Evaluation of Particle Swarm Optimization Algorithm for Precoding in Coordinated Multi-Point System

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Evaluation of Particle Swarm Optimization Algorithm for Precoding in Coordinated Multi-Point System

Amir Shehni

This thesis is presented as part of Degree of Master of Science in Electrical Engineering

Blekinge Institute of Technology Spring 2013

Blekinge Institute of Technology School of Engineering

Department of Electrical Engineering Examiner: Dr. Sven Johansson

Chalmers University of Technology Department of Signal and System

Supervisor: Associate Prof Tommy Svensson Advisor: Lic. Eng. Tilak Rajesh Lakshmana

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To

Baba and Maman

Azadeh and Alaleh

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Abstract

In future communication system, the constraint of recourse allocation are a challenging topic.

One way for observing such constraint is to apply frequency reuse factor of one in communication system.

Taking this factor to one causes the system to have serious problems such as small sum rate and interference specifically at cell-edge.

In 3GPP, LTE Advanced, one proposed technique for next generation of communication system is Coordinated Multi-Point which is called CoMP. One of the main part of this technique is a unit of precoder which prepares beamforming weight. Suitable precoding in CoMP and its subset which is called Joint Processing is a way to compensate weak performance in mutual border due to taking frequency reuse factor one.

Previously some linear methods have been proposed for precoding in CoMP. Recently the literature is shifted for studying the capability of stochastic algorithms for precoding in CoMP system.

In this master thesis we evaluate the Particle Swarm Optimization to form precoding matrix and compare its variations (Basic PSO, Random PSO and Multi-Start PSO) with each other and a linear technique which is known as Zero Forcing as well.

The result shows for precoding, the Multi-Start PSO has better performance in comparison to other types of algorithms and techniques in different system sizes and different objective functions.

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Acknowledgments

First of all I want to express my deepest gratitude to my supervisor Associate Professor Tommy Svensson, to give me the opportunity to perform this master thesis in department of signal and system at Chalmers University of Technology. It has been a pleasure for me to be in such a creative environment filled with very kind and helpful people and without support of my supervisor I could not use the facilities for progress of my research. I appreciate for all of his kindness.

I would also like to thank PhD student, Lic. Eng. Tilak Rajesh Lakshmana for giving his valuable time to guide me in the correct direction. I would not forget his technical and moti- vational discussion with me throughout my thesis. He encouraged me deeply understand the analytical things while doing mathematical or numerical manipulations and to keep me on the right track. I also admire his technical expertise in related softwares and I am thankful for all of his support.

It is also pleasure to acknowledge Dr. Sven Johansson who accepted to be my examiner at BTH and thank for solving some administrative problems due to being me in another university for performing master thesis.

Special thanks to my close friends who were my family in Sweden. I do not know whithout them, how I can tolerate for being far from my family in Iran. Thanks to Meysam, Maboud, Nasrin, Maryam and Davoud in Karlskrona and my friends in thesis room at Chalmers, Sarmad, Shilan and Cristian.

And all of my love to my family; Baba, Maman, Azadeh and Alaleh which being of them is the alone excuse of my life.

Amir Shehni Spring 2013, G¨oteborg

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List of Figures

2.1 Traditional Cellular Systems. . . 5

2.2 A CoMP system with multi-cell joint transmission . . . 5

2.3 CoMP transmission schemes,a CoMP scheduling and joint transmission . . . 7

2.4 The process which performs for precoding in CoMP. . . 8

2.5 A simplified illustration of precoding.. . . 8

3.1 An example for movement of swarm, a flock of birds . . . 13

4.1 Convergence RPSO with different Reset Particle(s) Position . . . 22

4.2 Convergence MSPSO with different Step Size . . . 22

4.3 Convergence RPSO vs BPSO . . . 23

4.4 Convergence MSPSO . . . 23

4.5 Comparison CDFs for deriving best value, change cognitive components (C1) . . . . 25

4.6 Comparison CDFs for deriving best value, change social components (C2) . . . 25

4.7 Comparison CDFs for deriving best value, change Max of inertia weight . . . 26

4.8 Comparison CDFs for deriving best value, change Min of inertia weight . . . 27

4.9 Comparison CDFs for deriving best value, change number of particle(s) . . . 28

4.10 Comparison CDFs for deriving best value, change step size of re-initializing . . . 29

4.11 Comparison CDFs for deriving best value, change number of particle(s) reset position 30 4.12 Comparison CDFs for getting best scenario in System Size 3x3 . . . 32

4.13 Comparison CDFs for getting best scenario in System Size 6x6 . . . 32

4.16 Different scenarios in different system sizes . . . 34

4.21 Different scenarios in different system sizes . . . 38

5.1 *A sample of output in our meetings during master thesis* . . . 46

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List of Tables

1.1 LTE Advanced requirements. . . 2 4.1 Optimum Values . . . 30 4.2 Mean Value in different sizes with regard to objective function Sum Rate Maximization 34 4.3 Mean Value in different sizes with regard to objective function Weight Interference

Minimization . . . 38

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Chapter 1

Introduction

Rel-8 LTE have made improvement for performance of system in coverage and capacity, higher data rate for user experience, reduced operating cost, reduced-latency deployment and seamless integration with existing system.

Extra enhanced requirements, however approved in 2008 to allow LTE to be approved as a radio technology for international Mobile Telecommunication-Advanced (IMT-Advanced).

IMT-Advanced requirements are defined by the International Telecommunication Union, which is an organization that prepares internationally approved standards for telecommunication. This effort for progress of LTE is known as LTE Advanced (LTE- A).

The LTE Advanced requirements are shown in Table 1.1 and focus primarily on progress in system throughput and latency reduction. From Table 1.1, it can be observed that the user spectral efficiency have increased significantly. Peak data rate of 1Gbps in the downlink and 500 Mbps in the uplink must be supported. Target latencies have been reduced significantly as well. In addition to progress in system throughput, deployment and operating- cost-related goals were also introduced. They contain support for power efficiency, efficient backhaul, open interfaces and minimized maintenance tasks.

In LTE Advanced, Coordinated Multi-Point (CoMP) schemes are a promising technique to increase the spectral efficiency of cellular networks. In the 3GPP terminology in the development of the LTE-Advanced specifications, we distinguish between the CoMP schemes Joint Processing,, Coordinated Beamforming and Coordinated Scheduling. The common aspect of all these schemes is to improve the signal-to-interference ratio at the user terminals by coordinated resource allocation and scheduling among neighboring base stations. This improvement is achieved either by choosing orthogonal resources for interfering transmis- sions in the time, frequency or space domain, or by simultaneous transmission of the same data from several base stations. In the following, we focus on Coordinated Joint Processing downlink of an LTE-A system.

In chapter two we focus on Coordinated Multi-Point in more details. We discuss about the categories of CoMP in different terms, transmission and reception, joint processing or

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coordinated beamforming /scheduling or hybrid and in each of them the different types of subsets will be discussed. In further some part of CoMP technique that is precoder which prepares beamforming matrix (the object of this thesis) will be considered. Furthermore some linear method like Zero Forcing for linear precoding will introduce.

In chapter three, stochastic algorithm with emphasis on Particle Swarm Optimization will be considered. The topics such as stochastic algorithms, power of such algorithm, PSO algorithm, the reason for choice of PSO, the variations of PSO include Basic PSO, Random PSO, Multi-Start PSO are introduced and discuses in more details. In addition the beamforming with regard PSO and proper mapping for such beamforming will discuss.

In chapter four the simulation result will be indicated. We will present the convergence conditions in different scenarios of PSO algorithms. In addition we will select the optimum values for parameters which play roles in PSO algorithm. Furthermore we consider defined objective functions Sum rate maximization and Weight interference minimization and perform our analysis in terms of those functions. Moreover we evaluate the performance of such objective functions while the sizes of system change.

Last chapter will be dedicated to conclusion and future work that occur in the context of CoMP and/or stochastic algorithm for instance PSO.

Table 1.1: LTE Advanced requirements

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Chapter 2

Coordinated Multi-Point

In current cellular system the reuse frequencies has been applied for increasing coverage and capacity. In these systems, the neighbor cells must use different frequencies however there is no problem with two cells sufficiently far to use same frequency. On the other hand future communication system must consider resource allocation. By keeping the frequency reuse factor one, it is inevitable that the system has intra cluster interference specifically on mutual border or cell-edge. The system has also inter cluster interference and traditional techniques for combating such imperfection have focused on either allocating orthogonal radio resources to different transmit signals, for example, frequency reuse, cell sectoring, or canceling interference via signal processing [1, 2].

Cooperative transmission and reception among multiple nodes has been proposed as a new technique for LTE Advanced [3]. This technique has the capability of improvement in the system spectrum efficiency and interference mitigation as well as cell-edge throughput.

There are several cooperative communication techniques have been acquired much interest, such as ”network coordination” [4], ”multi-cell processing” [5], ”network multiple-input multiple-output (MIMO)” [2], ”multicell multiuser MIMO” [6, 7, 8], ”group cell” [9] and

”distributed antenna systems” [10].

In standard development organization 3GPP, all of these subjects are categorized under the concept of Coordinated Multi-Point transmission/reception or CoMP [3]. It is desir- able and useful to improve cell edge throughput by converting an interfering signal to a desired signal. CoMP does not use the middle nodes for transmitting data to destination nodes rather it itself directly take parts to data transmission or making decision on scheduling/beamforming in time-frequency resource.

CoMP is broadly classified as downlink CoMP or uplink CoMP [3]. The downlink or the forward link is referred as CoMP transmission while the uplink or reverse link is referred as CoMP reception. In further CoMP, according to multiple geographically separated transmission points is defined as two types of networks, Homogeneous networks and Heterogeneous networks [11].

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CoMP reception has the less complexity in comparison to CoMP transmission because the BSs can estimate the channel state information (CSI) in uplink and there is no transceiver modifications in reverse link. In the case of CoMP transmission, the UE fed back the CSI to their serving BSs. According to collected data in BSs, a unit which its task is to precode, performs proper precoding weight according to defined objective functions [11].

CoMP transmission for a given time-frequency resource is categorized based on the avail- ability of user data in different BSs. Briefly, If the user data is available at more than one BS and several BSs perform cooperation for transmitting data simultaneously, it is referred as Joint processing and if the data is available in only one BS it is referred as coordinated scheduling or coordinated beamforming [3].

An alternative is Hybrid category that uses both coordinated beamforming/scheduling and joint processing [11].

2.1 Joint Processing

In the CoMP joint processing approach, user data is available simultaneously at all transmission points within the CoMP cluster. With sharing both the CSI and the data of all users in the cluster, coordinated multiple points can act as a single and distributed antenna array.

Joint processing could be categorized as dynamic point selection and joint transmission [3]. In dynamic point selection, the serving BSs dynamically changes over time-frequency resource. In this method the data is available in each user but transmitting is only from one BS. On the other hand in joint transmission approach, the data available in each BS and the transmission action is served from several BSs. In other words Joint transmission can be broadly described as a simultaneous transmission of data to a user from multiple cooperating BSs.

The transmission could be coherent or non-coherent and aims to improve the overall system throughput [13]. Non-coherent joint transmission might use methods such as single frequency network (SFN) or cyclic delay diversity (CDD) for diversity aims and increasing the transmitted power to user. On the other hand, coherent transmission would be based on spatial CSI from two or one BSs. However well synchronization and less timing error differ- ences between transmission points are needed to realize the full potential gains of coherent joint transmission schemes [13].

The amount of feedback from inter-base information exchange and users, while the number of BSs and users increase, is one of the main drawbacks for implementing joint processing.

So the design of suitable and efficient algorithm for overcoming the complexity of such sit- uations has great interest. for achieving such target, solutions that observe the use of joint processing techniques to limited number of BSs have been proposed. In this approaches the

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Figure 2.1: Traditional Cellular Systems

Figure 2.2: A CoMP system with multi-cell joint transmission

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network is typically divided into clusters of cells, and the joint processing schemes are implemented within the BSs in each clusters. The formation cluster can be dynamic or static.

Three types of scenarios that result in several degree of joint processing between BSs have been defined [14].

2.1.1 Centralized Joint Processing

In this scheme, transmitter side have the global CSI and with regard such CSI, the BSs within the cluster jointly carry out the power allocation and the design of precoding [3, 4].

2.1.2 Distributed Joint Processing

In this scheme, local CSI is available at each BS. Therefore, the power allocation and the precoders are locally calculated at each BS (distributed) but the user may receive its data from several BSs (joint processing), depending on its given channel conditions. In a first step, a multibase scheduling algorithm is required in order to assign users to BSs [14].

2.1.3 Partial Joint Processing

Another scheme which defines various stages of joint processing between BSs. In this scheme new parameter is applied which referred as sctive set. Joint processing stages or degrees determine such parameter or subsets of BSs for each user in the cluster area. Therefore users can only receive their data from subset of BSs included in its active set [15].

2.2 Coordinated Scheduling/Beamforming

In the coordinated scheduling/beamforming approach, as shown in Figure 2.3, the transfer- ring data symbols to destination nodes is only available at and transmitted from one point in CoMP cluster on a time-frequency resources. However in order to control ICI, the given CSI of all users among several transmission nodes is needed which user scheduling and beamforming can be coordinated. It is good to know which this approach can only mitigate ICI rather than exploiting it [3].

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Figure 2.3: CoMP transmission schemes,a CoMP scheduling and joint transmission

2.3 Precoding, a PHY Layer Design

In CoMP the multi-cell channel state information (CSI) is fed back from each terminal to the serving BS.

Precoding is a matrix-vector multiplication which is formed from CSI that are collected from BSs. The incoming IQ signal constellations of data streams from all BSs in a CoMP transmission are multiplied with a weight matrix before the signals are passed into IFFT at each antenna [16]. In JP, user equipment (UE) need to feed back the CSI of their BS-UE links. In centralized joint processing (CJP) scheme, the CSI is collected at a node in the network called central coordination node (CCN), to form an aggregated channel matrix [17, 18]

and based on this aggregated channel matrix, the CCN obtains the precoding weights, con- sisting of the beamforming weights after power allocation. As the CSI is reported to CCN, the link is referred as active link and unreported CSI is reffered as inactive link [19]. The process is shown in Figure 2.4.

For understanding precoding, look at the simplified communication system in Figure 2.5.

In (a) it has been observed the receiver carry out channel inversion for overcoming the effect of the channel via equalization. If the channel inversions perform at transmitter side, it leads that the complexity of receiver can be reduced as shown in figure (b). It would be seen that receiver need to feed back the CSI to transmitter as shown in figure by green arrow. In figure (c), the precoding weights are multiplied just before the symbol are upconverted, amplified and transmitted. In CoMP systems, the transmitter is located in several geographically separated BSs and the precoder reside in CCN.

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Figure 2.4: The process which performs for precoding in CoMP.

Figure 2.5: A simplified illustration of precoding.

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2.4 Zero Forcing

Multiple input/multiple output systems have great potential to achieve high throughput in wireless system.

One way to serve multiple users simultaneously is to use a coding scheme called Dirty Paper Coding (DPC) [20], which is a multi-user encoding strategy based on interference mitigation. Particularly when the number of users exceeds the number of transmit antenna, the capacity of system increases linearly by using DPC. For MIMO systems, in fact DPC is the optimal strategy in downlink from BS to user.

The drawback of DPC is to implement this technique in practical system. It needs the high computational burden of successive encoding and decoding especially when the number of users is large [20].

The proposed method for compensating such complexity is beamforming (BF) [21]. It is a suboptimal strategy that can also serve multiple users at a time, but the complexity is reduced relative to DPC. In BF each user is stream coded independently and multiplied by beaamforming weight vector for multiple antennas. Proper selection of weight vector can reduce interference and thereby support multiple users simultaneously. Finding the optimal beamforming weight vectors, however, is still a problem.

A proposed strategy which has significant performance that can easily be implemented in practical but whose performance is comparable to DPC is a suboptimal beamforming strategy, Zero Forcing beamforming (ZFBF) [22] where the weight vectors are chosen to avoid interference. In this approach beamforming weight can be easily founded by inverting the composite channel matrix of the users. ZFBF is generally power inefficient due to beamforming weights are not matched to user channels [22]. It can be seen when the number of users is large, the sum-rate performance will be close to DPC.

2.5 Beamforming using Linear Method

With regard to the constraint of system orthogonality the multiplication of BSs and number of antennas should be larger or equal to the number of users

K.N_t≥ M (2.1)

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where K is the number of BSs and N_t and M is number of antenna and number of users respectively. The linear technique can be used for aggregated channel matrix. In the case of Zero-Forcing beamforming we have

W = H^H(HH^H)⁻1 (2.2)

which -1 is inverse matrix and each base station is constrained to a maximum transmit power, P_max. The suboptimal power allocation based on [6] is perform for Zero Forcing under per-BS constraint, where at least one of the BSs is transmitting at maximum power and it is defined as

W = v u u t

P_max max

k=1,...,K||W(K.N_t, :)||²_p . W (2.3)

Where K.Nt selects the rows of the BF matrix W of the Kth BS with its Nt antenna towards the MUTs. The SINR at the mth UT is given as

SIN R = ||h_mw_m||² Σ^M_j=1

j6=m

||hmwm||²+ σ² (2.4)

which h and w are elements of channel matrix and precoding matrix.

Finally the sum rate per cell in terms of bits per second per hertz per cell (bps/hz/cell) is given as

R_tot = 1 K .

M

X

m=1

log₂(1 + SIN R_m) (2.5)

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2.6 Problem Statement

As mentioned before, precoding is one of the main part of CoMP. Previously, some methods have been researched which lead to linear precoding. One of these linear precoding is to use pseudoinverse method for getting proper matrix and is known as Zero Forcing beamforming (ZFBF) [22].It is good and has nice throughput, but it only acts well under specific conditions. It would be nice choice only if we have aggregated matrix as channel matrix and the problems occur by having sparse matrix [19].

For JP, as long as the aggregated channel matrix at CNN has suitable situation to inverse, linear techniques such as ZFBF would be used for interference control. When the ZFBF is calculated based on sparse aggregated channel matrix, it would be seen that a link that has been defined as an inactive link may mapped to a non-zero BF weight for that link which means such link has been active, not inactive [19]. It makes some undesirable effect in system such as unnecessary backhauling. There is some method called partial ZF [23] which has better throughput relative to ZFBF in the case of sparse aggregated channel matrix and in terms of proper mapping of zero elements but it has other problems, for instance no good performance as well as ZFBF for equivalent backhaul load reduction.

On the other hand, in literature there is not any linear technique that could invert the spars aggregated channel matrix and preserve the zero elements in the transpose version of inverse, when the aggregated channel matrix is not diagonal or block-diagonal [19].

Therefore the direction of literature has shifted to studying other methods that have better performance in case of sparse matrix. The stochastic methods have been proposed and look to be a nice alternative.

There are a number of stochastic algorithms that we can use in communication system with regard to our aims and objective functions. PSO algorithm would be a nice choice because it has only two parameters, velocity and position [24]. So it can easily apply to system due to little number of computation.

Previously Basic PSO has been analyzed [19] but we would like to know whether PSO in its essence has the ability for getting better result. Now we evaluate the Global PSO and consider the effect of its implementing in communication system. We would like to observe whether it can have any significant performance regard to proposed theoretical methods.

We also like knowing that how much the difference is for system throughput between Global and Basic PSO. Furthermore for more comprehensive comparison we consider to ZFBF as a linear technique versus stochastic technique. Moreover we research about its throughput while the size of systems changes. In order to perform analysis, the main contribution is to observe the effect of Global PSO.

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Chapter 3

Particle Swarm Optimization

3.1 Stochastic Algorithms

Stochastic optimization algorithms have been growing rapidly in popularity over the last decade or two, with a number of methods for solving challenging optimization problems.

Nature provides a lot of inspiration to gain insights into the working forces around us.

Evolutionary algorithms are stochastic algorithms whose driving force is optimization. There are various evolutionary algorithms, in particular genetic algorithms,the swarming of birds and the movement of ants.

Stochastic algorithms have the capability to use in designing hardware. For example, PSO is would be used for designing chipsets to lower the heat dissipation or the run length of wires in a given circuitry [12]. It could also been used for antenna deign with a desired side-lobe level or the element of antenna positions in a non-uniform array [19].

It would be claimed that stochastic optimization may give a solution to a given problem;

however the solution may not guarantee a global optimum, except specific triggers are defined within the algorithm. Particularly, in PSO for achieving global optimum, triggers such as Random PSO or Multi-Start PSO need to be considered [12].

Since the solution which are based on linear algebra, is generally easy for implementing them on digital signal processor, technology such as Convex optimization is preferred over stochastic optimization in the field of communication engineering. Now we are faced with the question, regarding this subject why we would consider to stochastic optimizations. In other words it was mentioned that convex optimization is an easier method than stochastic so what the reason is for application of stochastic optimization [12].

The key point here is that one need to relax the problem or reformulate so that the problem becomes convex. Main problem for using the convex optimization is careful formu- lating the variables. Undesirable reformulating may lead to different problem which is not

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Figure 3.1: An example for movement of swarm, a flock of birds

equivalent the actual problem. One of the advantages of stochastic algorithm is the point that the problems solved by such algorithms will be the actual problem solved. However it could be difficult to prove that is optimal [12].

3.2 Biological background of PSO Algorithm

Swarm forming is a tendency which appears in different organisms, for example, fish and bird. Swarming manner make several advantages, for instance protection from predator.

It is obvious that an animal near the center of swarm has more chance to be alive after the predators attack to swarm. In addition the swarm members may confuse the predators through coordinated movement, such as rapid division into subgroups.

From the other point of view, it is not completely evident that swarming always has the profits. For example when the swarm was form, it can be seen easier by predator.

There are other reasons than protection from predator to form swarm. For example finding a mate is easier in the swarm. Of course this property has both the advantages and drawbacks because the competition will exist in such conditions. Swarming is a prerequisite for cooperation between living creatures like bees, ants and termites specifically in works like foraging (food gathering). Here, the simple reason is that many eyes are likely to find desired things that a single pair of eyes [25].

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3.3 Basic Particle Swarm Optimization

In this part we introduce Particle Swarm Optimization algorithm. The simplest type of such algorithm is referred as Basic Particle Swarm Optimization (BPSO).

Particle swarm optimization (PSO) is based on the swarm properties and try to get those properties of swarm that implement in optimization. Essentially PSO algorithm is identified by two parameters velocity and position in the search space as well as some methods for acting on two parameters to reach the optimum value [25].

For basic PSO which can be observed in algorithm, in first step the position x and velocity v initialize. The next step is the number of particle. It is not a constant number and can change from problem to problem but the common value for N is between 20 till 40.

Position are normally initialized randomly in a given range

xij = xmin+ r(xmaxxmin), i = 1, ..., N, j = 1, ..., n (3.1)

Where x_ij is defined as the j^th component of particle position of p_i and r is a uniform number in the range [0,1]. Size of the swarm is indicated by N and n is the problem dimen- sionality (the variables numbers).

Initializing for velocities is as follows

v_ij = α

∆t

−x_max− x_min

2 + r(x_max− x_min)

, i = 1, ..., N, j = 1, ..., n (3.2)

Where vij is defined as the j^th component of particle velocity of pi and r again is a uniform number in the range [0,1]. α often set to 1 (in general case, it is a constant in the range [0,1]). ∆t is the length of time step which commonly is set to 1. It can be seen that the computation will be reduced by taking x_min = −x_max. Therefore the velocity update reduce to

v_ij = αx_min+ r(x_max− x_min)

∆t (3.3)

Now the initialization is completed and next step is to perform evaluation of each particle. Of course the evaluation depends on each problem. Note that related to optimization and knowing that the aim is to maximize or minimize, the sign of inequalities could change.

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Next is to update the velocity and position for all particles. Two measures are stored and used in PSO, namely the best position x^pb_i of particle i and the best performance x^sbof any particle in the swarm. Now we refer to the algorithm, when the particle was evaluated, two described tests in step 3 algorithm are performed. The first one is straightforward and compares the current performance of particle to previous one. For second test according to algorithm, after the first evaluation of all particles, x^sb is set to the best position thus found.

x^sb is then stored, and it is updated only when the condition in step 3.2 of the algorithm is fulfilled [25].

The Basic PSO Method is defined in following algorithm.

Initialize positions and velocities of the particles p_i x_ij = x_min+ r(x_max− x_min), i = 1, .., N, j = 1, .., n

v_ij = _∆t^α −^x^max^−x₂ ^min + r(x_max− x_min) , i = 1, ..., N, j = 1, ..., n

Evaluate each particle in the swarm, i.e. compute f (x_i), i=1, . . . ,N.

Update the best position of each particle, and the global best position.

Thus, for all particles pi, i=1, . . . ,N if f (x_i) ≥ f (x^pb_i ) then

x^pb_i ← x_i end if

if f (xi) ≥ f (x^sb_i ) then x^sb_i ← x_i

end if

Update particle velocities and positions:

v_ij ← v_ij + c₁q

x^pb_ij−x_ij δt

+ c₂r_xsb j−x_ij

δt

, i = i, .., N, j = 1, .., n.

Restrict velocities, such that |vij| < v_max. x_ij ← x_ij + v_ijδt, i = 1, .., N, j = 1, .., n.

Return to step 2, unless the termination criterion has been reached.

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3.4 Global Particle Swarm Optimization

The aim of optimization is to reach the best optimum values; but in variation of BPSO, it is proved that does not satisfy the algorithm and convergence conditions for global search so it does not have guaranteed convergence to a global minimum/maximum [26]. Some methods have been proposed for getting the global minimum. In this thesis we focus on two methods in subset of Global PSO, Random PSO (RPSO) and Multi-Start PSO (MSPSO).

3.4.1 Random Particle Swarm Optimization

The update equations are the same as the BPSO and therefore RPSO does satisfy the algorithm condition for local search. RPSO differs from BPSO in that some of particles have their positions randomly reset. During each iteration at least one particles position is set to a new random position in swarm [26].

3.4.2 Multi-Start Particle Swarm Optimization

The lack of diversity is one of the major problem that exist in the basic PSO when the particle start to converge to the same point. It is indicated that such point even may not be local optimum. For overcoming the undesired premature stagnation of the basic PSO, several methods have been proposed to successively inject randomness or chaos into the swarm.

These approaches together referred to as Multi-Start methods [26].

The main objective of Multi-Start methods is to increase diversity that cause the larger search space for getting the optimum value explored. We should pay attention that successive injection of chaos position will prevent the swarm to reach an equilibrium state.

The first person who expresses the benefits of randomly reinitializing particles was Kennedy and the process referred to as craziness. Of course he did not mention the evaluation of such operators. Since then, a number of researchers have proposed different approaches to apply- ing such craziness operator for PSO [26].

We should answer to several questions when the decision is to add randomness to the swarm. These questions would be such as what parameters should be considered to randomized, when the action of randomization should occur and how it should be done and which members of the swarm will be affected.

The diversity of the swarm can be increased by randomly initializing position vectors and/or velocity vectors [26].

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By initializing positions, particles are relocated to a different random position in the search space. By keeping the position vectors constant and randomizing the velocity vectors, particles keep their memory and save current and previous best position but the drawback is to force the search in different random directions. If we apply the random reinitializing of the velocity vector of a particle and finding no better solution, the particle will again be attracted toward its personal best position.

By initializing position vectors thought should be considered to what should be done with personal best positions and velocity vectors. Total reinitialization will have a particle‘s personal best also initialize to the new random position. This action removes the memory of particle and prevents the particle from moving back towards its previously found best position. At the first iteration after reinitialization the new particle is attracted only towards the previous global best position of the swarm. When positions are reinitialized, velocities are usually initialized to zero, to have a zero momentum at the first iteration after reinitialization. We can also initialize the velocities to small random values. Some researchers initialize velocities to the cognitive component before reinitialization. This ensures a momentum back toward the personal best position.

Another important that we should consider for implementing Multi-Start is the time that we should reinitialize. We should be aware that If the action of reinitialization take place too soon, the particle that are affected by this action do not have the sufficient time to explore their current regions before being relocated and it may cause that algorithm cannot reach to optimum values in the space that it loses due to soon reinitialization. On the other hand if the time for reinitializing is too long, it may cause that all particles have already converged.

There are several approaches for identifying the time of reinitializing.

At fixed interval, is the proposed method within the mass extension PSO that developed by Xie et al. It is proved prematurely reinitialize a particle may occur due to fixed interval [26].

Probabilistic approach, in this approach the time of reinitializing is base on probability. In the dissipative PSO Xie, et al. reinitialize velocities and positions on chaos factors which serve as probabilities of introducing chaos in the system. A problem with using this approach is like continually injection of randomness and prevents the system to reach an equilibrium state. The technique that applies for overcoming this problem in probabilistic approach while still taking advantage of chaos injection is to start with large chaos factor which reduce over time. The initial large chaos factors increase diversity in the first phases of the search, allowing particles to converge in the final stages [26].

Approach based on some ”convergence condition” where specific events trigger reinitialization. Using this approach; particle could exploit their local regions before reinitializing [26].

In this thesis we have applied this approach and evaluate the result of simulation for MSPSO in the framework of convergence condition.

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While there is not any improvement over time for particle, Venter and Sobieszczanski- Sobieski and Xie etal proposed the initiate for reinitialization. They considered in particle fitness of current swarm for variation. The particle are centered in close proximity to the global best position if the variation is small. The particles are reinitialized when they are two standard deviations away from the swarm center. Xie et etal count for each xi 6= ˆy the number of times that f (xi) − f (ˆy) < . When this count exceeds a given threshold, the corresponding particle is reinitialized. The setting values for and the count is important.

By taking too large, reinitialization occurs before having the chance for exploiting their current regions.

Hope and re-hope is defined by Clerc. If there is still hope that objective could be reached, particles can continue in their current search directions. If not particles are reinitialized near the global best position by considering the local shape of the objective function [26].

A number of convergence tests are defined by Van den Berg for his Multi-Start PSO, referred as randomized swarm radius condition, the particle cluster condition and the objective function slope condition [26].

The last technique is in subset of convergence condition is proposed by Lovberg and Krink. This technique has the capability of self-organizing. Each particle retains an addi- tional variable, Ci, reffered as critically of the particle. By performing some computations the particle makes decision to explore more when it is too similar to other particles [26].

The next question is which particles are allowed to reinitialize. Definitely, reinitialization the global best particle would not be a good idea! From the discussion above, a number of selection methods have already been identified. Decision making for selecting particle to initialize in probabilistic method is based on a user-defined probability. The convergence methods use specific convergence criteria to identify particle for reinitialization. The random selection scheme was proposed by Van der Berg, where a particle is reinitialized each iterations. This approach allows each particle to explore its current region before being reinitialized.

And finally the last question is how particles are reinitialized. It was mentioned before that velocities and/or positions can be reinitialized. For reinitializing velocities, each velocity component set to a random value constrained by the maximum allowed velocity. Venter and Sobieszczanski-Sobieski considered to cognitive component after reinitialization of position vectors and setting the velocity vector to that. For the position vectors, they are usually initialized to a new position subject to boundry constraint; that is x_ij(t + 1) ∼ U (x_min, j, x_max).

There is another approach to Multi-Start PSO. In this approach particle are randomly initialized and a PSO algorithm is executed until the swarm converges. The randomly initializing all particle occur when convergence is detected and the best position was recorded.

The process is repeated until a stopping condition is satisfied, at which point the best

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recorded solution is returned.

A Multi-Start method are defined in following algorithm [26].

Repeat

if f (S.y) ≥ f (y) then y = S.y

end if

if theswarmShasconverged then Reinitializeallparticles;

end if

for each particle i=1,..,S.n_s do if f (S.x_i) ≥ f (S.y) then

S.y_i = S.x;

end if

if f (S.x_i) ≥ f (S.y) then S.y_i = S.x;

end if end for

Update velocities;

Update positions;

Until stopping condition is true;

Refine y using local search;

Return y as the solution;

We applied this algorithm in our research for simulation as a representative of Multi-Start PSO.

3.5 Beamforming using Stochastic PSO Algorithm

For using the algorithm we should consider to proper mapping of selected algorithm parts to system elements. In our research each bird in a swarm play a role of non-zero elements of BF matrix and has both real and imaginary parts, i.e. the i -th member of the swarm is the i -th particle that carries all the (n = 2 . K . Nt . M) BF coefficients. The digit ”2” in equation refers to real and imaginary parts that consist the complex BF coefficients and the mapping of these parts, during initializing, is only for expressing how the BF is translate to particle.

Map the BF to particle:

X(i,j) ← <W(l,m), l {1, .., KN_t} , m {1, .., M }

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X(i,j+1) ← =W(l,m)

which < and = are allocated to complex coefficients of BF matrix.

Demap the variables in a particle to form the BF matrix during optimization:

W(l,m) ← {X(i, j)} + i {X(i, j + 1)}

These steps can be omitted in the actual implementation [19].

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Chapter 4

Simulation

4.1 System Model

In our model, the downlink of a static cluster of KBS with N_t antenna serving M user is considered. The system can be model as

y = HWx + n (4.1)

wherein y, H, W x and n are discrete time signal, channel matrix, precoding matrix, transmitted signal and noise, respectively.

4.2 Convergence

The notion of convergence means the equilibrium state that happen in system after certain number of iteration.

For RPSO as it has been shown in Figure 4.1, while the number of particle for position resetting increase, the level of convergence will have worse performance and reach to upper digit in convergence.

In Case of MSPSO with considering Figure 4.2, It could be observed while the size of interval decreases the level of convergence has better performance and worst graph is for the largest interval which the graph is approximately like BPSO.

As it has been shown in Figure4.3, for BPSO and RPSO we can have some reasonable convergence after proper iterations. The result shows that we got worse situation for RPSO in comparison to BPSO. It could be occurred due to resetting in each iteration (according to definition of RPSO)

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Figure 4.1: Convergence RPSO with different Reset Particle(s) Position

Figure 4.2: Convergence MSPSO with different Step Size

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Figure 4.3: Convergence RPSO vs BPSO

Figure 4.4: Convergence MSPSO

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that might causes the system to far away from desired point.

In case of MSPSO, it is shown in Figure 4.4, we cannot see any level of convergence. It could be illustrated since we do inject randomness for increasing diversity it causes system to prevent for convergence.

4.3 Optimum Values for properties of PSO Algorithm

In this part we consider to parameters that play roles in action of optimization with PSO for obtaining the optimum values. The process of values optimization has been performed in software of Matlab, the size of system for deriving best values is 6X6 and cumulative distributed function (CDF)is the measurement that determines the better values. The process for getting the optimum value is to change the desired value with fixing other values.

The first parameter that we consider for getting best value is the cognitive component.

According to definition, cognitive component is the degree to which it trusts its own previous performance as a guide towards obtaining better results [27].

The next parameter is social component while it tells how much a given particle should rely on its neighbors. The proposed range for the values of cognitive and social components can change from 1 to 2 [27].

The third parameter is inertia weight which is employed to control the impact between the previous and current velocities [27].

The last one is the number of particle. It could be changed from problem to problem and the defined range for that is between 20 and 40.

The following figures indicate the best values for these parameters.

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Figure 4.5: Comparison CDFs for deriving best value, change cognitive components (C1) Figure 4.5 indicates the best value for cognitive component is ”1.65”.

Figure 4.6: Comparison CDFs for deriving best value, change social components (C2) As it is shown in Figure 4.6, the best value for social component is ”1.80”.

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Figure 4.7: Comparison CDFs for deriving best value, change Max of inertia weight The result in Figure 4.7 shows the best value for maximum of inertia weight is ”2.1”.

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Figure 4.8: Comparison CDFs for deriving best value, change Min of inertia weight The Figure 4.8 indicates the best derived value for minimum of inertia weight is ”0.28”.

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Figure 4.9: Comparison CDFs for deriving best value, change number of particle(s)

In Figure 4.9 for getting the optimum value for number of particle, we found the best range is between 28 to 32. We performed the test with much accurate by smoother curve MC=1000, 2000, 5000. Some of them shows the best value is 31 and some of them 32. We have selected the value of 32 for this parameter.

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Figure 4.10: Comparison CDFs for deriving best value, change step size of re-initializing

In Case of MSPSO as it is discussed before, we selected the convergence condition approach but we should also pay attention to the time of reinitialization. We determine different intervals for reinitializtion which in the beginning of each interval the action of reinitialization performs. It could be observed with decreasing the intervals size the system has better performance.

One may express that since the decrease of interval size leads to better performance, we should take the size as small as possible but it can not be applied in practical. For the reason it would be said that is correct idea and we tested the smaller values and even got better result but it should be note that since in each reinitialization the computations perform in whole of iteration, small intervals lead to having reinitializtion over and over and the calculations spend too much. It means we should a compromise between level of convergence and the time of computation. With regard to our information in Figure 4.10 the digit ”50” would be good choice.

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Figure 4.11: Comparison CDFs for deriving best value, change number of particle(s) reset position For RPSO, as it is shown in Figure 4.11we selected different number of particle for reset of their positions. We have determined the values of 1, 2, 5, 10, 15 for such action. Therefore the best value for number of position in RPSO is ”1”.

All of the derived optimum values are summarized in Table 4.1.

Number of Particles 32 BPSO Cognitive Component (C1) 1.65 RPSO Social Component (C2) 1.80 MSPSO Max of Inertia Weight 2.1

Min of Inertia Weight 0.28 RPSO Number of reset particle(s) 1 MSPSO Interval length for Re-initializing 50

Table 4.1: Optimum Values

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4.4 Objective Functions

4.4.1 Sum Rate Maximization

The presented algorithm for PSO consist minimization of the objective function. Therefore, to maximize the sum rate we should apply proper change in our objective and the objective function written as

f (X(i, :)) = −R_tot (4.2)

It means that for every iteration before evaluating objective function, the sum rate per cell as in equation 8 needs to be calculated.

For the system by size of 3x3, the three variations of PSO approximately have same values and of course MSPSO has better result by a little benefit. It has better performance in comparison to BPSO, RPSO and ZFBF by 6%, 8% and 53% on their averages respectively.

Figure4.12

For 6x6, the application of MSPSO shows more effect. MSPSO has better performance in comparison to BPSO, RPSO, ZFBF by 13%, 17% and 63% on their averages respectively.

Figure4.13

For system 9X9, the progress clearly could be seen. MSPSO has better performance in comparison to BPSO, RPSO and ZFBF by 16%, 20% and 57% on their averages respectively.

Figure4.14

For system 12x12, it is shown that MSPSO has better performance in comparison to BPSO, RPSO and ZFBF by 18%, 24% and 48% on their averages respectively. Figure 4.15 All of experimental results for this objective function is summarized in Table 4.2 and Figure4.16.

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Figure 4.12: Comparison CDFs for getting best scenario in System Size 3x3

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3x3 6x6 9x9 12x12 ZFBF 18.2447 28.6555 36.9980 45.3033 BPSO 26.3431 41.6931 50.0752 55.6012 RPSO 25.8910 40.2763 48.3694 53.1357 MSPSO 27.9287 47.0994 58.2142 65.5667

Table 4.2: Mean Value in different sizes with regard to objective function Sum Rate Maximization

Figure 4.16: Different scenarios in different system sizes

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4.4.2 Weighted Interference Minimization

With regard to our knowledge choosing a single direct objective function of minimizing only the interference, cause the PSO to prefer only good SINR users and leave out the week SINR users. This make to save the power at BS and lead to lowering the sum rate. Therefore the objective function should consider minimizing the interference and improve the SINR of the weakest users as well. This objective function is called weight interference minimization.

f (X(i, :)) = ||Of f Diag(HW)||_p

minSIN Ruser (4.3)

For System 3x3, the values for all PSO variations are approximately the same and they are better than ZFBF by 30% on their averages. Figure 4.17

For 6x6 systems MSPSO has better performance in comparison to BPSO, RPSO and ZFBF by 8%, 9% and 55% respectively. Figure 4.18

For 9x9, the MSPSO keeps its superiority on the others by better performance in comparison to BPSO, RPSO and ZFBF by 20%, 27% and 55%. Figure4.19

The interesting point is in system 12x12 where ZFBF has better performance in comparison to BPSO and RPSO. In this system MSPSO is better than BPSO, RPSO and ZFBF by 32%, 41% and 22%. In this system, ZFBF has better performance in comparison to BPSO and RPSO by 8% and 15% respectively. Figure 4.20

In our simulation it has been shown that in all systems and all objective functions MSPSO is best one and exception the system 12x12 by objective function Weight interference minimization that ZFBF is in the second place, in other simulation BPSO has better performance.

All of derived values are summarized in Table 4.3 and Figure 4.21.

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3x3 6x6 9x9 12x12 ZFBF 18.1975 28.9344 36.7843 44.6279 BPSO 23.6362 41.4361 47.4701 41.0891 RPSO 23.5859 41.0464 44.7261 38.7427 MSPSO 23.4469 44.8426 56.8925 54.6759

Table 4.3: Mean Value in different sizes with regard to objective function Weight Interference Minimization

Figure 4.21: Different scenarios in different system sizes

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Chapter 5

Conclusion

In this master thesis, the capability of Particle Swarm Optimization algorithm evaluated for Precoding in Coordinated Multi-Point system.

At the beginning of this research work since global methods gain the better optimum during optimization, we expected that two variations of Global PSO have better performance in comparison to Basic one. But the result showed only one of those subset of global optimization has better performance.

The simulation has indicated that Multi-Start PSO has better throughput relative to two other variations of PSO, Random PSO and Basic PSO. It has also better result relative to Zero Forcing which was a representative of linear precoding.

The interesting result during optimization was the point that Basic PSO has better performance in comparison to Random PSO. It means that the basic variation of some stochastic algorithm gain better result even relative to some global variation.

We did the simulation with regard to two defined objective functions, Sum Rate Maxi- mization and Weight Interference Minimization. Furthermore the simulation was performed in different system sizes, 3x3, 6x6, 9x9 and 12x12.

In objective function Sum rate maximization in all system sizes Multi-Start PSO has the best performance and other places were dedicated to Basic PSO, Random PSO and Zero Forcing respectively. In objective function Weight interference minimization, the first place was for Multi-Start PSO but the linear precoding Zero Forcing was in second place. The other places were dedicated to Basic PSO and Random PSO, repectively.

For Future work, one can test the other geographical deployment of Coordinated Multi- point system and the other different sizes of system. Related to PSO algorithm, we reinitialized both parameters position and velocity in our research and the effect of reinitializing for each of them can be considered in next studies. Moreover other kinds of stochastic algorithms can be evaluated in Coordinated Multi-Point system.

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Appendix

Acronyms

LTE Long Term Evolution

IMT International Mobile Telecommunication CoMP Coordinated Multi-Point

3GPP 3rd Generation Partnership Project PSO Particle Swarm Optimization BPSO Basic Particle Swarm Optimization RPSO Random Particle Swarm Optimization MSPSO Multi-Start Particle Swarm Optimization MIMO Multiple Input Multiple Output

BS Base Station

UE User Equipment

CSI Channel State Information IFFT Inverse Fast Fourier Transform

JP Joint Processing

CJP Centralized Joint Processing CCN Central Coordination Node

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DPC Dirty Paper Coding

BF BeamForming

ZF Zero Forcing

SINR Signal to Interference plus Noise Ratio

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Figure 5.1: *A sample of output in our meetings during master thesis*

Thank you dear Tilak!

Without your guidance I was not able to perform this master thesis.

Evaluation of Particle Swarm Optimization Algorithm for Precoding in Coordinated Multi-Point System