Estimation of Inter-Cell Interference in 3G Communication Systems

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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--11/4516--SE

Linköping 2011

Department of Electrical Engineering Linköpings tekniska högskola

Linköpings universitet Linköpings universitet

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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--11/4516--SE

Handledare: Ylva Jung

isy, Linköpings universitet

Graham Goodwin

cdsc, University of Newcastle

Katrina Lau

cdsc, University of Newcastle

Examinator: Thomas Schön

isy, Linköpings universitet

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Avdelning, Institution

Division, Department

Division of Automatic Control Department of Electrical Engineering Linköpings universitet

SE-581 83 Linköping, Sweden

Datum Date 2011-09-01 Språk Language Svenska/Swedish Engelska/English Rapporttyp Report category Licentiatavhandling Examensarbete C-uppsats D-uppsats Övrig rapport

URL för elektronisk version http://www.control.isy.liu.se http://urn.kb.se/resolve?urn=urn:nbn:se:liu:diva-71156 ISBN — ISRN LiTH-ISY-EX--11/4516--SE

Serietitel och serienummer

Title of series, numbering

ISSN

—

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 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.

Nyckelord

Keywords inter-cell interference, thermal noise, Decision Feedback Equalizer, estimation, 3G, Kalman filter, nonlinear model, quasi-decentralized estimator

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Abstract

In this thesis the telecommunication problem known as inter-cell interference is examined. Inter-cell interference originates from users in neighboring cells and af-fects 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 prin-ciples. 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 fac-tors depends on the positions of users which are unknown. A quasi decentralized method for estimating the coupling factors using measurements of the total inter-ference 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.

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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 ev-eryone 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 the-sis. 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 2011

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3.1.3 Experiment 3 . . . 36 3.1.4 Experiment 4 . . . 37 3.1.5 Experiment 5 . . . 38 3.2 Estimation experiments . . . 39 3.2.1 Experiment A . . . 40 3.2.2 Experiment B . . . 43 3.2.3 Experiment C . . . 46 3.2.4 Experiment D . . . 55 3.2.5 Experiment E . . . 58 4 Concluding remarks 61 4.1 Conclusions . . . 61 4.2 Future work . . . 62 Bibliography 63 A Abbreviations 65 B Notation 66

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Chapter 1

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.

1.1 Purpose & Goals

The purpose of this thesis is to better understand single cell and multi-cell in-terference 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

1. 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.

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1.2 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.

1.2.1 The Evolution of 3G communicators

Mobile telephony started to become international in the early 1980s with the in-troduction 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 1980s 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 e-mail 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 1990s 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 prod-ucts were two of the main driving forces behind the development of the third gen-eration 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.

1.2.2 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 · log₂(1 +P

N). (1.1)

In (1.1), 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

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1.2 Background 3

and additive. The signal power can be written as P = Eb· R, where Eb 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 = φ0· BW where φ0 is the constant

noise power spectral density (W/Hz). This gives the inequality

R ≤ T = BW · log₂(1 + Eb· R

φ0· BW

).

Defining the bandwidth utilization β = R/BW gives the expression

R ≤ T = BW ·log₂(1+Eb φ0 ·β) ⇔ β ≤ log₂(1+Eb φ0 ·β) ⇔ 2 β_{− 1} β ≤ Eb φ0 .

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 Eb/φ0 increases quickly, see Figure 1.1. [2]

10−1 100 101 −5 0 5 10 15 20 Power−limited region Band−limited region β Minimum E b / φ0 required (dB)

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1.2.3 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) · p0(k), where γ(k) is a scaling factor called the power grant and p0(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) = p0(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

Si(k) =

p_i(k)

Ii(k)

,

whereIi(k) is the interference to the user, and is given by

Ii(k) =

X

1≤j≤n,

j6=i

(1+γj(k))·pj(k)+α·(1+γi(k))·pi(k)+N0+Iother(k). (1.2)

In equation (1.2), (1+γj(k))·pj(k) is the interference from user j, α·(1+γi(k))·pi(k)

is the interference from the user to itself (α is a constant) and Iother(k) is the

in-terference from users in other cells. The parameter N0 is unknown and consists

of thermal noise and other sources of interference. [9]

By rewriting (1.2) as Ii(k) = C −(1−α)·(1+γi(k))·pi(k), where C =

P

1≤j≤n(1+

γj(k))·pj(k)+N0+Iother(k) it can be shown that N0+Iother(k) is a scaling factor

for the total interference. Therefore, by estimating N0+ Iother(k) it is possible to

get a better estimation of the interference Ii(k). This will give a better model to

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1.3 Limitations 5

1.3 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.

1.4 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 trans-fers in the 3G network. [2] also contains details concerning the upcoming LTE system. Relevant aspects and discussions on Universal Mobile Telecommunica-tion 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 [14] 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 [12] 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. [13] 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 per-formance gains relative to the decentralized strategies used today.

The basic theory for constructing mathematical and physical models for a pro-cess as well as model validation and simulation can be found in [10]. A basic framework for stability analysis of Multiple-Input Multiple-Output (MIMO) sys-tems and nonlinear models is treated in [4]. A popular method for estimating

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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) Parti-cle filter (MPF or RBPF) and a general framework of how PF can be applied to complex systems.

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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.

2.1 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 2.1, where S∗ is the desired SIR for the user (in dB), p0(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) = 1. An alternative which has been considered in some detail in the literature is

K(q) = 1/(1 + q−1 + ... + q−d+1) where d is the delay and the mobile station

G(q) = q−d/(1 − q−1), (see [9] and the references therein).

S∗

K(q) G(q) Controller Mobile station + − + + − + S(k) g(k) I(k) e(k) u(k) p0_(k) _p(k) Desired SIR Measured SIR

Figure 2.1. Power control loop for a single user.

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2.1.1 Single cell, n equal users

By assuming d = 1 and g(k) = 0 the following can be derived

p(k + 1) = 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 usersI(k) = n · (1 + γ(k)) · p(k) + (α · (1 + γ(k)) · p(k) + N0),

where α · (1 + γ(k)) · p(k) is the auto-interference or self-interference and N0 is

the thermal noise at the antenna. As noted in section 1.2.3 a bar denotes a linear quantity. This gives the expression p(k + 1) = p(k) + S∗− S(k) = p(k) + S∗₋

p(k) + I(k) = S∗+ I(k) which in linear scale becomes

p(k+1) = S∗·I(k) = S∗·(α·(1+γ(k))·p(k)+N0). (2.1)

Equation (2.1) is a linear state-space representation of the system and can be re-written as p(k + 1) = A · p(k) + b, where A = S∗· α · (1 + γ(k)) and b = S∗· N0.

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 val-ues 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 maximiz-ing 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

∗

· N0

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2.1 Power control loop description and analysis 9

For the case of a single user (n = 1) 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 = 212. 0 50 100 150 200 250 −2 −1 0 1 2

Real component of pole

0 50 100 150 200 250 −5 0 5 10 15x 10 −9 Power [mW] γ

Figure 2.2. Poles (top) and Power (bottom) for increasing power grant (γ) where

S∗= 1/64, α = 0.3 and n = 1.

Figure 2.4 shows the value of the power grantγ that gives the highest achievable

data-rate for n = 2. It also assumes that S∗₁=S∗₂= 1/64 andα1= α2= 0.3. Just

as before the two approaches give the same result. As seen in the figure instability is reached atγ1= γ2= 49 which is earlier compared to the single-user case.

2.1.2 Multiple cells, n equal users

For the case of users in neighboring cells Ii(k) = n · (1 + γi(k)) · pi(k) + αi· (1 +

γ_i(k)) · p_i(k) + N0+ Iotherji(k), where Iotherji(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 ICl is a gain factor for how much user l interferes with the other user and

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S∗ 1

K1(q) G1(q)

Controller Mobile station + − + + − + S1(k) g1(k) I1(k) e1(k) u1(k) p01(k) p1(k) Desired SIR Measured SIR S∗ 2 K2(q) G2(q)

Controller Mobile station + − + + − + S2(k) g2(k) I2(k) e2(k) u2(k) p02(k) p2(k) Desired SIR Measured SIR H2 H1 IC2 IC1 Iother21(k) Iother12(k) N0 N0

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 p1(k + 1) = p1(k) + u1(k), u1(k) = e1(k), e1(k) = S1∗− S1(k), S1(k) = p1(k) − I1(k), p2(k + 1) = p2(k) + u2(k), u2(k) = e2(k), e2(k) = S2∗− S2(k), S2(k) = p2(k) − I2(k).

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2.1 Power control loop description and analysis 11

The interference for the users, expressed in linear scale, are described by

I1(k) = α1· (1 + γ1(k)) · p1(k) + N0+ Iother21(k),

I2(k) = α2· (1 + γ2(k)) · p2(k) + N0+ Iother12(k),

where

Iother21(k) = IC2· (1 + γ2(k)) · p2(k),

Iother12(k) = IC1· (1 + γ1(k)) · p1(k).

In the same way as in the single-user case above, the linear recursive power equa-tions are given by

p₁(k + 1) = S∗₁· (α1· (1 + γ1(k)) · p1(k) + N0+ IC2· (1 + γ2(k)) · p2(k)), (2.3)

p2(k + 1) = S ∗

2· (α2· (1 + γ2(k)) · p2(k) + N0+ IC1· (1 + γ1(k)) · p1(k)). (2.4)

Rewriting (2.3) and (2.4) using vector notation gives the state-space equation

p_k+1= A · p_k+ b. (2.5)

To find the maximum achievable data-rate before reaching instability the eigen-values of the matrix A in (2.5) are examined for equal increase of the power grants

γ₁(k) and γ₂(k) as well as IC1= IC2= 1. 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.

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0 50 100 150 200 250 −2 −1 0 1 2

Pole 1 Pole 2 0 50 100 150 200 250 −1 0 1 2 3 4x 10 −9 Power [mW] γ

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2.2 Inter-cell model 13

2.2 Inter-cell model

To get a basic understanding of the effects of inter-cell interference, consider the simple example in Figure 2.5.

−400 −200 0 200 400 600 800 1000 1200 1400 −400 −200 0 200 400 UE 1 UE 2 UE 1 Cell 1 Cell 2 Distance x [m] Distance y [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∗ 11· α11· (1 + γ11(k)) S ∗ 11· (1 + γ12(k)) S ∗ 11· IC21−1· (1 + γ21(k)) S∗12· (1 + γ11(k)) S ∗ 12· α12· (1 + γ12(k)) S ∗ 12· IC21−1· (1 + γ21(k)) S∗21· IC11−2· (1 + γ11(k)) S ∗ 21· IC12−2· (1 + γ12(k)) S ∗ 21· α21· (1 + γ21(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 ICij−l, which is the interference gain

from user j in cell i to cell l, the power reaching the own base station, P1, and

the power reaching the other base station, P2, 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.

P1= Pij d2_ij−i , (2.6) P2= Pij d2ij−l , (2.7)

where Pij is the transmission power from user j in cell i, dij−iis the distance from

user j in cell i to the base station in cell i and dij−lis the distance from user j in

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The coupling factors can then be expressed as the ratio P2/P1 as in

ICij−l= (

dij−i

dij−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 pos-sible 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∗_ijare equal for all [i, j], no fading, αij are equal for all [i, j], γij 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 = 1. 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.

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2.3 Loads 15 0 50 100 150 200 250 −2 −1.5 −1 −0.5 0 0.5 1 1.5 2 γ

Figure 2.7. Pole placement for increasing γ values where S∗11 = S ∗ 12 = S ∗ 21 = 1/64, α11= α12= α21= 0.3 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 2.1 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

Li(k) for i = 1 . . . n where n is the number of users in the cell, as described in

(2.9). It is known that, for a single cell, ifPn

i=1Li(k) < 1, 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.

Li(k) = (1 + γi(k)) 1 Si(k) + (1 − αi) · (1 + γi(k)) (2.9)

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The stability criterion Pn

i=1Li(k) < 1 can be derived by looking at the

SIR-equation for user i in cell l

Sli(k) =

Pli(k)

Ili(k)

. (2.10)

Let Cl be the total interference power for cell l. Then Clis given by

Cl(k) = n1 X i=1 (1+γ_i(k))·Pli(k)+Iotherml(k)+N0l. (2.11) It follows that Ili(k) = Cl(k) − (1 − αli) · (1 + γli(k)) · Pli(k).

Equation (2.12) is then given by rearranging (2.10) as described below

Sli(k) = Pli(k) Cl(k) − (1 − αli) · (1 + γli(k)) · Pli(k) ⇔ Cl(k) − (1 − αli) · (1 + γli(k)) · Pli(k) = Pli(k) Sli(k) ⇔ Cl(k) = ₁ Sli(k) + (1 − αli) · (1 + γli(k) · Pli(k) ⇔ Pli(k) = Cl(k) 1 Sli(k) + (1 − αli) · (1 + γli(k)) . (2.12)

Let Iotherml(k) be the inter-cell interference from cell m to cell l described by

Iotherml(k) = X ∀m, m6=l nm X j=1 ICmj−l· (1 + γmj(k)) · Pmj(k). (2.13)

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2.3 Loads 17

Inserting (2.13) into equation (2.11) gives

Cl(k) = n1 X i=1 (1 + γ_li(k)) · Pli(k) +X ∀m, m6=l nm X j=1 ICmj−l· (1 + γmj(k)) · Pmj(k) + N0l (2.14)

Equation (2.15) is given by inserting (2.12) into (2.14).

Cl(k) = nl X i=1 (1 + γ_li(k)) · Cl(k) 1 Sli(k) + (1 − αli) · (1 + γli(k)) +X ∀m, m6=l nm X j=1 ICmj−l· (1 + γmj(k)) · Cm(k) 1 Smj(k) + (1 − αmj) · (1 + γmj(k)) +N0l ⇔ Cl(k) = nl X i=1 Lli(k) · Cl(k) + X ∀m, m6=l nm X j=1 ICmj−l· Lmj(k) · Cm(k) + N0l ⇔ nl X i=1 Lli(k) + X ∀m, m6=l nm X j=1 ICmj−l· Lmj(k) · Cm(k) Cl(k) + N0l Cl(k) = 1 (2.15)

Normally the contribution from the other cells, Iotherml(k), is not taken into

ac-count. It is then easy to see that the stability criterion Pn

i=1Li(k) < 1 is only

valid if Cl(k) is arbitrarily large. However, if Iotherml(k) is taken into account and

Cl(k) is assumed to be arbitrarily large the stability of cell m will be affected. The

reason for this is that the factor Cm(k)/Cl(k) in (2.15) will be inverted. Clearly

it is not possible to have Pn

i=1Li(k) arbitrarily close to 1 when the contribution

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2.4 Stability - Multi-cell constraints

This section describes different approaches to derive necessary and sufficient con-ditions for stability that consider inter-cell interference.

2.4.1 2-cell case

In this section the 2-cell case, which is a special case of (2.15) is examined. The total interference power for cell 1 and cell 2 are given by (2.16) and (2.17) respec-tively. C1(k) − a1 z_n }| { 1 X i=1 L1i(k) ·C1(k) − b1 z_n }| { 2 X j=1 IC2j−1· L2j(k) ·C2(k) = N01 (2.16) C2(k) − n2 X j=1 L2j(k) | {z } a2 ·C2(k) − n1 X i=1 IC1i−2· L1i(k) | {z } b2 ·C1(k) = N02 (2.17)

By writing (2.16) and (2.17) on matrix-form we get 1 − a1 −b1 −b2 1 − a2 | {z } A ·C1 C2 | {z } C =N01 N02 | {z } N

Solving this expression for C gives:

C = 1 det(A)· 1 − a2 b1 b2 1 − a1 | {z } A−1 ·N (2.18)

For a feasible solution to (2.18), which is necessary for stability, C > 0 must hold as it is not possible to have a negative total interference power. Since N, a1, a2, b1,

b2> 0 the solution is only positive if a1, a2< 1 and det(A) > 0. The determinant

is given by

det(A) = (1 − a1) · (1 − a2) − b1· b2.

The condition that det(A) > 0 gives the inequality

b1· b2< (1 − a1) · (1 − a2). (2.19)

Together with a1, a2 < 1, (2.19) forms a necessary and sufficient condition for

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2.4 Stability - Multi-cell constraints 19

If full coupling from the other cell is assumed, b1 = a2 and b2 = a1 hold. Using

this, the following sufficient condition for stability can be derived

a1· a2< (1 − a1) · (1 − a2)

⇔ a1· a2< 1 + a1· a2− a1− a2

⇔ a1+ a2< 1. (2.20)

The sufficient stability condition above also holds in the case of coupling less than one, since

b1· b2≤ a1· a2< (1 − a1) · (1 − a2).

If the coupling is greater than one a necessary condition for stability is that either

b1 or b2 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−1has to be elementwise positive. Using Perron-Frobenius theorem, see [11], it can be shown that

(I − B)−1≥ 0 iff ρ(B) = |λmax(B)| < 1, (2.21)

where B is a nonnegative matrix. It is not hard to see that the A-matrix can be written in the form

A =      I −a1 b1 b2 a2 | {z } B     

, where B is a nonnegative matrix.

The necessary and sufficient condition given by (2.21) 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−1 is positive and we have a feasible solution for C.

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The eigenvalues of B are calculated as below det(B − λ · I) = a1− λ b1 b2 a2− λ = 0 ⇔ (a1− λ) · (a2− λ) − b1· b2= 0 ⇔ a1· a2− a1· λ − a2· λ + λ2− b1· b2= 0 ⇔ (λ −a1+ a2 2 ) 2 − (a1+ a2 2 ) 2 + a1· a2− b1· b2= 0 ⇔ λ =a1+ a2 2 ± r (a1+ a2 2 ) 2_{− a} 1· a2+ b1· b2.

It can now be seen that the largest absolute eigenvalue is given for the case of full coupling. In this case,

λ1= a1+ a2,

λ2= 0.

Hence, λmax(B) ≤ a1+ a2. This together with Perron-Frobenius theorem gives

the sufficient condition a1+ a2< 1, which is the same as in (2.20).

2.4.2 3-cell case

In this section the 3-cell case is analyzed and stability conditions are derived. The calculations are similar to the ones in 2.4.1 but since there are 3 cells the calculations are a bit more tedious. C is calculated as in (2.18) but in this case the A-matrix is given by

A =   1 − a1 −b12 −b13 −b21 1 − a2 −b23 −b31 −b32 1 − a3  

By rewriting A on the following form, (2.21) can once again be used.

A =        I −   a1 b12 b13 b21 a2 b23 b31 b32 a3   | {z } B       

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2.4 Stability - Multi-cell constraints 21

The eigenvalues of B are calculated as below

det(B − λ · I) = −λ3+ (a1+ a2+ a3) · λ2

+ (−a1· a2− a1· a3− a2· a3+ b12· b21+ b13· b31+ b23· b32) · λ

+a1· a2· a3+ b21· b32· b13+ b31· b12· b23

− a1· b23· b32− a2· b13· b31− a3· b12· b21

= 0.

If full coupling is assumed the eigenvalues are

λ1= a1+ a2+ a3,

λ2= λ3= 0.

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 0 and 1) 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 [11].

In other words, it can be concluded that a1+ a2+ a3< 1 is a sufficient condition

for stability in the case of full coupling (or less).

2.4.3 n-cell case

Since the A-matrix always has the same structure, the necessary and sufficient condition given by (2.21) as well as the sufficient condition a1+ a2+ · · · + an< 1

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.

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2.5 Stability - conservatism versus failure-rate

The sufficient stability condition a1+ a2+ · · · + an< 1 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

a1+ µ · a2< , (2.22)

where µ is a scaling factor between 0 and 1 and is a threshold. Let µ be defined as µ_i= ni X j=1 ICij−l ni , (2.23)

where ICij−lis 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 30 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 equa-tion (2.23). This simulaequa-tion is then repeated for a total of 10 000 times. Using the simulation data an, as well as a false alarm probability, that gives a missed

detection probability of 0.1% can be calculated. The result of the simulation is

 = 0.97 and a false alarm probability of 0.45%.

From this we may conclude that knowing the µ-parameter of a cell is very useful when scheduling the loads in each cell.

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2.6 A linear model for estimating the coupling 23

2.6 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 (2.15) shows that the total interference power for a cell can be written as Cl(k) = nl X i=1 Lli(k) · Cl(k) + X ∀m, m6=l nm X j=1 ICmj−l· Lmj(k) · Cm(k) + N0l,

where nl and nmare 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

C1(k) = a1(k) z }| { n1 X i=1 L1i(k) ·C1(k) + µ2(k) · a2(k) z }| { n2 X j=1 L2j(k) ·C2(k) + N01, C2(k) = n2 X j=1 L2j(k) · C2(k) + µ1(k) · n1 X i=1 L1i(k) · C1(k) + N02. (2.24)

Since C1(k) and C2(k) can be measured and a1(k) and a2(k) are assigned by the

scheduler and therefore known, let

y(k) =C1(k)

C2(k)

be the measurements and (2.25)

x(k) =     µ1(k) µ₂(k) N01(k) N02(k)     be the states. (2.26)

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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

x1(k + 1) = ρ1· x1(k) + ω1(k),

x2(k + 1) =ρ2· x2(k) + ω2(k),

x3(k + 1) = x3(k) + ω3(k),

x4(k + 1) = x4(k) + ω4(k),

where ρ₁ and ρ₂ are constants between 0 and 1 and ωi(k), i = 1, ..., 4, denotes the

process noise.

This gives the linear state-space model

x(k + 1) =     ρ₁ 0 0 0 0 ρ₂ 0 0 0 0 1 0 0 0 0 1     | {z } A ·x(k) + ω(k), y(k) =   0 a2(k)·C2(k) 1−a1(k) 0 1 1−a1(k) 0 a1(k)·C1(k) 1−a2(k) 0 0 1 1−a2(k)   | {z } C ·x(k) + v(k), (2.27)

which will be used to estimate the coupling factors. Note that v(k) is the mea-surement noise.

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2.7 Interference model 25

2.7 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 simu-lating 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.

2.7.1 Matrix inversion model

This simple model is a matrix inversion based on equation (2.18) which solves the equation for C1(k) and C2(k) given a matrix A. In this case

A = 1 − a1(k) −µ2(k) · a2(k) −µ1(k) · a1(k) 1 − a2(k) (2.28) and we get the solution

C1(k) C2(k) = 1 det(A)· 1 − a2(k) µ2(k) · a2(k) µ₁(k) · a1(k) 1 − a1(k) | {z } A−1 ·N01(k) N02(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.

0 5 10 15 20 25 30 35 40 45 50 0 0.2 0.4 0.6 0.8 1 Sample Load

Load over time

Cell 1 Cell 2 0 5 10 15 20 25 30 35 40 45 50 1 2 3 4 5 6x 10 −14 Sample Power [W]

Total interference power over time

Cell 1 Cell 2

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2.7.2 Iterative model

Another way of modeling the interference is to iteratively use equation (2.24) as a time update.

C1(k + 1) = a1(k) · C1(k) + µ2(k) · a2(k) · C2(k) + N01

C2(k + 1) = a2(k) · C2(k) + µ1(k) · a1(k) · C1(k) + N02

This gives the dynamic step response shown in Figure 2.10. It can be seen that it takes the model some time to rise to the new value.

0 5 10 15 20 25 30 35 40 45 50 0 0.2 0.4 0.6 0.8 1 Sample Load

Load over time

Cell 1 Cell 2

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2.7 Interference model 27

2.7.3 Fast model

The model given by equation (2.30) below, solves equation (2.24) for C1 and C2

at each time instance and uses this as a time update,

C1(k + 1) = µ₂(k) · a2(k) · C2(k) 1 − a1(k) + N01 1 − a1(k) C2(k + 1) = µ₁(k) · a1(k) · C1(k) 1 − a2(k) + N02 1 − a2(k) . (2.30)

The step response for this model is shown in Figure 2.11. As shown the model reaches its final value faster than the model described in section 2.7.2. Therefore this model is more suitable for cases where the multi-cell system has a relatively fast response. 0 5 10 15 20 25 30 35 40 45 50 0 0.2 0.4 0.6 0.8 1 Sample Load

Load over time

Cell 1 Cell 2

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2.7.4 Model with fixed λ

This model, given by equation (2.31), can be seen as an extension of the model described in section 2.7.3 using a forgetting factor λ,

C1(k + 1) = old info z }| { λ · C1(k) + innovation z }| { (1 − λ) · (µ2(k) · a2(k) · C2(k) + N01 1 − a1(k) ) C2(k + 1) = λ · C2(k) + (1 − λ) · ( µ1(k) · a1(k) · C1(k) + N02 1 − a2(k) ). (2.31)

Figure 2.12 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.

0 5 10 15 20 25 30 35 40 45 50 0 0.2 0.4 0.6 0.8 1 Sample Load

Load over time

Cell 1 Cell 2

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2.8 State simulation 29

2.8 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 2.13 contains an example-run of the simulation showing the cells and the positions of the UEs.

−400 −200 0 200 400 600 800 1000 1200 1400 −400 −300 −200 −100 0 100 200 300 400 Cell 1 Cell 2 UE positions Distance x [m] Distance y [m]

Figure 2.13. 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 direc-tion.

The loads of the UEs in this example-run of the simulation are shown in Fig-ure 2.14. 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.

0 50 100 150 200 250 300 350 400 450 500 0 0.2 0.4 0.6 0.8 1

Loads over time in cell 1

Sample Load 0 50 100 150 200 250 300 350 400 450 500 0 0.2 0.4 0.6 0.8 1

Loads over time in cell 2

Sample

Load

Figure 2.14. 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 0.05 per sample.

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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 2.15. It is easy to see that this result is reasonable by looking at Figure 2.13 together with Figure 2.6.

0 50 100 150 200 250 300 350 400 450 500 0 0.2 0.4 0.6 0.8 1

Observed IC values in cell 1

Sample IC−value 0 50 100 150 200 250 300 350 400 450 500 0 0.2 0.4 0.6 0.8 1

Observed IC values in cell 2

Sample

IC−value

Figure 2.15. 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 mea-surements of the total interference power. The meamea-surements, without measure-ment noise, shown in Figure 2.16 are generated using the model described in section 2.7.2. 0 50 100 150 200 250 300 350 400 450 500 0 1 2 3 4x 10

−13 Measured total interference power over time in cell 1

Sample C1 0 50 100 150 200 250 300 350 400 450 500 0 1 2 3 4x 10

−13 Measured total interference power over time in cell 2

Sample

C2

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2.9 Kalman filter 31

2.9 Kalman filter

This section describes the discrete Kalman filter, which will be used for the es-timation 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 + 1) = 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 R1 and R2 respectively, the

cross-spectrum between ω(k) and v(k) is constant and equal to R12. The observer

that minimizes the estimation error x(k) − ˆx(k) is given by

ˆ

x(k + 1|k) = A · ˆx(k|k − 1) + B · u(k) + K · (y(k) − C · ˆx(k|k − 1) − D · u(k)).

The Kalman filter gain K, is given by

K = (A · P · CT + R12) · (C · P · CT+ R2)−1,

where P is the positive and semi-definite solution to the algebraic Riccati equation

P = A · P · AT + R1

− (A · P · CT+ R12) · (C · P · CT + R2)−1· (A · P · CT + R12)T.

This P is equal to the covariance matrix for the optimal estimation error x(k) − ˆx(k|k − 1). [4]

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2.10 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.[1] 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 theˆ feedback to the filter. The block diagram of the DFE is shown in Figure 2.17.

KF sat(·)

y(k) x(k)ˆ ˆˆx(k)

z−1

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Chapter 3

Results

In this chapter, the results and conclusions of the experiments performed during this thesis are presented.

3.1 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.

3.1.1 Experiment 1

In Figure 3.1, cell 1 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) = 30. In cell 1, 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 1, that gives instability the simulation is run 2000 times, using different positions. Since the positions of the UEs in cell 1 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 1 and thus always the same no matter how many simulations are run.

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−400 −200 0 200 400 600 800 1000 1200 1400 −400 −200 0 200 400 30 Cell 1 Cell 2 Distance x [m] Distance y [m]

Figure 3.1. Experiment 1, a UE close to the border.

Calculating the mean µ and variance σ2_{of the}_{γ(k), in cell 1, that causes instability}

over the 2000 simulations gives

µ = 44.8960, σ2= 5.9221.

If the power grant in cell 2 is increased from 30 to 40 the results are

µ = 43.5065, σ2= 10.3621.

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 3.1 again but increase the number of UEs in cell 1 from two to three. This gives the following results

µ = 25.4475, σ2= 1.0988.

This shows not only a decrease in the mean but more interestingly a large decrease in variance.

Reducing the radius of cell 1 from 500m to 250m gives the results below

µ = 43.6955, σ2= 5.8997.

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3.1 Stability experiments 35

These results are very similar to the original case in Figure 3.1, but more inter-estingly the value of IC21−1has increased from 0.67 to 2.25. This is equivalent to

adding 1.58 users, with the same grant as the user in cell 2, in cell 1. A possible explanation for the similar µ and σ values is that although the contribution from the user in cell 2 to cell 1 has increased the contribution from the users in cell 1 to cell 2 has decreased.

3.1.2 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 1 than the base station in cell 2, i.e. IC21−1> 1. −400 −200 0 200 400 600 800 1000 1200 1400 −400 −300 −200 −100 0 100 200 300 400 30 Cell 1 Cell 2 Distance x [m] Distance y [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 = 40.6455, σ2= 24.3940.

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3.1.3 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 1 contains two UEs.

−400 −200 0 200 400 600 800 1000 1200 1400 −400 −200 0 200 400 30 Cell 1 Cell 2 Distance x [m] Distance y [m]

Figure 3.3. Experiment 3, a UE far from the border.

The results of these changes are

µ = 48.0165, σ2= 0.2764.

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.

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3.1 Stability experiments 37

3.1.4 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 1 is still two.

−400 −200 0 200 400 600 800 1000 1200 1400 −400 −200 0 200 400 1 30 Cell 1 Cell 2 Distance x [m] Distance y [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 σ2

Consider all 44.3110 8.0503 Consider large only 44.6130 7.1928 Consider small only 48.3200 0.2547

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.

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3.1.5 Experiment 5

To further compare the influence of different power grants as well as different distances the setup in Figure 3.5 is used.

−400 −200 0 200 400 600 800 1000 1200 1400 −400 −200 0 200 400 1 30 Cell 1 Cell 2 Distance x [m] Distance y [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 σ2

Consider all 47.6655 0.5779 Consider large only 47.9975 0.2926 Consider small only 48.3600 0.2665

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.

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3.2 Estimation experiments 39

3.2 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 2.10.

M1 - the matrix inversion model in section 2.7.1.

M2 - the iterative model in section 2.7.2.

M3 - the fast model in section 2.7.3.

M4 - the model with fixed λ in section 2.7.4.

The simulation run in Sim has one user in each of the two cells. This makes the states µ₁and µ₂coincide with the IC-values calculated using equation (2.8). The UEs move around with a standard deviation of 1.5 m in each direction and their loads vary with a standard deviation of 0.05. The parameters N01 and N02 are

assumed constant with the value −107 dBm [12]. In order to make the estimated coupling factors more visible in the innovation the measurements are scaled with a factor of 1014 to have an order of magnitude one before they are passed to the C-matrix and the Kalman filter. Note that this implies that N01and N02are also

scaled and that the plots show the scaled versions. In order to avoid making the estimates too slow the constantsρ₁ and ρ₂ in the linear model are set to 0.98, a value found through trial and error. The matrices Q, R and P0mentioned in the

following sections refer to the covariance for the process noise, measurement noise and initial state covariance.

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3.2.1 Experiment A

The purpose of this experiment is to investigate the performance of the quasi-decentralized estimation strategy in ideal conditions. The setup is shown in Fig-ure 3.6. The loads are taken from the simulation and the coupling factors from the linear model. They are then used in the matrix inversion model to produce the measurements. Note that there is no measurement noise in this experiment. Since the C-matrix needs a measurement of the total interference power in the other cell in order to estimate the total interference power in the own cell, the measurements need to be passed to the C-matrix. The reason that the measure-ments are passed separately to the C-matrix and the Kalman filter is that in later experiments measurement noise will be added in between these.

KF y(k) ˆ x(k) z−1 M 1 Sim G l(k) µ(k) C

Figure 3.6. Block diagram of the experiment.

The matrices Q, R and P0 greatly influence the quality of the estimates. For

this setup the Q-matrix is chosen to match the process noise used in the model to generate the coupling factors. In order to make the Kalman filter put a lot of trust into the measurements the R-matrix is made small. Finally, since the initial guesses of the states are set to the true value of the states, the P0-matrix is made

small. Q =     2.5e-6 0 0 0 0 2.5e-6 0 0 0 0 1e-8 0 0 0 0 1e-8     , R =1e-4 0 0 1e-4 P0=     1e-1 0 0 0 0 1e-1 0 0 0 0 1e-1 0 0 0 0 1e-1    

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The results for this setup are shown in Figure 3.7, 3.8, 3.9 and 3.10.

0 50 100 150 200 250 300 350 400 450 500 0 0.2 0.4 0.6 0.8 1

Load over time in cell 1

Sample Load 0 50 100 150 200 250 300 350 400 450 500 0 0.2 0.4 0.6 0.8 1

Sample

Load

Figure 3.7. Loads over time in the two cells. The total loads are shown in black and

the loads of the UEs are shown in blue.

0 50 100 150 200 250 300 350 400 450 500 −0.4 −0.2 0 0.2 Sample µ1 µ

1 over time in cell 1

True µ 1 Estimated µ 1 0 50 100 150 200 250 300 350 400 450 500 −0.4 −0.2 0 0.2 Sample µ2

µ₂ over time in cell 2

True µ 2 Estimated µ₂

Figure 3.8. True (blue) and estimated (green) coupling factors using the matrix

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0 50 100 150 200 250 300 350 400 450 500 1.5 2 2.5 3 3.5

N₀₁ over time in cell 1

Sample N01 True N 01 Estimated N₀₁ 0 50 100 150 200 250 300 350 400 450 500 1.5 2 2.5 3 3.5 N

Sample

N02

True N₀₂ Estimated N

02

Figure 3.9. True (blue) and estimated (green) thermal noise power using the matrix

inversion model. 0 50 100 150 200 250 300 350 400 450 500 0 5 10 15 20 25

Measured total interference power over time in cell 1

Sample C1 0 50 100 150 200 250 300 350 400 450 500 0 5 10 15 20 25

Sample

C2

Figure 3.10. Measured total interference power over time using the matrix inversion

model.

Conclusions

As expected the estimates are quite good since the coupling factors are generated using the same Q-matrix as in the Kalman filter, i.e. the process noise is known. However, as seen in Figure 3.8 the estimates are sometimes outside the allowed interval, [0, 1]. A solution to this problem is to add a DFE to the setup.

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3.2.2 Experiment B

The purpose of this experiment is to determine how much the DFE affects the estimate. The only difference compared to experiment 1 is that the estimate of the state vector is passed to the DFE-loop. The setup of this experiment is shown in Figure 3.11. KF y(k) ˆ x(k) z−1 M 1 Sim G l(k) µ(k) sat(·) ˆˆx(k) C

The results for this setup are shown in Figure 3.12, 3.13, 3.14 and 3.15.

0 50 100 150 200 250 300 350 400 450 500 0 0.2 0.4 0.6 0.8 1

Sample Load 0 50 100 150 200 250 300 350 400 450 500 0 0.2 0.4 0.6 0.8 1

Sample

Load

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0 50 100 150 200 250 300 350 400 450 500 0 0.5 1 Sample µ1 µ

True µ 1 Estimated µ 1 0 50 100 150 200 250 300 350 400 450 500 0 0.5 1 Sample µ2 µ

True µ 2 Estimated µ

2

Figure 3.13. True (blue) and estimated (green) coupling factors using the matrix

in-version model. 0 50 100 150 200 250 300 350 400 450 500 0 1 2 3 4

N₀₁ over time in cell 1

Sample N01 True N 01 Estimated N₀₁ 0 50 100 150 200 250 300 350 400 450 500 0 1 2 3 4 N

Sample

N02

True N₀₂ Estimated N

02

Figure 3.14. True (blue) and estimated (green) thermal noise power using the matrix

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3.2 Estimation experiments 45 0 50 100 150 200 250 300 350 400 450 500 0 5 10 15 20 25

Sample C1 0 50 100 150 200 250 300 350 400 450 500 0 5 10 15 20 25

Sample

C2

Figure 3.15. Measured total interference power over time using the matrix inversion

model.

Conclusions

By comparing the estimates of this experiment, shown in Figure 3.13 with the estimates in Figure 3.8 it can be seen that the DFE solves the problem of esti-mates ending up outside the allowed limits. Therefore the DFE will be used in all future experiments. As expected the estimates are still quite good since the setup is basically the same as in Experiment A.

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3.2.3 Experiment C

In this experiment things are not as ideal as before since both the loads and the coupling factors are now taken from the simulation. To investigate the impact of the interference model this setup is simulated once with each of the four interfer-ence models. The setup is shown in Figure 3.16.

KF y(k) ˆ x(k) z−1 M 1− 4 Sim l(k) µ(k) sat(·) ˆˆx(k) C

To be able to make correct comparisons of the models, all measurements in this experiment are calculated using the same loads and coupling factors. The ran-dom walks of the two UEs are shown in Figure 3.17 and the loads are shown in Figure 3.18. −400 −200 0 200 400 600 800 1000 1200 1400 −400 −300 −200 −100 0 100 200 300 400 Cell 1 Cell 2 UE positions Distance x [m] Distance y [m]

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3.2 Estimation experiments 47 0 50 100 150 200 250 300 350 400 450 500 0 0.2 0.4 0.6 0.8 1

Sample Load 0 50 100 150 200 250 300 350 400 450 500 0 0.2 0.4 0.6 0.8 1

Sample

Load

the loads of the UEs are shown in blue.

In this experiment more thought was needed to be put into determining the Q-matrix since the process noise is no longer known. If an estimate of V ar(x1,2) is available, the expression in equation (3.1) provides good starting values for the first two diagonal elements in the Q-matrix, q1,2_{. Since the last two states are}

assumed to be constant their corresponding values in the Q-matrix are very small. The reasoning regarding R and P0 is the same as in the previous experiments.

V ar(x1,2) = q 1,2 1 − (ρ)2 (3.1) Q =     3e-8 0 0 0 0 3e-8 0 0 0 0 1e-8 0 0 0 0 1e-8     , R =1e-4 0 0 1e-4 P0=     1e-1 0 0 0 0 1e-1 0 0 0 0 1e-1 0 0 0 0 1e-1    

Estimation of Inter-Cell Interference in 3G Communication Systems

Institutionen för systemteknik

Department of Electrical Engineering

Examensarbete

Estimation of Inter-cell Interference in 3G

Communication Systems

Estimation of Inter-cell Interference in 3G

Communication Systems

Examensarbete utfört i Reglerteknik

vid Tekniska högskolan i Linköping

av

Abstract

Acknowledgments

Contents

Chapter 1

Introduction

1.1

Purpose & Goals

1.2

Background

1.2.1

The Evolution of 3G communicators

1.2.2

Rate constraints

1.2.3

Power Control and Interference Model

1.3

Limitations

1.4

Related research

Chapter 2

Theory

2.1

Power control loop description and analysis

2.1.1

Single cell, n equal users

2.1.2

Multiple cells, n equal users

2.2

Inter-cell model

2.3

Loads

2.4

Stability - Multi-cell constraints

2.4.1

2-cell case

2.4.2

3-cell case

2.4.3

n-cell case

2.5

Stability - conservatism versus failure-rate

2.6

A linear model for estimating the coupling

2.7

Interference model

2.7.1

Matrix inversion model

2.7.2

Iterative model

2.7.3

Fast model

2.7.4

Model with fixed λ

2.8

State simulation

2.9

Kalman filter

2.10

Decision Feedback Equalizer

Chapter 3

Results

3.1

Stability experiments

3.1.1

Experiment 1

3.1.2

Experiment 2

3.1.3

Experiment 3