model ρ x(k + ) = ρ 2 x(k) + ω(k), }{{} A a 2(k) C 2(k) y(k) = a (k) a (k) x(k) + v(k), (2.27) a (k) C (k) a 2(k) a 2(k) }{{} C which will be used to estimate the coupling factors. Note that v(k) is the measurement noise.

37 2.7 Interference model Interference model In order to test the estimator, the system needs to be simulated. This involves generating grants for the users, simulating the positions of the users, converting the grants and positions to loads and coupling factors, respectively, and then simulating the total interference response (for each cell) to the given loads and coupling factors. In this section, four models which describe the interference response for a given set of loads and coupling factors will be presented and evaluated. To make it easier to compare the different models and draw conclusions there is only one UE in each cell Matrix inversion model This simple model is a matrix inversion based on equation (2.8) which solves the equation for C (k) and C 2 (k) given a matrix A. In this case [ ] a A = (k) µ 2 (k) a 2 (k) µ (k) a (k) a 2 (k) (2.28) and we get the solution [ ] C (k) = C 2 (k) [ ] det(a) a2 (k) µ 2 (k) a 2 (k) µ (k) a (k) a (k) }{{} A [ ] N (k) N 2 (k) (2.29) To examine the dynamics of this model a step test was performed. As seen in Figure 2.9 the response is instantaneous which is a desired property. Load Load over time Cell Cell x 4 5 Total interference power over time Power [W] Cell Cell Figure 2.9. Step response with coupling factors µ = µ 2 =.2.

38 26 Theory Iterative model Another way of modeling the interference is to iteratively use equation (2.24) as a time update. C (k + ) = a (k) C (k) + µ 2 (k) a 2 (k) C 2 (k) + N C 2 (k + ) = a 2 (k) C 2 (k) + µ (k) a (k) C (k) + N 2 This gives the dynamic step response shown in Figure 2.. It can be seen that it takes the model some time to rise to the new value. Load Load over time Cell Cell x 4 5 Total interference power over time Power [W] Cell Cell Figure 2.. Step response with coupling factors µ = µ 2 =.2.

39 2.7 Interference model Fast model The model given by equation (2.3) below, solves equation (2.24) for C and C 2 at each time instance and uses this as a time update, C (k + ) = µ 2(k) a 2 (k) C 2 (k) a (k) C 2 (k + ) = µ (k) a (k) C (k) a 2 (k) + N a (k) + N 2 a 2 (k). (2.3) The step response for this model is shown in Figure 2.. As shown the model reaches its final value faster than the model described in section Therefore this model is more suitable for cases where the multi-cell system has a relatively fast response. Load Load over time Cell Cell x 4 5 Total interference power over time Power [W] Cell Cell Figure 2.. Step response with coupling factors µ = µ 2 =.2.

40 28 Theory Model with fixed λ This model, given by equation (2.3), can be seen as an extension of the model described in section using a forgetting factor λ, old info {}}{ C (k + ) = λ C (k) + innovation {}}{ ( λ) ( µ 2(k) a 2 (k) C 2 (k) + N ) a (k) C 2 (k + ) = λ C 2 (k) + ( λ) ( µ (k) a (k) C (k) + N 2 ). (2.3) a 2 (k) Figure 2.2 shows the step response for this model. The value of the forgetting factor determines how much the previous value affects the current value. A low value makes the model more suitable for cases where the loads vary fast but also more sensitive to noise. A high value gives the opposite result. Load Load over time Cell Cell x 4 5 Total interference power over time Power [W] Cell Cell Figure 2.2. Step response with coupling factors µ = µ 2 =.2 and λ =.5.

41 2.8 State simulation State simulation As mentioned in section 2.7, simulations of the states are needed in order to produce measurements of the total interference power. These measurements are later used in the estimation of the states. The simulation has two neighboring cells, each containing a number of UEs. Both the positions and the grants of the UEs are random walks from random starting positions. Figure 2.3 contains an example-run of the simulation showing the cells and the positions of the UEs. UE positions 4 Cell Cell Distance y [m] Distance x [m] Figure 2.3. Example-run of the simulation showing how the three UEs in each cell move around. The UE positions have a standard deviation of 4 m/sample in each direction. The loads of the UEs in this example-run of the simulation are shown in Figure 2.4. The magenta line shows the sum of the loads in the own cell. The black line shows the total load, which is the sum of the loads in the own cell plus the coupled loads from the other cell. Loads over time in cell Load Load Loads over time in cell Figure 2.4. Example-run of the simulation showing the total load, sum of the loads and the loads of the three UEs in each cell. The UE loads have a standard deviation of.5 per sample.

42 3 Theory Using the position information of the UEs and the geometry described in equation (2.8) the coupling factors for each UE can be calculated. The result is shown in Figure 2.5. It is easy to see that this result is reasonable by looking at Figure 2.3 together with Figure 2.6. Observed IC values in cell.8 IC value Observed IC values in cell 2 IC value Figure 2.5. The resulting coupling factors for each UE in the two cells. One of the models described in section 2.7 can now be used to produce measurements of the total interference power. The measurements, without measurement noise, shown in Figure 2.6 are generated using the model described in section Measured total interference power over time in cell x C Measured total interference power over time in cell 2 x C Figure 2.6. Measurements of the total interference power in each cell.

43 2.9 Kalman filter Kalman filter This section describes the discrete Kalman filter, which will be used for the estimation of the states. For Gaussian noise the Kalman filter is the blue (Best Linear Unbiased Estimator) [6] and is therefore a good first candidate for the state estimation. For a system written in state-space form as x(k + ) = A x(k) + B u(k) + N ω(k), y(k) = C x(k) + D u(k) + v(k), where ω(k) and v(k) are white noise with covariance R and R 2 respectively, the cross-spectrum between ω(k) and v(k) is constant and equal to R 2. The observer that minimizes the estimation error x(k) ˆx(k) is given by ˆx(k + k) = A ˆx(k k ) + B u(k) + K (y(k) C ˆx(k k ) D u(k)). The Kalman filter gain K, is given by K = (A P C T + R 2 ) (C P C T + R 2 ), where P is the positive and semi-definite solution to the algebraic Riccati equation P = A P A T + R (A P C T + R 2 ) (C P C T + R 2 ) (A P C T + R 2 ) T. This P is equal to the covariance matrix for the optimal estimation error x(k) ˆx(k k ). [4]

44 32 Theory 2. Decision Feedback Equalizer There is always a risk that the estimates computed by the Kalman filter (KF) do not lie within the allowed limits of the states. This can lead to problems in future estimates. To ensure that the estimates lie between the limits a Decision Feedback Equalizer (DFE) can be used.[] The DFE takes the estimated state vector, ˆx(k), from the filter and if a state lies outside its allowed limits, it is saturated to the closest of these limits. This new value, ˆx(k), is set as the output and is also the feedback to the filter. The block diagram of the DFE is shown in Figure 2.7. y(k) ˆx(k) ˆx(k) KF sat( ) z Figure 2.7. Block diagram of the DFE.

45 Chapter 3 Results In this chapter, the results and conclusions of the experiments performed during this thesis are presented. 3. Stability experiments To get a better understanding of inter-cell interference a number of experiments have been conducted. The purpose of these experiments is to investigate how grant size, position and number of UEs affect the stability of the cell. 3.. Experiment In Figure 3., cell contains 2 UEs with random positions and cell two contains one UE close to the border. The UE in cell 2 has a large power grant, γ(k) = 3. In cell, the power grants of the two UEs are increased equally until instability is reached. To get a good approximation of the γ(k), in cell, that gives instability the simulation is run 2 times, using different positions. Since the positions of the UEs in cell change between each simulation they are not shown in the figure. As a reference to the results in this section, instability is reached for γ(k) = 49 for the single-cell case with two UEs. The result of this special case is independent of the positions of the UEs in cell and thus always the same no matter how many simulations are run. 33

46 34 Results 4 Cell Cell 2 Distance y [m] Distance x [m] Figure 3.. Experiment, a UE close to the border. Calculating the mean µ and variance σ 2 of the γ(k), in cell, that causes instability over the 2 simulations gives µ = , σ 2 = If the power grant in cell 2 is increased from 3 to 4 the results are µ = , σ 2 =.362. By comparing the results from the two cases above it is seen that with an increase in the power grant in cell 2 there is also in increase in the variance. Consider the case in Figure 3. again but increase the number of UEs in cell from two to three. This gives the following results µ = , σ 2 =.988. This shows not only a decrease in the mean but more interestingly a large decrease in variance. Reducing the radius of cell from 5m to 25m gives the results below µ = , σ 2 =

47 3. Stability experiments 35 These results are very similar to the original case in Figure 3., but more interestingly the value of IC 2 has increased from.67 to This is equivalent to adding.58 users, with the same grant as the user in cell 2, in cell. A possible explanation for the similar µ and σ values is that although the contribution from the user in cell 2 to cell has increased the contribution from the users in cell to cell 2 has decreased Experiment 2 To investigate how the position of the cells affect the stability, consider the case in Figure 3.2 where a UE belonging to cell 2 is placed in the intersection of the two cells. Note that the UE is closer to the base station in cell than the base station in cell 2, i.e. IC 2 >. Distance y [m] Cell Cell Distance x [m] Figure 3.2. Experiment 2, a UE belonging to cell 2 in the intersection of the two cells. The results of this case are shown below. Experiments show that the variance increases even more if the overlap is made bigger while the mean decreases. µ = , σ 2 =

48 36 Results 3..3 Experiment 3 Let us now see what happens if the position of the UE in cell 2 is changed in accordance with to Figure 3.3 and cell contains two UEs. 4 Cell Cell 2 Distance y [m] Distance x [m] Figure 3.3. Experiment 3, a UE far from the border. The results of these changes are µ = 48.65, σ 2 = This shows that the position of the UE has a significant impact on both mean and variance i.e. the contribution from a UE to the inter-cell interference decreases with the distance from the neighboring base station.

49 3. Stability experiments Experiment 4 To investigate how the size of the power grant affects the inter-cell interference the setup in Figure 3.4 is used. The number of UEs in cell is still two. 4 Cell Cell 2 Distance y [m] Distance x [m] Figure 3.4. Experiment 4. Two UEs, with different power grants, at the same distance from the other base station. The table below shows the mean and variance for the case where both UEs in cell 2 are included, only the UE with a large power grant and finally only the user with a small power grant. µ σ 2 Consider all Consider large only Consider small only As expected the UEs with a large power grant have a greater influence on both mean and variance compared to the UEs with a small power grant.

50 38 Results 3..5 Experiment 5 To further compare the influence of different power grants as well as different distances the setup in Figure 3.5 is used. 4 Cell Cell 2 Distance y [m] Distance x [m] Figure 3.5. Experiment 5, two UEs with different positions and power grants. The results of the experiment are shown in the table below. µ σ 2 Consider all Consider large only Consider small only This experiment shows that it is not only the size of the power grant, but also the distance to the opposite base station, that determines the final contribution to the inter-cell interference. Therefore it is important that also UEs with small power grants and short distances are taken into account.

51 3.2 Estimation experiments Estimation experiments There are many parameters that affect the accuracy of the estimation of the state vector. To be able to evaluate the result of the quasi-decentralized estimation strategy, a series of tests has been conducted. The notation used for the functions in the experiments in these sections is explained in the list below. Sim - the simulation described in section 2.8. G - the linear model in section 2.6. C - the output matrix derived in section 2.6. KF - the Kalman filter described in section 2.9. DFE - the DFE explained in section 2.. M - the matrix inversion model in section M2 - the iterative model in section M3 - the fast model in section M4 - the model with fixed λ in section The simulation run in Sim has one user in each of the two cells. This makes the states µ and µ 2 coincide with the IC-values calculated using equation (2.8). The UEs move around with a standard deviation of.5 m in each direction and their loads vary with a standard deviation of.5. The parameters N and N 2 are assumed constant with the value 7 dbm [2]. In order to make the estimated coupling factors more visible in the innovation the measurements are scaled with a factor of 4 to have an order of magnitude one before they are passed to the C-matrix and the Kalman filter. Note that this implies that N and N 2 are also scaled and that the plots show the scaled versions. In order to avoid making the estimates too slow the constants ρ and ρ 2 in the linear model are set to.98, a value found through trial and error. The matrices Q, R and P mentioned in the following sections refer to the covariance for the process noise, measurement noise and initial state covariance.