Applied Science, Faculty of
Engineering, School of (Okanagan)
DSpace
UBCO
Alwazani, Hibatallah Asem M.N
Master of Applied Science - MASc
University of British Columbia
Distributed intelligent reflecting surfaces (IRS)s deployed in wireless systems promise improved system performance, while spurring on diverse challenges such as channel estimation (CE), average signal-to-interference and noise (SINR) analysis, and IRS-user association. This is due to the passive nature of the reflecting elements and large multi-user system dimensions. In light of these challenges, we undertake the CE problem for the distributed IRSs-assisted multi-user MISO system. An optimal CE protocol requiring relatively low training overhead is developed using Bayesian techniques under the practical assumption that the base-station (BS)-IRSs channels are dominated by the line-of-sight (LoS) components. Simulation results corroborate the normalized MSE (NMSE) analysis and establish the advantage of the proposed protocol as compared to a benchmark scheme in terms of training overhead. In addition, we derive the average SINR for the distributed IRSs system with perfect channel state information (CSI) using a sub-optimal IRS reflecting configuration. After that, a successive refinement method is developed to find IRS-user association for the formulated max-min SINR problem which motivates user-fairness. Simulations validate the average SINR analysis while confirming the superiority of deploying a distributed IRSs scheme as well as an optimized IRS-user association as opposed to a centralized IRS deployment and random assignment.
Channel Estimation and UserAssociation for Distributed IntelligentReflecting Surfaces Assisted MISOCommunicationsbyHibatallah Asem M.N AlwazaniB.Sc. Hons., Effat University, 2019A THESIS SUBMITTED IN PARTIAL FULFILLMENT OFTHE REQUIREMENTS FOR THE DEGREE OFMASTER OF APPLIED SCIENCEinTHE COLLEGE OF GRADUATE STUDIES(Electrical Engineering)THE UNIVERSITY OF BRITISH COLUMBIA(Okanagan)March 2021© Hibatallah Asem M.N Alwazani, 2021The following individuals certify that they have read, and recommendto the College of Graduate Studies for acceptance, a thesis/dissertation en-titled:Channel Estimation and User Association for Distributed IntelligentReflecting Surfaces Assisted MISO Communicationssubmitted by Hibatallah Asem M.N Alwazani in partial fulfilment ofthe requirements of the degree of Master of Applied ScienceProf. Anas Chaaban, School of EngineeringSupervisorProf. Hossain Jahangir, School of EngineeringSupervisory Committee MemberProf. Loic Markley, School of EngineeringSupervisory Committee MemberProf. Abbas Milani, School of EngineeringUniversity ExamineriiAbstractDistributed intelligent reflecting surfaces (IRS)s deployed in wireless sys-tems promise improved system performance, while spurring on diverse chal-lenges such as channel estimation (CE), average signal-to-interference andnoise (SINR) analysis, and IRS-user association. This is due to the passivenature of the reflecting elements and large multi-user system dimensions. Inlight of these challenges, we undertake the CE problem for the distributedIRSs-assisted multi-user MISO system. An optimal CE protocol requiringrelatively low training overhead is developed using Bayesian techniques un-der the practical assumption that the base-station (BS)-IRSs channels aredominated by the line-of-sight (LoS) components. Simulation results cor-roborate the normalized MSE (NMSE) analysis and establish the advantageof the proposed protocol as compared to a benchmark scheme in terms oftraining overhead. In addition, we derive the average SINR for the dis-tributed IRSs system with perfect channel state information (CSI) usinga sub-optimal IRS reflecting configuration. After that, a successive refine-ment method is developed to find IRS-user association for the formulatedmax-min SINR problem which motivates user-fairness. Simulations validatethe average SINR analysis while confirming the superiority of deploying adistributed IRSs scheme as well as an optimized IRS-user association asopposed to a centralized IRS deployment and random assignment.iiiLay SummaryFor a traditional wireless communication system, the transmitter sendsinformation carried by electromagnetic (EM) waves across the radio environ-ment, in free-space, to a particular receiver. The transmitter and receiver arecontrolled and optimized for this transmission, while the radio environmentis largely random and uncontrolled. To meet the ever-increasing demandsof communications, engineers converged on the idea of deploying distributedintelligent reflecting surfaces (IRS)s in the radio environment as a way tooptimize the transmission over it. What the distributed IRSs do essentiallyis add more degrees of freedom into the wireless system presented as re-configurable entities that enable the radio environment to be smart. Wefocus on the updated wireless system which includes the distributed IRSs,then we provide the theoretical foundations for this distributed IRSs assistedwireless system. Moreover, we operate the distributed IRSs by pairing eachIRS with a receiver then re-configuring that IRS to steer the informationsignals to the intended receiver. Finally, we formulate a conclusion on theusage of distributed IRSs in the wireless system based on the simulationresults.ivPrefaceProf. Anas and Dr. Qurrat Ul Ain (Annie) supervised this research.Moreover, both have reviewed the thesis and validated the theoretical deriva-tions and numerical results. They have also authored with me the followingpublications. This thesis extends the system model defined in Chapter 2from the journal paper titled (Intelligent Reflecting Surface-Assisted Multi-User MISO Communication: Channel Estimation and Beamforming Design)and published in 2020 IEEE Open Journal of the Communication Society(OJCOM) [1]. Chapter 3 of this thesis is in the conference paper titled(Channel Estimation for Distributed Intelligent Reflecting Surfaces AssistedMulti-User MISO Systems) and published during 2020 in IEEE GLOBE-COM [2]. A journal paper version of Chapter 4 along with some of theother chapters in the thesis is prepared for submission at a prestigious IEEEjournal such as IEEE Transactions in Wireless Communication (TWC).vTable of ContentsAbstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iiiLay Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ivPreface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vTable of Contents . . . . . . . . . . . . . . . . . . . . . . . . . . . viList of Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ixList of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xList of Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . xiiList of Abbreviations . . . . . . . . . . . . . . . . . . . . . . . . xiiiAcknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . xvDedication . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xviChapter 1: Introduction . . . . . . . . . . . . . . . . . . . . . . . 11.1 Beyond 5G and towards 6G . . . . . . . . . . . . . . . . . . . 11.2 Smart Radio Environments . . . . . . . . . . . . . . . . . . . 21.3 What are Intelligent Reflecting Surfaces? . . . . . . . . . . . . 21.3.1 Architecture . . . . . . . . . . . . . . . . . . . . . . . 31.3.2 Anomalous vs Specular Reflection . . . . . . . . . . . 41.4 State of the Art . . . . . . . . . . . . . . . . . . . . . . . . . . 41.4.1 Relays vs. Intelligent Reflecting Surfaces . . . . . . . . 51.4.2 Passive Beamforming . . . . . . . . . . . . . . . . . . 51.4.3 Explicit Channel Estimation . . . . . . . . . . . . . . 61.4.4 Distributed Intelligent Reflecting Surfaces: An Exten-sion . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7viTABLE OF CONTENTS1.5 Problem Statement . . . . . . . . . . . . . . . . . . . . . . . . 71.6 Objectives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81.7 Contributions . . . . . . . . . . . . . . . . . . . . . . . . . . . 81.8 Outline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9Chapter 2: System Model and Preliminaries . . . . . . . . . . 102.1 Transmission Mode . . . . . . . . . . . . . . . . . . . . . . . . 112.2 Fading Direct and IRS-User Channels . . . . . . . . . . . . . 122.2.1 Spatial Correlation . . . . . . . . . . . . . . . . . . . . 152.2.2 Channel Estimation . . . . . . . . . . . . . . . . . . . 152.3 BS-IRS l Channel . . . . . . . . . . . . . . . . . . . . . . . . . 162.4 Maximum Ratio Transmission Precoding . . . . . . . . . . . . 182.5 Downlink Transmission . . . . . . . . . . . . . . . . . . . . . 192.6 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19Chapter 3: Channel Estimation . . . . . . . . . . . . . . . . . . 213.1 Proposed Protocol . . . . . . . . . . . . . . . . . . . . . . . . 223.2 Noise Variance Minimization . . . . . . . . . . . . . . . . . . 243.3 Minimum Mean Squared Error Estimation . . . . . . . . . . . 263.4 Normalized Mean Squared Error Analysis . . . . . . . . . . . 273.5 Benchmark protocol . . . . . . . . . . . . . . . . . . . . . . . 293.6 Simulations and Discussion . . . . . . . . . . . . . . . . . . . 30Chapter 4: Average SINR and IRS-User Association . . . . . 354.1 Downlink Transmission . . . . . . . . . . . . . . . . . . . . . 364.2 Average SINR . . . . . . . . . . . . . . . . . . . . . . . . . . . 414.2.1 Performance Under Maximum Ratio Transmission . . 434.2.2 Main Results . . . . . . . . . . . . . . . . . . . . . . . 444.3 IRS-User Association Optimization Problem . . . . . . . . . 494.3.1 Exhaustive Search . . . . . . . . . . . . . . . . . . . . 504.3.2 Successive Refinement Algorithm . . . . . . . . . . . . 504.4 Simulations and Discussion . . . . . . . . . . . . . . . . . . . 51Chapter 5: Conclusion . . . . . . . . . . . . . . . . . . . . . . . . 605.1 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 605.2 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 605.3 Limitations of the Study . . . . . . . . . . . . . . . . . . . . . 615.4 Future Directions . . . . . . . . . . . . . . . . . . . . . . . . . 61Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63viiTABLE OF CONTENTSAppendix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68Appendix A: Proof of Theorem 3.1 . . . . . . . . . . . . . . . . . . 69Appendix B: Proof of Theorem 3.2 . . . . . . . . . . . . . . . . . . 71Appendix C: Proof of Lemma 4.3 . . . . . . . . . . . . . . . . . . . 73Appendix D: Proof of Lemma 4.6 . . . . . . . . . . . . . . . . . . . 79viiiList of TablesTable 3.1 Estimation accuracy per unit time 1NMSE×τC (s−1) . . 33Table 4.1 Simulation parameters. . . . . . . . . . . . . . . . . . . 43ixList of FiguresFigure 1.1 IRS passive beamforming . . . . . . . . . . . . . . . . 3Figure 1.2 Anomalous vs. specular reflection . . . . . . . . . . . 4Figure 2.1 Distributed IRSs MISO system model in uplink anddownlink. . . . . . . . . . . . . . . . . . . . . . . . . . 10Figure 2.2 Time Division Duplexing (TDD) vs Frequency Divi-sion Duplexing (FDD) transmission . . . . . . . . . . 12Figure 2.3 Rayleigh fading with no LoS path . . . . . . . . . . . 13Figure 2.4 Channel coherence interval . . . . . . . . . . . . . . . 14Figure 2.5 LoS BS-IRS l channel H1,l visualization . . . . . . . . 17Figure 3.1 Distributed Intelligent Reflecting Surfaces multi-userMISO system uplink transmission. . . . . . . . . . . . 21Figure 3.2 Normalized Mean Square Error vs noise . . . . . . . . 31Figure 3.3 NMSE vs path loss factor for direct channel. . . . . 32Figure 3.4 NMSE vs path loss factor for IRS assisted channels . 32Figure 3.5 Comparison of proposed and benchmark protocols. . 33Figure 4.1 Distributed IRSs MISO system model. Red lines arefor downlink transmission. . . . . . . . . . . . . . . . 36Figure 4.2 Maximizing a squared norm with unit-modulus con-straints. . . . . . . . . . . . . . . . . . . . . . . . . . . 40Figure 4.3 Tightness of approximation for γ¯k . . . . . . . . . . . 42Figure 4.4 Upper and lower bounds of γ¯k . . . . . . . . . . . . . 48Figure 4.5 No Association Set-up . . . . . . . . . . . . . . . . . . 52Figure 4.6 Nearest Rule Association . . . . . . . . . . . . . . . . 53Figure 4.7 Successive Refinement Association . . . . . . . . . . 53Figure 4.8 Minimum user SINR against N . . . . . . . . . . . . 54Figure 4.9 Minimum user SINR against L . . . . . . . . . . . . . 55Figure 4.10 Minimum user SINR for centralized and distributedIRSs scenarios against N . . . . . . . . . . . . . . . . 56xLIST OF FIGURESFigure 4.11 Distributed IRSs with L = 16 deployment scenario . . 57Figure 4.12 Centralized IRSs with L = 16 deployment scenario . . 57Figure 4.13 Minimum user SINR vs number of iterations in SRalgorithm . . . . . . . . . . . . . . . . . . . . . . . . . 58Figure 4.14 Number of iterations to converge vs L . . . . . . . . . 58xiList of Notation(·)H the Hermetian of a complex vector or matrix.(·)† the pseudo inverse of a matrix.∠· phase of a complex number for vectors it istaken element-wise.〈·, ·〉 the inner product of two vectors.E[·] expected value of random variables.X a matrix is written in bold upper-case lettersor symbols.x a vector is written in bold lower-case lettersor symbols. Hadamard product.⊗ Kronecker product.tr(·) trace of a matrix.x ∼ CN (µ, σ2) x is a circularly symmetric complex Gaussianwith mean µ and variance σ2.x∗ the conjugate of a complex number x.| · | the absolute value of a number, for vectors itis taken element-wise.|| · || the Euclidean norm of a vector.x ∈ CM×1 x is a complex M -dimensional vector.diag(·) creates a diagonal matrix out of a vector.Var(·) variance of a random variable.vec(·) creates a vector out of a diagonal matrix.xiiList of AbbreviationsAoA Angle-of-Arrival.AoD Angle-of-Departure.BS Base-station.CE Channel estimation.CSI Channel state information.DFT Discrete Fourier transform.EM Electromagnetic.IRS Intelligent reflecting surfaces.LoS Line-of-sight.LS Least squares.MIMO Multiple-input multiple-output.MISO Multiple-input single-output.MMSE Minimum mean square error.MRT Maximum Ratio Transmission.NLoS Non line-of-sight.NMSE Normalized mean square error.xiiiList of AbbreviationsSINR Signal-to-interference and noise ratio.SNR Signal-to-noise ratio.TDD Time-division duplexing.ULA Uniform linear array.UPA Uniform planar array.xivAcknowledgementsTo Prof. Anas and Dr. Annie, each of your presence has made this re-search roller-coaster full-filling, meaningful, and forever sketched into mem-ory. Not only your incomparable guidance and your incredible insight, butalso your understanding and patient natures in the transfer and teaching ofthis knowledge proved invaluable to a student stepping foot into the onceesoteric research community. With such a prime view into what it is to beexcellent researchers, dedicated teachers, and patient supervisors, my list ofrole models has definitely been updated.To Prof. Jahangir, Prof. Markley, and Prof. Milani, I am doubly gratefulto you as committee members and course instructors, where key conceptstaught to me during the past year were crucial for building my researchfoundations. I am also thankful for your time and constructive feedback.I am thankful to the wonderful friends I gained here at UBCO whohave helped me during my research, special thanks to Bayan Alnahhas forfirst introducing this exciting subject to me. Also huge thanks to SarahBahanshal, my dearest friend and roommate who patiently listened to myresearch problems and always cheered me up. To Kelowna ducks everywhere,whenever I felt down I went to Okanagan lake and your presence made mefeel lighter and put things into perspective, thank you.I am extremely thankful to my mom whose prayers are the reason Iapproached this capacity in knowledge. All such strides former and future,I channel to you. I am also grateful towards uncle Hussain, whose supportcannot be measured. Finally, I am thankful to my sisters and brothers whowere always there for me and have given me motivation to pursue higherstudies.xvDedicationTo my little niece and nephews, I dedicate this thesis to you.xviChapter 1IntroductionThere are two uncannily accurate and related trends that trace out themeteoric pace of technological evolution in the past decades: Moore’s lawand Cooper’s law. The former, extrapolated by inventor and engineer Gor-don Moore in 1965, is related to the doubling of transistors per chip everytwo years, while the latter, discovered by cell-phone inventor Martin Cooperis about the doubling of data rates per link every two years. Recently,Moore’s law, an exponentially increasing trend, has hit a plateau sendingshock waves across the technological world [3]. Since these laws are depen-dent to some extent, communication engineers must discover cost-effectiveways to prolong the inevitable plateau of Cooper’s law, each generationstandard in cellular communication has managed to stay on track with theexponentially increasing trend and increase data rates by introducing newkey technologies or by densifying old technologies. The next sections intro-duce a novel idea tossed around in beyond 5G and towards 6G discussionsto tackle the increasing data rate demand by smartly re-configuring radioenvironments.1.1 Beyond 5G and towards 6GWith 5G underway, significant technologies such as massive multiple-input multiple-output (MIMO) and milimeter (mm)Wave have been steadilycommercialized. Inherent limitations within these technologies are beingbrought to the fore [4], one of which is the inability to control the radio en-vironment while being energy-cognizant [5]. These limitations are spurringon the development of the next generation 6G wireless system which is ex-pected to be a convergence of several technology trends including that ofsmart environments and smart surfaces examined below.11.2. Smart Radio Environments1.2 Smart Radio EnvironmentsThe goal of future wireless systems is not only speed and connectivity asit was before. In fact, future wireless systems achieve several objectives toserve several information and communication technology applications likecomputing, sensing, and distributed intelligent communications. Two fun-damental limitations exist in current communication standards [6]:− High power consumption of wireless systems. With the expected ad-dition of millions of devices, the resultant power consumption will beprohibitive.− Interruptions in connectivity due to random wireless channels, limitedmmWave propagation, and susceptibility to blockage.Usually, to work around these limitations and enable supporting high wire-less traffic, wireless networks operators densify the network by deployingmore base stations (BS)s , more antennas at the BS, or use more spectrum.Every generation of wireless networks has incurred more power consump-tion and more radio wave emissions to fulfill its specification (rate, latency,number of connected devices, etc.).Fleshing out the concept of smart-radio environment might just be theneeded alternative of providing better data rates with lower power consump-tion. Note that current radio environments are not completely controlled,due to random fluctuating channels being the cause for interruptions in con-nectivity limitation, leading to excessive attenuation, fading, blockage, etc.in some scenarios. To mitigate these effects, smart-radio environments cre-ate a tunable space playing an active role in customizing and transferringinformation signals.The fundamental limitations stated above need to be resolved with sev-eral emerging solutions being considered under the umbrella of smart-radioenvironments. Considering the high power consumption limitation, one ofthe more cost-effective enablers of smart-radio environments is called intel-ligent reflecting surfaces (IRS)s discussed next.1.3 What are Intelligent Reflecting Surfaces?To enable smart radio environments, communication engineers convergedon the idea of deploying software-controlled surfaces, i.e. IRSs, on objects’surfaces in the environment such as buildings. Intelligent reflecting surfaces(IRS)s are a seemingly recent concept, built on decades of work [7]. Each21.3. What are Intelligent Reflecting Surfaces?Figure 1.1: IRS passive beamforming for a certain user with an incident EMplane-wave.IRS contains a large number of economical, passive, and scattering elements,where each element can change the amplitude and/or a phase shift of the im-pinging electromagnetic (EM) waves to achieve reflect beamforming gains.The distributed smart surfaces or more accurately distributed IRSs wouldreflect and/or scatter the incoming incident wave to achieve a desired ob-jective. This objective can be in the form of increasing the received powerof the desired signal at a receiver, or decreasing the power of interference,or increasing the ratio between the two. An IRS can help to achieve theseobjectives via beam-focusing, which is tuning the magnitude of the radi-ation pattern in a certain direction, or beam-steering which is modifyingthe direction of the beam (see Fig. 1.1). There are many use cases whereIRSs may find their niche [5], including: interference mitigation, enhancedcoverage, benefiting Tetrahertz bands communication, etc.1.3.1 ArchitectureIRSs are imagined to be reflect antenna arrays or meta-surfaces coat-ing walls in the environment, electrically reconfigured to affect propagatingelectromagnetic (EM) waves. Currently, IRS research is veering towardsmeta-surfaces, which are man-made surfaces made up of sub-wavelengthsized meta-atoms [8]. These meta-atoms reconfigure and modify the overallimpedance of the surface thus manipulating the impinging EM wave andachieving polarization control, beam-steering, or beam-focusing [9]. The31.4. State of the ArtFigure 1.2: Anomalous reflection occurring through continuous changingsurface impedance vs specular reflection with constant surface impedancemeta-surface can be implemented with switching diodes which produces abinary approach to changing the amplitude or phase which would have twostates, and with varactors (voltage-controlled variable capacitors) which givecontinuous variations for these properties.1.3.2 Anomalous vs Specular ReflectionFig. 1.2 illustrates the difference between anomalous and specular reflec-tion. The former is defined as reflecting to the user by shaping the scatteredfield so that the main beam is in a desired direction, the color gradientindicates continuously changing surface impedance. While the latter en-compasses Snell’s law with angle of incidence equaling angle of reflectionand has constant surface impedance.1.4 State of the ArtIn this section, the state of the art, focusing on comparisons with relayingtechnology, passive beamforming, channel estimation, and distributed IRSsdeployment is reviewed and the limitations are presented.41.4. State of the Art1.4.1 Relays vs. Intelligent Reflecting SurfacesThe IRS fills a gap in the area of relay-assisted wireless systems, whichis namely the question of achieving full-duplex relaying with low power con-sumption. An IRS can be considered as a full-duplex transparent relay withtransparent relays re-transmitting after amplifying signals such as Amplify-and-Forward (AF) relays, while non-transparent relays re-transmitting afterregenerating signals such as Decode-and-Forward (DF) relays. Another dis-tinguishing feature is the mode of operation for relays, classical active relaysusually operate in half-duplex, i.e. re-transmission and reception of signalsis separated temporally, whereas full-duplex has no such separation. Here,IRSs passively reflect signals without generating new radio signals and aretherefore considered energy-efficient.[10] compares energy efficiency of AF relays with IRSs where IRSs arefound to perform better in that regard since they enhance received powerstrength without using power amplifiers but by appropriate phase shiftsdesign.On the other hand, [5] makes a valid argument about the uncertain en-ergy consumption of IRSs since their control interface would likely consumelots of power, and until efficient CE protocols and real-time re-configurabilityof the IRS are solidified, one cannot make such concrete statements abouttheir energy efficiency or the lack thereof. Another meaningful compari-son between a DF relay which is non-transparent vs an IRS concludes thatIRSs need a large number of reflecting elements to remain competitive toDecode-and-Forward relaying technology [11].1.4.2 Passive BeamformingDesigning IRS phase shifts as to shape impinging electromagnetic (EM)waves has been studied before extensively, [10, 12–14]. These works assumethat the IRS is an array of passive elements able to independently addphase shifts onto the incident EM wave to reflect passively in desired ways,illustrated in Fig. 1.1. Adjusting these phase shifts at the IRSs is referredto as passive beamforming or reflect beamforming. Additionally, some ofthese works [10, 12, 13] focus on jointly optimizing transmit beamformingat the BS and passive beamforming at the IRS to meet a certain goal suchas maximize energy efficiency [10], maximize minimum user SINR subjectto a transmit power constraint [12], or minimize transmit power subject toquality of service constraints [13].Prior works deal with optimizing phase shifts at the IRS while amplitude51.4. State of the Artcoefficients are assumed to have unity magnitude, on the other hand, [13]tackles this assumption and introduces a phase shift model that includespractical phase-dependent amplitude responses of the IRS elements. Fur-thermore, [14] paves the way for coordinated passive beamforming for thedistributed IRS-assisted wireless systems, and jointly optimizes the powerat the BS and the phase shifts at each IRS to maximize sum-rate.1.4.3 Explicit Channel EstimationMost existing work on designing different IRS schemes assume perfectCSI available at the BS and the IRS. However, this assumption is imprac-tical. The passivity of the IRS translates to an inability to estimate thechannels or assist the BS in estimating the channels by transmitting pilotsymbols, which poses the biggest challenge in developing CE protocols forthese systems [12, 15, 16]. Given the IRS phase shifts have to be updated ineach coherence interval, CSI needs to be acquired for all IRS-assisted linksat the pace of channel variation, which overburdens the system.The first estimation protocol that appeared for a single-user IRS-assistedMISO system, known as the ON/OFF protocol [17], estimates all IRS-assisted links one-by-one by serially turning one element ON while keepingthe others OFF. This protocol was significantly improved in terms of thenormalized mean squared error (NMSE) performance in [18] for a single-user system and in [1] for a multi-user system, where an optimal solutionfor IRS phase shifts vectors in the CE phase was developed.However, the CE time scales linearly with the number of IRS elements inall these works, imposing prohibitively high training overhead. To combatthis, [19] and [20] introduce the idea of grouping adjacent IRS elements intosub-surfaces, which will decrease the training overhead but also reduce thereflect beamforming gains promised by using a large number of IRS elements.Other solutions exploit channel sparsity that exists in mmWave and massiveMIMO channels to develop low-overhead algorithms for CE [21, 22].A two-timescale CE framework is proposed in [23] that exploits the prop-erty that the BS-IRS channel is quasi-static, while the IRS-user channel isdynamic. The assumption that BS-IRS channel is LoS and fixed is also madein many other works such as [1, 12, 15, 21, 24] and is quite practical givenBS and IRS have fixed positions with few obstacles around. The authors in[25] envision a novel IRS architecture where a few IRS elements are activeand can sense the channels to aid in CE. They then propose compressivesensing and deep learning based methods to design the system.61.5. Problem Statement1.4.4 Distributed Intelligent Reflecting Surfaces: AnExtensionThe few existing works on CE and beamforming all deal with singleIRS-assisted MISO systems and still impose large training overheads. In amulti-user MISO system, severe blockages between the base station (BS) andusers can downgrade the overall system performance. Adding a single IRSwith a sufficient number of elements provides coverage for a certain area, yetthe number of degrees of freedom the IRS adds to the MISO system dependsheavily on the IRS placement which can reduce to only one under certainchannel conditions. Introduced first in [26], the rank deficiency problem inthe BS-IRS link states that the degrees of freedom offered by the channelthrough the IRS when the BS-IRS channel is LoS will equal to one yieldinggains only when there is one user in the system.Recently, the idea of deploying multiple IRSs in a distributed manner hasbeen considered to overcome several signal blockages by coating the blockingstructures with IRSs. Moreover, distributed IRSs are a promising solutionto the rank-deficiency problem stated earlier, since the overall BS-IRS linkin the distributed case would be the sum of multiple rank one channelswhich guarantees higher rank channels [27], thereby benefiting the multi-user setting. So far, the couple of works that have appeared on distributedIRSs assume perfect CSI and design beamforming to increase coverage andperformance [27, 28].Only one paper so far handles the IRS-user association problem [29]which is about assigning IRSs to users to optimally balance the passivebeamforming gains among different BS-user links. They derive the signal-to-interference-and-noise (SINR) expression at each user in closed form, andcreate an SINR balancing optimization problem to find the IRS-user associ-ation parameters. However, their system model assumes single-antenna BSsto simplify analysis and thus is impractical in a world where massive MIMOis being deployed quickly.1.5 Problem StatementThis research extends current single-IRS MISO systems to the distributedIRSs MISO systems with a focus on channel estimation, theoretical anal-ysis for average signal-to-interference-and-noise (SINR) at each user, andoptimization of IRS-user association. The key aspect of this work is the de-velopment of statistical channel estimation protocol for the distributed IRSsassisted MISO system where each BS-IRS link is dominated by the LoS link71.6. Objectiveswhile all other links undergo fast fading. A novel closed-form average SINRexpression for the system is derived and validated through Monte Carlosimulations. Utilizing the derived average SINR expression, the IRS-userassociation problem is solved to maximize the minimum user SINR usinga successive refinement algorithm. Simulation results illustrate the positiveeffect of deploying IRSs in a distributed fashion with optimized associationparameters.1.6 ObjectivesThe main objectives are stated in the following:− To provide a holistic, unprecedented discourse of distributed IRSs asan enabling technology for beyond 5G (B5G) and 6G, and pinpointthe advantages and disadvantages of deploying distributed IRSs.− To provide a practical system model and a channel estimation protocolwith relatively low channel training overhead for distributed IRSs as-sisted MISO system under LoS BS-IRS channels and Rayleigh fadingIRS-user and direct channels.− To analyze the average SINR as a performance metric of the dis-tributed IRSs assisted MISO system under LoS BS-IRS channels andRayleigh fading IRS-user and direct channels.− To optimize over IRS-user association parameters for a formulatedmax-min average SINR problem.1.7 ContributionsThe main contributions can be summed up as:− Tackling the unprecedented CE problem for a distributed IRSs-assistedMISO system. To the best of our knowledge, this research marks thefirst contribution in developing a CE scheme for the distributed IRSscase (published in [2]).− Deriving average SINR expression for the distributed IRSs assistedMISO system under MRT precoding and with IRS-user associationparameters.81.8. Outline− Designing an effective yet subptimal successive refinement algorithmfor the IRS-user association with lower complexity than existing meth-ods. This algorithm finds optimized IRS-user association parametersand performs closely to full enumeration.− Drawing insights from resulting simulations on the CE protocol ascompared to a benchmark protocol as well as on the effect of optimizedIRS-user association on the system performance.1.8 OutlineThe rest of this thesis is partitioned as follows: Chapter 2 gives a glimpseonto some wireless communication concepts used in later chapters and in-troduces the distributed IRSs assisted MISO system model with Rayleighfading channels and LoS BS-IRS channel. Chapter 3 provides a novel chan-nel estimation protocol to estimate the downlink channel using uplink train-ing. Chapter 4 derives the ergodic analysis for the SINR and then solves anassociation problem with max-min average SINR as an objective. Chapter5 includes the conclusion, limitations, and future directions.9Chapter 2System Model andPreliminariesIn this chapter, we cover the system model utilized in this work andreview some key concepts regarding wireless channels. The channels aremodelled as Rayleigh fading for the direct and IRS-user links, and as arank one matrix containing array responses of the BS and each IRS l for theBS-IRS l line-of-sight (LoS) link. The below figure depicts a distributed IRS-assisted MISO system, with IRSs installed on high-rise buildings, a multi-antenna BS, and single-antenna users. Furthermore, there is a centralizedcontrol interface between the BS and all the other IRSs.Figure 2.1: Distributed IRSs MISO system model in uplink and downlink.Consider M antennas at the BS serving K single-antenna users. Thecommunication between the BS and users is assisted by L IRSs, each equipped102.1. Transmission Modewith N elements (see Fig. 2.1). The IRSs are deployed in the environmentin a distributed manner with fixed positions and their operation is controlledby IRS controllers that communicate with the BS over a backhaul link [12].The collective channel between the BS and user k in the uplink is givenbyhk = hd,k +L∑l=1H1,lΘlh2,l,k, (2.1)where H1,l ∈ CM×N is the channel between BS and IRS l, h2,l,k ∈ CN×1is the channel between IRS l and user k, and Θl = diag(αl,1 exp(jθl,1), . . . ,αl,N exp(jθl,N )) ∈ CN×N is the reflection matrix for IRS l, where θl,n ∈[0, 2pi] is the phase-shift applied by element n of IRS l and αn,l ∈ [0, 1] isthe amplitude reflection coefficient. Also hd,k ∈ CM×1 is the direct channelbetween BS and user k.The distributed IRSs assisted MISO are a generalization of the single IRSscenario, therefore, we build on [26] and extend the system model discussedto distributed surfaces while retaining the similar transmission conditionsexplained in details below.2.1 Transmission ModeUplink and downlink transmissions can be separated in time or fre-quency. The uplink is described as the transmission from users to the BS,while the downlink is the transmission of signals from BS to users. Timeseparation, i.e. time-division duplexing (TDD), occurs when the downlinktransmission operates in a different times slot than the uplink transmis-sion but both occupy the same frequency band, whereas frequency division-duplexing (FDD) happens when the downlink operates in a different fre-quency band than the uplink at the same time as depicted in Fig. 2.2.While FDD is deployed in practice, TDD has become part of the canoni-cal model for massive multiple-input multiple-output (MIMO) technologymainly because the channel estimation for TDD has less training time [30].112.2. Fading Direct and IRS-User ChannelsFigure 2.2: Time Division Duplexing (TDD) vs Frequency Division Duplex-ing (FDD) transmissionIn this IRS-assisted MISO system model, we consider a TDD transmis-sion mode due to the similarities between IRS-assisted wireless systems andmassive MIMO. One of which is the prohibitively large system dimensionsergo large numbers of channels to estimate (cf. sec. 2.2.2). Here, TDD ex-ploits channel reciprocity and thus eliminates the need to estimate channelsboth in the uplink and downlink (for example hHk in the downlink is theHermetian of hk defined in (2.1) in the uplink). In the next section, we layout the fading model for the direct channel hd,k and IRS l-user k channelh2,l,k, in addition, we present some important definitions regarding fadingchannels.2.2 Fading Direct and IRS-User ChannelsElectromagnetic waves propagating through wireless channels are suffi-ciently yet incompletely characterized by multipath fading (depicted in Fig.2.3) and shadowing among other effects. This is due to the complicatedpropagation through dynamic moving terminals in the environments or torich scattering that cannot be mathematically fully described. However,considerable work has been done to statistically describe and model theserandom fading channels in certain settings. One of the simpler models isRayleigh fading which captures the multipath component found in wirelesschannels [31].122.2. Fading Direct and IRS-User ChannelsFigure 2.3: Rayleigh fading with no LoS pathDefinition 2.1. Multipath fading is characteristic of wireless channelswhere the propagating wave is randomly scattered, delayed, reflected, anddiffracted which creates multiple paths in the environment due to scatter-ers that create these additional paths, thus leading to destructive and/orconstructive interference at the terminal.Fading is modeled as random channels with some distribution, one ofwhich is the Rayleigh distribution for a fading channel h, so thath ∈ CN (0,C) (2.2)where C ∈ CM×M is the covariance matrix, and M is the number of anten-nas. One practical scenario which is often modelled using Rayleigh fading ismobile systems, where no line-of-sight (LoS) path exists between the trans-mitter and receiver. Considering the system model illustrated in Fig. 2.1,the ground users will likely be blocked by surrounding objects. Thus, themodelling of Rayleigh fading for the direct and IRS-user channels is justi-fied. Moreover, we assume the fading these channels undergo is classified asblock-fading and flat-fading.Definition 2.2. Block fading occurs when the response of the channelis constant (static) for a certain period of time. This period is known ascoherence interval.The length of a coherence interval depends on the fading channel char-acteristics and the mobility of the user, assuming that the transmitter isfixed. Higher mobility of the user quickens the pace of channel variations,and lessens the period of static behavior. For a TDD transmission modepreviously defined in sec. 2.1, portions for uplink and downlink transmis-sion are reserved in the coherence period, where a fraction of the uplink is132.2. Fading Direct and IRS-User Channelsalso reserved for pilot training to achieve channel estimation (CE), demon-strated in Fig. 2.4. This poses a time constraint on the CE overhead, andeffectively on the number of pilots (preamble signals used for CE) a user cansend.Figure 2.4: Channel coherence interval divided into uplink training phase,uplink phase, and downlink phase.As for the flat-fading assumption for the h2,l,k and hd,k channels, wenote that these channels remain frequency non-selective under certain con-ditions, which implies no channel induced inter-symbol interference (ISI).More formally, we present the following definition.Definition 2.3. Flat-fading channels have the property that all receivedmultipath components (see Definition 2.1) of a symbol arriving withinthe duration of that symbol [32], in other words, when the bandwidth of thechannel is larger than the bandwidth of the signal.Now, we model statistically h2,l,k and hd,k channels by correlated Rayleighfading represented ash2,l,k =√β2,l,kR1/2IRSl,kzl,k, (2.3)hd,k =√βd,kR1/2BSkzd,k, (2.4)where RIRSl,k is the spatial correlation matrix of the channel from IRS l touser k, RBSk is the spatial correlation matrix from the BS to user k, andzl,k ∼ CN (0, IN ) and zd,k ∼ CN (0, IM ) describe the fast fading vectors ofthe IRS-user link and the BS-user link, respectively. The path loss factor isdenoted as βd,k for the direct channel and β2,l,k for the IRS-user k link. Thenotion of spatial correlation is explained below in sec. 2.2.1, and it arises inthese vector channel models because the antennas at the BS as well as thereflecting elements in the IRSs are arranged compactly.142.2. Fading Direct and IRS-User Channels2.2.1 Spatial CorrelationSpatial correlation is a property related to multiple antennas at the trans-mitter and/or receiver. In layman words, it arises in practice when receivedpower is stronger in certain directions than others. More formally, spatialcorrelation happens due to two factors, one of which is propagation environ-ment, where the channel vector is more probable to point in some directionsmore than others. The other is because of insufficient spacing between an-tennas [33]. Take RIRSl,k in (2.3) for instance, the off-diagonal elements willrepresent the correlation between the antenna elements, while the diagonalelements are all set to one.Definition 2.4. Spatially Correlated Channels. A fading channel vec-tor h is Rayleigh distributed if it follows h ∼ CN (0,C), where C is thecovariance matrix of the distribution (also its correlation matrix). Whenthe channel’s direction h/‖h‖ and its magnitude ‖h‖ are independent ran-dom variables then h is said to be spatially uncorrelated. Otherwise, it isspatially correlated [30].2.2.2 Channel EstimationChannel estimation (CE) is a process of sending pilots (known sequenceof symbols between transmitter and receiver) through an unknown channel.The transmission coefficients which are complex-valued channel gains canthen be estimated by correlating the received signal with the known pilotsat the receiver’s end.Since the IRSs are completely passive and do not have any signal pro-cessing capabilities, the BS has to estimate all channels and compute thedesired configurations for all the IRSs then share this information with theIRSs’ micro-controllers over back-haul connections. Each IRS then appliesthe required reflection configuration Θl, l = 1, . . . , L.The reason for CE is to negate the detrimental effects of channels or atleast suppress those effects, to do that one needs to sense these channelsfirst. If we model the effects of the channels as linear, we may use a linearestimator, one of which is the least squares (LS) estimator which is anoptimal linear estimator given no prior information of the channels. On theother hand, if the BS ascertains the large-scale statistics of the Rayleighfading channels given in (2.4) and (2.3), we can shift to a Bayesian channelestimation method called minimum mean squared error (MMSE) estimationwhich is optimal for Rayleigh-distributed fading channels.152.3. BS-IRS l ChannelIn the following section, we present the channel model for H1,l, which weshow next to be of a LoS nature and known a priori at the BS. Note that sincewe make the assumption of knowing H1,l, the number of channels to estimatein the system model, illustrated in Fig. 2.1, is reduced to (M+NL)K instead(M + +NL)K + MNL. Although the number of unknowns is reduced wemay incur high training time dedicated for channel estimation especially inthe multi-user setting and large M,N,L system dimensions.2.3 BS-IRS l ChannelPreviously, we established practical models for the direct and IRS-userk channels. Recall there are two channels which make up each IRS cascadedchannel, one of which is the BS-IRS l channel H1,l which is defined here.Many works on IRS-assisted systems, for example: [1, 12, 15, 21, 24], assumethe BS-IRS channel to be LoS dominated. This assumption is quite practicaland is supported in literature with the following remarks:− First, there will always exist a LoS path between the BS and IRS. Asthe BS tower is generally elevated high and the IRS is also envisionedto be integrated onto the walls of (high-rise) buildings, so both willhave few obstacles around. Given the positions of BS and IRSs arefixed, a stable LoS channel between the BS and each IRS will exist andcan be constructed at the BS using directional (LoS angle of departure(AoD) and angle of arrival (AoA)) information.− Second, the path loss in NLoS paths is much larger than that in LoSpath loss in the next generation systems, resulting in any NLoS pathsconstituting the BS-IRS channel to be much weaker. In fact it is notedthat in mmWave systems, the typical value of Rician factor (ratio ofenergy in LoS to that in NLoS) is 20dB and can be as large as 40dB insome cases [24], which is sufficiently large to neglect any NLoS channelcomponents. Under these remarks, we assume that each BS-IRS lchannel is LoS.We make the next assumption that the BS would have prior knowledgeof H1,l because of the LoS dominant nature of each IRS l and the BS linkexplained earlier. The BS can have knowledge of the locations, orientations,and array parameters of the IRS to construct the BS-to-IRS channel.Assuming a uniform linear array (ULA) at the BS and uniform planararray (UPA) at the IRSs depicted in Fig. 2.5, the LoS BS-IRS l channel can162.3. BS-IRS l ChannelFigure 2.5: Model of h1,l,n ∈ CM×1 which is the nth column vector of ofLoS BS-IRS l channel matrix H1,l with multiple transmit antennas at theBS and multiple elements at IRS l. The signals from the transmit antennasarrive almost in parallel at the nth receive element of the IRS.be written asH1,l =√β1,lalbHl . (2.5)Here, β1,l the path loss factor for BS-IRS l channel and al is the arrayresponse vector for the BS, while bl is the array response vector for IRS l.Since we assume a ULA at the BS, the steering vector al ∈ CM×1 can bewritten asal = [1, e−jkdBS cos(θ), . . . , e−jkdBS(M−1) cos(θ)]T , (2.6)where dBS is the inter-antenna spacing, k = 2pi/λc is the wave number, λcis carrier wavelength, θ is the elevation angle-of-departure (AoD) from theBS to IRS l. [34].The IRS is envisioned to be a planar array of N = NxNz elements,where Nx denotes the number of horizontally placed elements and Nz is thenumber of vertically placed elements. The array response vector bl ∈ CN×1for a UPA at IRS l is expressed asbl = bTl,x ⊗ bTl,z (2.7)172.4. Maximum Ratio Transmission PrecodingHere, ⊗ denotes the Kronecker product, andbl,x = [1, e−jkdIRS sin(ϕ) cos(ϑ), . . . , e−jkdIRS(Nx−1) sin(ϕ) cos(ϑ)], (2.8)bl,z = [1, e−jkdIRS cos(ϕ), . . . , e−jkdIRS(Nz−1) cos(ϕ)], (2.9)where dIRS is the inter-antenna spacing, and ϑ(ϕ) denote the azimuth (el-evation) angle-of-arrival AoA of the path from BS to IRS l [35], [36]. Itis apparent that each BS-IRS l channel matrix has a rank one structure(because it is made up of the outer product of two vectors), thus H1,l is arank-one matrix with a unique non-zero singular value λH1,l =√β1,l [34].The rank one assumption is accurate since the elements at both the BS andIRSs are co-located, as both the BS and IRSs are elevated such that no ob-stacles lie in-between, and the IRSs are assumed to be in the far-field, suchthat the propagation distance dBS−IRSl between the BS and IRS l is muchbigger than the largest dimension of that IRS. To ensure far-field assumptionholds true, we have thatdBS−IRSl ≥2D2λc(2.10)and D is the largest dimension of the antenna array. Note that D in theULA case is the total length of the array. While D for UPA is the diagonallength of the UPA array. Although a channel with rank one means onlyone degree of freedom for the system, with distributed surfaces the overallchannel between BS and each user will at least have rank L, which allowsmultiple users to be served efficiently.2.4 Maximum Ratio Transmission PrecodingIn sec. 2.2, we establish the definition of multipath fading, here weattempt to mitigate it. Multipath fading is thought to be the most adversepropagation effect for wireless communications, but a way to combat it is touse antenna diversity techniques, one of which is known as maximum ratioprecoding (MRT) [37].Note that to achieve precoding, one needs at least two transmit dimen-sions in general, since one can precode in time or frequency in addition tospace (antennas). Define an MRT digital precoding vector fk ∈ CM×1, withM being the number of antennas at the transmitter, written asfk =hk√E[‖hk‖2](2.11)182.5. Downlink Transmissionwith hk stated in (2.1) and similar to [38].With precoding such as (2.11) used at the BS, the transmit signal x ∈CM×1 is formulated asx =K∑k=1√pkfksk, (2.12)where pk and sk ∈ CN (0, 1) are the allocated power and data symbol of userk, respectively.Given sk’s are independently and identically distributed (i.i.d.) CN (0, 1)variables, x has to satisfy the average Tx power constraint asE[||x||2] = tr (PFHF) ≤ Pmax, (2.13)where Pmax > 0 is the power constraint at the BS, P = diag(p1, . . . , pK) ∈CK×K is the power allocation matrix and F = [f1, . . . , fK ] ∈ CM×K is theprecoding matrix.MRT precoding maximizes signal-to-noise ratio (SNR) which is optimalfor single-user wireless systems, yet for the multi-user setting this is not theoptimal precoder as this precoding technique does not factor in interferencefrom other users. However, it is simpler than other precoding techniquesand thus utilized in the theoretical analysis in Chapter 4.2.5 Downlink TransmissionAssuming flat-fading channels (cf. Definition 2.3), the received base-band signal in the downlink at user k is defined asyk = hHk x + nk, (2.14)and nk ∼ CN (0, σ2) is the noise at the receiver with noise variance σ2, x isthe transmit signal determined in (2.12), and hk defined as in (2.1). Theexpression for (2.14) is used in Chapter 4 to compute the average signal-to-interference and noise (SINR) at each user.2.6 SummaryIn this chapter, we described the system model, the channels’ models,and the transmission scheme. To realize an optimized transmission, es-timated channels are obtained at the BS which is covered in Chapter 3192.6. Summarybuilding on the system model and concepts covered here to find a novel CEprotocol for the distributed IRSs assisted MISO system. After this, the BScan use a precoding scheme such as MRT to send precoded signals to theusers in the downlink. Since this is a TDD system, the BS can compute theaverage SINRs for all the users. The SINR is then used as an importantperformance metric at the BS to optimize transmission through associat-ing IRSs to users and configuring IRSs optimally discussed in Chapter 4.The reason SINR is chosen to represent the system holistically is that otherperformance metrics such as transmission rate, outage probability, and biterror rate are all functions of SINR.20Chapter 3Channel EstimationThis chapter presents the proposed CE protocol and derives analyti-cal expressions of the MMSE estimates and NMSE. We consider correlatedRayleigh fading channels between the BS and users, and the IRSs and users,while each BS-IRS channel is LoS. The LoS BS-IRSs channel matrices canbe computed a priori at the BS using the locations of the IRS, which arefixed and can be used to compute the LoS angle of departure (AoD) andangle of arrival (AoA). Leveraging this, we develop an optimal CE proto-col using the Bayesian technique of minimum mean squared error (MMSE)estimation, that promises low CE error and imposes a significantly lowertraining overhead than what we need when we extend the conventional pro-tocols from [1, 17, 18] to this setup. The normalized mean squared error(NMSE) in the MMSE estimates is analytically derived and studied usingsimulations.Figure 3.1: Distributed Intelligent Reflecting Surfaces multi-user MISO sys-tem uplink transmission.213.1. Proposed Protocol3.1 Proposed ProtocolIn this research, we assume a time division duplexing (TDD) systemwhere the downlink channels are estimated using the received uplink pilotsymbols by exploiting channel reciprocity. Since the IRSs are passive, theBS has to estimate all channels and determine optimal IRS configurations.We assign an uplink training phase of τC seconds (sec) in the coherenceinterval (cf. Fig. 2.4) which is further broken down into S sub-phases oflength τS =τCS sec. Throughout the CE phase, users transmit mutuallyorthogonal pilot sequences of length TS symbols, with TS =τSτ˜ where τ˜is the duration of each symbol. The pilot sequence of user k is denotedas xp,k = [xp,k,1, . . . , xp,k,TS ]T ∈ CTS×1, such that xHp,kxp,l = 0, for k 6= l,k, l = 1, . . . ,K and xHp,kxp,k = PCTS τ˜ = PCτS Joules, where PC is thetransmit power of each user. The IRS l applies the reflect beamformingmatrix Θl,s = diag[φl,s,1, . . . , φl,s,N ]T ∈ CN×N in sub-phase s, where φl,s,n =αl,s,n exp(jθl,s,n). The received training signal at the BS, Ytrs ∈ CM×TS insub-phase s is then given asYtrs =K∑k=1(hd,k +L∑l=1H1,lΘl,sh2,l,k)xHp,k + Ntrs (3.1)where Ntrs ∈ CM×TS is the matrix of noise vectors at the BS, with eachcolumn distributed independently as CN (0, σ2IM ). After correlating thereceived training signal with the training sequence of user k, the BS obtainsthe observation vector, rtrs,k ∈ CM×1, for user k in sub-phase s asrtrs,k = (hd,k +L∑l=1H1,lΘl,sh2,l,k) + ntrs,k, (3.2)which can be compactly written asrtrs,k = hd,k + H1Θsh2,k + ntrs,k, k = 1, . . . ,K, (3.3)where h2,k = [hT2,1,k, . . . ,hT2,L,k]T ∈ CNL×1 is the concatenated vector of all LIRS-user k channels, Θs = diag(Θ1,s, . . . ,ΘL,s) ∈ CNL×NL is the reflectionmatrix of all the IRSs in sub-phase s, H1 = [H1,l, . . . ,H1,l] ∈ CM×NL is theconcatenation of all L BS-IRS channel matrices, and ntrs,k =Ntrs xp,kPCτS∈ CM×1is the noise.Since we remarked earlier that H1,l ∈ CM×N can be perfectly known atthe BS using locations of the IRSs, we will focus only on the estimation of223.1. Proposed Protocolhd,ks and h2,ks. To this end, we defineH¯1 = diag(√MIM ,h1,1,1, . . . ,h1,1,N ,h1,2,1 . . . ,h1,2,N , . . . , (3.4)h1,L,1, . . . ,h1,L,N ) ∈ CM(NL+1)×NL+M ,as the block diagonal matrix of h1,l,ns, where h1,l,n is the nth column ofH1,l. Aggregating the observation vectors across the S sub-phases as rtrk =[rtrT1,k , . . . , rtrTS,k ]T ∈ CSM×1, we obtainrtrk = (Vtr ⊗ IM )H¯1hd,2,k + ntrk , k = 1, . . . ,K, (3.5)where,Vtr =1 vT1,1 · · · vTL,1.........1 vT1,S · · · vTL,S ∈ CS×(NL+1) (3.6)hd,2,k = [1√MhTd,k,hT2,k]T ∈ C(NL+M)×1 cascades the direct and IRS-userchannels and ntrTk = [ntrT1,k , . . . ,ntrTS,k ]T ∈ CSM×1 is the concatenated noiseacross all sub-phases. Note that vl,s = vec(Θl,s) is the phase shifts vectorfor IRS l in sub-phase s.Define V˜tr = (Vtr ⊗ IM )H¯1 ∈ CSM×NL+M . To guarantee the existenceof left pseudo-inverse of V˜tr (given as V˜†tr = (V˜HtrV˜tr)−1V˜Htr ), we must haveSM ≥ NL+M. (3.7)Rewriting the condition to accentuate only the parameter S, we obtainS ≥ NLM+ 1. (3.8)Because of prior knowledge of H1,l, the proposed CE protocol estimateschannels sequentially while training time is decreased by a factor of M dueto having M copies of the h2,k link, 1 sub-phase remains for the directchannel, while for cascaded IRS channels it becomes NL/M since partialknowledge of the cascaded IRS links is present at the BS along with large-scale statistics of the unknown part of the link. Thus, the minimum numberof CE sub-phases is actually reduced by a factor of M , as compared to theconventional protocols. For example: as compared to the protocol in [18] fora single IRS (L = 1) system, which requires N + 1 sub-phases to estimatethe channels, we will require N/M + 1 sub-phases, which is a substantialreduction given next generation systems use large numbers of BS antennas.233.2. Noise Variance MinimizationMultiplying (3.5) by V˜†tr results inr˜trk = hd,2,k + (V˜HtrV˜tr)−1V˜Htrntrk . (3.9)We denote by n˜trk = (V˜HtrV˜tr)−1V˜Htrntrk ∈ C(NL+M)×1 the measurement noiseand by Cn˜trk= E[n˜trk n˜trHk ] ∈ C(NL+M)×(NL+M) its covariance matrix. Next,we minimize the noise variance in (3.9) and derive the MMSE estimates.3.2 Noise Variance MinimizationNext step is to design Vtr such that variance of the noise is minimizedand the noise across the estimated channels is uncorrelated.Recalling that ntrs,k =Ntrs xp,kPCτSwe obtainCn˜trk= (V˜HtrV˜tr)−1V˜HtrE[ntrk ntrHk]V˜tr(V˜HtrV˜tr)−1, (3.10)=σ2PCτS(PCτS)2(V˜HtrV˜tr)−1 =σ2PCτS(V˜HtrV˜tr)−1, (3.11)where,V˜HtrV˜tr = H¯H1 (VHtrVtr ⊗ IM )H¯1. (3.12)We note that by construction H¯H1 H¯1 = MΣ, where Σ = diag(IM , β1,1IN ,..., β1,LIN ) ∈ CNL+M×NL+M . Therefore, to ensure uncorrelated channel es-timates and equal noise variance in each estimate, Vtr must have equallyscaled orthogonal columns such that (VHtrVtr)−1 = ζINL+1. Minimizing thenoise variance is equivalent now to minimizing ζ, under the constraints:1. Vtr has the structure in (3.6).2. αl,s,n ∈ [0, 1].3. θl,s,n ∈ [0, 2pi].4. (VHtrVtr)−1 = ζINL+1.The last constraint can be written asζ =NL+ 1tr(VHtrVtr)=NL+ 1∑Ss=1∑NL+1n=1 |[Vtr]s,n|2. (3.13)243.2. Noise Variance MinimizationUsing the second constraint in (3.13), we see thatζ ≥ 1S. (3.14)A solution that meets the lower bound in (3.14) with equality is the NL+ 1leading columns of an S × S DFT matrix as expressed below [1, 18].[Vtr]s,n = w(n−1)(s−1), n = 1, . . . , NL+ 1, s = 1, . . . , S, (3.15)where w = e−j2pi/S is the primitive Sth root of unity. This design of Vtrresults in (VHtrVtr)−1 = 1S INL+1 and therefore achieves the minimum noisevariance in (3.14) since ζ = 1S . Using this we can simplify (3.12) asV˜HtrV˜tr = SH¯H1 H¯1 = SMΣ, (3.16)and (V˜HtrV˜tr)−1 = Σ−1SM .To derive the MMSE estimates, we refer back to (3.9) and simplify itwith the optimal DFT design in (3.15) to obtainr˜trk = hd,2,k +1SMΣ−1V˜Htrntrk , k = 1, . . . ,K, (3.17)Recalling that hd,2,k = [1√MhTd,k,hT2,k]T , we can write r˜trk = [r˜trT0,k , r˜trT1,k , . . . , r˜trTL,k ]T ∈C(M+NL)×1, where r˜tr0,k ∈ CM×1 is used to estimate the direct channel andr˜trl,k ∈ CN×1 is used to estimate the channel from IRS l to user k.The relationship between r˜0,k and hd,k for user k can be written using(3.17) and definition of V˜tr asr˜tr0,k =1√Mhd,k +1SMΣ−10√MIM (vtr1 ⊗ IM )Hntrk=1√Mhd,k +1S√M(vtr1 ⊗ IM )Hntrk . (3.18)where√MIM corresponds to the first M×M diagonal matrix of H¯1, Σ−10 =IM is the first M ×M diagonal matrix of Σ−1 and vtr1 is the first S × 1column of Vtr. This observation vector can be scaled by a factor of√M atthe BS to obtainr˜tr0,k = hd,k +1S(vtr1 ⊗ IM )Hntrk . (3.19)Note that (3.19) is actually the LS estimate of hd,k which we denote ashˆLSd,k.253.3. Minimum Mean Squared Error Estimation3.3 Minimum Mean Squared Error EstimationThe BS can compute the MMSE estimate of hd,k as stated in the follow-ing lemma.Theorem 3.1. The MMSE estimate hˆd,k of hd,k is given ashˆd,k = βd,kRBSk(βd,kRBSk +σ2IMSPCτS)−1r˜tr0,k, (3.20)which is distributed as hˆd,k ∼ CN (0,Ψd,k) whereΨd,k = β2d,kRBSk(βd,kRBSk +σ2IMSPCτS)−1RHBSk . (3.21)Proof. The proof follows from noting that the MMSE estimate is given asas hˆd,k = Wr˜tr0,k, where W = E[hd,kr˜trH0,k ](E[r˜tr0,kr˜trH0,k ])−1. The expressionsfor E[hd,kr˜trH0,k ] and E[r˜tr0,kr˜trH0,k ] can be derived by noting that ntrk and hd,kare independent Gaussian vectors. More details can be found in AppendixA.The uncorrelated estimation error h˜d,k = hˆd,k − hd,k is also statisticallyindependent of hˆd,k since both vectors are jointly Gaussian. Moreover, it isdistributed as h˜d,k ∼ CN (0, Ψ˜d,k), where Ψ˜d,k = βd,kRBSk −Ψd,k.Next we estimate h2,l,k ∈ CN×1 using the observation vector r˜trl,k. Weextract H¯1,l = diag(h1,l,1, . . . ,h1,l,N ) ∈ CMN×N from H¯1 and note thatH¯H1,lH¯1,l = MΣl = β1,lMIN for l = 1, . . . , L. The relationship betweenr˜trl,k ∈ CN×1 and h2,l,ks can now be written using (3.17) asr˜trl,k = h2,l,k +1SMΣ−1l H¯H1,l(Vtrl ⊗ IM )Hntrk= h2,l,k +1SMβ1,lH¯H1,l(Vtrl ⊗ IM )Hntrk , (3.22)where Vtrl = Vtr(:, [N(l− 1) + 2 : Nl+ 1]) ∈ CS×N for l = 1, . . . , L and (:, ·)denotes all the rows for a particular column or set of columns of a matrix.Note that (3.22) is actually the LS estimate of h2,l,k which we denote ashˆLS2,l,k.Theorem 3.2. The MMSE estimate hˆ2,l,k of h2,l,k is given ashˆ2,l,k = β2,l,kRIRSl,k(β2,l,kRIRSl,k+σ2INβ1,lSMPCτS)−1r˜l,k, (3.23)263.4. Normalized Mean Squared Error Analysisfor l = 1, . . . , L, k = 1, . . . ,K. It is distributed as hˆ2,l,k ∼ CN (0,Ψ2,l,k),whereΨ2,l,k = β22,l,kRIRSk(β2,l,kRIRSk+σ2INβ1,lSMPCτS)−1RHIRSk . (3.24)Proof. The proof is similar to that of hˆd,k and found in Appendix B.The uncorrelated estimation error h˜2,l,k = h2,l,k − hˆ2,l,k is also statisti-cally independent of hˆ2,l,k since both vectors are jointly Gaussian. Moreover,it is distributed as h˜2,l,k ∼ CN (0, Ψ˜2,l,k), where Ψ˜2,l,k = β2,l,kRIRSk−Ψ2,l,k.In order to compute the MMSE estimates, the BS must have previousknowledge of the correlation matrices at the IRSs and the BS. However, it iswell-known that the correlation matrices vary slowly and remain static overmany coherence intervals. The BS can easily obtain the prior knowledge ofthese matrices since possessing second-order channel statistics at the BS ispervasive in massive MIMO literature [1].3.4 Normalized Mean Squared Error AnalysisHere we analyze the NMSE in the LS and MMSE estimates under thedeveloped CE protocol. Note that normalization occurs with respect to thechannel power which is defined for the direct channel as tr(E[hd,khHd,k]). Thereason we compute the normalization is to enable comparison between thedirect channel and the IRS-user channels and to provide a better visualiza-tion. We define the NMSE (e.g. for the direct channel) asNMSE(hˆd,k) =tr(E[h˜d,kh˜Hd,k])tr(E[hd,khHd,k])=tr(Ψ˜d,k)tr(E[hd,khHd,k]). (3.25)The NMSEs for the LS estimates in (3.19) and (3.22) are independent ofthe structure of RIRSl,k and RBSk . The NMSE in the MMSE estimatesdoes depend on these matrices, but we assume them to be identity matricesfor the purpose of this section to obtain closed-form expressions and yieldinsights. The effect of correlation on NMSE is shown in the simulationslater.First, the NMSE in the MMSE estimate of hd,k is derived using the fact273.4. Normalized Mean Squared Error Analysisthat tr(E[hd,khHd,k]) = tr(βd,kIM ) = Mβd,k asNMSE(hˆd,k) =βd,kMβd,k(M − βd,ktr(βd,kIM +σ2IMSPCτS)−1)=1Mβd,kMβd,kσ2SPCτS(βd,k +σ2SPCτS) = σ2SPCτSβd,k +σ2SPCτS. (3.26)The expression reveals that the NMSE increases to 1 as σ2 grows large orS, PC , τS grow small (using L’Hopital’s rule). Moreover by increasing S,the quality of estimation can be improved but the training time will alsoincrease. Similarly, the NMSE increases as βd,k decreases.The NMSE in the LS estimate of hd,k in (3.19) is obtained using E[ntrk ntrHk ] =σ2PCτS(PCτS)2IMS and vtrH1 vtr1 = S asNMSE(hˆLSd,k) =σ2PCτStr((vtr1 ⊗ IM )H(vtr1 ⊗ IM ))Mβd,kS2(PCτS)2(3.27)=σ2βd,kSPCτS. (3.28)We see that the NMSE in LS estimate will grow unboundedly if either σ2grows large, or the argument in the denominator grows too small. Thedifference in the NMSE of MMSE and LS estimates can be straightforwardlycalculated to beNMSE(hˆLSd,k)−NMSE(hˆd,k)=σ4(βd,kSPCτS)2 + βd,kSPCτSσ2≥ 0, (3.29)for all βd,k, PC , τS , and S. Therefore the MMSE estimate of the directchannel will outperform the LS estimate. We can also observe the growthbehavior of both NMSEs with respect to noise and see that the NMSE inMMSE estimate grows slower than that in the LS estimate, making MMSErobust even at low values of signal-to-noise ratio (SNR).The NMSE in the estimates of h2,l,ks can be derived similarly by not-ing that from (2.3) that under RIRSl,k = IN , we have tr(E[h2,l,khH2,l,k]) =tr(β2,l,kIN ) = Nβ2,l,k. Therefore NMSE(hˆ2,l,k) =1Nβ2,l,ktr(Ψ˜2,l,k) and we283.5. Benchmark protocolget NMSE(hˆ2,l,k) =β2,l,kNβ2,l,k(N − β2,l,ktr(β2,l,kIN +σ2INβ1,lSMPCτS)−1),=σ2SMPCτSβ2,l,kβ1,l +σ2SMPCτS. (3.30)Also, the NMSE in LS estimate in (3.22) can be shown to beNMSE(hˆLS2,l,k) =1β2,l,kβ1,lσ2SMPCτS. (3.31)The performance gap between the MMSE and LS estimate of h2,l,k isNMSE(hˆLS2,l,k)−NMSE(hˆ2,l,k) ==σ4(β2,l,kβ1,lSMPCτS)2 + β2,l,kβ1,lSMPCτSσ2≥ 0, (3.32)because β2,l,k, β1,l and M are all non-negative. Therefore the MMSE esti-mate of h2,l,k outperforms the LS estimate.3.5 Benchmark protocolAs a benchmark (BM), we extend the protocol in [1] for a single IRSto the distributed IRSs setup. The derivation is similar to [1], and isrepeated here for completeness. Under this protocol, we derive the esti-mates of the cascaded IRS-assisted channels, i.e. the columns of H0,l,k =H1,ldiag(h2,l,k) ∈ CM×N , without exploiting full knowledge of H1,ls. As aresult, (3.3) is written asrBMs,k = (hd,k + H0,kvs) + ntrs,k, k = 1, . . . ,K, (3.33)where H0,k = H1diag(h2,k) ∈ CM×NL is the cascaded matrix of all IRS-assisted channel vectors and vs = diag(Θs) ∈ CNL×1 is the concatenatedbeamforming vector for all the IRSs in subphase s. Note that H1, h2,k andΘs are defined after (3.3). We let h¯d,2,k = [hTd,k,hT0,1,1,k, . . . ,hT0,1,N,k,hT0,2,1,k,. . . ,hT0,L,N,k]T ∈ CM(NL+1)×1, where h0,l,n,k is the nth column of H0,l,k, andcollect the observation vectors across S subphases to obtainrBMk = (Vtr ⊗ IM )h¯d,2,k + ntrk , k = 1, . . . ,K, (3.34)293.6. Simulations and Discussionwhere Vtr is given as (3.6) and ntrk is the same as in (3.5). To ensure thatthe left pseudo-inverse exists for Vtr, we must have S ≥ NL+ 1. Comparedwith the proposed protocol’s condition of S ≥ NLM +1, the benchmark wouldimpose almost an M times larger training overhead which will compromisethe downlink performance of the system.With optimal design for Vtr found to be the DFT matrix, we can performthe pseudo-inverse operation on (3.34) and getr˜BMk = h¯d,2,k +1S(Vtr ⊗ IM )Hntrk . (3.35)Write r˜BMk = [r˜BMT0,k , r˜BMT1,1,k , . . . , r˜BMT1,N,k , . . . , r˜BMTL,1,k , . . . , r˜BMTL,N,k]T ∈ CM(NL+1)×1,we can obtain the MMSE estimates of hd,k and h0,l,n,k as followshˆBMd,k = βd,kRBSkQd,kr˜BM0,k , (3.36)where Qd,k =(βd,kRBSk +σ2SPCτSIM)−1, andhˆBM0,l,n,k = rl,n,kβl,kh1,l,nhH1,l,nQl,n,kr˜BMl,n,k, (3.37)for n = 1, . . . , N , l = 1, . . . , L, k = 1, . . . ,K, whereQl,n,k =(rl,n,kβl,kh1,l,nhH1,l,n +σ2SPCτSIM)−1, (3.38)and βl,k = β1,lβ2,l,k, rl,n,k is the (n, n)th entry of the matrix RIRSl,k andh1,l,n is the nth column of β−1/21,l H1,l.Note that with HˆBM0,l,k = [hˆBM0,l,1,k, . . . , hˆBM0,l,N,k] ∈ CM×N , the expression forhˆBM2,l,k can be recovered under the benchmark protocol through the relation-ship HˆBM0,l,k = H1,ldiag(hˆBM2,l,k), when H1,l is known apriori.This is done for comparison with the proposed protocol where we directlyestimate h2,l,ks.3.6 Simulations and DiscussionIn the simulation, we set N = 32, L = 4, M = 8, PC = 1W , andτS = 50µs. Denoting (x, y) metres (m) as the coordinates, the BS is placedat (0,0)m and the four IRSs are placed at (0,100), (100,0), (-100,0) and(0,-100)m. The path loss factors are computed as βd,k =10−2.8d3.67BS−user, β2,l,k =303.6. Simulations and DiscussionFigure 3.2: Normalized Mean Square Error vs noise. Th. refers to thetheoretical expressions given. Corr. refers to the scenario of correlatedRayleigh fading channels.10−2.8d3.67IRSl−userwhich are non-LoS (NLoS) path loss factors computed at 2.5 GHzcarrier frequency for the 3GPP Urban Micro (UMi) scenario from TR36.814(also found in Section V of [1]), and β1,l =10−2.6d2.2BS−IRSlis the LoS path lossfactor, where d(.) stands for Euclidean distance for a particular link. Thenumber of sub-phases is set as S = NLM + 1 = 17 for the proposed protocol.Fig. 3.2 shows the NMSE curves for both MMSE and LS estimates of hd,kand h2,l,k (for the latter we plot1L∑Ll=1 NMSE(hˆ2,l,k)). The Monte-Carlosimulated values match the theoretical analysis in Sec. III-C. As expectedthe MMSE estimates performs better than LS estimates especially in thehigh noise regime. Also, the NMSE in h2,l,k is higher than that in hd,k dueto the multiplicative path loss factor β1,lβ2,l,k in (3.30) and (3.31).The orange curves in the figure show the NMSE for the correlated sce-nario where [RIRSl,k ]i,j = η|i−j| (similar definition for RBSk) and η = 0.95.The NMSE in the LS estimates is unaffected (not plotted for clarity of fig-ures) while that in the MMSE estimates is actually less for the correlatedcase [1].313.6. Simulations and DiscussionFigure 3.3: NMSE vs path loss factor for direct channel. Dashed linesrepresent a smaller number of sub-phases S than the dotted lines.Figure 3.4: NMSE vs path loss factor for IRS assisted channels.The NMSE vs the path loss factor is also plotted for the direct channeland IRS assisted channel in Fig. 3.3 and Fig. 3.4 respectively, where itcan be seen that the LS estimate performs worse than the MMSE estimatesfor all values of path loss factor. Increasing S reduces the NMSE in bothLS and MMSE estimates. However, increasing S means increasing trainingoverhead, which reduces the time left for downlink transmission.323.6. Simulations and DiscussionTable 3.1: Estimation accuracy per unit time 1NMSE×τC (s−1)L 2 6 8 10Proposed (×105) 1.809 1.784 1.776 1.782Benchmark (×105) 1.767 1.777 1.766 1.769Figure 3.5: Comparison of proposed and benchmark protocols. The blackdots are the Monte Carlo simulations of NMSE. Pink lines correspond tochannel estimation time for the proposed protocol (dashed) and benchmarkprotocol (solid). Blue lines correspond to NMSE for both protocols.In Fig. 3.5, we compare the proposed protocol which needs S = NLM + 1sub-phases to the benchmark protocol which needs S = NL+ 1 sub-phases,in terms of NMSE and training time versus L. To ensure a fair comparison,both protocols are tested for the NMSE for the cascaded IRS-assisted chan-nel. The NMSE in the benchmark protocol is lower due to it using a larger Sfor estimation, with the gap between the two NMSE curves becoming smallerwith L. On the other hand, the time required for CE grows linearly withthe number of IRSs in the benchmark protocol, which is detrimental for theoverall downlink performance, while that for the proposed protocol growswith a much smaller slope. In fact, the additional gap in required trainingtime for the BS relative to the proposed scheme becomes significantly largeas L (or N) increase. To better capture this trade-off, we tabulate in Tab.3.1 a figure of merit defined as 1NMSE×τC (s−1), which can be interpreted asthe measure of estimation accuracy per unit training time. Ideally, its value333.6. Simulations and Discussionshould be high. We observe that the proposed protocol actually performsbetter in terms of estimation accuracy per unit training time for all L. Thismakes the proposed protocol scalable to large-scale distributed deploymentof IRSs.In the next chapter, we derive the average SINR and formulate an op-timization problem dedicated to solving for optimized IRS-user associationparameters.34Chapter 4Average SINR and IRS-UserAssociationWith distributed IRSs deployed in a multi-user system, a natural nextstep is to associate certain IRSs to different users in an optimal man-ner to achieve a performance objective such as max-min average signal-to-interference and noise ratio (SINR). The association implies each IRS willbe tuned to a particular user. The motivation behind single-user associationwith an IRS is that it is user-centric and simplifies the optimal phase shiftsdesign since the design can be based on the local channel state information(CSI) of the associated user, thus reducing complexity.This chapter focuses on the average (ergodic) analysis of the SINR ateach user in the distributed IRSs-assisted multi-user MISO system, whereeach IRS is associated with a user in the system. For a particular IRS-userassociation pair, we choose the design for IRS phase shifts that would maxi-mize the received signal strength at that user. Under that design, we utilizestatistical tools to obtain a closed form expression for the average SINR ateach user. Then, we develop average SINR expression under IRS-user as-sociation parameters, optimized IRS passive beamforming, and maximumratio transmission (MRT) precoding at the BS. Next, we formulate and solvea max-min average SINR optimization problem to find optimal IRS-user as-sociation.While the CE scheme provided in the previous chapter can be applied intheory to any channel model, in this chapter, we focus on uncorrelated fadingat the BS and IRS to obtain tractable closed-form expression for the averageSINR under the umbrella of perfect CSI. However, this analysis is extendible(with some tedious algebraic manipulations) to correlated Rayleigh fadingand imperfect CSI. We formally state the assumptions made on the systemmodel below.Assumption 1. The correlation matrices of the direct and IRS-user channelare set to identity i.e. (RIRSl,k = IN and RBSk = IM ) which shifts the modelto uncorrelated Rayleigh fading.354.1. Downlink TransmissionFigure 4.1: Distributed IRSs MISO system model. Red lines are for down-link transmission.Assumption 2. The analysis is done under the assumption that the BS hasperfect CSI to design the precoder and IRS reflect beamforming vectors.Assumption 2 is made to ensure a tractable analysis since the distributedIRSs MISO system is complex where many links are involved. The resultswe obtain are relevant since they pave the way for imperfect CSI analysisand can serve as meaningful upper bounds for imperfect CSI scenarios.Assumption 3. The reflection coefficients αl,n = 1,∀n, ∀l.Under idealized conditions, metasurfaces can achieve unity magnitudeamplitude in the scattered field even with arbitrary phase [39]. Also, mostcurrent work on IRSs make this assumption.4.1 Downlink TransmissionRecall from Chapter 2 that in the downlink, the kth user receivesyk = hHd,kx +L∑l=1hH2,l,kΘHl HH1,lx + nk, (4.1)364.1. Downlink Transmissionwhere nk ∼ CN (0, σ2) is the channel noise and the channels are given in(2.3), (2.4), and (2.5), as well as the transmit signal vector x has the struc-ture shown in (2.12). The downlink transmission scheme is described elab-orately in Chapter 2 and illustrated in Fig. 4.1. To better visualize theinstantaneous SINR expression γk for user k, we can separate the receivedsignal yk into a desired term ykD and interference and noise term ykINwritten asykD =√pkhHd,kfksk +√pkL∑l=1hH2,l,kΘHl HH1,lfksk, (4.2)ykIN =K∑t6=k,t=1(√pthHd,kftst +√ptL∑l=1hH2,l,kΘHl HH1,lftst)+ nk, (4.3)The direct and IRS cascaded channels in the downlink can be rewrittencollectively ashHk = hHd,k +L∑l=1hH2,l,kΘHl HH1,l (4.4)= hHd,k +L∑l=1vHl HH0,l,k, (4.5)whereH0,l,k = H1,ldiag(h2,l,k) =√β1,lalbHl diag(h2,l,k) (4.6)Note that H0,l,k ∈ CM×N and vl = vec(Θl) ∈ CN×1, where vec(·) is usedto create a vector from a diagonal matrix, and diag(·) creates a diagonalmatrix out of a vector.At this stage, we can introduce association variables λl,k ∈ {0, 1} whichare binary variables denoting association between the lth IRS and kth user.The collected channel (4.4) with association variables can be reformulatedashHk (λk) = hHd,k +L∑l=1λl,khH2,l,kΘklHl HH1,l +L∑l=1(1− λl,k)hH2,l,kΘlHHH1,l, (4.7)= hHd,k +L∑l=1λl,kvklHl HH0,l,k +L∑l=1(1− λl,k)vHl HH0,l,k, (4.8)374.1. Downlink Transmissionwhere λk ∈ B1×L is the row vector of associated or non-associated IRSswith respect to user k. To make the association between the optimized(tuned) beamforming vector vl to a particular user k, we replace vl by vkllhenceforth, where kl is the value of k for which λl,k = 1. In addition, wecan rewrite (4.1) with association parameters, simply asyk = hHk (λk)x + nk. (4.9)We find optimal vklHl for IRS l associated with user k such that λl,k = 1 as tomaximize the channel gain via that IRS to that user. Note that for the multi-user case (which we have), there is no known closed-form optimal solutionfor vklHl that maximizes the SINR. Furthermore, finding vklHl for each IRSl such that SINR at its associated user is maximized will result in a highlyintractable joint optimization problem involving the beamforming vectorsof all IRSs and their association parameters with the users. Therefore,we assume that for an IRS l associated with user k the channel gain ismaximized via that IRS to the user, similar to [29]. Hence, the passivebeamforming optimization problem for an IRS l associated with user k canbe defined as(P0) maxvl‖hHd,k + vHl HH0,l,k‖2 (4.10)s.t. |vl,n| = 1, ∀n. (4.11)Note that P0 finds the optimized beamforming vector vkll for user k, whenλl,k = 1. Expanding the objective function yields ‖hHd,k‖2+2〈vHl HH0,l,k,hHd,k〉+‖vHl HH0,l,k‖2, where we can drop the first term since it does not depend onthe optimization variable to get(P0′) maxvl2〈vHl HH0,l,k,hHd,k〉+ ‖vHl HH0,l,k‖2 (4.12)s.t. |vl,n| = 1, ∀n. (4.13)To make this more general, we drop the subscripts and define(P0”) maxv2〈vHHH ,hH〉+ ‖vHHH‖2 (4.14)s.t. |v| = 1, ∀n. (4.15)Proposition 4.1. Under Assumptions 2 and 3, and by copying the structureof H = abHdiag(c) as in (4.6) and c =√β1,ldiag(h2,l,k), we can formulatethe optimal solution of (P0”) asvkl = exp(j∠diag(cH)b) exp(j∠aHh) (4.16)384.1. Downlink TransmissionProof. To find the angle of a complex (Hermitian) inner product, it sufficesto find <(〈vHHH ,hH〉) = ∑Mm=1 |gm||hm| cos(∠gm −∠hm). Where <(〈·, ·〉)denotes the real part of the inner product of two complex vectors, andgm (hm) is the mth component of g = Hv (h). This expression achievesits maximum value when ∠gm = ∠hm, equivalently (∠vHdiag(cH)baH =∠hH) and ∠· for element-wise phase of the vector. Noting that when wechoose vH as in (4.16) we satisfy the relation, since we achieve ( ∠αvh =∠h), where αv = ‖diag(cH)b‖2‖a‖2 is a scalar.Similar argument can be done for the second term in the objective func-tion (4.14), where maxv‖vHHH‖2= maxvvHdiag(cH)baHabHdiag(c)v (4.17)= maxv‖a‖2vHdiag(cH)bbHl diag(c)v, (4.18)= maxv‖a‖2|vHdiag(cH)b|2, (4.19)= maxvn‖a‖2∣∣∣∣ N∑n=1|vn||cn||bn| exp(j(∠v∗n + ∠c∗n + ∠bn))∣∣∣∣2, (4.20)= ‖a‖2(N∑n=1|cn||bn|)2, (4.21)which achieves it maximum value with optimal reflection beamforming vec-tor given in (4.22). Note that c∗n is the complex conjugate of the complexnumber cn. Moreover, bn, v∗n, and cn denote the nth elements of b, vH , andc, respectively.We can now write the general result in (4.16) with proper notation forthe beamforming vector of IRS l dedicated to user k, such that λl,k = 1, asvkll = exp(j∠diag(hH2,l,k)bl) exp(j∠aHl hd,k). (4.22)To shed some light on how to maximize a squared norm given unit-modulus constraint, a the second term in the objective function (4.12) isplotted in Fig. 4.2 with optimal reflection beamforming vector and (4.20) isalso plotted against −1 ≤ exp(j(∠v∗n + ∠c∗n + ∠bn)) ≤ 1 encompassing thesum of phases of all the terms inside the squared norm. Furthermore, (4.21)is plotted as a validation step.394.1. Downlink TransmissionFigure 4.2: Maximizing a squared norm with unit-modulus constraints. Thisshows that the solution for reflection beamforming vector is the optimalsolution.Once we substitute the optimized beamforming vector given in (4.22) in(4.8) we obtainhk(λk) = |hHd,k| · ej∠hHd,k+ (4.23)L∑l=1λl,kej∠hHd,kalN∑n=1|h2,l,k,n||bl,n|aHl+L∑l=1(1− λl,k)vHl HH0,l,kwhere ∠ and |.| refer to element wise phase and magnitude of the vectors. Anintuitive observation to make is that this solution for vkll negates the phasesfrom the the cascaded IRS channel HH0,l,k and lies in-phase with the directchannel hHd,k which indicates that the signal coming from the direct channeland from the cascaded IRS channels at the associated user will combinecoherently and constructively.404.2. Average SINR4.2 Average SINRThe SINR γk at user k under the transmission model just described withIRS-user association parameters is given asγk =pk‖hHk (λk)fk(λk)‖2∑Kt6=k,t=1 pt‖hHk (λk)ft(λt)‖2 + σ2. (4.24)In general, this SINR can not be perfectly computed at the user except whenperfect CSI is available at the receiver which is a moot possibility. The down-link SINR is an important metric of performance which can be fed back tothe BS, or the BS can discover it given sufficient information of the channels,and aids in the decision-making process for certain hyper-parameters of thedistributed IRS assisted MISO system: BS active beamforming, power con-trol at the BS, and IRSs passive beamforming. In this work, we focus on thelatter which can split into IRSs re-configurability and IRS-user association,and assume the previous two are given or fixed.Since practical wireless channels are often modelled having random fad-ing (c.f. sec. 2.2), we direct our attention to average SINR γ¯k which can betracked at the user and depends only on large-scale statistics of the chan-nels. Large-scale statistics such as path loss factor and correlation matriceschange slowly as compared to the channels themselves. Therefore, the userwill not need to compute the instantaneous SINR in each coherence interval(more on coherence interval is found in sec. 2.2) and instead use γ¯k as metricfor performance. By using γ¯k as a performance metric, the association pa-rameters will not need to be updated on the coherence time scale but aftermultiple coherence intervals depending on the varying large-scale statisticsof the channels. Each IRS will then use local CSI of the associated user todetermine its optimal configuration as in (4.22).Next, we formulate Lemma 4.2 which conveys the expectation of a ratioof terms defining the structure of SINR.Lemma 4.2. The average SINR γ¯k can be approximated asE[γk] ≈ pkE[‖hHk (λk)fk(λk)‖2]∑Kt6=k,t=1 ptE[‖hHk (λk)ft(λt)‖2] + σ2(4.25)Proof. For average SINR, we obtain the expectation of a ratio, where E[γk] =E[XY ]. Now assume E[X] = µX and E[Y ] = µY , we can expand the ratio(using the bivariate first-order Taylor series expansion) around the point414.2. Average SINRFigure 4.3: Tightness of approximation for γ¯k(µX , µY ) so thatXY=µXµY+1µY(X − µX)− µXµ2Y(Y − µY ) +R (4.26)where R is the remainder of the the other order terms in the expansion.Taking the first moment of this ratio providesE[XY]= E[µXµY+1µY(X − µX)− µXµ2Y(Y − µY ) +R](4.27)≈(i)E[µXµY]+ E[1µY(X − µX)]− E[µXµ2Y(Y − µY )](4.28)=(ii)E[X]E[Y ](4.29)where the last line is a ratio of expectations, (i) follows from dropping thelesser order terms R, and (ii) follows since the expectation of the secondand third terms is zero.Remark 1 This approximation for E[γk] is justified as follows: since Xand Y contain quadratic forms, the variance relative to the mean of theserandom variables is small [40], [41], [42]. Hence, this approximation is tight(as also verified numerically using the same set-up in Tab. 4.1 and illustratedin Fig. 4.3).424.2. Average SINRTable 4.1: Simulation parameters.Parameter ValueArray parameters:BS configuration Uniform linear arrayIRS configuration Uniform planar arrayAntenna gain 5dBidBS , dIRS 0.5λNoise level −60dBmPath Loss:Model 10−C/10dαC (Fixed loss at d = 1m) 25dB (β1), 30dB (β2,k, βd,k)α (Path loss exponent) 2.2 (β1), 3.67 (β2,k, βd,k)Penetration Loss:hd,k 20dBh2,l,k 5dBSystem Dimensions:N 8, 16L 8K 4M 164.2.1 Performance Under Maximum Ratio TransmissionWith maximum ratio transmission (MRT) and using Assumption 2, thedigital precoder is found to befk(λk) =hk(λk)√E[‖hk(λk)‖2], (4.30)Note that MRT is optimal for single-user systems (refer to sec. 2.4). Nev-ertheless, it is shown in [43] that for large system dimensions such as whenwhen M becomes large MRT reaches the performance of optimal linear pre-coding due to channel hardening and favorable propagation. Another reasonMRT is chosen as the BS precoding technique is because its simpler thanits counterparts (for example. ZF precoding). Since this work focuses onIRSs (and not on BS beamforming techniques), we resort to MRT to find434.2. Average SINRtractable expressions. Substituting the MRT precoder in (4.25) yieldsγ¯k =pkE[‖hHk (λk)‖4]E[‖hHk (λk)‖2]∑Kt6=k,t=1 ptE[‖hHk (λk)ht(λt)‖2]E[‖hHt (λt)‖2]+ σ2(4.31)4.2.2 Main ResultsIn this section, we derive all the terms in (4.31). This would require usto find the first and second moments of a complex Gaussian quadratic formabbreviated as (CGQF). Furthermore, we combine these results to obtainthe average SINR expression under MRT with association parameters. Wecan see that hk ∈ CM×1 ∼ CN (0,Rk) since it contains the addition ofcomplex Gaussian vectors. As a result, hk is a complex Gaussian vectorwith zero mean and correlation matrix Rk, which will be worked out later.Expanding the gain of the collective channel in (4.8) givesE[‖hk(λk)‖2] = E[hHd,khd,k + 2L∑l=1λl,kvklHl HH0,l,khd,k+ (4.32)L∑l=1L∑l¯=1λl,kλl¯,kvklHl HH0,l,kH0,l¯,kvkll¯+L∑l=1L∑l¯=1(1− λl,k)(1− λl¯,k)vHl HH0,l,kH0,l¯,kvl¯+ 2L∑l=1(1− λl,k)vHl HH0,l,khd,k + 2L∑l=1L∑l¯=1(1− λl,k)λl¯,kvHl HH0,l,kH0,l¯,kvkll¯].Notice that some of these terms cancel due to independence betweenchannels h2,l,k and hd,k. Moreover, vkll is of the form given in (4.22). Now,we shift our attention to computing this expectation which is demonstratedin the following lemma.Lemma 4.3. After computing the expectation, (4.32) is found to beE[‖hk(λk)‖2] = Mβd,k +L∑l=1λl,k(√MN√β1,lβ2,l,kβd,kpi2(4.33)+ tr(HH1,lH1,lΣv˜∗l )− β2,l,ktr(HH1,lH1,l))+L∑l=1β2,l,ktr(HH1,lH1,l)444.2. Average SINRNote that E[‖hk(λl,k)‖2] = tr(Rk).Proof. Details found in Appendix C.It can be seen that average channel gain increases with respect to M . Asfor L and N , it depends on the user association. For instance, if user k doesnot have any IRSs associated so that λl,k = 0, ∀l, then we still gain somethingfrom all the non-associated IRSs. Lemma 4.4 showcases a significant resultin CGQF and thus completes the analysis of the numerator for γ¯k.Lemma 4.4. The fourth moment E{‖hk(λl,k)‖4} in the numerator of γ¯k(4.31) is derived using a result of a CGQF’s second moment [40] asE[‖hk(λk)‖4] = tr(R2k) + tr(Rk)2. (4.34)Proof.E[‖hk(λk)‖4] = V ar{‖hk(λk)‖2}+ (E{‖hk(λk)‖2})2 (4.35)= V ar{‖hk(λk)‖2}+ tr(Rk)2 (4.36)= tr(R2k) + tr(Rk)2, (4.37)where V ar{‖hk(λk)‖2} is taken from the following result of the second mo-ment of the CGQF [44]V ar{‖hk(λl,k)‖2} = tr(R2k). (4.38)One can see from Lemma 4.4 that this result relies heavily on the cor-relation matrix Rk, as the size of Rk increases so does the channel gain.Increasing M increases quadratically the fourth-order moment in (4.36).However, increasing the size of Rk also affects the interference seen below.Lemma 4.5. In the interference term∑Kt6=k,t=1 ptE[‖hHk (λk)ht(λt)‖2]E[‖hHt (λt)‖2], the nu-merator is found to beE[‖hHk (λk)ht(λt)‖2] = tr(RtRk). (4.39)Proof.E[‖hHk (λk)ht(λt)‖2] = E[hHk (λk)ht(λt)hHt (λt)hk(λk)]. (4.40)454.2. Average SINRSince ht(λt) and hk(λk) are independent, we can use conditional expectationasEhk [hHk (λl,k)Eht [ht(λl,k)hHt (λl,k)|hk(λl,k)]hk(λl,k)], (4.41)= Ehk [hHk (λl,k)Rthk(λl,k)], (4.42)= Ehk [tr(hHk (λl,k)Rthk(λl,k))], (4.43)= Ehk [tr(Rthk(λl,k)hHk (λl,k))], (4.44)= tr(RtEhk [hk(λl,k)hHk (λl,k)]), (4.45)= tr(RtRk). (4.46)It is straightforward to note that when Rk increases in size, so does theoverall interference. Intuitively, increasing the channel gain for one user,increases the interference sensed at another. Previously, Corollaries 4.8 and4.9 inspected the impact of variations of the correlation matrices and thelevel of diversity between correlation matrices for different users. The con-clusion is that having diverse channels lends to a higher overall SINR, whilethe presence of similar channels, or similar correlation matrices leads to adegradation in SINR performance.To obtain expressions in (4.46) and (4.37) we need to find the expressionfor Rk which is next computed. Define the correlation matrix for hk to beRk =E[hd,khHd,k + 2L∑l=1λl,khd,kvklHl HH0,l,k (4.47)+L∑l=1L∑l¯=1λl,kλl¯,kH0,l,kvkll vklHl¯HH0,l¯,k+L∑l=1L∑l¯=1(1− λl,k)(1− λl¯,k)H0,l,kvlvHl¯ HH0,l¯,k],where other terms are zeros as shown in Appendix C. The expression givenin the next lemma for Rk unifies the effect of the system dimensions M,N,L,path loss factors β1,l, β2,l,k, βd,k, and association parameters λl,k on γ¯k whichdepends on Rk. Sec. 4.3 optimizes the association parameters thus tuningRk, k = 1, . . . ,K as to maximize the minimum γ¯k.Lemma 4.6. Using Assumption 1, the correlation matrix for the collectivechannel in (4.8) is shown to be464.2. Average SINRRk =βd,kIM +L∑l=1λl,k(2√β1,lβ2,l,kβd,kNpi4√Mej∠H1,lHH1,l (4.48)+ H1,lΣv˜kllHH1,l − β2,l,kH1,lHH1,l)+L∑l=1β2,l,kH1,lHH1,l.Proof. The proof is postponed to Appendix D.Now, we can combine these results in the following Theorem 4.7 to finda closed-form expression for the SINR at user k, note that this closed-formexpression manifests due to the Rayleigh fading nature of the direct andIRS l-user k channels, and the MRT precoding used at the BS. Moreover, itwould be more accurate to call this an approximation of the average SINRfollowing with Lemma 4.2.Theorem 4.7. Using the results from Lemmas 4.3, 4.4, 4.5, the ergodicSINR under MRT with user association parameters is given asγ¯k =pktr(Rk)(tr(R2k) + tr(Rk)2)∑Kt6=k,t=1pttr(Rt)tr(RtRk) + σ2. (4.49)Without loss of generality (WLOG), set pk = cktr(Rk)∀k, where ck isa coefficient that will keep this expression from violating the average powerconstraint in (2.13) to getγ¯k =ck(tr(R2k) + tr(Rk)2)∑Kt6=k,t=1 cttr(RtRk) + σ2. (4.50)Corollary 4.8. When the correlation matrices of different users are orthog-onal, i.e. RtRk = 0M . This average SINR in (4.50) is simplified toγ¯k,Up =ck(tr(R2k) + tr(Rk)2)σ2(4.51)This hardly occurs in practice but implies that the spatial correlationmatrices are vastly different between users, which also means that the num-ber of M effective antennas for each user reduces to M/K in order to supportthe induced orthogonality [30]. This special case essentially decomposes themulti-user channel into K orthogonal single-user channels without interfer-ence.474.2. Average SINRFigure 4.4: Upper and lower bounds of γ¯k. The black solid line is linear inσ2 since the upper bound sets the interference to zero which changes SINRto signal-to-noise ratio (SNR). This figure is simulated using Tab. 4.1Corollary 4.9. For the worst case when the correlation matrices of theusers are all identical, i.e., Rt = Rk [30], we obtainγ¯k,Low =1 + tr(Rk)2/tr(R2k)(K − 1) + σ2/cktr(R2k). (4.52)This is also an extreme case since it implies that the users are not suf-ficiently physically separated. Essentially, Corollaries 4.8 and 4.9 can bethought of as meaningful bounds for γ¯k, where we can check that γ¯k,Low ≤γ¯sk ≤ γ¯k,Up.Proof. To check that γ¯sk ≤ γ¯k,Up, it suffices to check that tr(RtRk) is non-negative, one can see that the definition of tr(RtRk) in (4.42), is the expecta-tion of a non-negative value, since hHk (λl,k)Rthk(λl,k) ≥ 0, for Rt is a covari-ance matrix which is positive semi-definite matrix. Hence, tr(RtRk) ≥ 0.We can also check that interference is a power term which can only be non-negative.As for γ¯k,Low ≤ γ¯sk, we need to make a fair comparison so we assumethat the channel gains ‖hk‖2 for all users, k = 1, . . . ,K are equal.484.3. IRS-User Association Optimization ProblemThus, it suffices to show thatK∑t6=k,t=1tr(RtRk) ≤ (K − 1)tr(R2k), (4.53)=⇒K∑t6=k,t=1hHk (λk)ht(λt) ≤ (K − 1)hHk (λk)hk(λk), (4.54)=⇒K∑t6=k,t=1hˆHk (λk)hˆt(λt) ≤ (K − 1)hˆHk (λk)hˆk(λk), (4.55)which is true, since the inner product will rely on the angle between thechannels, where hˆk and hˆt are unit vectors denoting the directions of hk andht, respectively.Lastly, Fig. 4.4 illustrates the tightness of the lower bound and upperbound as noise variance σ2 is relatively large.In the next section, we formulate the max-min SINR problem using theaverage SINR derived in here to find the optimal IRS-user association pairs.4.3 IRS-User Association Optimization ProblemFor this multi-user system, we consider max-min SINR as the metric toimprove the performance and fairness of the system. This objective for anIRS-assisted system was previously tackled in [12]. We can formulate themax-min average SINR problem which promotes user fairness as(P1) maxΛminkγ¯k (4.56)s.t. ‖λl‖1 = 1, 1 ≤ l ≤ L, (4.57)‖λk‖1 ≤ L, 1 ≤ k ≤ K, (4.58)λl,k ∈ {0, 1}, 1 ≤ l ≤ L, 1 ≤ k ≤ K, (4.59)Here, Λ ∈ BK×L denotes the binary association matrix between all users andIRSs, λl ∈ BK×1 is a column vector which one element equalling 1 and therest zeros, the index of 1 in the vector points to the associated user k for thelth IRS, and λk ∈ B1×L is a row vector of associated and non-associated LIRSs for a particular user k. Constraint (4.57) indicates that each IRS mustbe assigned to only one user, where ‖x‖1 =∑i |xi| is the l1-norm, and xisare elements of a vector x. On the other hand, constraint (4.58) considers494.3. IRS-User Association Optimization Problemthe user’s perspective, as any user has at most L IRSs associated to it.In addition, the binary restriction (4.59) makes this problem combinatorialthus hard to tackle. This is classified as a non-convex mixed-integer non-linear programming (MINLP) problem, which is NP-hard. We can find theoptimal solution for the association by exhaustive search, but the complexityis prohibitive and in the order O(KL). In the next subsections, we definethe search space (codebook) for the exhaustive search method and define asuccessive refinement algorithm.4.3.1 Exhaustive SearchIn exhaustive search, we run over every possible IRS-user associationcombination taken from a specific codebook. To create the codebook C ∈BK×L×KL that can generate all possible KL Λ combinations, we need totransform the problem into a simpler and already solved base conversionproblem. First, define the rules according to (P1):1. Each IRS l is associated to a only one user k.2. User k has at most L IRSs associated to it.3. λl,k is a binary variable.We assign a number N ∈ Z+, where N can take values from 1, . . . ,KL touniquely represent each matrix ΛN . We can then find the base K represen-tation of N , which we store as a row vector r ∈ FK , where FK is a finiteGalois field with order K. Each element k in r, k = 1, . . . ,K denotes theposition index of 1 in λl ∈ BK×1 where the rest of its elements are all zeros,because of (4.57). Thus, L of the λl columns are assembled as ΛN .4.3.2 Successive Refinement AlgorithmIn a bid to observe the gains of user-associations, one low-complexitymethod referred to as successive refinement (SR) is used solve this optimiza-tion problem. This method has a complexity in the order of O(L) which isbetter than the well-known branch and bound algorithm for mixed integerlinear programming problems which can reach a complexity as high as thatof exhaustive search O(KL). SR algorithm matches closely to the optimal504.4. Simulations and Discussionsolution given by exhaustive search as shown next in the simulations.Algorithm 1: Successive Refinement Algorithm to Solve (P1)Result: Find desired IRS-user association Λ∗Initialize Λ association matrix based on nearest rule;Initialize v∗l using (4.22); Compute γ¯ks in (4.49) based on Λ and v∗l ;Set the iteration number i = 1, Set state = true ;while state is true dok¯ = mink(γ¯k) bottleneck user;for l = 1 : L doif λl,k¯ = 0 thenSave previous association as j; λl,k¯ = 1, λl,j = 0;Update all γ¯ks ;γ˜(l) = mink(γ¯k) bottleneck user SINR;reset λl,k¯ = 0, λl,j = 1;endendFind l¯ = maxl(γ˜) best assigned IRS for bottleneck user k¯ ;λl¯,k¯ = 1, λl¯,j = 0; i = i+ 1;Update all γ¯ks based on current IRS-user association.;if γ¯k¯(i) < γ¯k¯(i− 1) thenstate = false;γ¯k¯(i) = γ¯k¯(i− 1);λl¯,k¯ = 0, λl¯,j = 1;endend4.4 Simulations and DiscussionThe simulation parameter values are taken from Table 4.1. The BSis located at (0, 0)m coordinates. The IRSs are deployed in an arc withradius 100m. The users are set at an arc with radius 85m with one userset further away at a radius of 130m. The path loss factors are computedat 2.5 GHz carrier frequency for the 3GPP Urban Micro (UMi) scenariofrom TR36.814 (also found in Section V of [1]). The LoS channel modelwas used for H1,l and the non-LOS (NLOS) channel model was used togenerate path loss factors for h2,l,k and hd,k, where d in the expression for10−C/10dα denotes the Euclidean distance between different links. We alsoassume penetration losses given in Tab. 4.1 for the direct and IRS-user514.4. Simulations and DiscussionFigure 4.5: No Association Set-uplinks due to obstacles in the environment, since the IRSs are envisionedto be elevated the penetration loss is significantly less than that of groundusers. The distributed IRSs deployment is illustrated in Fig. 4.5 where theusers and IRSs are spread on arc as described earlier and a user (numbered2) is deliberately positioned further away from the BS and from the IRSs.This deployment scenario highlights the effect of IRSs in helping edge users,as the algorithm would need to assign more IRSs to that minimum userwhich is by inspection assumed to be user 2.524.4. Simulations and DiscussionFigure 4.6: Nearest Rule AssociationFigure 4.7: Successive Refinement AssociationIn Fig. 4.6, the associations are updated based on nearest rule (NR),where IRSs with the shortest distance to a particular user would be assigned.4.7 shows the association resulting from the SR algorithm, which focuses onelevating the SINR for the bottleneck user. We noticed earlier by inspectionthat user 2 was the bottleneck user. Indeed, SR managed to capture thatobservation and assign more IRSs to user 2. We will next see that by doingso, the system achieved a higher overall minimum user average SINR. Plot-ting the minimum user SINR with SINR derived in (4.49) against N in Fig.4.8, we notice that successive refinement algorithm explained in Algorithm1 achieves a close performance to that of exhaustive search. With exhaus-tive search having complexity in the order of O(KL) it does not scale wellwith many users in the system and many more IRSs. Hence the successiverefinement (SR) algorithm which scales linearly with L is sufficient to solvethe IRS-user association problem. Moreover, the minimum user SINR im-534.4. Simulations and DiscussionFigure 4.8: Minimum user SINR against number of elements N in each IRSwith exhaustive search, successive refinement, nearest rule, random assign-ment, and no IRSs.proves with increasing number of elements N but not in the order of N2.Generally for single user systems where there is no interference, deployingIRSs commonly achieves gains in the order N2 in the receiver’s SNR whichis a combination of a gain for the received power collected from the N IRSelements and a beamforming gain factor N collected after passive beam-forming [12], [5]). The reason we do not see an N2 gain is when we increaseN we not only benefit the channel gain but also heighten the interferencefor the minimum user SINR.Note that nearest rule (NR) and random assignment perform similarly,which is attributed to the fact that they do not necessarily help the mini-mum user. Thus, even when N increases, the performance remains nearlyconstant. However, NR and random assignment do perform better thanhaving no IRSs in the system, as we mentioned before, non-associated IRSsstill contribute to users which is seen in Lemma 4.3 and the discussion un-derneath it.544.4. Simulations and DiscussionFigure 4.9: Minimum user SINR against number of distributed IRSs L.Both theoretical result in Theorem 4.7 and Monte Carlo results are plotted.Solid lines represent N = 16, while dashed lines represent N = 8.The Monte-Carlo simulated values averaged over 1000 channel realiza-tions match the theoretical analysis in sec. 4.2.2 as shown in Fig. 4.9. Thisvalidates Theorem 4.7, and proves the ergodic analysis is accurate. Sinceexhaustive search does not scale well with increasing L, it was discardedfor the feasibility of this plot. However, it was already established in theprevious figure that SR performs closely to that of exhaustive search whichis optimal (also discussed in [29]). Increasing number of IRSs is beneficial,since there are more ways to assign them to users and improve performance.Performance can be further improved if the locations of the IRSs are care-fully selected as to broaden coverage. However, that is not the focus of thiswork.The solid lines represent a doubled number of IRS elements than thatrepresented by the dashed lines and perform better for the minimum userSINR as expected. Moreover, we see that random assignment is yet againclosely matched with NR but they both increase the minimum user SINRalthough at a slower rate than SR. Deploying more and more IRSs meanseven edge users are probable to have nearby IRSs assigned to them.554.4. Simulations and DiscussionFigure 4.10: Minimum user SINR for centralized and distributed IRSs de-ployment scenarios against N. Solid lines represent L = 16, dashed linesrepresent L = 8.Fig. 4.10 depicts the marginal improvement the minimum user SINRgains when the IRSs are distributed as opposed to having the IRSs be inone unit or in a centralized deployment. This is because of the spatialdiversity the IRSs distributed placement offers. Centralized deployment ofIRSs compacts the L IRSs which can be effectively represented by a singlelarge IRS unit with NL elements. In reality, the IRSs and BS are co-located(explained in Chapter 2 sec. 2.3) because of the LoS nature of the BS-IRS l channel, the far-field assumption, and lack of nearby scatterers, whichlessens the degrees of freedom especially when the IRSs are deployed closetogether. Having the IRSs spread out increases the degrees of freedom to atleast L which enhances the effect of the IRSs on the system.564.4. Simulations and DiscussionFigure 4.11: Distributed IRSs with L = 16 deployment scenarioFigure 4.12: Centralized IRSs with L = 16 deployment scenarioTo visualize this deployment scenario for L = 16, we plot in Fig. 4.11and Fig. 4.12 the SR optimized association for both the distributed andcentralized IRSs, respectively. One can see that the centralized IRSs arenot majorly assigned to the previously inspected minimum user which wasassumed to be 2, since there are high penetration losses between the BS andthe other users through the direct link hd,ks. Hence, the lack of nearby IRSsis severe on the system performance, and the minimum user SINR bouncesback and forth between the users furthest from the IRSs in each iteration ofSR algorithm largely due to this deployment scenario.Now, we turn our attention to the convergence of the SR algorithm.Looking at Fig. 4.13, we observe how each updated iteration of the SRalgorithm matches an improvement in the minimum user SINR as expected.574.4. Simulations and DiscussionFor L = 8, the convergence actually occurs at the ninth iteration. Thishappens when there is no further IRS-user re-association that could improvethe system’s minimum SINR.Figure 4.13: Minimum user SINR vs number of iterations in SR algorithmFigure 4.14: Number of iterations needed for SR to converge vs number ofIRSs L. With impact of user location on convergence rate of SR algorithmWe next plot the effect of the user 2 location on the convergence rate ofthe SR algorithm. When user 2 is at 125m radius, then some IRSs wouldbe already assigned due to nearest rule initialization, thus leading to lessnumber of iterations needed to converge. As the second user moves away,the iterations increase as the user requires more IRSs to achieve similar per-584.4. Simulations and Discussionformance to other users. Finally, we notice that the trend remains linear inL, which is the motivation behind the suboptimal SR algorithm as opposedto other existing high-complexity methods.59Chapter 5Conclusion5.1 SummaryThis work proposes a low-complexity CE protocol for the distributedIRSs-assisted MISO system and derives both LS and MMSE estimates ofthe BS-users direct channels and IRSs-users channels. The NMSE in alldeveloped estimates is analytically studied. By leveraging the LoS natureof the BS-IRSs channel matrices, the proposed protocol achieves a betterestimation accuracy per unit training time as compared to the benchmarkprotocol.The work also contains the derivation of the average SINR under MRTprecoding with IRS-user association parameters. Under the assumption ofuncorrelated Rayleigh fading and perfect CSI, a tractable average SINR ex-pression was derived using statistical tools. We formulate a max-min averageSINR problem to optimize the IRS-user association parameters. We solvethe max-min average SINR problem using a low-complexity algorithm calledsuccessive refinement. Simulations results show that successive refinementachieves similar performance as exhaustive search. Moreover, we study adeployment scenario where the effect of increasing the number of IRSs L,increasing the number of elements in each IRS N , the physical distributionof the IRSs, and user location in the system is presented.5.2 ResultsThe main results of this work are summarized below:− Chapter 3 provides a novel channel estimation (CE) protocol for dis-tributed IRSs.− Chapter 3 provides a comparison between a benchmark CE protocoland the proposed CE protocol for channel estimation that shows thatour proposed protocol has higher estimation accuracy per unit timeby about 1%.605.3. Limitations of the Study− Chapter 4 provides an average SINR expression under MRT precodingthat is validated using Monte Carlo simulations.− Chapter 4 compares between deploying the IRSs in a centralized ordistributed manner, showing the minimum user SINR gets quadrupledfor the distributed case as opposed to centralized for N = 40.− Chapter 4 shows a linear gain for minimum user SINR as L or Nincrease.− Chapter 4 shows that the sub-optimal successive refinement algorithmperforms closely to the optimal solution given by exhaustive search.5.3 Limitations of the Study− The NLoS part of the BS-IRS channel is not taken into considerationin the CE protocol scheme.− MRT precoding at the BS only maximizes the SNR and often used fora single-user system, the analysis needs to consider ZF precoding atthe BS.− Strict assumptions are made to ensure a tractable analysis of the er-godic SINR. However, some assumptions are impractical such as un-correlated Rayleigh fading or perfect CSI. We need to extend the sys-tem model to a more practical implementation and bridge the gap toindustry.5.4 Future DirectionsAn important future direction is to develop low-overhead CE protocolsfor scenarios where the NLoS paths in BS-IRS l channels are non-negligible.Another is to extend the theoretical work for average SINR for imperfectCSI case and thus make the connection to the developed CE protocol. Onepossible direction to take is to derive the average SINR without first impos-ing a solution for reflect beamforming vector at each IRS, and then optimizeover both association parameters and reflect beamforming.Another exciting direction is the study of practical implementations ofIRSs from the communication perspective, for example defining path lossmodels based on physical optics theory, as well as modelling practical phase615.4. Future Directionsshifts at the IRS elements based on electromagnetic theory. 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Since ntrk and hd,kare independent Gaussian vectors we obtainE[hd,kr˜trH0,k ] = E[hd,khHd,k] = βd,kRBSk , (A.3)= E[hd,k(hd,k +1S(vtr1 ⊗ IM )Hntrk)H],(A.4)and E[r˜tr0,kr˜trH0,k ] =E[hd,khHd,k] +(vtr1 ⊗ IM )HE[ntrk ntrHk](vtr1 ⊗ IM )S2(A.5)= βd,kRBSk +σ2S2(PCτS)(vtr1 ⊗ IM )HISM (vtr1 ⊗ IM ),= βd,kRBSk +1Sσ2IMPCτS, (A.6)69Appendix A. Proof of Theorem 3.1where the last two steps follow from observing thatE[ntrk ntrHk]= E[ntrs,kntrHs,k ⊗ IS](A.7)=1(PCτS)2E[Ntrs xp,kxHp,kNtrHs]⊗ IS=PCτS(PCτS)2ISE[Ntrs NtrHs ]⊗ IS=σ2(PCτS)ISM ,and (A.6) follows from vtrH1 vtr1 = S under the optimal DFT design for Vtr.Therefore using (A.3) and (A.6) in (A.1) we obtain (Σ−10 )2)hˆd,k = βd,kRBSk(βd,kRBSk +σ2IMSPCτS)−1r˜tr0,k. (A.8)Since hˆd,k is a complex Gaussian vector, we can compute its covariancematrix Ψd,k as followsE[hˆd,khˆHd,k] = βd,kRBSk(E[r˜tr0,kr˜trH0,k ])−1E[r˜tr0,kr˜trH0,k ](E[r˜tr0,kr˜trH0,k ])−1βd,kRHBSk= β2d,kRBSk(βd,kRBSk +σ2IMSPCτS)−1RHBSk . (A.9)This concludes the proof.70Appendix BProof of Theorem 3.2Proof. In a similar fashion, the MMSE estimate for h2,l,k for the observedtraining signal in (3.22) is equivalent tohˆ2,l,k = Wr˜l,k, (B.1)In whichW = E[h2,l,kr˜Hl,k](E[r˜l,kr˜Hl,k])−1 (B.2)Since ntrk and h2,l,k are independent and uncorrelated random Gaussianvectors we can solve asE[h2,l,kr˜Hl,k] = E[h2,l,k(h2,l,k +1SMΣ−1l H¯H1,l(Vtrl ⊗ IM )Hntrk)H],= E[h2,l,khH2,l,k] = β2,l,kRIRSl,k , (B.3)Moreover E[r˜l,kr˜Hl,k] == E[h2,l,khH2,l,k] +Σ−1l H¯H1,l(Vtrl ⊗ IM )HE[ntrk ntrHk](Vtrl ⊗ IM )H¯1,lΣ−1lS2M2(B.4)= β2,l,kRIRSl,k +σ2S2M2(PCτS)Σ−1l H¯H1,l(Vtrl ⊗ IM )HISM (Vtrl ⊗ IM )H¯1,lΣ−1l ,(B.5)= β2,l,kRIRSl,k +σ2S2M2(PCτS)Σ−1l H¯H1,l(VtrHl Vtrl ⊗ IM )H¯1,lΣ−1l , (B.6)= β2,l,kRIRSl,k +σ2SM2(PCτS)Σ−1l H¯H1,l(IMN )H¯1,lΣ−1l , (B.7)= β2,l,kRIRSl,k +σ2SM(PCτS)Σ−1l ΣlΣ−1l , (B.8)= β2,l,kRIRSl,k +σ2SM(PCτS)Σ−1l (B.9)71Appendix B. Proof of Theorem 3.2where some steps and details are redundant and glossed over in thisproof. The expression in (B.7) is exacted by the observation that VtrHl Vtrl =SIM under the DFT protocol for the original undistributed matrix Vtr.Moreover, (B.9) is a result of the known BS-IRS l channel matrix multi-plication H¯H1,lH¯1,l which results in MΣl. Therefore, the gathered MMSEestimate using (B.9) and (B.3) in (B.1) for h2,l,k is realized to behˆ2,l,k = β2,l,kRIRSl,k(β2,l,kRIRSl,k +σ2Σ−1lSM(PCτS))−1r˜l,k, (B.10)The covariance matrix for hˆ2,l,k follows the same steps as hˆd,k’s and can bedirectly computed.72Appendix CProof of Lemma 4.3GivenE[‖hk(λk)‖2] = (C.1)E[hHd,khd,k + 2L∑l=1λl,kvklHl HH0,l,khd,k, (C.2)+L∑l=1L∑l¯=1λl,kλl¯,kvklHl HH0,l,kH0,l¯,kvkll¯, (C.3)+L∑l=1L∑l¯=1(1− λl,k)(1− λl¯,k)vHl HH0,l,kH0,l¯,kvl¯, (C.4)+ 2L∑l=1(1− λl,k)vHl HH0,l,khd,k, (C.5)+ 2L∑l=1L∑l¯=1(1− λl,k)λl¯,kvHl HH0,l,kH0,l¯,kvkll¯]. (C.6)With vkll defined in (4.22) and hk(λk) defined in (4.8). We can separatethis since expectation operation is linear, hence we obtain the sum of sixexpectations terms (E1, . . . , E6) where we defineE1 = E[hHd,khd,k] = Mβd,k, (C.7)which comes from Assumption 1 and the model for direct channel statedin (2.4). Moreover, E2 is interpreted as the inner product of the cascadedchannel and the direct channel which we maximized through the optimized73Appendix C. Proof of Lemma 4.3configuration for vkll , we can thus computeE2 = 2L∑l=1λl,kE[vklHl HH0,l,khd,k], (C.8)= 2√β1,lL∑l=1λl,kEN∑n=1|bl(n)||h2,l,k(n)| · aHl hd,k], (C.9)= 2√β1,lL∑l=1λl,kE[N∑n=1|h2,l,k(n)| · |aHl hd,k|], (C.10)= 2√β1,lL∑l=1λl,kE[N∑n=1|h2,l,k(n)| · |aHl hd,k|], (C.11)= 2√β1,lL∑l=1λl,kN∑n=1pi4√aHl al, (C.12)=√β1,lβd,kβ2,l,k2L∑l=1λl,kpi4√MN, (C.13)where H0,l,k is given as in (4.6) and h2,l,k(n) is the nth element in h2,l,k.Given |X| ∼ Rayleigh(σ), then the meanµX =√pi4σ, (C.14)where σ is the standard deviation. The extra-looking 1√2factor appears dueto the fact that the Rayleigh distributed random variable is an absolute valueof a complex normal random variable X. Note that |h2,l,k(n)| and |aHl hd,k|are statistically independent Rayleigh distributed random variables in whichtheir mean values are given as√piβ2,l,k/2 and√piβdkaHl al/2, respectively.We can split the third expectation term as followsE3 = E[L∑l=1L∑l¯=1λl,kλl¯,kvklHl HH0,l,kH0,l¯,kvkll¯], (C.15)=L∑l=1λl,kE[vklHl HH0,l,kH0,l,kvkll ]+ (C.16)L∑l=1L∑l¯ 6=lλl,kλl¯,kE[vklHl HH0,l,kH0,l¯,kvkll¯], (C.17)74Appendix C. Proof of Lemma 4.3=L∑l=1λl,kE[vklHl HH0,l,kH0,l,kvkll ], (C.18)=L∑l=1λl,kE[ej∠hHd,kH0,l,kHH0,l,kH0,l,kej∠HH0,l,khd,k ], (C.19)=L∑l=1λl,kE[‖H0,l,kvkll ‖2]. (C.20)The sum for l 6= l¯ cancels to zero, due to independence between the differentIRS l-user k channels. Now, we can use a result from CGQF to solve forthis asE3 =L∑l=1λl,kE[ej∠hHd,kH0,l,kHH0,l,kH0,l,kej∠HH0,l,khd,k ], (C.21)=L∑l=1λl,kE[v˜klHl Av˜kll ] (C.22)Where v˜kll = diag(|h2,l,k|)ej∠HH1,lhd,k , andA = HH1,lH1,l = β1,l‖al‖2blbHl (C.23)is a symmetric, deterministic matrix. We can now writeE3 =L∑l=1λl,kE[v˜klHl Av˜kll ] =L∑l=1λl,k(tr(AΣv˜kll) + µHv˜kllAµv˜kll). (C.24)At this point, we can introduce Lemma C.1 which is relevant in untan-gling the expectation term Σv˜kll= E[v˜kll v˜klHl ] in (C.24).Lemma C.1. The expectation of a random diagonal matrix D multiplied bya deterministic matrix A multiplied by DT is a quadratic form defined to beE[DADT ] = E[diAi,jdTj ], (C.25)Proof. We can expand E[DADT ] = E[[DADT ]i,j ] = E[diAi,jdTj ] and ob-serve the following for the diagonal and off-diagonal terms of the expressionsuch that,E[diAi,idTi ] = (Var(di) + E[di]2)Ai,i, (C.26)75Appendix C. Proof of Lemma 4.3for the diagonal entries, and for i 6= j,E[diAi,jdTj ] = E{di}Ai,jE[dj ]. (C.27)Using Lemma C.1, the covariance matrix Σv˜kllis found to beΣv˜kll= E[v˜kll v˜klHl ] (C.28)= E[diag(|h2,l,k|)ej∠HH1,lhd,kej∠hHd,kH1,ldiag(|h2,l,k|)] (C.29)= Eh2,l,k [diag(|h2,l,k|)Ehd,k [ej∠blaHl hd,kej∠hHd,kalbHl ]diag(|h2,l,k|)] (C.30)= Eh2,l,k [diag(|h2,l,k|)ej∠blej∠bHl ]diag(|h2,l,k|)], (C.31)= Eh2,l,k [diag(|h2,l,k|)ej∠HH1,lH1,ldiag(|h2,l,k|)] (C.32)= (1− pi4+pi4)β2,l,kej∠HH1,lH1,l IN + piβ2,l,k4ej∠HH1,lH1,l (1N − IN ),(C.33)= β2,l,kIN +piβ2,l,k4ej∠HH1,lH1,l (1N − IN ), (C.34)= β2,l,kIN +piβ2,l,k4ej∠blbHl (1N − IN ). (C.35)In the last step, we recall the model of H1,l in (2.5), some of these stepsrely on concepts defined earlier such as the mean of a Rayleigh distributionshown in (C.14). Moreover, the mean µv˜kllin the quadratic form (C.24) isderived asµv˜∗l= E[v˜kll ] = E[diag(|h2,l,k|)ej∠HH1,lhd,k ] (C.36)= E[diag(|h2,l,k|)]E[ej∠HH1,lhd,k ]] (C.37)= 0N×1. (C.38)This follows from the independence between the direct and IRS-user chan-nels, and from the fact that the phase distribution of a complex normal isuniform. Since hd,k follows a circularly symmetric Gaussian distribution,the mean of the phase term will equal the center point, which is zero. De-noting θ = ∠HH1,lhd,k, we can expand the definition of the expectation toachieveE[ejθ] =∫ 2pi0ejθ2pidθ =(ej2pij2pi− 1j2pi)1N×1 = 0N×1. (C.39)76Appendix C. Proof of Lemma 4.3As for the fourth term, recall that vl is the non-associated lth IRS beam-forming vector for which λl,k = 0. Where it is instead optimized for anotheruser t, such that one can refer to (4.22) to create for an IRS l associatedwith a particular user tvl = ej∠diag(h2,l,tbl)ej∠alhd,t) (C.40)Using Assumption 1 and observing that the covariance matrix Σvl = IN ,we obtainE4 = E[L∑l=1L∑l¯=1(1− λl,k)(1− λl¯,k)vHl HH0,l,kH0,l¯,kvl¯], (C.41)=L∑l=1(1− λl,k)E[vHl HH0,l,kH0,l,kvl] = β2,l,ktr(A), (C.42)with the structure of A stated in (C.23) and similar to the process of com-puting E3 in (C.15).The fifth term is equal to zero due to the conditional expectation of thedirect channel hd,k for user k and independence to other channels for usert, where t 6= k. In realizing this independence, we can writeE5 = E[2L∑l=1(1− λl,k)vHl HH0,l,khd,k}, (C.43)= 2L∑l=1(1− λl,k)E[vHl HH0,l,kE[hd,k|hd,t,h2,l,k,h2,l,t]] = 0 (C.44)Finally, the sixth term is also equal to zero because conditioning over theother independent random channels yieldsE6 = E[2L∑l=1L∑l¯ 6=l(1− λl,k)λl¯,kvHl HH0,l,kH0,l¯,kvkll¯ ], (C.45)= 2L∑l=1L∑l¯ 6=l(1− λl,k)λl¯,kE[vHl HH0,l,kH0,l¯,kvkll¯ ], (C.46)= 2L∑l=1L∑l¯ 6=l(1− λl,k)λl¯,kE[vHl HH0,l,kH1,l¯|diag(h2,l,k)|× (C.47)Ehd,k [ej∠HH1,lhd,k |hd,t,h2,l,k,h2,l,t]] = 0.77Appendix C. Proof of Lemma 4.3Thus, we complete the proof and find that collecting these terms yields theresult in Lemma (4.3), stated here for completeness.E[‖hk(λk)‖2] = Mβd,k +L∑l=1λl,k(√MN√β1,lβ2,l,kβd,kpi2(C.48)+ tr(HH1,lH1,lΣv˜∗l )− β2,l,ktr(HH1,lH1,l))+L∑l=1β2,l,ktr(HH1,lH1,l).78Appendix DProof of Lemma 4.6In Lemma (4.6), we defined Rk as the covariance matrix of hk(λk), nowwe are interested in computing the steps needed to achieve Lemma 4.6, notethatRk =E[hd,khHd,k + 2L∑l=1λl,khd,kvklHl HH0,l,k (D.1)+L∑l=1L∑l¯=1λl,kλl¯,kH0,l,kvkll vklHl¯HH0,l¯,k+L∑l=1L∑l¯=1(1− λl,k)(1− λl¯,k)H0,l,kvlvHl¯ HH0,l¯,k],= Rk,1 + Rk,2 + Rk,3 + Rk,4 (D.2)where other terms are zeros as shown in Appendix C. Since the expecationis a linear operator, we find thatRk,1 = E[hd,khHd,k] = βd,kIM (D.3)which comes from Assumption 1 and the model for hd,k defined in (2.4).Now for Rk,2, let us use the definition of HH0,l,k in (4.6) to write outRk,2 =E[2L∑l=1λl,khd,kvklHl√β1,ldiag(hH2,l,k)blaHl ] (D.4)without loss of generality, we can drop the unimportant terms and defineR˜k,2 =E[hd,kvklHl diag(hH2,l,k)blaHl ] (D.5)One way to tackle such a problem is to dissect it into its element form asfollows[R˜k,2]m,m¯ =E[hd,k(m)vklHl diag(hH2,l,k)blal(m¯)∗] (D.6)79Appendix D. Proof of Lemma 4.6where al(m¯)∗ is the m¯th element in aHl and hd,k(m) is the mth element inhd,k. We can further simplify by noticing that a row vector times a diagonalmatrix times a column vector yields sum of product of their elements andusing the optimized reflection vector as in (4.22) to obtain[R˜k,2]m,m¯ =E[hd,k(m)N∑n=1ej∠hHd,kalej∠bl(n)∗h2,l,k(n) (D.7)× h∗2,l,k(n)bl(n)al(m¯)∗].Here, h2,l,k(n) is the nth element in h2,l,k, and bl(n) is the nth element in bl.Thus, the phases cancel, the expectation of E[|h2,l,k(n)|] =√piβ2,l,k/4 sinceit is the mean of a Rayleigh distribution defined in (C.14), and |bl(n)| = 1, ∀nto yield[R˜k,2]m,m¯ =N∑n=1√piβ2,l,k4al(m¯)∗E[hd,k(m)ej∠hHd,kal ]. (D.8)Define R¯k,2 as follows to focus on the remaining expectation,[R¯k,2]m,m¯ = E[hd,k(m)ej∠hHd,kal ], (D.9)= E[|hd,k(m)|ej∠hd,k(m)hHd,kal ]. (D.10)Define θhd,k = ∠hd,k(m)hHd,kal and expandθhd,k = ∠hd,k(m)M∑i=1h∗d,k(i)al(i) (D.11)= ∠hd,k(m)h∗d,k(m)al(i) + hd,k(m)M∑i 6=mh∗d,k(i)al(i) (D.12)= ∠|hd,k(m)|2al(m)(1 +∑Mi 6=m h∗d,k(i)al(i))h∗d,k(m)al(m))(D.13)= ∠|hd,k(m)|2al(m)) + ∠(1 +∑Mi 6=m h∗d,k(i)al(i))h∗d,k(m)al(m))(D.14)= ∠al(m) + ∠(1 +∑Mi 6=m h∗d,k(i)al(i))h∗d,k(m)al(m))(D.15)80Appendix D. Proof of Lemma 4.6All these steps came from the fact that the angle of product is sum of angles.And if we have an angle of a scalar times a complex number it is equivalentto the angle of the complex number. LetX =∑Mi 6=m h∗d,k(i)al(i))h∗d,k(m)al(m)(D.16)Y = 1 +X (D.17)where X ∼ CN (0, M−1|hd,k(m)|2 ) and Y ∼ CN (1,M−1|hd,k(m)|2 ). We can update[R¯k,2]m,m¯ = ej∠al(m)E{|hd,k(m)|ej∠Y }. (D.18)At this point we have[R˜k,2]m,m¯ =N√pi4ej∠al(m¯)∗al(m)E[|hd,k(m)|ej∠Y ], (D.19)=N√pi4ej∠al(m¯)∗al(m)Cm. (D.20)Computing the expectation, the diagonal entries of Rk,2 are equal to theiraverage values, now we use (C.12) which is equivalent to E2 = tr(Rk,2) todeduce that Cm =√βd,kpi4M and obtain[Rk,2]m,m¯ =2L∑l=1λl,k√β1,lβ2,l,kβd,kNpi4√Mej∠a∗l (m¯)al(m). (D.21)As for Rk,3, we can check thatRk,3 = E[L∑l=1L∑l¯=1λl,kλl¯,kH0,l,kvkll vklHl¯HH0,l¯,k] (D.22)=L∑l=1λl,kE[H0,l,kvkll vklHl HH0,l,k] (D.23)=L∑l=1λl,kβ1,lalE[bHl diag(h2,l,k)vkll vklHl diag(hH2,l,k)bl]aHl (D.24)=L∑l=1λl,kβ1,lR˜k,3 (D.25)81Appendix D. Proof of Lemma 4.6The summation reduces due to independence between the different IRS-userschannels. Using vkll = exp(j∠diag(hH2,l,k)bl) exp(j∠aHl hd,k), we can find anexpression for R˜k,3 as followsR˜k,3 = E[bHl diag(h2,l,k)vkll vklHl diag(hH2,l,k)bl], (D.26)=N∑n=1N∑n¯=1E[b∗l (n)h2,l,k(n)vkll (n)vkl∗l (n¯)h∗2,l,k(n¯)bl(n¯)], (D.27)where b∗l (n), h2,l,k(n), and vkll (n) are elements in bHl ,h2,l,k, and vkll , respec-tively. Note that we can decompose this double sum asR˜k,3 =N∑n=1E[b∗l (n)h2,l,k(n)vkll (n)vkl∗l (n)h∗2,l,k(n)bl(n)]+ (D.28)N∑n=1N∑n¯ 6=nE[b∗l (n)h2,l,k(n)vkll (n)vkl∗l (n¯)h∗2,l,k(n¯)bl(n¯)].The first sum yields NE[|h2,l,k(n)|2] = NV ar(|h2,l,k(n)|) + NE[|h2,l,k(n)|]2in which the variance of a Rayleigh distributed random variable |h2,l,k(n)| isgiven as V ar(|h2,l,k(n)|) = (1− pi4 )β2,l,k, the mean is given by (C.14) whichresults in E[|h2,l,k(n)|]2 = pi4β2,l,k. Putting it all together, NE[|h2,l,k(n)|2] =Nβ2,l,k.The argument of the second sum is b∗l (n)bl(n)E[|h2,l,k(n)|]E[|h2,l,k(n¯)|] =b∗l (n)bl(n)β2,l,kpi4 . Following the result for Σv˜kllin (C.35), and plugging(D.28) in (D.25) yieldsRk,3 =L∑l=1λl,kH1,lΣv˜kllHH1,l (D.29)Finally, Rk,4 is written asRk,4 = E[L∑l=1L∑l¯=1(1− λl,k)(1− λl¯,k)H0,l,kvlvHl¯ HH0,l¯,k] (D.30)=L∑l=1(1− λl,k)E[H0,l,kvlvHl HH0,l,k] (D.31)=L∑l=1β1,l(1− λl,k)E[albHl diag(h2,l,k)vlvHl diag(hH2,l,k)blaHl ] (D.32)82Appendix D. Proof of Lemma 4.6=L∑l=1β1,l(1− λl,k)R˜k,4 (D.33)The summation reduces due to independence between the different IRS-users channels. Using vl = exp(j∠diag(hH2,l,t)bl) exp(j∠aHl hd,t), for t 6= kand defining R˜k,4 we getR˜k,4 = E[albHl diag(h2,l,k)ej∠diag(hH2,l,t)blej∠blHdiag(h2,l,t)diag(hH2,l,k)blaHl ](D.34)= albHl E[diag(h2,l,k)ej∠diag(hH2,l,t)blej∠blHdiag(h2,l,t)diag(hH2,l,k)|h2,l,t],(D.35)= ‖bl‖2alaHl . (D.36)Notice that E[diag(h2,l,k)ej∠diag(hH2,l,t)blej∠blHdiag(h2,l,t)diag(hH2,l,k)|h2,l,t] canbe seen as random diagonal matrix diag(hH2,l,t) multiplied by a deterministicmatrix ej∠diag(hH2,l,t)blej∠blHdiag(h2,l,t) multiplied by the same diagonal ma-trix. Therefore, we have the right set-up to apply Lemma C.1 and get thefollowing: the off-diagonals are equal to zero, while the diagonal terms areequal to β2,l,k. Thus, we achieve a scaled identity for this expectation termand which simplifies our expression when plugging R˜k,4 for Rk,4 to findRk,4 =L∑l=1(1− λl,k)β1,lβ2,l,k‖bl‖2alaHl (D.37)=L∑l=1(1− λl,k)β2,l,kH1,lHH1,l (D.38)Thus, we have completed the proof of Lemma 4.6, and the expressionfor Rk is validated.Rk =βd,kIM +L∑l=1λl,k(2√β1,lβ2,l,kβd,kNpi4√Mej∠H1,lHH1,l (D.39)+ H1,lΣv˜kllHH1,l − β2,l,kH1,lHH1,l)+L∑l=1β2,l,kH1,lHH1,l.83
Thesis/Dissertation
eng
Electrical Engineering
Vancouver : University of British Columbia Library
Attribution-NonCommercial-NoDerivatives 4.0 International
Graduate
Channel estimation and user association for distributed intelligent reflecting surfaces assisted MISO communications
Text