Performance analysis and power allocation in MU-MIMO systems with imperfect channel state information

(1)

Performance Analysis and Power

Allocation in MU-MIMO Systems

with Imperfect Channel State

Information

Author:

Daniele Davoli

Supervisors:

Prof. Marco Moretti

Prof. Luca Sanguinetti

Prof. Gabor Fodor

A thesis presented for the Master degree in

Telecommunication Engineering

Department of Information Engineering

University of Pisa

(2)

Abstract

The performance of the uplink of multi-user multiple-input multiple-output (MU-MIMO) systems depends on the receiver architecture. The current approach is to design receivers that minimize the mean squared error (MSE) of the received data symbols without taking into account the channel state information (CSI) errors in the minimization process. In this project we analyze a new minimum mean squared error (MMSE) receiver that exploits the instantaneous channel state information of each user to minimize the MSE in the presence of CSI errors. We analyze the performance of the proposed receiver assuming either Gaussian channels with Rayleigh fading and with or without uncorrelated antennas or the Winner II channel model. We compare the performance of the newly proposed CSI error-aware receiver with previously proposed receivers in terms of MSE and sum-rate. We also derive an analytical expression to evaluate the signal-to-interference-plus-noise ratio (SINR) of the proposed receiver when operating with Gaussian channels and use this expression to develop pilot-to-data power ratio setting algorithms for sum-rate maximization and max-min fairness. Simulation results indicate that the proposed receiver and associated algorithms outperform previously proposed benchmark schemes using sub-optimal receivers.

(3)

Acknowledgments

With full gratitude I would like to thank prof. Marco Moretti and prof. Luca Sanguinetti my thesis supervisors at the University of Pisa and prof. Gábor Fodor, my supervisor at Ericsson Research they both guided me during this thesis work and helped me overcome the difficulties that I encountered. It has been a pleasure and a honor to work with them. I want to thank dr. Göran Klang, Sector Manager of Concepts and Performance Research Area in the Radio Department at Ericsson Research, he gave me the great opportunity to develop my thesis at Ericsson Research. I would also like to thank the others PhD and master thesis students in Ericsson Research, they gave me great insight on my work. Finally, I must express my very profound gratitude to my parents and to my friends for providing me with unfailing support and continuous encouragement throughout my years of study and through the process of researching and writing this thesis. This accomplishment would not have been possible without them.

(4)

List of Figures

2.1 Plot patterns . . . 9

3.1 Channel coefficient generation procedure . . . 14

3.2 Winner II Channel modeling process . . . 15

5.1 Mean squared error over pilot-to-data power ratio in Rayleigh fading with a user . . . 26

5.2 Sum-spectral efficiency over pilot-to-data power ratio in Rayleigh fading with a user . . . 27

5.3 Squared error CDF in Rayleigh fading with a user . . . 27

5.4 Mean squared error over pilot-to-data power ratio in Rayleigh fading with 7 users . . . 29

5.5 Sum-spectral efficiency over pilot-to-data power ratio in Rayleigh fading with 7 users . . . 29

5.6 Squared error CDF in Rayleigh fading with 7 users . . . 30

5.7 Multi-Users Winner II Urban Micro Scenario . . . 32

5.8 Multi-Users Winner II Urban Macro Scenario . . . 33

6.1 Nelder-Mead flow-chart . . . 37

6.2 Gaussian channel with Rayleigh fading with 7 users . . . 41

6.3 Gaussian channel with Rayleigh fading with 7 and 19 users . . . 42 6.4 Gaussian correlated channel with Rayleigh fading with 3 and 7 users 44

(7)

Chapter 1 Introduction

1.1 Introduction

The performance of the uplink of MU-MIMO systems depends critically on the receiver architecture and on the quality of the acquired channel state information at the receiver (CSIR). A popular approach is to propose receivers schemes at the base station that are designed to minimize the MSE of the uplink received data symbols assuming the availability of perfect CSIR. Indeed, the currently available MU-MIMO receivers are designed assuming that perfect CSIR will be available and that channel state information errors will cause only a minor performance degradation.

In this thesis work, we test and analyze a newly proposed MMSE receiver especially applicable in a massive MU-MIMO system. This design makes use of the non-perfect instantaneous channel estimates available for all users at the base station to minimize the MSE and take into account the dependence of the channel estimation quality and the receiver performance on the pilot-to-data power ratio (PDPR). By invoking the theory of random matrices, we calculate the

signal-to-interference-plus-noise ratio as a function of the number of antennas, the pilot signal power level and the data signal power level of all users and investigate the performance of the new receiver. We compare the performance of the new receiver with the previously proposed covariance-based multi-user MMSE receiver and with the known naive MU-MIMO MMSE receiver.

In this project, we investigate the performance of MU-MIMO receiver structures which take into account both the inherent trade-off between the resources spent on CSIR acquisition and data transmission and the inherent statistical errors in the available instantaneous and long term CSIR. This receiver design is especially applicable in situations in which the coherence bandwidth and coherence time are limited and the optimization of resource distribution between CSI acquisition and

(8)

data transmission is critical.

To prove the benefit of such approach we compare the new design to the common used naive MU-MIMO MMSE receiver that exploits the instantaneous channel estimates available for all users at the base station to minimize the MSE but doesn’t take into account the channel estimation quality and doesn’t try to suppress the channel estimation errors. We also propose an algorithm capable of maximizing the performance of the receiver in terms of sum-rate among all the users or in terms of max-min fairness by adjusting the PDPR of each user.

1.2 Thesis Outline

This thesis is divided into 6 chapters: Chapter 2 explains how the MU-MIMO system is modeled; Chapter 3 explains how the wireless channel is modeled and the channel estimation method, also it discusses important assumptions that we use throughout the whole project; Chapter 4 presents how the different employed receivers are done with a focus on the new proposed MMSE receiver; Chapter 5 illustrates the test made on the new receiver and comparison with the older receivers that we use as reference; Chapter 6 explains the new algorithms and prove theirs capabilities; Chapter 7 draw the conclusion and our plan to extend our work.

(9)

Chapter 2 System Model

2.1 System model

In this project we consider the uplink transmission of a MU-MIMO wireless system consisting of K single antenna mobile stations and a base station with Nr antennas.

The K mobile stations transmit simultaneously in the same frequency channels and each transmission consist in two stages: training phase and data transmission phase. In the training phase each mobile station transmit a pilot signal to the base station that uses the signal to estimate the CSI in the coherence interval. In the data transmission phase the base station receives the data signal from all the mobile stations and implement receivers that are constructed based on the CSI derived during the training phase to estimate the data symbols that each mobile station transmit. We also suppose that, in our scenario, the communication bandwidth is much smaller than the reciprocal of the delay spread of the channel, so that we can consider the channel non-frequency selective.

We have three different possible pilot arrangements: block type arrangements which dedicates all frequency channels within a given time slot to either channel estimation or data transmission, comb pilot pattern employs pilot and data symbol mixed in the frequency domain within a single time slot and mixed type arranges the pilot and data sub-carriers discontinuously in time and frequency domains. In our work we focus on the comb type pilot arrangement since is better suitable for non-frequency selective channels as shown in [6].

In order for the base station to be able to discriminate between the simultane-ously transmitted pilot signals of each mobile station we assume to use orthogonal pilot sequences s = [s1, . . . , sτp]

T _{∈ C}τp×1_{, in which each pilot symbol is scaled as}

|si|2 = 1, for i = 1, . . . , τp. The pilot symbols are assumed orthogonal when K ≤ τp,

such orthogonality can be obtained by using a Zadoff-Chu (ZC) sequence. In most of our simulations we will also assume that K _Nr. Since we use a comb type

(10)

(a) Frequency continuous-time spaced pilot allocation (Block type)

(b) Time continuous-frequency spaced pilot allocation (Comb type)

(c) Time spaced-frequency spaced pilot allocation

Figure 2.1: Plot patterns

arrangement of the pilot symbols, given F sub-carrier in the coherence bandwidth, a fraction of τp sub-carriers are allocated to the pilot and Fd= F − τp sub-carriers

are allocated to the data symbols; in practice, this type of arrangement is suitable for time varying channels, so that channel estimation is facilitated at the same time instant that is used for data transmission. Each mobile station transmits at a constant power Ptot and the transmissions power can be distributed unequally

in each sub-carrier. In particular, considering a transmitted power Pp for each

pilot symbols and P for each data symbols transmission, the sum constraints of τpPp+ (F − τp)P = Ptot is enforced.

This model can represent a multi-carrier Long Term Evolution (LTE) system, in which the available bandwidth is organized into physical resource blocks (PRBs) comprising F sub-carriers where the available bandwidth is distributed. Each PRB represents a coherent bandwidth chunk and obtains its own channel estimate.

(11)

Chapter 3 Channel Model

In this system, the quality of channel estimation and data reception are important aspects of the system performance. Specifically, the distribution of the transmitted pilot and data power plays a crucial role in the optimization of the overall per-formance of our system in term of sum-rate and MSE. In this chapter we present three common system models: Gaussian channel with Rayleigh fading, Gaussian channel with Rician fading and Winner II channel mode. In our new MMSE receiver the channel estimation method changes its structure hence we present the two common channel estimation method, least square (LS) and minimum mean squared error and we illustrate how they affect the component of the new proposed MMSE receiver.

3.1 Channel modeling

The multi-path channel can be characterized by two different types of fading: the large-scale fading and the small-scale fading. The large-scale fading is used to describe the signal level at the receiver after traveling over a large area (hundreds) of wavelengths), it is the signal attenuation due to the distance between transmitter and receiver and the effect of diffraction and reflection due to the objects in the propagation path. The large-scale fading is normally composed by the path-loss, the attenuation of the signal due to the distance between the transmitter and the receiver and the effects of reflection of the signal on the objects on the path, and the shadowing, the attenuation of the signal due to the effects of diffraction over the objects in the path. The small-scale fading is used to describe the rapid fluctuations of the amplitudes, phases, or multi-path delays of radio signal over a short period of time or distance, so that large-scale fading effects may be ignored, it is caused by the interference between the different versions of the transmitted signal due to the multi-path that arrive at the receiver at slightly different times. The small-scale

(12)

fading is mostly caused by three effects: rapid changes in signal strength over a small travel distance or time interval; random frequency modulation due to varying Doppler shifts on different multi-path signals; time dispersion caused by multi-path propagation delays [16]. For our simulation environment we have exploited three different channel models that model both large-scale and small-scale fading effects:

1. Gaussian channel with Rayleigh fading; 2. Gaussian channel with Rician fading; 3. Winner II channel;

The first two models use a stochastic approach to describe the small-scale fading while the large-scale fading is obtained from the log-normal model given the distance between the base station and the mobile stations and the path-loss coefficient the shadowing effects are not considered in this models.

The third model uses a geometric approach and tries to evaluate all the possible variables to generate a more realistic channel environment based on the Third Gen-eration Partnership Project (3GPP) channel model. This model jointly evaluates the large-scale, both path-loss and shadowing, and the small-scale parameters.

3.1.1 Gaussian channel with Rayleigh fading

The Gaussian channel with Rayleigh fading is a simple stochastic model that can be used to describe non line of sight (NLOS) multi-path MU-MIMO wireless channels with no dominant path and a large number of reflective path, in this case, the envelope of the received signal can be described by the Rayleigh PDF. With these assumptions the channel can be statistically described as a circular symmetric complex normal distribution with zero mean, since there are no dominant paths, and a given variance σ2_:

h_{∼ CN(0, σ}2) (3.1)

In our system we consider a receiving base station with Nr receiving antennas, in

this case, the Gaussian channel with Rayleigh fading can be modeled as:

h _{∼ CN(0, C)} (3.2)

We choose to normalize the power of the channel to one and the correlation matrix, since the channel is zero mean is equal to the covariance matrix, C is set to the identity matrix INr in most of our simulations.

(13)

As stated before the stochastic model used to describe the amplitude variations of the small-scale fading in this model is the Rayleigh probability distribution function with parameter σ:

f (x_{|σ) =} x σ2exp x2 sσ2 ! (3.3) while the phase variations are uniformly distributed between [0, 2π].

The large-scale fading, in this model we considered only the path-loss, is modeled as a log-normal model with the attenuation in decibel Lk= 20 log10(4πλ) +

10β log₁₀(dk) where λ is the wavelength, β is the path-loss exponent and dk is the

distance of the k-th user to the base station.

3.1.2 Gaussian channel with Rician fading

The Gaussian channel with Rician fading is a simple stochastic model that can be used to describe both NLOS and LOS multi-path MU-MIMO wireless channels with one dominant component, much stronger than the others, which is typically is the LOS path and a large number of reflective paths; in this case the envelope of the received signal can be described by the Rician PDF.

With these assumptions the channel can be statistically described as a circular symmetric complex normal distribution with non-zero mean, due to the dominant path, and a given variance σ2_:

h_{∼ CN(ν exp(jθ), σ}2) (3.4)

where θ is the angle of arrival of the LOS component.

The stochastic model used to describe the amplitude variations of the small-scale fading is the Rician distribution with parameters ν2 = _K+1K Ω and σ2 = _2(K+1)1 Ω:

f (x_{|ν, σ) =} x σ2exp − (x2+ ν2) 2σ2 ! I0 _xν σ2 = = 2(K + 1)x Ω exp −K − (K + 1)x2 Ω ! I0  2 s K(K + 1) Ω x   (3.5)

where K = _2σν22 is the Rician K-factor which describe the ratio between the power

of the dominant component and the power of the other components; Ω is the total power from both components Ω = ν2_{+ 2σ}2 _{and can be used with K to describe the} distribution; Io(·) is the zero order modified Bessel function of the first kind. The

phase variations of the small-scale fading is described as a uniformly distributed random variable between [0, 2π].

(14)

When ν = 0 the dominant component disappear, in this condition we can see that the Rayleigh fading is a particular case of the Rician fading.

When the system has multiple receiver antennas the Rician fading channel can be modeled in the following way as shown in [3],

h = s K K + 1hLOS+ s 1 K + 1hN LOS (3.6)

where hLOS = [1, ej2πd cos θ, . . . , ej2π(Nr−1)d cos θ]T, in which θ is the angle of arrival

of the LOS component and d is the antenna spacing in function of the wavelength;

hN LOS is a column of complex Gaussian random variable, hN LOS ∼ CN(0, C), as

in the Rayleigh fading channel. In the case of Rician fading channel with multiple receiving antennas the channel can be described as:

h_{∼ CN(} s K K + 1hLOS, C K + 1) (3.7)

The large-scale fading, in this model the path-loss, is modeled as a log-normal model with the attenuation in decibel Lk = 20 log10(4πλ ) + 10β log10(dk) where λ is

the wavelength, β is the path-loss exponent and dk is the distance of the k-th user

to the base station.

3.1.3 WINNER II channel model

The Winner II channel model, unlike the previous two channel models, is a geometry based stochastic channel modeling approach, which allows creating of an arbitrary double directional channel model for link level and system level simulations of local area, metropolitan area, and wide area wireless communication systems; geometry based modeling of the radio channel enables separation of propagation parameters and antennas. The channel parameters, including delay spread, delay values, angle spread, shadow fading and cross-polarization ratio, for individual snapshots are determined stochastically, based on the statistical distributions extracted from channel measurements. Transmitted signals can be reflected, refracted and diffracted by the environment during the propagation process hence, channel realizations are generated with geometrical principle by summing contributions of rays with specific small-scale parameters like delay, power, angle of arrival and angle of departure [10]. There are several different scenarios that can be exploited: indoor office/residential, indoor to outdoor, typical urban micro-cell, bad urban micro-cell, large indoor hall, outdoor to indoor micro-cell, LOS stationary feeder rooftop to rooftop, LOS stationary feeder street-level to street-level, LOS stationary feeder below-rooftop to street-level, NLOS stationary feeder above rooftop to street-level, feeder link base station to fixed relay station at approximately rooftop to rooftop

(15)

Set scenario, net-work layout and antenna parameters

Assign propaga-tion condipropaga-tion

(NLOS/LOS) Calculate pathloss

Generate correlated large scale parameters

(CS,AS,SF,K) Generate

clus-ter powers Generate delays Generate arrival and

departure angles Perform random

coupling of rays Generate XPRs

Draw random

initial phases Generate chan-nel coefficient Apply path lossand shadowing General parameters:

Small scale parameters:

Coefficient generation:

Figure 3.1: Channel coefficient generation procedure

level, suburban, typical urban macro-cell, bad urban macro-cell outdoor to indoor macro-cell, rural macro-cell, base station to mobile relay station in rural macro-cell, mobile relay station to mobile station in rural macro-cell. These scenarios are modeled using the same method, but with different parameters. The parameters of the propagation scenarios in both LOS and NLOS conditions are obtained through several measurement campaigns. In this thesis project, we will exploit two different scenarios of this channel model for our tests: urban micro-cell and urban macro-cell. The general procedure for generating the channel coefficients is illustrated, step-by-step, in figure 3.1. While in figure 3.2 is illustrated the procedure used to model the channel.

Urban Micro-Cell

In urban micro-cell scenarios, the height of the antennas at the base station and at the mobile stations are assumed to be well below the tops of surrounding buildings. The antennas are assumed to be outdoors in an area where streets are laid out in a Manhattan-like grid. The streets in the coverage area are classified as “the main street”, where there is the LOS from all locations to the BS, with the possible exception in cases where the LOS is temporarily blocked by traffic (e.g. trucks and buses) on the street. Streets that intersect the main street are referred to as perpendicular streets, and those that run parallel to it are referred to as parallel streets. This scenario is defined for both the LOS and the NLOS cases. Cell shapes are defined by the surrounding buildings, and energy reaches NLOS streets as a result of the propagation around corners, through buildings, and between them.

(16)

WINNER II D1.1.2 V1.2

Page 28 (82) account several aspects – e.g. channel sounder setup, measurement route, link budget. Channel measurements are done according to the campaign planning and documented accurately. Measurement data is stored onto a mass memory (e.g. magnetic tape or hard disk).

The second phase of the channel modelling process concentrates on data analysis. Depending on the required parameters, different analysis methods and items are applied. Output of data post-processing could be, e.g., a set of impulse responses, path-loss data, or extracted multidimensional propagation parameters. For the post-processed data, statistical analysis is done to obtain parameter PDFs.

The third phase of the channel modelling process covers the items required in simulation. Parameters are generated according to the PDFs, by using random number generators and suitable filters. MIMO transfer matrix is obtained by using the generated parameters, and information about the antennas. In our approach MIMO transfer matrices are generated by using the sum-of-rays method. Generated impulse responses are called channel realisations, which are then used in simulations.

generic model measurement data measurement data measurement data Campaign planning Channel measurements parameter PDFs parameter PDFs parameter PDFs data post-processing / analysis parameter generation MIMO transfer matrix generation parameter PDFs parameter PDFs channel realisations array responses Simulations 1 2 3 generic model measurement data measurement data measurement data Campaign planning Channel measurements measurement data measurement data measurement data Campaign planning Channel measurements parameter PDFs parameter PDFs parameter PDFs data post-processing / analysis parameter generation MIMO transfer matrix generation parameter PDFs parameter PDFs channel realisations array responses Simulations 1 1 2 3

Figure 3-2 WINNER channel modelling process

3.3 Network layout

WINNER MIMO radio channel model enables system level simulations and testing. This means that multiple links are to be simulated (evolved) simultaneously. System level simulation may include multiple base stations, multiple relay stations, and multiple mobile terminals as in Figure 3-3. Link level simulation is done for one link, which is shown by blue dashed ellipse. The short blue lines represent channel segments where large scale parameters are fixed. System level simulation consists of multiple links. Both link level and system level simulations can be done by modelling multiple segments, or by only one (CDL model).

Figure 3.2: Winner II Channel modeling process

Urban Macro-Cell

In a typical urban macro-cell, the mobile stations are located outdoors at street level and the fixed base station is clearly above the surrounding building heights. As for propagation conditions, NLOS or obstructed LOS is a common case, since the street level is often reached by a single diffraction over the rooftop. The building blocks can form either a regular Manhattan-like grid or have more irregular locations. Typical building heights in urban environments are over four floors. Buildings height and density in typical urban macro-cell are mostly homogeneous.

3.2 Channel estimation

In order to perform the channel estimation, we must first describe the form of the pilot signal at the base station. The base station is equipped with N r antennas so that the N r_{× τ}p matrix Y(p) of the received pilot signal at the base station can be

written as: Y(p) = K X k=1 αk q Ppkhks T k + N (3.8)

Where αk account for the propagation loss of user k, hk is the small-scale fading

channel, sk is the pilot symbol of the k-th user and N ∈ CNr×τp is the additive

white gaussian noise (AWGN) with N_{∼ CN(0, σ}2

(17)

For the base station to be able to estimates the channel of each user we assume that it knows or that it is able to retrieve: the large-scale fading parameter of each user αk, the transmitted pilot power of each user Ppk and the transmitted pilot symbols

of each user sk. Hence, exploiting the pilot sequences orthogonality and knowing

additional information on the noise or on the channel, is possible to estimate the channel through the linear LS or the linear MMSE channel estimation methods.

3.2.1 Least square channel estimation

The goal of the channel least square estimator is to minimize the square distance between h, the actual channel, and ˆh, the estimated channel. The LS channel estimator for the k-th user can be defined as:

ˆ hksTk = ((αk q Ppk) H_(α k q Ppk)) −1 (αk q Ppk) H_Y(p) ₌ 1 αk q Ppk Y(p) ˆ hksTks ∗ k(sTks ∗ k) −1 = 1 αk q Ppk Y(p)s∗_k(sT_ks∗_k)−1 ˆ hk = 1 αk q Ppkτp Y(p)s∗_k = hk+ 1 αk q Ppkτp Ns∗_k (3.9)

where sk ∈ Cτp×1 is the vector of pilot symbols for user k and (sTks∗k) = τp. As

shown in equation (3.9), 1

αk√P_pkτp

Ns∗_k is the error term. It follow that the channel estimation error, ek is also normally distributed:

ek ∼ CN(0, Cek) where Cek = σ2 p α2 kPpkτp INr (3.10)

Rayleigh fading channel

When hk ∼ CN(0, Ck), the estimated channel ˆhk is a circular symmetric complex

normal distributed vector ˆhk ∼ CN(0, Rk), with Rk, E{ˆhkhˆ H k} = Ck+ Cek = Ck+ σ2 p α2 kPpkτp INr (3.11)

where INr is the identity matrix of size Nr×Nr. As it is shown in [5], the distribution

of the channel realization hk conditioned on the estimate ˆhkis normally distributed

as:

(hk|ˆhk)∼ Dkhˆk+ CN (0, Qk) (3.12)

where Dk , CkR−1k and Qk , Ck− CkR−1k Ck. Is important to note that the

circular complex CN (0, Q_k) term, that characterize the channel estimation error, can be thought as a zero mean estimation noise, which can be made small by sufficiently increasing the pilot power.

(18)

3.2.2 MMSE channel estimation

The goal of the linear MMSE channel estimator is to minimize the MSE between h, the actual channel, and ˆh, the estimated channel. Since we are exploiting the pilot orthogonality the channel of every user can be estimated by itself hence the estimated channel of the k-th user can be defined as:

ˆ hksTk = E{hksTk} + CHY(p)_h ksT_kC −1 Y(p)(Y (p)_{− E{Y})} _(3.13) substituting CH_Y(p)_h ksTk = αk q PpkτpChk and CY(p) = α 2 kPpkτpChk+ σ 2 pINr we have: ˆ hksTk = E{hk}sTk + αk q PpkτpChk(α 2 kPpkτpChk+ σ 2 pINr) −1 (Y(p)_{− E{Y}) (3.14)} At last we can obtain:

ˆ hk = E{hk} + Chk Chk + σ2 p α2 kPpkτp INr !−1 ·  h_k− E{h_k} + 1 αk q Ppkτp Ns∗_k   (3.15)

The MMSE channel estimation perform better than the LS channel estimation when the pilots signal-to-noise ratio (SNR) is low but it requires the prior knowledge of the channel covariance matrix Chk

Rayleigh fading channel

When hk∼ CN(0, Ck) and Ck= INr we can simplify the equation of the MMSE

channel estimator: ˆ hk= Ck Ck+ σ2 p α2 kPpkτp INr !−1 1 αk q Ppkτp Y(p)s∗_k = Ck Ck+ σ_p2 α2 kPpkτp INr !−1 h_k+ 1 αk q Ppkτp Ns∗_k   (3.16)

In this case we can observe that when Ck

Ck+ σ2 p α2 kPpkτp INr −1 = 1 the MMSE channel estimator coincide with the LS channel estimator, this means that if the pilot SNRk =

α2

kPpkτp

σ2

p is high enough, the LS and the MMSE channel estimation

(19)

The covariance matrix of the estimated channel using the MMSE estimator is given as Rk , E{ˆhkhˆ H k} = Ck Ck+ σ2 p α2 kPpkτp INr ! (3.17) As in the LS estimator, the distribution of the channel realization hk conditioned

on the estimate ˆhk is normally distributed as:

(hk|ˆhk)∼ Dkhˆk+ CN (0, Qk) (3.18)

(20)

Chapter 4 Receivers design

To introduce the different considered receivers we first define the structure of the received data signal at the base station. This signal can be written as:

y = αlhl q Plxl+ K X k=1,k6=l αkhk q Pkxk+ nd (4.1)

where Pk is the transmitted data power per symbols of the k-th mobile station and

xk is the respective data symbols, scaled such as |xk|2 = 1. K is the number of

mobile stations and, for each mobile station hk represent the normalized channel

obtained from the previously described models and αk account for the power of the

channel and include the effects of the path-loss and eventual shadowing. nd is the

AWGN noise with nd∼ CN(0, σd2) and σd2 is the noise power. We assume that that

the base station applies a linear receiver Gl ∈ C1xNr to the received data signal

in order to estimate the transmitted data symbols of user-l. In the case of perfect knowledge of the channel gains, the MSE of the received data symbols, averaged over the transmitted symbols and the thermal noise, is defined as [2]:

M SE(Gl, H) = Ex,nd{|Gly− xl| 2 } = = Glαlhl q Pl− 1 2 + K X k=1,k6=l Pk|Glαkhk| 2 + σ2_dGlGHl = = 1_{− α}l √ PlGlhl− αl √ PlhHl G H l + + Gl K X k=1 α2_kPkhkhHk + σ 2 dINr ! GH_l (4.2)

where H = [h1, . . . , hK]∈ CNrxK is the matrix collecting the channel gains for all

the K users.

In this thesis our hypothesis is the presence of channel estimation errors hence, in order to evaluate the linear receiver structure that minimizes the MSE in presence

(21)

of channel estimation errors, we need to express the MSE as a function of the receiver Gl and the estimated channel ˆH rather than the actual channel H. To

obtain this new MSE formula we take the (4.2) and we average over hk|ˆhk:

M SE(Gl, ˆH) = Ehk|ˆhk{MSE(Gl, H)} = = 1_{− α}l √ PlGlDlhl− αl √ PlhHl D H l G H l + + Gl K X k=1 α2_kPk(DkhkhHkD H k + Qk) + σ 2 dINr ! GH_l (4.3)

In this new expression of the MSE we can see that the matrices D and Q, illustrated in the previous chapter, introduce the effect of the channel estimation errors. With these two formulae we can evaluate the following receiver:

1. The naive single-user MMSE receiver minimize the (4.2) when K = 0; 2. The covariance based MMSE receiver minimize the (4.3) when K = 0; 3. The naive multi-user MMSE receiver minimize the (4.2);

4. The robust MMSE receiver minimize the (4.3);

4.1 Naive single-user MMSE receiver

The first receiver analyzed is the so called single-user naive MMSE receiver, this receiver aims to minimize the MSE when there are no channel estimation errors due to the noise on the pilot signal and when there is only one user in the cell (4.2). The receiver can be defined in this way:

Gnaive_l = αl q Plhˆ H l α2_lPlhˆlhˆ H l + σ 2 dINr −1 (4.4) For this receiver is possible to find a close formula for evaluating the MSE in a single-user system when Rk, Dk, Qk are diagonal, this happens when the channel

correlation matrix Ck is diagonal and when the channel is modeled with the Rayleigh fading channel h _{∼ CN(0, C) [7]. The following formula represent the} unconditional MSE: E(MSE) = d2lNr G(σ2_d, 1 + Nr) + prG(1 + Nr, 1 + Nr)− 1 + + b pr G(σ_d2, Nr) + prG(Nr, Nr)− 1 − 2d (prG(Nr, 1 + Nr)) + 1 (4.5) where p = α2 lP and G(x, y) , pr1e a prxE in y,_pra with Ein(n, z) , R∞ 1 e −zt tn dt that is

(22)

programming environments. It is important to note that q and d, with D = dI and

Q = qI, carry the dependency on the pilot power and p carries the dependency

on the data power in the formula (4.5). With this formula is possible to evaluate theoretically the performance of the receiver in function of the data power Pl

and the pilot power Pp and eventually find the best PDPR at the transmitter to

minimize the MSE of the data symbols.

4.2 Covariance based MMSE receiver

The covariance based MMSE receiver minimize the MSE when there are channel estimation errors due to the noise on the pilot signals and there is only one user in the cell (4.3). This receiver address to the channel estimation errors problem with the matrices Dk and Qk, second order statistic related to the channel correlation

matrix Ck, this two matrices carry information about the CSIR errors and their

structure depends from the chosen channel estimation method. This receiver exploits the channel correlation matrix Ck in order to partially reduce the

multi-user interference. The channel correlation matrix is a second order statistic of the channel and this receiver does not employ the actual channel estimates of the other users hk in order to reduce the multi-user interference hence it is not the MMSE

receiver for the multi-user case. This receiver was firstly described in [4] and is defined in this way:

GCMMSE_l = αl q Plhˆ H l D H l ·  α2_lP_l( D_lhˆ_lhˆ H l DHl + Ql | {z }

CSI errors compensation ) + K X k6=l α2_kPkCk+ σd2INr | {z } MU-MIMO interference plus noise   −1 (4.6)

For this receiver is possible to obtain a theoretical closed form for the unconditional MSE in the case of proper antenna spacing and when the channel is modeled with the Rayleigh fading channel hk ∼ CN(0, Ck) where Ck = INr. If the channel

correlation matrices are equal the identity matrix, it implies that Dk = dkINr and

(23)

symbols of the user-l can be written as: MSEl = bl eslrlbl _(b_l_{+ N}_r_s_l_r_l_)E_in_N_r_, bl slrl − slrl s2 lr2l + Nr eslrlbl _(b_l_{+ (1 + N}_r_)s_l_r_l_)E_in_{1 + N}_r_, bl slrl − slrl slrl − 2eslrlbl _N_r_E_in _{1 + N}_r_, bl slrl ! + 1 (4.7) where Ein(n, z) , R∞ 1 e −zt

tn dt is the standard exponential integral function,

com-monly available in numerical programming environments. With the theoretical formula for evaluating the unconditional MSE is possible to develop an algorithm that minimize the MSE in function of the PDPR [3].

4.3 Naive multi-user MMSE receiver

The naive multi-user MMSE receiver is an improved version of the naive single-user MMSE receiver that minimize the MSE (4.2) also in a multi-user system and not only in a single-user system like the previous receiver. This receiver to reduce the multi-user interference exploits the instantaneous channel estimates of all the users ˆhk while the single-user receiver would use only the channel estimate of the

user-l. As expected, this receiver has the same performances of the naive single-user MMSE receiver in a single user system while has far more better performance in a multi-user environment since it correctly address to the multi-user interference. This receiver is defined in this way:

Gnaive_l = αl q Plhˆ H l K X k=1 α2_kPk(ˆhkhˆ H k) + σ 2 dINr !−1 (4.8)

4.4 Robust MMSE receiver

The robust MMSE receiver is an improved version of the covariance based MMSE that address to its problem: to do not use the instantaneous channel estimates of all the users to compensate for the multi-user interference. This new receiver is able to minimize the MSE in presence of both CSI errors and multi-user interference. This receiver, firstly described in [2] and it is defined in this way:

GIMMSE_l = αl q Plhˆ H l D H l K X k=1 α2_kPk(Dkhˆkhˆ H kD H k + Qk) + σ 2 dINr !−1 (4.9)

(24)

The receiver exploit the instantaneous channel estimations of all the users ˆhk in

order to reduce the multi-user interference and use Dk and Qk of all the users to

detect and compensate the CSI errors on the channel estimates of every user. If we assume to have perfect CSIR, i.e setting to zero the noise on the pilots such as σ2_p = 0, we have Rk = Ck and this implies that Dk , CkR−1k = INr and

Q_k _{, C}k− CkR−1k Ck = 0. If we set Dk= INr and Qk = 0 in the robust MMSE

receiver we obtain: GIMMSE_l = αl q Plhˆ H l K X k=1 α2_kPk(ˆhkhˆ H k ) + σ 2 dINr !−1 (4.10) the same equation of the naive multi-user MMSE receiver. This assure us that the new receiver will have the same performance of the naive multi-user MMSE receiver when we assume perfect channel estimation while we expect it to have better performance when we introduce errors in the channel estimation.

For this receiver is possible to obtain the theoretical expression of the SINR when we assume that the channel hk∈ CNrx1 is a complex Gaussian channel with

zero mean and NrxNr covariance matrix Ck, where each Ck is Hermitian. The

form of the expression depends on whether we consider a scalar correlation matrix or a generic one. At the moment of writing, we are still working on the generic form of the expression so, in this thesis, we only show the particular form expression in the case of the scalar correlation matrix while for the generic case we only show some simulation results of the formula applied to the sum-rate algorithm.

4.4.1 Scalar correlation matrix

In this case, we consider Ck= ckI in particular with c = 1, this case is representative

when there is no correlation between the different path between each mobile station and the base station and every path has the same power as for the Rayleigh fading channel. The first step is to evaluate the theoretical expression for finding the asymptotic average of the SINR of this receiver [2]:

K X k=1 α2_kPkqk+ σd2 = Nrα2lPldl γ_l − K X k=1,k6=l α2 kPkdk 1 + γ_lα2kPkdk α2 lPldl (4.11)

In this equation, our objective is to find γ_l that is the SINR of the l-th user. Is important to note that this equation is valid only when Rk, Dk, Qk are diagonal,

so that Rk = rkINr, Dk = dkINr and Qk = qkINr. In this expression, the SINR

of the l-th user is an implicit function of the pilot powers of all the users Ppk and

of the data powers of all the users Pk. This formula has a fundamental role for

(25)

optimal theoretical PDPR of each user in order to maximize the sum-rate or the fairness. These formulae use the random matrix theory with the assumption that Nr, K → inf with K/Nr fixed hence its accuracy it’s affected by the number of

users and the number of antennas and the approximation become more accurate with the increase of the number of antennas and users.

4.4.2 Non-scalar correlation matrix

For this case, we will produce only simulation results since the theoretical formula is still to be published. This new formula has the same importance of the previous formula since it gives the SINR in function of the pilot powers and of the data powers and makes possible to create an algorithm that finds the optimal theoretical PDPR of each user in order to maximize the sum-rate or the fairness.

(26)

Chapter 5 Receivers Test

In this chapter we expose the results of We tested the newly proposed robust MMSE receiver with different channel models in order to prove its performance in comparison with the other presented receivers used as a reference. For evaluating the sum-spectral efficiency SSE =PK

k=1log2(1 + SINR) we define the SINR of the considered user-l as:

SINRl= α2_lPl GlDl ˆ hl 2 Pk k=1,k6=lα2kPk GlDk ˆ hk 2 +PK k=1α2kPkGlQkG H l + σd2GlGHl (5.1)

This equation is not the common defined SINR: SINRl = α2 lPl|Glhl| 2 Pk k=1,k6=lα2kPk|Glhk|2+ σd2GlGHl (5.2)

This SINR equation is the one estimated from the base station after the receiver when we also consider the effects of the channel estimation errors, this SINR definition can be found in [2].

The MSE of the considered user l is defined as E{|xl− ˆxl|2} where xl is the

transmitted data symbol of the user-l and ˆxl = Gly is the received data symbol of

the user-l after the receiver filter and y is the data signal at the base station (4.1).

5.1 Gaussian Channel with Rayleigh Fading

The first set of tests are performed on a simulation environment based on the Gaussian channel with small scale Rayleigh fading, in this environment hk ∼

CN (0, Ck). The different mobile stations are uniformly randomly distributed inside

(27)

the distance and defined as:

Lk = 20 log10( 4π

λ ) + 10β log10(dk)[dB] (5.3)

where λ is the wavelength, β is the path-loss exponent and dk is the distance of the

k-th user to the base station. In these environments, we compare the four receivers previously described in terms of MSE and sum-spectral efficiency.

5.1.1 Single-user scenario

In this first scenario we evaluate the performance of the different receivers in the single-user case, our aim is to show the importance of the PDPR tuning to maximize the performance in terms of both MSE and sum-spectral efficiency also we want to prove that the robust MMSE receiver and the covariance MMSE receiver have better performance than the two naive receiver. In the following table are summarized the main parameters used for running the MATLAB simulations.

Parameter Value

Number of receivers antennas Nr = 10, 70

7 Number of users K = 1

Users path-loss Lk = 20 log10(4πλ) + 10β log10(dk)

Radius of the cell r = 5000

Numbers of pilots and data symbols τp = 1; τd = 23

Power budget per mobile station Ptot = τpPp+ τdP = 250mW

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 −20 −18 −16 −14 −12 −10 −8 −6 −4 −2

Pilot power/Total Power

MSE[dB]

Robust Covariance Nr=70

Naive SU Naive MU Nr=10

Figure 5.1: Mean squared error over pilot-to-data power ratio in Rayleigh fading with a user

(28)

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 1 2 3 4 5 6 7

SSE[bit/s/Hz]

Robust Covariance Nr=70 Naive SU Naive MU Nr=10

Figure 5.2: Sum-spectral efficiency over pilot-to-data power ratio in Rayleigh fading with a user −20 −15 −10 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 SE[dB] CDF Robust Covariance Nr=70 Naive SU Naive MU Nr=10

(29)

The simulation results show the importance of the PDPR tuning in order to obtain the best performance from the receivers. The second notable results that we can see is that the two CSI aware receivers, the covariance based MMSE receiver (4.6) and the robust MMSE receiver (4.9), performs better in terms of MSE than the two naive receivers, single-user (4.4) and multi-user (4.8), this is due to the fact that in the single-user scenario there are errors on the channel estimation caused by the noise on the pilot signals hence the CSI aware receivers have an advantage in reducing the channel estimation errors, their MSE gain increase with the increase of the number of antennas. Also, in the single-user scenario, there is no need of multi-user interference suppression method hence the two naive receivers and the two CSI aware receivers have, respectively, the same performance.

5.1.2 Multi-user scenario

In the second scenario, we switch from the single-user scenario to the multi-user scenario hence there is more than one user. The mobile stations are uniformly distributed inside the cell of the given radius. Since we are in the multi-user case the receivers will have to address to the multi-user interference. As shown in the previous section the four receivers use different approaches to control this interference: the naive single-user MMSE receiver (4.4) doesn’t account at all for the multi-user interference, the covariance based MMSE receiver (4.6) uses the channel correlation matrices to address to the multi-user interference and both the multi-user MMSE receiver and the robust MMSE receivers use the instantaneous channel estimates of all the users to compensate the multi-user interference. Since the single-user MMSE receiver doesn’t account for the multi-user interference it has been excluded from the figures due to its lack of performance when compared to the other receivers; hence we compare the performance of the three receivers that manage to reduce the multi-user interference. The Theoretic results in the sum-spectral efficiency figure are the theoretical results obtained from the use of the implicit formula of the SINR of the robust MMSE receiver (4.11) applied to the sum-spectral efficiency formulae. In the following table are summarized the main parameters used for running the MATLAB simulations.

Parameter Value

7 Number of users K = 7

Users path-loss Lk = 20 log10(4πλ) + 10β log10(dk)

Radius of the cell r = 5000

(30)

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 −20 −15 −10 −5 0

MSE[dB] _Robust _Covariance _Nr=70

Naive MU Nr=10

Figure 5.4: Mean squared error over pilot-to-data power ratio in Rayleigh fading with 7 users 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0 5 10 15 20 25 30 35 40 45 50

SSE[bit/s/Hz]

Robust Theoric Nr=70

Covariance Naive MU Nr=10

Figure 5.5: Sum-spectral efficiency over pilot-to-data power ratio in Rayleigh fading with 7 users

(31)

−20 −15 −10 −5 0 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 SE[dB] CDF Robust Covariance Nr=70 Naive MU Nr=10

Figure 5.6: Squared error CDF in Rayleigh fading with 7 users

In these figures, we can clearly see that the two receivers that use the instan-taneous channel estimates for compensating for the multi-user interference have better performances than the one that uses only the channel’s correlation matrices. The CSI errors compensation made by the robust MMSE doesn’t provide with much gain over the multi-user naive MMSE receiver due to the simplicity of the channel, in a more complex channel we expect a stronger performances difference. The theoretical approximation of the sum-spectral efficiency of the robust MMSE receiver obtained using the theoretical approximation of the SINR (4.11) becomes, as expected, more accurate with the increase of the number of antennas.

(32)

5.2 Winner II Channel

To further test the performances of the new robust MMSE receiver in a more realist scenario, we decided to use a more complex and accurate channel model. This channel model, called Winner II, it is built from the 3GPP specifications and it is described in Chapter 3. From this channel model, we will use two particular scenarios: Urban macro and Urban micro. In these simulations we use the naive multi-user MMSE receiver as a comparison with our proposed receiver, the others two analyzed receivers, naive single-user MMSE and covariance MMSE, have been omitted because of their very low performances in comparison with the two more advanced receivers in these two environments. In the following table are summarized the main parameters used for running the MATLAB simulations. All the documentation of the MATLAB implementation of the Winner II channel model can be found here [11].

Parameter Value

Number of users K = 7

Users path loss Evaluated from the model

Power budget per mobile station Ptot = τpPp+ τdP = 250mW

Environment Urban Macro and Urban Micro

In the figures we can see that, while in the Gaussian channel the CSI errors compensation of the robust MMSE receiver provide almost no improvement on the performances, in this more complex channel model it is important especially when the pilots power is low and so the channel estimation has more errors; in this case the robust MMSE receiver perform far more better than the naive multi-user receiver both in terms of MSE and sum-spectral efficiency. The performance gain, especially in terms of MSE, seems to increase with the increase of the number of antennas.

Is also interesting to notice that the formulae to obtain the D and Q matrices that the robust MMSE receiver exploit to compensate the channel estimation errors are derived under the assumption that the channel is Gaussian with zero mean, this means that the D and Q matrices are not made for this model and this new receiver is not the real MMSE receiver in the Winner II model. Nonetheless, the robust MMSE receiver performs better than the naive multi-user MMSE receiver both in terms of MSE and sum-spectral efficiency; even if the D and Q matrices are not perfectly tuned to the specific channel model they still help the new receiver to compensate the channel estimation errors.

(33)

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 −20

−15 −10 −5

MSE[dB] Robust Nr=70 Naive MU Nr=10 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 10 15 20 25 30 35 40 45

SSE[bit/s/Hz] −22 −20 −18 −16 −14 −12 −10 −8 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.91 SE[dB] CDF

(34)

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 −20

−15 −10 −5

MSE[dB] Robust Nr=70 Naive MU Nr=10 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 10 15 20 25 30 35 40 45 50

SSE[bit/s/Hz] −22 −20 −18 −16 −14 −12 −10 −8 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.91 SE[dB] CDF

(35)

Chapter 6 Pilot Power Optimization

One of the main objectives of this master thesis project is to improve the sum-rate (or the sum-spectral efficiency) and the fairness in a MU-MIMO system when exploiting the robust MMSE receiver. In this chapter we illustrate in details our optimization tasks and the solutions that we found.

For our optimization task we define the two objective function: max

Pp

K

X

k=1

log₂(1 + SINRk) when Ppk + Pk= Ptotk (6.1)

max Pp,Pd

min

k log2(1 + SINRk) when Ppk + Pk = Ptotk (6.2)

where Pp = [Pp1, Pp2, . . . , PpK]∈ R

1×K

+ is the vector of the pilot symbols powers for all the stations and K is the number of users that transmit at the same time. In the first objective function (6.1) there are K variables to tune since the sum between pilots power and data power for each station is fixed and equal to Ptot,

in this case our aim is to maximize the sum-rate without considering any form of fairness so, in theory, could be possible for a mobile station to put all the transmitted power on the pilot signal.

In the second objective function (6.2) there are K variables to tune as since the sum between pilots power and data power for each station is fixed and equal to Ptot, in this case we will try to maximize the fairness of the system, the fairness is

obtained by maximizing the minimum rate among all the user equipment.

One fundamental key of our optimization task is the implicit function (4.11) for evaluating the SINR of the robust MMSE receiver, without this formula wouldn’t be possible to evaluate the SINR theoretically and find the best combination of PDPR for every user in order to succeed in our two objectives. As stated in chapter 4 this formula is only valid when the channel correlation matrix is scalar and when the channel has zero mean; we are currently working on an extended version of the

(36)

formula that will cover all the possible channel correlation matrix. In our work we focus on the two different channel correlation matrices:

1. The channel correlation matrix is scalar C = cI 2. The channel correlation matrix is non-scalar C_{6= cI}

As stated before, for the non-scalar correlation matrix case we provide only some initial results and not the theoretical SINR formula that we plan to provide in a future paper.

6.1 Fairness

The fairness is an important aspect in most performances studies especially in distributed systems where the resources are shared between the users; a system like ours, a single-cell wireless system with one base station and several mobile stations. Often the fairness it’s specified only qualitatively but, in our case, it’s important to find a quantitative method to measure the fairness in the system in order to compare the different algorithms that we employ. In our experiments, we decided to measure the fairness using the with the Jain’s fairness index. This index is defined as: (PK k=1log2(1 + SIN Rk))2 K_·PK k=1(log2(1 + SIN Rk))2 (6.3) With this index, the results can vary between 1

K, the worst case, and 1, the best

case and it is maximum when all the users achieve the same rate. The Jain’s fairness index was firstly defined [15]. As stated before we maximize the fairness by trying to reach the max-min fairness that happens when the minimum rate among all the users is maximized [13][17].

6.2 Optimal Power Allocation Algorithm

The first approach to the two optimization problems is to solve one flaw of the SINR theoretic formula of the robust MMSE: this formula is in a semi-closed form hence is difficult to find a simple expression for its derivative. For this reason, we decided to avoid gradient oriented minimization methods and to use a derivative-free optimization method. Between the different minimization methods we opted to use the Nelder-Mead’s simplex method, this method is simple to understand and to implement and uses only basic operation to find the minimum of a given function, it’s also computationally efficient when the number of the minimization variables in low [14].

(37)

6.2.1 Nelder-Mead’s simplex method

The simplex method of Nelder and Mead is an unconstrained minimization method that performs a search in K-dimensional space using heuristic ideas. It is also known as the nonlinear simplex. Its main strengths are that it requires no derivatives to be computed and that it does not require the objective function to be smooth. The weakness of this method is that it is not very efficient, particularly for problems with more than about 10 design variables; above this number of variables convergence becomes increasingly difficult and the number of the iterations necessary for the convergence increases.

A simplex is a structure in K-dimensional space formed by K+1 points that are not in the same plane, hence a line segment is a 1-dimensional simplex, a triangle is a 2-dimensional simplex and a tetrahedron forms a simplex in 3-dimensional space. The simplex, a generalized triangle in K-dimension, is also called a hyper-tetrahedron. The Nelder-Mead algorithm starts with a simplex (K+1 sets of design variables x) and then modifies the simplex at each iteration using four simple operations. The sequence of operations to be performed is chosen based on the relative values of the function, that we want to minimize, at each of the generated points. After generating the initial simplex, we have to evaluate the function at each of its vertices in order to identify three key points: the points associated with the highest ("worst") the second highest ("lousy") and the lowest ("best") values of the objective. The Nelder-Mead algorithm performs five main operations to the simplex: reflection, expansion, outside contraction, inside contraction and shrinking. Each of these operations generates a new point and the sequence of operations performed in one iteration depends on the value of the function at the new points relative to the value of the function in the others key points. Let xw, xl

and xb denote the worst, the second worst and best points, respectively, among all

K+1 points of the simplex.

The first operation is to take the average of all the point excluding the worst: xa= 1 K K+1 X k=1,k6=w xk (6.4)

with the average we can perform the five main operation listen below:

1. Reflection: xr = xa+ α(xa− xw) where α is the reflection factor and is

usually set to 1

2. Expansion: xe = xr+ γ(xr − xa) where γ is the expansion factor and is

usually set to 1

3. Inside contraction: xc= xa− β(xa− xw) where β is the contraction factor

(38)

Initialize K-simplex, evaluate K+1 point

Rank vertices: best, lousy and worst

Is the stop criterium met? Compute the average

Reflect _{better than best point?}Is reflected point Expand

The best point is the solution

Is expanded point better than reflected point?

Accept

reflection

Accept

expansion

Is reflected point worse than worst point? Perform

inside contraction

Is reflected point worse than the worst point?

Accept

contraction Shrink

Is reflected point worse than lousy point?

Accept

reflection

Perform

outisde contraction

Is new point worse than the worst point?

Accept contraction Shrink no yes no yes yes no no yes no yes _no yes no yes

Figure 6.1: Nelder-Mead flow-chart

4. Outside contraction: xo = xa+ β(xa− xw) where β is the contraction factor

and is usually set to 0.5

5. Shrink: _{∀k = 1, . . . , K, k 6= b, x}k = xb+ ρ(xk− xb) where ρ is the shrink

factor and is usually set to 0.5

We can define various convergence criterion:

1. The difference of the values of the best and the worst points is less than a given tolerance ε1

(fw− fb) < (ε1) (6.5)

2. The size of the simplex is less than a given tolerance ε2 s =

K

X

k=1

(39)

3. The standard deviation is less than a given tolerance ε σ = v u u t PK+1 k=1(fi− ¯f )2 K + 1 < ε (6.7)

In the flowchart 6.1 are summarized all the iterations of the Nelder-Mead’s simplex method.

6.2.2 The Nelder-Mead Algorithm

This method it’s not perfect for our maximization task since it is built for un-constrained optimization, to avoid that the algorithm explores points outside the constraints we include checks and operations that avoid that the points of the simplex, that are the pilots powers of each user, assume values outside the interval [0, Ptot]. The two functions that are used as input in the algorithm are:

1. ₋PK

k=1log2(1 + SINRk) for the sum-rate maximization task

2. _{− min}klog2(1 + SINRk) when for the fairness maximization task

The minus is necessary since the Nelder-Mead simplex is a minimization method and our aim is to maximize the two functions.

When the convergence condition of the algorithm is met we have the optimal set of pilots powers and the expected sum-rate. In our algorithm, we decided to use the (6.7) as the termination condition that gives more reliable results compared to the other two termination conditions especially when we consider the second objective function (6.2). The algorithm is illustrated in 1.

(40)

Algorithm 1 Nelder-Mead’s simplex method Input: Initial guess, x0

1: Random generate n+1 points between [0, Ptot]inordertocreateasimplex

2: repeat

3: Identify the highest (Xw: worst), second highest (Xl: lousy) and

lowest (Xb: best) value points with function values f (Xw), f (Xl), f (Xb)

respectively

4: Evaluate Xa, the average of the points in the simplex excluding Xw.

5: Perform reflection to obtain Xr, evaluate f (Xr).

6: if f (Xr) < f (Xb) then

7: Perform expansion to obtain Xe, evaluate f (Xe)

8: if f (Xe) < f (Xr) then 9: Xw ← Xe, f (Xw)← f(Xe) (Accept expansion) 10: else 11: Xw ← Xr, f (Xw)← f(Xr) (Accept reflection) 12: end if 13: else if f (Xr)≤ f(Xl) then 14: Xw ← Xr, f (Xw)← f(Xr) (Accept reflection) 15: else 16: if f (Xr) > f (Xw) then

17: Perform an inside contraction and evaluate f (Xc)

18: if f (Xc) < f (Xw) then

19: Xw ← Xc, f (Xw)← f(Xc) (Accept inside contraction)

20: else

21: Shrink the simplex

22: end if

23: else

24: Perform an outside contraction and evaluate f (Xo)

25: if f (Xo)≤ f(Xw) then

26: Xw ← Xo, f (Xw)← f(Xo) (Accept outside contraction)

27: else

28: Shrink the simplex

29: end if

30: end if

31: end if

32: Random generate the points of Xw that are not in the interval [0, Ptot]

33: until r PK+1 k=1(fi− ¯f ) 2 K+1 < ε Output: Optimum, x?

Performance analysis and power allocation in MU-MIMO systems with imperfect channel state information