Applied Science, Faculty of
Engineering, School of (Okanagan)
DSpace
UBCO
Alnahhas, Bayan
Master of Applied Science - MASc
University of British Columbia
The recent concept of intelligent reflecting surfaces (IRS) enabled wireless communication can overcome the adverse effects of the wireless channel by manipulating the propagation of the radio waves in the environment. This emerging technology, which relies on low-cost passive reflecting elements, is shown to yield promising performance results in recent studies that often focus on Rayleigh fading environments and assume the availability of perfect channel state information (CSI) at the base station (BS) and the IRS. This work considers the practical setting where only imperfect CSI is available at the BS and the IRS, and studies the downlink sum-rate performance of a multi-user IRS-assisted multiple-input single-output (MISO) system in the asymptotic regime where the number of BS antennas, IRS elements and users grow large. We first derive the MMSE estimates of the IRS-assisted and direct channels under a protocol that ensures that the IRS element stay passive. Then we formulate the signal-to-interference-plus-noise ratio (SINR) and sum rate expressions under maximum ratio transmission (MRT) precoding, and obtain their asymptotically tight deterministic approximations using tools from random matrix theory. The asymptotic performance limits are also derived that reveal that in Rayleigh fading environments, IRS only yields an array gain and loses on the reflect beamforming gain promised by existing works. Simulation results show the tightness of the asymptotic expressions for moderate system dimensions and reveal important insights into how beneficial the IRSs are when the environment has rich scattering as represented in Rayleigh fading. It turns out that IRS introduces high sum- rate gains in noise-limited scenarios whereas the gains start to diminish in interference-limited scenarios.
IRS-ASSISTED RAYLEIGH FADING MISO SYSTEMS: BEAMFORMING ANDASYMPTOTIC ANALYSISbyBayan AlnahhasB.Sc. Hons., Effat University, 2018A 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 2020© Bayan Alnahhas, 2020The following individuals certify that they have read, and recommend to the College of GraduateStudies for acceptance, a thesis/dissertation entitled:IRS-ASSISTED RAYLEIGH FADING MISO SYSTEMS:BEAMFORMING AND ASYMPTOTIC ANALYSISsubmitted by Bayan Alnahhas in partial fulfilment of the requirements of the degree ofMaster of Applied ScienceProf. Anas Chaaban, School of EngineeringSupervisorProf. Md. Jahangir Hossain, School of EngineeringSupervisory Committee MemberProf. Chen Feng, School of EngineeringSupervisory Committee MemberProf. Mahmudur Fatmi, School of EngineeringUniversity ExamineriiAbstractThe recent concept of intelligent reflecting surfaces (IRS) enabled wireless communication canovercome the adverse effects of the wireless channel by manipulating the propagation of the radiowaves in the environment. This emerging technology, which relies on low-cost passive reflectingelements, is shown to yield promising performance results in recent studies that often focus onRayleigh fading environments and assume the availability of perfect channel state information(CSI) at the base station (BS) and the IRS. This work considers the practical setting where onlyimperfect CSI is available at the BS and the IRS, and studies the downlink sum-rate performanceof a multi-user IRS-assisted multiple-input single-output (MISO) system in the asymptotic regimewhere the number of BS antennas, IRS elements and users grow large. We first derive theMMSE estimates of the IRS-assisted and direct channels under a protocol that ensures that theIRS element stay passive. Then we formulate the signal-to-interference-plus-noise ratio (SINR)and sum rate expressions under maximum ratio transmission (MRT) precoding, and obtain theirasymptotically tight deterministic approximations using tools from random matrix theory. Theasymptotic performance limits are also derived that reveal that in Rayleigh fading environments,IRS only yields an array gain and loses on the reflect beamforming gain promised by existingworks. Simulation results show the tightness of the asymptotic expressions for moderate systemdimensions and reveal important insights into how beneficial the IRSs are when the environmenthas rich scattering as represented in Rayleigh fading. It turns out that IRS introduces high sum-rate gains in noise-limited scenarios whereas the gains start to diminish in interference-limitedscenarios.iiiLay SummaryWith the inevitable increase of interconnected devices and wireless communication demands, itbecomes important to consider a new platform for data transmission. The current deployed multi-user systems help accommodate our daily demands of data rates and day-to-day mobile applicationuse. Nevertheless, the future requirements are expected to exponentially increase. Consequently,there is a strong need to improve our current networks to help realize the future vision of data drivensociety. Intelligent reflecting surface (IRS) comes as an effective and highly potential solution toenvision a radically new platform of communication. In an IRS-assisted communication, theenvironment becomes a controllable factor which IRS can manipulate in favor of user reception byaltering the reflection signals in direction of intended users. Such technology is seen as a promisingstep towards realizing a smart radio environment that optimizes the quality of service for users andachieves high data rate limits in cellular networks.ivPrefaceThis thesis is based on research work conducted under the supervision of Prof. Anas Chaabanand Qurrat-UI-Ain Nadeem (Annie), who is a Post Doctoral Fellow with Prof. Anas Chaaban.The contents of this work along with the numerical results were proofread and validated by Prof.Chaaban and Annie.vTable of ContentsAbstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iiiLay Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ivPreface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vTable of Contents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viList of Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ixList of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xAcknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xiiChapter 1: Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.1 Background and Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.1.1 MIMO Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31.1.2 FDD/TDD Systems and Channel Estimation . . . . . . . . . . . . . . . . . 81.1.3 Precoding Techniques . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121.1.4 Asymptotic Analysis and Random Matrix Theory . . . . . . . . . . . . . . 141.2 Intelligent Reflecting Surfaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . 161.2.1 Background and Motivation . . . . . . . . . . . . . . . . . . . . . . . . . 171.2.2 State of the Art . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 201.3 Thesis Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 211.3.1 Objectives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 211.3.2 Contribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22viTABLE OF CONTENTS1.3.3 Outline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23Chapter 2: IRS-Assisted MISO Systems . . . . . . . . . . . . . . . . . . . . . . . . . . 242.1 System Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 242.2 Channel Estimation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 262.2.1 State of the Art . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 262.2.2 Proposed Protocol . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 272.3 Beamforming and Achievable Rate . . . . . . . . . . . . . . . . . . . . . . . . . . 302.3.1 Maximum-Ratio Transmission Precoding . . . . . . . . . . . . . . . . . . 302.3.2 SINR and Achievable Rate . . . . . . . . . . . . . . . . . . . . . . . . . . 302.4 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31Chapter 3: Asymptotic Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 333.1 Motivation and Useful Lemmas . . . . . . . . . . . . . . . . . . . . . . . . . . . . 333.2 Deterministic Equivalence Results . . . . . . . . . . . . . . . . . . . . . . . . . . 343.2.1 IRS Assisted System with a Direct Channel . . . . . . . . . . . . . . . . . 353.2.2 IRS Assisted System with a Blocked Direct Channel . . . . . . . . . . . . 373.2.3 IRS Assisted System Under Perfect CSI . . . . . . . . . . . . . . . . . . . 383.2.4 Special Case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 393.3 Reflect-Beamforming for Performance Optimization . . . . . . . . . . . . . . . . . 413.4 Conclusion and Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43Chapter 4: Numerical Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 454.1 Deterministic Equivalent Validation . . . . . . . . . . . . . . . . . . . . . . . . . 464.2 IRS Performance with Respect to the Conventional System . . . . . . . . . . . . . 474.2.1 Performance Comparison . . . . . . . . . . . . . . . . . . . . . . . . . . . 474.2.2 Impact of Channel Estimation Time . . . . . . . . . . . . . . . . . . . . . 494.2.3 Impact of Channel Noise Under an IRS-Simplified Channel Model . . . . . 504.2.4 Impact of Cell Radius . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 514.3 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52Chapter 5: Conclusion and Future Direction . . . . . . . . . . . . . . . . . . . . . . . 53viiTABLE OF CONTENTS5.1 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 535.1.1 Takeaways . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 535.1.2 Limitations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 545.2 Future Direction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56Appendix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64Appendix A: Proof of the Channel Correlation . . . . . . . . . . . . . . . . . . . . . . 64A.1 Proof of (2.2) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64A.2 Proof of (2.13) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64A.3 Proof of (2.16) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65A.4 Proof of (2.17) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66Appendix B: Proof of Theorem 3.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68Appendix C: Proof of Corollary 3.4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72Appendix D: Proof of Lemma 3.4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73D.1 Proof of Derivative Expression for ∂nk∂θn . . . . . . . . . . . . . . . . . . . . . . . . 73D.2 Proof of (3.26) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74viiiList of TablesTable 1.1 The minimum number of samples consumed for pilot transmission and CSIfeedback in terms of total number of per coherence interval for three MIMOvariants, while assuming M > K . . . . . . . . . . . . . . . . . . . . . . . 10ixList of FiguresFigure 1.1 Multiuser cellular network . . . . . . . . . . . . . . . . . . . . . . . . . . 3Figure 1.2 Multiuser MIMO system setup . . . . . . . . . . . . . . . . . . . . . . . 6Figure 1.3 Beamforming representation . . . . . . . . . . . . . . . . . . . . . . . . . 7Figure 1.4 Channel estimation in TDD system . . . . . . . . . . . . . . . . . . . . . 9Figure 1.5 Isotropic scattering, approximated as a rayleigh fading . . . . . . . . . . . 15Figure 1.6 IRS-assisted wireless transmission: IRS facilitates beamforming by direct-ing a beam centered at the user end . . . . . . . . . . . . . . . . . . . . . 18Figure 1.7 IRS-assisted system depicting two practical scenarios . . . . . . . . . . . . 19Figure 2.1 Sketch of IRS-assisted MISO system. . . . . . . . . . . . . . . . . . . . . 24Figure 4.1 Multiuser IRS-assisted system setup . . . . . . . . . . . . . . . . . . . . 45Figure 4.2 Numerical validation of the deterministic (det.) equivalent of the SINRagainst the Monte-Carlo (MC) values, under perfect (per.) and imperfect(imper.) CSI., where ρ is abstracted as 1/σ2. . . . . . . . . . . . . . . . . 47Figure 4.3 Sum-rate comparison of IRS-assisted (IRS-asst.) MISO system undergradient ascent algorithm optimized (G.A.O) phases and exhaustive searchoptimized phases, and comparison with sum-rate of conventional (conv.)MISO systems. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48Figure 4.4 Sum-rate comparison of IRS-assisted (IRS-asst.) and conventional (conv.)MISO system under varying channel estimation time, where τc is thechannel estimation time seconds. . . . . . . . . . . . . . . . . . . . . . . 49Figure 4.5 Performance of IRS-assisted system under the channel in (3.20), where ρis abstracted as 1/σ2. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50xLIST OF FIGURESFigure 4.6 Performance analysis of IRS-assisted system under varying cell radius withM = 50 and K = 32 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51xiAcknowledgementsI would like to thank Prof. Anas Chaaban for his guidance and support throughout this work.His continuous follow up on work progress and thorough feedback on results were great assets indirecting this work. I would also like to thank him for granting me the freedom in my studies andproviding me with strong academic flavor in various research directions. I extend my sincere thanksto Annie who closely supervised this work and validated the results. Annie has been an exceptionalsource of advice and support amid this work and I would sure attribute a lot of the success to herand her diligent help.My thanks also goes to my colleagues and friends who were an additional fortunate source ofsupport and advice throughout the journey. Finally, deepest thanks to my parents and family whohave always and remain a major advocate for my academic journey and an endless source of loveand support.xiiChapter 1Introduction1.1 Background and MotivationThe amount of wireless data communication has been increasing in an exponential pace over thepast decades. The significant number of connected devices, the fulfilled demands for real-time videostreaming and cell coverage, and the achieved high data rates for advanced mobile applications havebecome some of the main features that characterize our current wireless networks [BHS17]. Overthe past years we have witnessed a major explosion of wireless connectivity and streaming mediarevolution, where audio and video are delivered on demand over the internet everywhere aroundthe globe. From smart appliances, to smart businesses, traffic systems and industries, our currentwireless network, such as the Long Term Evolution (LTE) based cellular wide area networks andsatellite services, paved the way to revolutionizing our communication experience. Nevertheless,future demands are always increasing and wireless networks are envisioned to connect a yet greaternumber of terminals as well as to provide a new competitive data rate limits with ubiquitous cellcoverage. In fact, according to Cisco [BDRDR+19], the overall data traffic is expected to increase to77 exabytes per month by 2022 with number of interconnected devices reaching up to 28.5 billion.As a result, wireless networks seem to consistently move towards a higher need of bandwidth tosatisfy the emerging vision of interconnected wireless networks. This exponential trend motivatedresearchers to find potential solutions that address some fundamental wireless network limitations.Two timeless facts are evident in wireless networks , demands on wireless throughput willalways grow, and the available electromagnetic spectrum will never increase. The performance ofwireless networks is typically limited at its physical layer, that is the number of bits transferredbetween two points depends on the spectrum availability, laws of electromagnetic propagation andprinciples of information theory [Mar15, RPL+13, Ngo15]. As shown in Fig. 1.1, a typical cellular11.1. Background and Motivationnetwork consists of a number of cells, where in each cell, a base station equipped with M antennasserves K number of user terminals. To improve the wireless communication performance, researchtypically approaches network problem through three basic solution directions: (i) deploying moreaccess points per cell (known as cell densification), (ii) using more spectral bands, or (iii) increasingspectral efficiency, that is the number of bits that are conveyed in a given resource block or unitbandwidth [MLYN16]. While the stringent expectations of emerging wireless networks are likelyto require an ever-increasing use of bandwidth to realize the future vision of internet of things (IoT)and emerging wireless networks such as 5G and beyond, numerous leading research work helpdevelop solutions that make an efficient use of the available spectrum band while at the same timeprovide a reliable wireless throughput throughout a designated area [KUV19, BSW+19, AH16].In the light of this, we will take a step back and introduce a revolutionary technology, known asmultiple input and multiple output (MIMO), where a BS is equipped with large number of antennasserves many users in the same time-frequency resource block [Ngo15, GKH+07]. The technologyhad significantly changed the vision of wireless networks for decades and paved the way to realizingour current deployed network infrastructure. In the following subsections, we will provide a briefbackground on the MIMO technology and how it works. We will then follow this with importanttechniques utilized, such as beamforming, which helps leverage the additional spatial dimensionsprovided in MIMO systems by directing a beam to intended user and thus improving user’s signalreception. Finally, we’ll introduce randommatrix theory (RMT) that is used to study the asymptoticperformance of the system and help us realize the benefits of large system dimensions in solvingimportant optimization problems.21.1. Background and MotivationM antenna BSK user terminalscell l cell jUT kuplinkdownlinkdownlinkFigure 1.1: Multiuser cellular network1.1.1 MIMO SystemsMultiple input multiple output (MIMO) technology has caught major attention over the pastdecades and was seen as a viable approach to substantial improvement in spectral efficiency. Theevolutionary technology is known to enhance the reliability of communication by providing sub-stantial multiplexing and diversity gains. In wireless communication, the transmitted signal suffersfrom fading due to multipath propagation and shadowing due to large obstacles between the trans-mitter and receiver. By deploying multiple antennas at the BS and/or receiver, multiple streamscan be sent over the channel, and thus data may be spatially multiplexed onto a MIMO channelproviding an R-fold capacity increase, where R is the number of independent channels determinedby the rank of the channel matrix [TV05, Gol05]. MIMO technology is now relatively mature andis incorporated in the wireless broadband standards such as LTE-Advanced to support our currentdemand of bandwidth [JKL+17]. In this subsection, we will highlight the evolution of MIMOtechnology while discussing the performance at each stage.MIMO systems are mainly categorized into three categories, namely point to point MIMO,Multiuser MIMO and Massive MIMO [MLYN16]. We will provide a brief outline for every classi-31.1. Background and Motivationfication, as the process of evolving from one classification to another will pave the way to a strongerunderstanding on the main work of this dissertation, i.e. IRS-assisted MIMO systems [BSW+19].A point-to-point MIMO link consists of a BS equipped with a concentrated array of M antennascommunicating with a user equipped with concentrated array of K antennas. The fundamentalidea of point to point systems is that in every channel use, the BS transmits a signal vector, and theuser receives a signal vector that is a linear combination of transmitted signal and the combiningcoefficients of the channel matrix, which are determined by the channel propagation [Gol05].By Shannon’s theorem, the uplink and downlink capacity in bits/s/Hz of a MIMO point-to-pointsystems with Gaussian noise can be defined as:C =maxDvlog2 |IK + ρGDvGH | (1.1a)s.t. tr(Dv) ≤ 1, (1.1b)where the optimization is over the positive semi-definite covariance matrix Dv ∈ CM×M , and G isan K × M channel gain of the channel matrix between the BS and K user terminals, IK is a K × Kidentity matrix, the superscript H denotes to the conjugate transpose, and ρ is the signal to noiseratio (SNR) which is proportional to the corresponding total radiated powers. Using singular valuedecomposition (SVD), we can rewrite the channel matrix as G = UAVH , where U is an K × Kunitary matrix, V is an M × M unitary matrix and A is K × M diagonal matrix with non-negativeentries. The diagonal elements σi of matrix A are the singular values of channel matrix G, whereG is of rank R with R positive singular values, satisfying R ≤ min(M,K). The SVD allows us toconvert the MIMO system into parallel, non interfering channels by pre-multipying the input withmatrix V (i.e transmit precoding) and post-multiplying the output by the matrix UH [BCC+07].The result is R parallel non-interfering channels, also known as eigenmodes of the channel. Usingwaterfilling algorithm, we can optimally allocate powers over the parallel channels, such that:Pi = (µ − 1ρσ2i)+, 1 ≤ i ≤ R (1.2)41.1. Background and Motivationwhere Pi is the power associated with the i’th transmitted symbol, (·)+ defines max(·,0), and µ isthe water level chosen such that∑Ri=1 Pi = 1. The optimal covariance matrix, (i.e Dv that achievesthe maximum in (1.1)) is Dv = VPVH , where P = diag(P1, . . . , PR, 0, . . . , 0) ∈ CM×M . A moredetailed explanation on SVD and waterfilling algorithm can be found in [TV05, BCC+07].Notice that under certain channel assumptions, such as sufficiently large value of signal to noiseratio (SNR) and a rich scattering environment modeled by independent Rayleigh fading, where thechannel matrix G becomes of full rank, such that R = min(M,K), and under the use of asymptoticrandom matrix theory, C in (1.1) scales linearly with min(M ,K) and logarithmically with SNR[MLYN16, NLM13]. This can be approximately expressed as:C ∝ min(M,K) log2(ρ), (1.3)where M and K are the antenna array size of BS and receiver, respectively. Therefore, the spectralefficiency can be simultaneously increased by deploying large arrays in the transmitter and receiver,making M and K large.Nevertheless, in practice, point to point MIMO systems fall back in several factors. First, theequipment is complicated and requires independent RF chains per antenna as well as the use ofadvanced digital signal processing to separate the data streams. Second, scaling up the number ofantennas requires a proportional amount of training time (time required for the receiver to learn thechannel matrix). Third, to realize the benefits in equation shown in (1.3), the propagation channelmust support at least min (M ,K) independent channels. This is not typically the case in propagationenvironments, as line of sight (LOS) condition presents a particular challenge for compact arrays,wherein the channel matrix would have a minimum rank of one thus permitting only one datastream [Mar15, RPL+13].Multiuser (MU) MIMO system help mitigate some drawbacks of point-to-point MIMO system.MU-MIMO is essentially equivalent to starting with a MIMO point to point link, and breaking thesingle K antenna user terminal to K autonomous single antenna users [LLS+14]. The illustrationshown in Fig.1.2 shows a conventional MU-MIMO system, where a BS equipped with M antennascommunicates with K single antenna users. It is worth noting that though the K users do not51.1. Background and MotivationPrecoding 1Mdata stream 1data stream KUT 1UT KDecodingDecodingDecodingM antenna BSCSICSICSICSIdata stream 1data stream 2data stream KFigure 1.2: Multiuser MIMO system setupcooperate in a mutiuser system, this does not compromise on the overall capacity or sum throughputof the system. The Shannon’s sum capacity in the uplink and downlink links can be expressed asfollows [VT03]:Cul = log2 |IM + ρulGGH |, (1.4)Cdl =maxvm≥0log2 |IM + ρdlGHDvG| (1.5a)s.t.M∑m=1vm ≤ 1, ∀m, (1.5b)where Dv is a diagonal covariance matrix with diagonal elements v = [v1 . . . vM], G is an M × Kchannel matrix, and ρdl/ρul is the downlink/uplink SNR. According to the problem formulatedin (1.5), the downlink capacity forMUMIMO requires a solution of a convex optimization problem.MU-MIMO systems possess two major practical limitations. Achieving the capacity expressedin (1.5) requires a complicated coding technique known as dirty paper coding, whose complexitygrows exponentially with the system dimension at both the BS and user ends. Additionally, inthe downlink, the BS and user terminals must both know the channel matrix G, which requires61.1. Background and Motivationconsiderable amount of resource for pilot transmission. Nevertheless, MU-MIMO systems holdtwo important advantages over point to-point MIMO systems, that is MU-MIMO systems are lesssensitive to assumptions on the propagation environment, and do not require multiple antennas atthe user end [Mar15, GKH+07].To address the scalability issue that MU-MIMO faces, Massive MIMO comes as a potentialsolution and is considered as a salable version of MU-MIMO. Massive MIMO systems breaksthe scalability barrier by using simple linear precoders while at the same time achieving a betterperformance than a non-Massive MIMO system [Mar15]. Each BS in a Massive MIMO systems isequipped with large number of M antennas serving large number of K users. Under time divisionduplexing (TDD) operation, only the BS is required to learn the channel matrix G, where the timerequired to obtain the CSI is independent of BS antenna array size. Furthermore, as number of BSantennas increases, the propagation environment becomes favorable [TV05], i.e channels becomemutually orthogonal, therefore simple linear precoding/decoding techniques are found optimal andthe system’s performance approaches Shannon’s limit.Intended user terminalM antenna BSFigure 1.3: Beamforming representationIncreasing the antenna array size has several advantageous factors that are worth noting. In thedownlink, the BS ensures that each user terminal receives only the signal intended for it. This isaccomplished by creating a beam with a narrow angular window centered in the direction of theterminal. Larger the number of BS antennas are, the narrower the radiated beam is. By properly71.1. Background and Motivationchoosing the transmitted waveforms, signals’ scattering and reflecting components will experienceconstructive superposition at the intended terminal and destructive superposition at the non-intendedterminals [TV05]. We refer to this technique used to produce directed beams as “beamforming",as depicted in Fig.1.3. The more antennas deployed at the BS, the more sharply focused thebeam power is at the intended end. Another important consequence of deploying large number ofantennas is phenomenon known as channel hardening. The significance of channel hardening liesin the fact that small scale fading effects tend to disappear as M grows large [WCDS12a]. To clarifythis point, consider an M dimensional channel response vector h and beamforming vector a. AsM grows large, and by the law of large numbers, the scalar channel gain hTa becomes close to itsexpected value, i.e. E[hTa] , which is a deterministic value. The resulting scalar effective channelfacilitates the use of simple schemes for resource allocation and power control. This holds greatadvantage with regards to solving optimization problems, as we’ll see later in this thesis. Section1.1.4 in this chapter will further elaborate on the benefits of studying the systems in an asymptoticregime and the benefit of using random matrix theory (RMT) in system analysis.1.1.2 FDD/TDD Systems and Channel EstimationIn this section, we will highlight two known duplexing modes which MIMO systems operatein, namely Frequency Division Duplexing (FDD) and Time Division Duplexing (TDD) modes.In a practical system, the BS does not know the exact channel state information (CSI) and has toestimate it instead. Depending on the duplexing mode, the channel estimation schemes differ. Wewill briefly discuss how is the channel estimation run in each mode and follow this to introducea reliable channel estimation technique known as linear minimum mean square error (LMMSE)estimation.• Time Division Duplexing (TDD) operation: In the TDD mode, the uplink and downlinktransmission use the same frequency spectrum, but different time slots. In addition, theuplink and downlink channels are reciprocal. The BS learns the uplink channel from uplinkpilots and uses it as a legitimate estimate for the downlink channel. More precisely, theprocess is as follows: during the uplink transmission, K users transmit K orthogonal pilots tothe BS. The BS estimates the uplink channel based on the received pilots and uses it to detect81.1. Background and Motivationthe signals transmitted. During the downlink transmission, the BS uses the channel estimatedin the uplink to precode the transmit symbols. The users will also need to know the effectivechannel gain to detect their signals, therefore the BS sendsK beamformed pilots and each usercan estimate the effective channel gain based on the received pilot [TV05, Gol05, Ngo15].• Frequency Division Duplexing (FDD) operation: In the FDD mode, the uplink and downlinkchannels lack reciprocity. The uplink and downlink channels uses different frequency spec-trum. To obtain the CSI in the downlink transmission, the BS transmits M orthogonal pilotsto K users, where each user estimates the channel and feeds it back to the BS through theuplink. Similarly, in the uplink transmission, K users transmit K orthogonal pilots sequencesto the BS for channel estimation purpose [TV05, Ngo15].Uplink Pilots(K orthogonal pilots)Uplink DataTransmissionDownlinkpilotsT (symbols)K (symbols) K (symbols)Downlink DataTransmissionFigure 1.4: Channel estimation in TDD systemIt is important note that sending pilots consumes resources. Specifically speaking, duringthe time-frequency period in which the channel is static (known as the coherence period), eachtransmitting antenna needs to be assigned a unique pilot waveform and they need to be orthogonal.In other words, ifK antennas transmitK orthogonal pilots in uplink, at leastK slots in the coherenceperiod have to be designated for pilot transmission, thus resource consuming [Mar06]. The symbolslot structure shown in Fig. 1.4 illustrates the training process where it is assumed that the channelstays constant over T symbols and K < T . In this illustration, at least K symbols are needed asuplink pilots and K symbols for downlink beamformed pilots, thus at least 2K symbols from Tsymbols are dedicated for training process. With the continuous increase in number of wirelesslyconnected users, it is worth considering the resources used in different MIMO system settings.Table 1.1 summarizes the amount of resources needed for pilots and CSI feedback for each Point-to-point MIMO, MU MIMO and Massive MIMO operating in both the TDD and FDD schemes.91.1. Background and MotivationTo explain the resource use shown in the table, let’s look at the conventional multi-user MIMOas an example, and compare the TDD and FDD mode resource usage. Recall that in FDD mode,the BS learns the downlink channel by communicating M pilots to user terminals, which estimatethe downlink channel and communicate the estimated CSI coefficients back to BS in the uplink.The BS also learns the uplink channel by listening to K pilots sent by user terminal via the uplink.This sums up to K pilots and M CSI coefficients resource usage in the uplink and M pilots in thedownlink. Now since the TDD mode leverages reciprocity advantage, the BS can learn the uplinkchannel by listening to the K pilots sent by terminals in the uplink and use it a legitimate estimatefor downlink channel. The user terminal also requires the knowledge of effective channel gain fordecoding, thus M pilots are required for the downlink. This indeed sums up for K pilots in theuplink and M pilots in the downlink. A similar scenario occurs in Massive MIMO under FDDmode with same resource usage, but we can notice that in TDD mode the large system dimensionadvantage kicks in, and due to the favorable condition, no pilots are required in the downlink.FDD TDDUplink Downlink Uplink DownlinkPoint-to-point (M × K) MIMO K pilots M pilots K pilots M pilotsConventionalMultiusersMIMOK pi-lots+ MCSI-coeff.M pilots K pilots M pilotsMassive MIMOK pi-lots+ MCSI-coeff.M pilots K pilots noneTable 1.1: The minimum number of samples consumed for pilot transmission and CSI feedback interms of total number of per coherence interval for three MIMO variants, while assuming M > KTherefore, massiveMIMO system operating in TDD stands out, as the number of pilots requiredis independent of the number of BS antennas, and the CSI feedback is not required. For thisreason, Massive MIMO systems are seen as a promising candidate given its unlimited scalability[LETM14, RPL+13].101.1. Background and MotivationChannel EstimationThe performance of wireless network heavily depends on the quality of channel estimates.Estimating a random channel is often a complicated process, and the popular Bayesian estimators(MMSE and MAP) are difficult to determine in closed form as well as computationally expensiveto implement [Kay97]. For this reason, it is a practical solution to retain the MMSE criterionbut constrain the estimator to be linear. In this subsection, we will introduce the linear minimummean square estimation (LMMSE) criterion. To begin, let’s consider a conventional and simplifiedMU-MIMO system, wherein an M-antenna BS is communicating with K single antenna users andoperating in a TDD scheme. In such a topology, the received signal at user k can be expressed asfollows:yk = hHk x + nk, (1.6)where hk ∼ CN(0, βkIM) ∈ CM×1 is the channel between user k and BS modeled as an i.i.d.Rayleigh fading channel with pathloss parameter βk , x ∈ CM×1 is the data signal vector, andnk ∼ CN(0, σ2) [MSI17, WCDS12a].During the training phase, users transmit K mutually orthogonal pilot sequences to BS in theuplink. The BS essentially aims at estimating hk from the observed signal by first correlating itwith the pilot sequence to obtain a training signal, which it uses to calculate the channel estimate.The training signal can be expressed as follows:ytrk = hk + nULk , k = 1 . . .K, (1.7)where nULk ∼ CN(0, 1ρtr IM) and ρtr is the training SINR. The channel estimate hˆk can be computedfrom ytrk such that, hˆk = Aytrk where A is some transformation weight matrix that operates onthe training signal to obtain the channel estimate [MJ19a]. Therefore, the BS aims at finding theoptimal weighting coefficients that minimizes E[|hˆk − hk |2]= E[|Aytrk − hk |2] to the following:hˆk = E[ytrk hHk ](E[htrk htrHk ])−1ytrk , (1.8)where E[htrk htrHk ] is the covariance matrix of hk and E[ytrk hHk ] is a cross-covariance matrix [Kay97].When estimating a Gaussian vector, i.e. hk in our example, the MMSE estimator becomes linear.111.1. Background and MotivationTherefore, to find the MMSE estimator for a Gaussian vector, it is sufficient to find the LMMSEestimator. Notice that MMSE estimate in (1.8) requires the second order statistical properties of thedata in a random process, therefore it is considered as a reliable and practical estimation approachfor all realizations in a random process.1.1.3 Precoding TechniquesComplex signal processing techniques such as maximum likelihood detection are required toprovide an optimal performance for wireless system signal detection performance. Nevertheless, ithas been shown in [RPL+13, Mar10], that for a BS with large number of antennas, linear processingschemes are found nearly optimal. This significantly reduces the computational complexity, whichadds another advantage for massive MIMO systems. In this subsection, we’ll introduce some mainlinear precoding techniques used in MIMO systems.We will consider a similar setup as that introduced in the previous section, where a BS with Mantennas is communicating with K single antenna users while using linear precoders. The signaltransmitted from M-antenna BS, x is a linear combination of symbols intended for K users. Thelinearly precoded signal can be defined as:x =√αWs, (1.9)where s ∈ CK×1 is the transmitted source information before precoding,W ∈ CM×K is the linearprecoding matrix, and α = 1tr(WWH ) is the normalization constant to satisfy the power constraintE[| |x| |2] = 1.In TDD mode, the downlink signal received at the k’th user terminal can be expressed asfollows:ydlk =√αρdlhTkWs + nk, (1.10)=√αρdlhTkwk sk +√αρdl∑l,khTkwlsl + nk, (1.11)where nk ∼ CN(0, σ2) is the Gaussian noise, hk ∈ CM×1 and ρdl is the downlink SINR.Therefore the SINR at the k’th user can be expressed as follows:121.1. Background and MotivationSINRk =αρdl |hTkwk |2αρdl∑l,k |hTkwl |2 + σ2. (1.12)Maximum Ratio TransmissionMaximum Ration Transmission (MRT) also known as Matched Filter (MF) is simply theconjugate of the downlink channel, i.e:WMRT = H∗, (1.13)where H = [h1, . . . , hK] ∈ CM×K ,hk ∈ CM×1, and H∗ = [h∗1, . . . , h∗K] ∈ CM×K .Therefore the received signal at user k can be expressed as:ydlk =√αρdlhTkH∗s + nk, (1.14)The MRT precoder is known to maximize the signal gain at the intended end, but, as shown in[BSHD15], for close values of M and K , the performance of massive MIMO degenerates due tothe strong inter-user interference. When increasing the number of BS antennas, the channel vectorsin H become close to mutually orthogonal and thus the term HTH∗ approaches a diagonal matrix.This asymptotic property cancels out the inter-user interference and makes the MRT precoder anoptimal solution [FHX+18]. We will be using this technique in our proposed system model in thiswork.Zero-ForcingZero-forcing (ZF) precoding is a precoding technique that aims at eliminating the interferenceat the expense of channel capacity. This precoder eliminates the interference by transmitting thesignal towards the intended user while nulling in direction of other users. The Zero-forcing precoderobtained is expressed as follows:WZF = H∗(HTH∗)−1. (1.15)Therefore the corresponding received signal is defined as follows:131.1. Background and Motivationydlk =√αρdlhTkH∗(HTH∗)−1s + nk . (1.16)This technique is optimal in the absence of noise, but in the presence of noise, it amplifies thenoise power.Regularized Zero ForcingThe Regularized Zero Forcing (RZF) is a precoding technique that holds popularity in its abilityin trading off the advantages of MF and ZF precoders. According to [FHX+18, BSHD15], the RZFprecoder can be defined as follows:WRZF = H∗(HTH∗ + X + λIK)−1. (1.17)The above definition of an RZF precoder can be seen as a ZF precoder regularized by anHermitian non-negative matrix X and regularization factor λ.The received signal can thus be defined as follows:ydlk =√αρdlhTkH∗(HTH∗ + X + λIK)−1s + nk . (1.18)Notice that the choice of X and λ affect the property of RZF precoder. Specifically, if X = 0then (1.17) becomes an MF precoder when λ → ∞, while an ZF precoder when λ → 0. Thedesign of these parameters typically depends on the system’s optimization scheme and design[FHX+18, Ngo15].1.1.4 Asymptotic Analysis and Random Matrix TheoryIn this sectionwewill highlight the significance of studying the wireless system in an asymptoticregime, an approach that is considered in this thesis when analyzing the asymptotic performanceof an IRS-assisted MIMO system.141.1. Background and Motivationλ212MM-antenna BasestationFigure 1.5: Isotropic scattering, approximated as a rayleigh fadingLet us consider a setting where a massiveMIMO system is deployed in a single cell and operatesin an isotropic scattering environment, i.e. channels are modeled are i.i.d. Rayleigh fading, asdepicted in Fig.1.5. The antenna separation is chosen so as to satisfy the spatial correlation functionof the wave equation r(d) = sinc(2d/λ) were d is the spatial separation of the two locations atwhich the field is sampled [MBB+00, 3Gp10]. The channel distribution, Rayleigh fading, canbe justified by the central limit theorem by assuming that each antenna receives a superpositionof many waveforms that originates from independent scattering [MLYN16]. Since a wirelessfading channel is practically random in nature, system’s propagation analysis involves studyingthe network’s performance on average. For an i.i.d. Rayleigh fading model with channel matrixH = [h1, . . . , hK] ∈ CM×K , E[|hmk |2] = βk , and E[hm∗k hml ] = 0 for l , k, where hmk is the m’thelement in channel vector hk , then by the law of large numbers:1M| |hk | |2 → βk, M →∞, k = 1, . . . ,K, (1.19)1MhHk hl → 0, M →∞, k , l . (1.20)Therefore, asymptotically, an independent Rayleigh fading channel attains a favorable propaga-tion, i.e. the user’s channel vectors are almost mutually orthogonal [MLYN16].Generally speaking, multiple antenna systems heavily rely on signal processing and thus aremore concerned with vectorial inputs rather than scalar inputs. Therefore, it is of great research and151.2. Intelligent Reflecting Surfacessystem analysis interest to show that a functional of a random matrix H can be well approximatedby a deterministic quantity which only depends on the statistical property of H [CD11, TV04]. Tofurther elaborate on this point, consider a sequence of K i.i.d. random vectors randomly drawnfrom an M-variate zero mean random process, such that x1, x2 . . . , xK ∈ CM×1, where K/M is afixed ratio, then,RK =1KK∑i=1xixHia.s.−−−−→K→∞ R , E[x1xH1 ], (1.21)where a.s. stands for almost surely, and RK is the empirical covariance matrix computed from ob-served vector samples. The convergence property holds for any matrix norm, i.e. | |R − RK | | → 0with probability 1. As K is chosen very large for a fixed M (keeping K/M fixed), RK is a goodapproximation of R, and the convergence property holds. These deterministic quantities derivedfrom random matrices are usually given in a concise form as (1.21) shows, and helps draw someinsights about important system parameters as well as provide a strong tool to build simple schemesfor resource allocation and power control for large dimension MIMO system.Therefore with the increase of system’s dimension, i.e. as the number of antennas grow large,the randommatrix theory (RMT) advantage kicks in, and what was random start to behave in deter-ministic manner [WCDS12b, RPL+13]. This becomes particularly important when formulating thesignal to interference noise ratio (SINR) and other system’s performance metrics in an asymptoticregime, as we will see in Chapter 3. We will make use of large system dimension to obtain insightson system’s behavior. More specifically, using tools from random matrix theory (RMT), we studythe asymptotic performance of the IRS-assisted system, where IRS is equipped with N reflectingelements, by deriving the deterministic approximations of the SINR and the sum-rate, which aretight in the limit M, N,K →∞.1.2 Intelligent Reflecting SurfacesThe emerging future wireless networks including fifth generation (5G) and beyond are not onlyanticipated to fulfill the challenging demands of increased connectivity, data reliability and speed,161.2. Intelligent Reflecting Surfacesbut also to successfully realize a distributed intelligent communication platform that meets the re-quirements of the current growing wireless communication direction. The new user requirements,new application and communication trends will bring about challenging problems that motivateresearchers to envision a radically different communication platform [YXXL19, Ren19, BRdR+19].Sixth-G is in fact expected to accommodate a society that is typically data driven, thus enablingautonomous systems with unlimited wireless connectivity and negligible latency [CSAJ19]. Never-theless, asworks on previouswireless generations have shown, the advent of a new generation is usu-ally accompaniedwith the inevitable high energy consumption and considerable hardware/spectrumdeployment cost [BIK+16, WLC+17, MIA16]. Under these concerns and over the past few decadesa growing interest in investigating new technological techniques that help meet the challengingdemands of emerging wireless networks while at the same time provide an improved spectral andenergy consumption has notably been increasing in wireless communication research.1.2.1 Background and MotivationVarious technological advances, including massive multiple-input multiple-output (MIMO),millimeter wave (mmWave) communication, and network densification, are leading the emergenceof 5G networks. Their advantages are indisputable, but they face two main practical limitations.First, they consume a lot of power and incur high hardware/spectrum deployment costs which arecritical issues for practical implementation [BIK+16, WLC+17, MIA16], and second, they struggleto provide quality of service guarantee in harsh propagation environments, due to the lack of controlover the wireless channel. Wireless research has always focused on the design and optimizationthe communication end-points and the transmission protocols that can adapt to the time-varyingcharacteristics of the propagation environment. The wireless environment (or channel) itself isgenerally assumed uncontrollable and often acts as an adversary to the communication process.For example, high path loss especially at mmWave frequencies limits the network coverage andresults in blockages, multi-path propagation results in fading, poor scattering results in rank-deficient MIMO channels while anomalous reflections and refractions from environmental objectsresult in uncontrollable interference.In light of these limitations, the biggest challenge has been to find a way to control the propaga-171.2. Intelligent Reflecting Surfacestion of radio waves in the environment in an energy-efficient way to improve the system performancewithout increasing the power consumption, resulting in green, sustainable and pervasive futurewire-less networks. A transformative concept that addresses this need is that of a smart propagationenvironment enabled by intelligent reflecting surfaces (IRSs), also known as reconfigurable in-telligent surfaces (RISs) [WZ19c, Ren19], that can be deployed on the environmental objects tocustomize the propagation of radio waves in the environment through controlled reflections. Inthe recent literature on this concept, IRS is abstracted as a planar array of large number of passivereflecting elements, where each element can be remotely and independently reconfigured to inducea desired phase shift onto the incident electromagnetic waves. By properly adjusting the phaseshifts induced by all passive elements of the IRS, the reflected electromagnetic waves can addconstructively at the desired user to achieve high received signal power and add destructively at thenon-intended users to mitigate the interference effect.signalM antenna BSDirected BeamIRSIRScontrollerFigure 1.6: IRS-assisted wireless transmission: IRS facilitates beamforming by directing a beamcentered at the user endCurrent implementations of IRSs include reflect-arrays and reconfigurablemeta-surfaces, whichcan manipulate the incident waves in a passive manner without generating new radio signals andtherefore without incurring any notable power consumption. Hence, as Fig. 1.6 shows, an IRScan be seen as array of meta-atoms that each scatter the incoming signal by a controllable phase toproduces a directed beam as a product of the joint effect of all induced phase-shifts [ZBM+19].181.2. Intelligent Reflecting SurfacesM antenna BSUTIRS-assisted channelIRSDirect Channel(a) An IRS-assisted system with adirect channel path between theBSand user terminalM antenna BSIRS-assisted channelIRSUTBlockage(b) An IRS-assisted system with ablockage between the BS and userterminalFigure 1.7: IRS-assisted system depicting two practical scenariosTo conceptualize the benefit of IRS in MISO systems, let us consider two practical settings asshown in Fig. 1.7. First setting is in which the information signal is transmitted to user via twopaths, an IRS-assisted path and a direct BS-user path, as shown in Fig. 1.7(a). The BS typicallyapplies the beamforming technique on the transmitted signal to direct the beam power to the user.The user in turn also benefits from the desirable IRS-multipath effect, in which it receives signalreflected from the IRS towards its direction and achieves a sum of strong signal power at its end.Another practical setting is shown in Fig. 1.7(b). In this case, IRS deployment advantage standsout, where its presence helps the user realize the benefit of the IRS reflected beam and receivethe intended information despite the blockage present between it and the BS. In this scenario theuser is still capable of receiving the intended information and communicating with the BS. It isworth noting that IRS technology possesses an additional advantage, that is the practicality indeployment. IRS technology can be easily mounted onto the walls of existing buildings allowingtheir integration into the already built-in networks, without requiring any change in the hardware ofthe BS or user terminals. With such convenience and potential in deploying the IRS, this technologyhas gained considerable attention in theoretical and practical-based research as will be highlightedin the following section.191.2. Intelligent Reflecting Surfaces1.2.2 State of the ArtMotivated by the remarkable advantages of IRS technology, various works on IRS-assistedsystem have appeared in the last year that focus on the design of the reflect beamforming matrix atthe IRS to meet different objectives. In [GLCL19], the authors formulate and solve the sum-ratemaximization problem for the IRS-assisted multiple-input single-output (MISO) system by jointlyoptimizing the active beamforming at the BS and passive reflect beamforming at the IRS. Theyassume the availability of perfect CSI to design the active beamforming while the reflect beam-forming is designed by modeling the true channel coefficients as realizations from the sample spacethat is dominated by the knowledge of the imperfect CSI. In [WZ19b, WZ18], Wu et. al. optimizethe precoding at the BS and reflect beamforming at the IRS under the criteria of minimizing thetotal transmit power at the BS, subject to users’ individual SINR constraints. They further extendtheir work in [WZ19a, WZ19d] to solve a similar optimization problem under the more practicalassumption that IRS elements can only apply discrete phase shifts instead of continuous phaseshifts. The work in [NKC+19b] aims at maximizing the minimum SINR of an IRS-assisted MISOsystem in the asymptotic regime, under the scenarios where the BS-to-IRS channel matrix is arank-one line of sight (LoS) matrix and a full-rank LoS matrix. IRSs have also found applicationsin [LDY+19] and [MLG+19] in UAV communication and NOMA systems respectively. Moreover,the use of IRSs to enable indoor communication in the presence of blockages has been the subjectof [TSJP16, TSKJ18].Most of the current literature on IRS-assisted systems focuses on solving optimization prob-lems while assuming perfect channel state information (CSI) to be available at the BS and the IRS.Channel estimation is a big practical issue in IRS-assisted communication systems since the IRSelements have no radio resources of their own to send or receive pilot symbols and no signal pro-cessing capability to estimate the IRS-assisted links. Recently some works have proposed channelestimation protocols to estimate the IRS-assisted channels taking into account the radio limitationsof the IRS [MJ19b, LD19, YZZ19a, CLCY19, HY19]. Least-square channel estimates based onchannel training have been proposed in [MJ19b, LD19, YZZ19a] for a single-user system whilechannel estimation algorithms that exploit the sparsity of the cascaded channel matrix comprising201.3. Thesis Workof the BS-to-IRS link and the IRS-to-user link have been proposed for a multi-user system in[CLCY19, HY19]. However, all these works only focus on developing the channel estimationprotocols and do not utilize the derived estimates or algorithms in the design and analysis of theIRS-assisted downlink system. Additionally, up to the authors’ knowledge, while there has beena remarkable focus on the development of beamforming optimization algorithms for IRS-assistedsystems, there has been no notable work studying the performance limits of an IRS-assisted system.Although simulations results of existing works show that it is always beneficial to deploy an IRSbut these conclusions need to be backed by theoretical analysis and tested in a more practical settingwith imperfect CSI.1.3 Thesis WorkIn this section, we will highlight this work’s main objectives and contributions, and provide abrief outline of the thesis.1.3.1 ObjectivesIn light of the research gaps highlighted in the previous section, the main objectives of this workcan be summarized as follows:− Study the performance of a multi-user IRS-assisted MISO system in a Rayleigh fadingenvironment while assuming imperfect channel information at the BS. This includes thefollowing:– Derive the MMSE estimates of the channels using a protocol that takes into account thepassive nature of the IRS elements.– Design MRT procoders at the BS for downlink transmission and formulate the corre-sponding downlink SINR and sum rate expressions.− Study the downlink sum-rate performance of a multi-user IRS-assisted MISO system in anasymptotic regime, that is where the system dimensions grow large. This includes:211.3. Thesis Work– Obtain the asymptotically tight approximation of SINR and sum rate expressions underMRT precoding using random matrix theory (RMT) tools.– Draw insights on the impact of IRS phase shift on the overall asymptotic performanceof the system.− Provide numerical results that prove the following:– The derived asymptotic sum rate expression is tight even for moderate system dimen-sions.– The array gain achieved by the deployment of IRS in MISO systems while loss ofIRS-beamforming gains in the asymptotic regime under Rayleigh fading environments.– The usefulness of IRS in interference limited scenarios.1.3.2 ContributionIn this thesis, we consider a multi-user IRS-assisted MISO system where an IRS, comprisingof N passive reflecting elements, is deployed in the LoS of an M-antenna BS to assist it in commu-nicating K single-antenna users. We formulate the downlink signal to interference plus noise ratio(SINR) and sum rate expressions under maximum ratio transmission (MRT) precoding and thedeveloped MMSE channel estimation protocol. Using tools from random matrix theory (RMT),we study the asymptotic performance of the IRS-assisted system by deriving the deterministicapproximations of the SINR and the sum-rate, which are tight in the limit M, N,K → ∞. Thederived deterministic equivalents are simplified for several special cases to gain insights into theimpact of M and N on the system performance. We find that the IRS phase shift values do notmatter asymptotically in the perfect CSI scenario and in the scenario with imperfect CSI and nodirect channel. The phase shifts do appear in terms involving the error in the estimation of the directchannel that gets propagated to the estimation of IRS-assisted links under the proposed estimationprotocol, and a project gradient ascent algorithm is proposed to optimize them. However, the termsin which these phase shifts appear are actually negligible and we find that even in the generalsetting with imperfect CSI and both IRS-assisted and direct channels, the IRS phase shifts playno significant role asymptotically. The asymptotic performance limit derived for fixed M and K ,221.3. Thesis Workand growing N reveals that IRS only yields an array and no beamforming gain in the asymptoticlimit. This restricts the usefulness of IRS under Rayleigh fading to noise-limited scenarios. Weconclude the thesis with numerical results that show the tightness of the developed deterministicequivalents for moderate system dimensions, confirm the insights drawn from theoretical analysisabout the usefulness of the IRS and highlight the high sensitivity of IRS-assisted systems to channelestimation errors.1.3.3 OutlineThe rest of the thesis will be divided as follows: Chapter 2 presents the system model, outlinesthe channel estimation protocol and derives the MMSE estimates as well as introduces the utilizedprecoding technique. Chapter 3 carries out a rigorous asymptotic analysis of the SINR and sum-rate under imperfect CSI and simplifies the derived deterministic equivalents for different operatingregimes. The chapter also concludes with presenting a gradient ascent algorithm to optimize thedeterministic equivalent expressions. Chapter 4 includes the simulation results that support ourwork. Chapter 5 concludes the thesis and provide potential future direction.23Chapter 2IRS-Assisted MISO SystemsIn this chapter we will introduce the IRS-assisted system model and propose a practical channelestimation protocol that takes into account the passive property of IRS elements. We derive theMMSE channel estimate, which BS uses to design MRT precoders for transmission. The finalsignal to interference and noise ratio (SINR) and sum rate expressions are also provided withinsights on the effect of channel estimate quality. This chapter provides essential channel modelexpressions, which will help pave the way to proceeding chapters.2.1 System ModelIn this section, we present the transmission model for the considered IRS-assisted downlinkmulti-user MISO system. We also develop an MMSE based channel estimation protocol to obtainCSI at the BS.Figure 2.1: Sketch of IRS-assisted MISO system.242.1. System ModelSignal ModelAs shown in Fig. 2.1, we consider an IRS-assisted MISO communication model where a BSequipped with M antennas, communicates with K single antenna users. An IRS composed of Npassive reflecting elements is installed in the LoS of the BS and assists it in communicating with theusers. Equipped with a smart controller and using feedback from the BS, the IRS can dynamicallyadjust the phase shift induced by each reflecting element on the impinging electromagnetic waves.Additionally, we consider a time-division duplex protocol and assume transmission over quasi-staticflat-fading channels. The received baseband signal yk at user k is given as,yk = (hd,k +H1Θh2,k)Hx + nk, (2.1)hereH1 ∈ CM×N is the LoS static channel between the IRS and the BS, and h2,k ∼ CN(0, β2,kIN ) ∈CN×1 and hd,k ∼ CN(0, βd,kIM) ∈ CM×1 are the Rayleigh block fading channel vectors betweenuser k and the IRS and user k and the BS respectively, where CN(m,R) denotes the multi-variatecomplex Gaussian distribution with meanm and covariance matrixR and β2,k , βd,k are the channelattenuation coefficients. Also Θ = diag(θ1, θ2...θN ) represents the response of the IRS, whereθn = αn exp( jφn), where φn ∈ [0, 2pi] is the phase shift introduced by the element n and αn ∈ [0, 1]is the amplitude reflection constant. Note that β2,k and βd,k are the channel attenuation coefficientsfor the IRS to user and BS to user links, respectively. The channel attenuation coefficient of BSto IRS link is captured in H1 and is represented by β1. The channel from user k to the BS, i.e.hk = hd,k +H1Θh2,k is statistically equivalent tohk = A1/2k zk, (2.2)where zk ∼ CN(0, IM) and Ak = βd,kIM + β2,kH1ΘΘHHH1 . Note that ΘΘH = diag(α), whereα = [α21, . . . , α2N ]. The derivation of Ak is provided is Appendix A.Moreover, x = ∑Kk=1√pkgk sk is the Tx signal vector, where gk ∈ CM×1, pk and sk ∼ CN(0, 1)are the precoding vector, signal power and data symbol for user k respectively. The Tx vectorsatisfies the average Tx power per user constraint as,E[| |x| |2] = tr (PGHG) ≤ Pmax, (2.3)252.2. Channel EstimationwherePmax > 0 is theTxpower constraint at theBS,P = diag(p1, . . . , pK) andG = [g1, g2, . . . , gK] ∈CM×K . The downlink SINR at user k is defined as,γk =pk |hHk gk |2∑i,k pi |hHk gi |2 + σ2, (2.4)where,hk = hd,k +H1Θh2,k, (2.5)denotes the overall channel between the BS and user k. The user rate is defined as,Rk = log2(1 + γk), (2.6)and the sum rate is obtained asRsum =K∑k=1log2(1 + γk). (2.7)2.2 Channel EstimationOne issue that is accompanied with the IRS technology is the channel estimation. Giventhe large number of elements the IRS contains, the number of links to be estimated increases.Additionally, the IRS elements are passive in nature, that is the elements lacks radio resources toundergo any signal transmission or processing, and therefore the CSI can only be estimated at theBS.2.2.1 State of the ArtCurrent literature on IRS-assisted systems’ channel estimation involves several promising ap-proaches to tackling the above limitations. In [TAA19a], Taha et al. proposed an algorithm thatuses compressive sensing and deep learning techniques to tackle the estimation problem. In theirsetting, some IRS elements are active and rest are passive, where the deep learning techniques andcompressive sensing are used to estimate the channels of all IRS elements via active IRS units. In[HY20], He et al. proposed a two-stage algorithm, which includes a sparse matrix factorizationstage and matrix completion stage to derive the cascaded BS-IRS and IRS-user channel estimates.The estimation is done by keeping one element (or group of elements) active at each stage of262.2. Channel Estimationestimation process. Another approach which involves minimum variance unbiased estimator isintroduced in [JC19], where Jensen et al. proposed a channel estimation scheme that is based onminimizing Cramer Rao Lower Bound (CRLB) under certain constraint. You et al. in [YZZ19b]proposed a channel training protocol where the IRS elements are divided into M groups, such thateach group shares the same reflection coefficient. The user sends an (M + 1) pilot symbols tothe AP, where the AP stacks the consecutive received signals and estimates the extended channelh. Motivated by the above works, in the following section, we will propose a new and practicalchannel estimation protocol that basically works by turning ON one IRS element at a time whilekeeping the rest OFF during the training process to estimate each channel individually. The processinvolves estimating a total of N + 1 channels.2.2.2 Proposed ProtocolChannel estimation is necessary to compute the precoding vectors at the BS and the reflectbeamforming matrix Θ at the IRS. The real difficulty is in the estimation of H1 and h2,k , since theIRS has no radio resources of its own to transmit pilot symbols that enable the BS to estimateH1 orto sense h2,k using the pilot signals received from the users. Therefore, the BS has to estimate all thechannels and share this information with the IRS controller. Given the passive nature of the IRS, weadopt the TDD protocol and exploit channel reciprocity in estimating the downlink channels at theBS using the received uplink pilot signals from the users. For this purpose, we divide the channelcoherence period of τ sec into an uplink training phase of τc and a downlink transmission phaseof τd . Throughout the uplink training phase, the users transmit mutually orthogonal pilot symbols.After correlating the received training signal with the pilot sequence of user k, the BS estimatesthe channel vector hk based on the received observation vector ytrk ∈ CM×1 [HtD13], given as,ytrk = (hd,k +H1Θh2,k) + nULk , k = 1, . . . ,K, (2.8)where nULk ∼ CN(0, 1ρtr IM) is the noise in the uplink and ρtr > 0 is the effective training SNR. Ingeneral, ρtr depends on the pilot transmit power and the length of the pilot sequences. Here, weassume it to be a given parameter. To this end, note that with a change in variables, we can obtain,H1Θh2,k = H0,kv, (2.9)272.2. Channel Estimationwhere H0,k = H1diag(hT2,k) and v = [θ1, θ2, . . . , θN ] ∈ CN×1. With this representation, we havecascaded H1 and h2,k as H0,k ∈ CM×N , where H0,k = [h0,1,k, . . . , h0,N,K] is a matrix of N columnvectors. Each vector h0,i,k ∈ CM×1 (shown in red curved arrows in Fig. 2.1) can be interpreted asthe channel from the user to the BS through the IRS when only element i of the IRS is ON i.e.αi = 1, φi = 0 and αn = 0, n , i. We will focus on the MMSE estimation of h0,i,k , i = 1, . . . , N andhd,k for k = 1, . . . ,K at the BS. The received observation vector in (2.8) can be represented using(2.9) as,ytrk = (hd,k +H0,kv) + nULk , (2.10)= (hd,k +N∑i=1h0,i,kθi) + nULk , k = 1, . . . ,K, (2.11)where h0,i,k = R1/20,i,kqk . Note that qk ∼ CN(0, IM), R0,i,k = E[h0,i,khH0,i,k] = β2,kH1(:, i)H1(:, i)H ,and H1(:, i) is column i of H1 . The latter is obtained by using the fact that h0,i,k = H1(:, i)h2,k(i).The channel estimation interval is divided into N + 1 sub-phases of length τs = τcN+1 . TheIRS reflecting elements are controlled such that during the first sub-interval, all IRS elements areturned OFF, i.e. αn = 0, ∀n. Therefore in this sub-phase the BS will only receive the pilot signalstransmitted over the direct channel and will be able to estimate hd,k . In the following (i + 1)sub-phases, where i = 1, . . . , N , only the ith IRS element is turned ON in the full reflection mode(i.e. αi = 1, φi = 0). Therefore in sub-phase i, the BS receives pilot symbols over the direct channeland the channel via IRS element i. This will facilitate the estimation of h0,i,k .The received training signal in sub-phase 1 of the channel estimation phase is,rtr1,k = hd,k + nUL1 , k = 1, . . . ,K . (2.12)The MMSE estimates of hˆd,k , k = 1, . . . ,K based on (2.12) are derived to be,hˆd,k = Rd,kQd,krtr1,k, (2.13)where Rd,k = βd,kIM , Qd,k = (βd,kIM + 1ρtr IM)−1 and ρtr is the training SNR. The proof of (2.13)is provided in Appendix A.2.282.2. Channel EstimationThe received observation vector during the sub-phase (i + 1) of the channel estimation intervalis given as,rtri,k = hd,k + h0,i,k + nULi , k = 1, . . . ,K, i = 1, . . . , N . (2.14)Now since the BS already has the estimate of hd,k , it can subtract it from the observation vectorand obtain,r˜tri,k = h˜d,k + h0,i,k + nULi , (2.15)where h˜d,k = hd,k − hˆd,k . The MMSE estimates of hˆ0,i,k based on the observation vector in (2.15)are derived to be,hˆ0,i,k = R0,i,kQi,k r˜tri,k, (2.16)where R0,i,k = β2,kH1(:, i)H1(:, i)H ,Qi,k =(CHh˜d,k h˜d,k + β2,kH1(:, i)H1(:, i)H + 1ρtr IM)−1andCh˜d,k h˜Hd,k = Rd,k + Rd,kQd,kRd,k . The derivation of (2.16) is provided in Appendix A.3.Note that both hˆd,k and hˆ0,i,k follow Gaussian distributions. Also under the orthogonalityproperty of the MMSE estimates, the estimation error in the direct channel, denoted as h˜d,k =hd,k − hˆd,k , follows Gaussian distribution and is uncorrelated with hˆd,k (as well as independent dueto joint Gaussianity of both vectors) . Similarly, the estimation error vector in each IRS-assistedchannel, denoted as h˜0,i,k = h0,i,k − hˆ0,i,k , also follows Gaussian distribution and is uncorrelated(and independent) with hˆ0,i,k . Using the results derived above, it can be easily seen that hˆk =hˆd,k +∑Ni=1 hˆ0,i,kθi will also follow a Gaussian distribution and can be expressed as a correlatedRayleigh channel as follows.Lemma 2.1. The channel estimate hˆk = hˆd,k +∑Ni=1 hˆ0,i,kθi can be represented as,hˆk = C1/2k rk, (2.17)where rk ∼ CN(0, IM) and,Ck = Rd,kQd,kRd,k +N∑i=1R0,i,kQi,k(β2,kH1(:, i)H1(:, i)H + 1ρtrIM)QHi,kRH0,i,k+N∑i=1N∑j=1θiR0,i,kQi,kCh˜d,k h˜Hd,kQHj,kRH0, j,kθ∗j . (2.18)292.3. Beamforming and Achievable RateProof. The proof of Lemma 2.1 is provided in Appendix A.4.The derived MMSE estimates of the composite channel hk can be used to design transmitprecoders and to evaluate user’s SINR as well as the achievable sum rate expressions. This will befurther elaborated in the following section.2.3 Beamforming and Achievable RateIn this section, we’ll use the derived MMSE channel estimates to design the maximum ratiotransmission (MRT) precoder and formulate the SINR expression. The formulated expressions willnote the effect of the channel estimation error on the overall achievable sum rate.2.3.1 Maximum-Ratio Transmission PrecodingWith TDD operation, BS utilizes the MMSE estimates of hk from (2.17) to implement MRTprecoding. By adopting MRT, the symbol of each user is precoded by the Hermitian of its channelvector. MRT reduces the computational complexity greatly as compared to RZF/ZF precoding,which involves the inversion of the Gram matrix of joint users’ channel matrix. With MRTprecoding, the precoding vector is given as gk = ζ hˆk and the precoding matrix is given as,GH = ζHˆ, (2.19)where Hˆ =[hˆ1, hˆ2, · · · , hˆK]H ∈ CK×M , and ζ ensures that the power constraint in (2.3) is satisfiedas,ζ2 =Pmaxtr(PHˆHˆH) = PmaxΨ, (2.20)where Ψ = tr(PHˆHˆH).2.3.2 SINR and Achievable RateTo compute the SINR in the downlink, the user needs to have CSI too. A well-known featureof large-scale MIMO systems is channel hardening, i.e. the effective useful channel hHk gk of a user302.4. Discussionconverges to its average value when M grows large. Hence, it is sufficient for each user to haveonly the statistical CSI and the resulting performance loss vanishes in the large system limit. Usingthis idea, an ergodic achievable user rate can be computed using a technique from [Med00], widelyapplied to large-scale MIMO systems [HtD13, NKDA19]. The main idea is to decompose yk in(2.1) using the definition of x in Sec. II-A as yk =√pkE[hHk gk]sk +√pk(hHk gk − E[hHk gk])sk +∑l,k√plhHk glsl + nk and assume that the average effective channel E[hHk gk] is perfectly known atuser k. By treating the interference and channel uncertainty as worst-case Gaussian noise, the userk can achieve the ergodic rate,Rk = log2(1 + γk), (2.21)without knowing the instantaneous values of hHk gk . The parameter γk can be interpreted as theeffective average downlink SINR of user k and is defined under MRT as,γk =pkE[|hHk hˆk |2]pkvar[hHk hˆk] +∑l,k plE[|hHk hˆl |2] + Ψρ, (2.22)where ρ = Pmaxσ2. The ergodic achievable sum rate is given as:Rsum =K∑k=1log2(1 + γk). (2.23)2.4 DiscussionIn this chapter we derived the MMSE estimate of the composite channel hk through a proposedprotocol that involves estimating N + 1 channels. More precisely, N channels account for thechannels between the BS and user k via the i’th IRS element for i = 1 . . . N , and one direct channelaccounts for the channel between the BS and user k. Therefore the composite channel estimatehˆk = hˆd,k +∑Ni hˆ0,i,kθi relies on the N + 1 channel estimates to compute. Once the BS computesthe composite MMSE channel estimate hˆk , it utilizes linear processing technique to design MRTprecoders, which help optimize the performance gain at the user and leverage the benefits of MIMOsystems as explained in section 1.1.3. Under the channel hardening effect, user’s SINR expressioncan be expressed as in (2.22)312.4. DiscussionIt is important to note the effect of the channel estimate accuracy on the overall system’s per-formance. As shown in (2.23), the sum rate depends on the SINR at user terminals, whose valuein turn depends on the channel estimate hˆk . Therefore, the channel estimation technique and theestimate quality play a major role in the SINR achieved at user terminal k. In fact, consideringthe SINR expression in (2.22), we can note the effect of the power of channel estimation errorin the denominator terms, var[hHk hˆk] and∑l,k plE[|hHk hˆl |2], where a dominating error power cansignificantly drop the system’s performance.Additionally by considering the expression in (2.22) in the asymptotic limit, the effective averagedownlink SINR can be analyzed under an asymptotic regime and expressed as a deterministicequivalent. These expressions can be very useful to solve optimization problems as will be seen inthe following chapter.32Chapter 3Asymptotic AnalysisIn this section we derive the deterministic approximations of users’ SINR and ergodic rateexpressions underMRTprecoding for IRS assisted systems [WZ18,NKC+19b,MLG+19, TAA19b].This assumption seems realistic with the advancement in the design and fabrication of loss-lessmeta-surfaces [BMR17, EE16]. Section 3.1 provides a brief motivation behind studying theasymptotic performance of the wireless system followed with some useful lemmas utilized in ourwork. Section 3.2 will provide the assumptions made and the derived expressions.3.1 Motivation and Useful LemmasAn important performance metric in communication systems is the users’ ergodic rates, whichare generally difficult to study for finite system dimensions. In fact, the SINR expression in (2.22)yields no insights into the performance of the system. With the advent of 5G technology and thedeployment of large-dimension networks to serve multiple users, the network performance analysisunder the asymptotic regime has become an area of interest. In the large M, N,K regime, the users’SINRs and ergodic rates tend to approach deterministic quantities. These quantities depend only onthe statistics of the channels and are referred to as deterministic equivalents. These deterministicequivalents are almost surely (a.s.) tight in the asymptotic limit, i.e. as the system dimensionsgrow to infinity, the approximations yielded by these deterministic equivalents tend to the actualvalues with probability one. These equivalents depend only on the statistics of the channel and arevery useful to solve important optimization problems as studied in [WCDS12a, HtD13, NKDA19,NKC+19b]. Given the resulting optimization solutions rely only on the knowledge of channel’slarge-scale statistics which vary very slowly, so solving the optimization problems at the pace offast fading is not required.Under this motivation, we exploit the statistical distribution of hk and large values of M, N,K to333.2. Deterministic Equivalence Resultscompute the deterministic approximations of user’s SINR and ergodic rates. These approximationswill yield very important insights into how the presence of IRS impacts the performance of thesystem.The derivations in this section will exploit the following lemmas.Lemma 3.1. Let A ∈ CN×N and x ∈ CN×1 be a random vector of i.i.d entries independent of A.Assume the components of x have mean zero, variance 1 and A satisfies lim supN | |A| | < ∞. Then,1NxHAx − 1Ntr(A) a.s−−−−→N→∞ 0. (3.1)Lemma 3.2. LetA be as defined in lemma 3.1 and x and y be i.i.d vectors with the same distributionas defined in lemma 3.1 as well. Then,1NxHAy a.s−−−−→N→∞ 0. (3.2)Lemma 3.3. Let A1,A2 . . . be an infinite sequence of matrices with AN ∈ CN×N , be deterministicwith bounded spectral normandB1,B2 . . . withBN ∈ CN×N , be randomHermitianwith eigenvaluesλBN1 ≤ λBN2 · · · ≤ λBNN such that with probability 1, there exists ε > 0 for which λBN1 > ε for alllarge N . Then for v ∈ CN1NtrANB−1N −1NtrAN (BN + vvH)−1 a.s−−−−→N→∞ 0. (3.3)One intuitive way to look at the above lemmas is to relate it to the law of large numbers (LLN).For simplicity, let us consider A be an identity matrix, i.e. A = IN . By the LLN, xHAx approachesE[xHAx] as N → ∞. For a finite N ,we can show that E[xHAx] = 1N tr(A) [CD11], which isequivalent to saying E[xHINx] = 1N tr(IN ) for the simplified scenario, where tr(IN ) is the sum of xentries variances. Therefore, lemma 3.1 can be concluded. A similar approach can be taken whenlooking at lemma 3.2. The above lemmas and their rigorous proofs can be found in [CD11, TV04]3.2 Deterministic Equivalence ResultsThe main theorem proposed in this section and the corollaries that follow are the major contri-butions of this work as they use the IRS-assisted channel in (2.2) and its estimate in (2.17) to derive343.2. Deterministic Equivalence Resultsthe deterministic approximation of each user’s SINR under MRT precoding. Before we introducethese results, we will require the following assumptions:Assumption 1. M , N and K grow large with a bounded ratio as 0 < lim infM,K→∞ KM ≤lim supM,K→∞ KM < ∞ and 0 < lim infM,N→∞ MN ≤ lim supM,N→∞ MN < ∞.Assumption 2. The LoS channel matrix H1 satisfies,lim supM→∞| |H1HH1 | | < ∞. (3.4)Assumption 3. The entries of the power matrix P = diag(p1, p2, . . . , pK) are of order O(1/K).The above assumptions are needed to help ensure a bounded norm on channel matrix, dimensionratio and power as the system’s dimensions grow. This is required in the derivation of the deter-ministic equivalents of the SINR and sum rate to ensure a bounded limit and tight approximationas system grows large.We now present the deterministic equivalent of the SINR of user k defined in (2.22) in the followingTheorem.3.2.1 IRS Assisted System with a Direct ChannelIn this subsectionwewill provide the deterministic equivalents of the SINR and sum expressionsof an IRS-assisted system with a composite channel comprised of both an IRS-assited channel anda direct channel, as shown in Fig. 2.1.Theorem 3.1. Under Assumptions 1, 2 and 3, the SINR of user k defined in (2.22), for the channelin (2.2) and its estimate in (2.17) converges as,γk,MRT − γ◦k,MRTa.s−−−−−−−−→M,N,K→∞ 0, (3.5)353.2. Deterministic Equivalence Resultswhere γ◦k,MRT is defined as:γ◦k,mrt =pkK |∑Ni=1 θitr(Ch˜d,k h˜d,kR0,i,kQi,k) +β2d,kMβd,k+1ρtr+∑Ni=1 tr(R0,i,kR0,i,kQi,k)|2∑l,kplK tr(ClAk) +pkK∑Kk=1 trCkρ,(3.6)where Ak = β2,kH1HH1 + βd,kIM and Cl is defined in Lemma 2.1 in (2.18).Proof. The proof of Theorem 3.1 is provided in Appendix BTwo important insights can be drawn from the expression in (3.6). First, the phase shifts appliedby the IRS elements, i.e. vn = exp( jθn), do not appear anywhere except for the terms involving theMMSE covariance matrix Ch˜d,k h˜d,k of hd,k . The error in the estimation of hd,ks propagates in theestimation of the subsequent h0,i,ks under the proposed protocol as shown in (2.16) making hd,kand r˜tri,k , i = 1, . . . , N dependent due to the presence of h˜d,k in r˜tri,k . If the direct channel is estimatedaccurately, the phase shifts will not matter asymptotically in an IRS-assisted system and IRS willnot yield any reflect beamforming gain. This phenomenon is caused by the spatial isotropy thatholds upon the IRS-assisted channel, which is insensitive to the beamforming betweenH1 and h2,kunder Rayleigh fading h2,ks1. Second, the IRS can still yield an array gain due to the sum of Ncontributions of the IRS elements in the numerator. However, there are also some terms with sumof N contributions of the IRS elements in the denominator. We will see whether the IRS gain inthe desired signal energy is larger than the gain in the interference in subsection 3.21 .In the next corollary, we present the deterministic equivalents of the users’ ergodic rates in (2.21).Corollary 3.1. Assume that Assumption 1, 2 and 3 hold true. From the continuous mappingtheorem it follows that the individual downlink rates Rk of the users converge as,Rk − R◦ka.s−−−−−−−−→M,N,K→∞ 0, (3.7)where,R◦k = log(1 + γ◦k), (3.8)where γ◦k is given by (3.6).1The same conclusion holds if both H1 and h2,k undergo Rayleigh fading.363.2. Deterministic Equivalence ResultsAn approximation for the average sum rate can be obtained as follows:R◦sum =K∑k=1log(1 + γ◦k), (3.9)such that,1K(Rsum − R◦sum)a.s−−−−−−−−→M,N,K→∞ 0. (3.10)The next subsection considers the special case where the direct channel from the BS to the user isblocked.3.2.2 IRS Assisted System with a Blocked Direct ChannelLet us now consider an IRS assisted transmission scenario in Fig. 2.1, where the direct channelis blocked, i.e. hd,k = 0. For such a scenario, the channel model in (2.2) is simplified to,hk = H1Θh2,k . (3.11)The channel can be equivalently written as,hk = A1/2k zk, (3.12)where zk ∼ CN(0, IM) and Ak = β2,kH1HH1 .The channel estimation phase will only have N sub-phases where the IRS elements are switchedON one-by-one. There is no estimation of the direct channel involved and the estimates of h0,i,k ,i = 1, . . . , N in (2.16) are simplified as,hˆ0,i,k = R0,i,kQi,krtri,k, (3.13)whereR0,i,k is the same as defined in Section II-B, rtri,k = h0,i,k+nULi andQi,k =(β2,kH1(:, i)H1(:, i)H+ 1ρtr IM)−1.Moreover the channel estimate hˆk =∑Ni=1 hˆ0,i,kθi can be represented as,hˆk = C1/2k rk, (3.14)373.2. Deterministic Equivalence Resultswhere rk ∼ CN(0, IM) and,Ck =N∑i=1R0,i,kQi,kR0,i,k . (3.15)The deterministic equivalent of the SINR defined in (2.22) is presented in the following Corol-lary.Corollary 3.2. Let Assumption 1, 2 and 3 hold true. Then the SINR of user k defined in (2.22), forthe channel in (3.11) and its estimate in (3.14) converges as,γk,MRT − γ◦k,MRTa.s−−−−−−−−→M,N,K→∞ 0, (3.16)where γ◦k,MRT is defined as:γ◦k,MRT =1K pk |trCk |21K∑l,k pltrClAk +pkK∑Kk=1 trCkρ, (3.17)where Ck =∑Ni=1 R0,i,kQi,kR0,i,k and Ak = β2,kH1HH1 .Notice that due to the lack direct channel between the BS and user terminal, the IRS phaseshifts, which appeared along side the direct channel error in Theorem 3.1, do not show up under thissetting. Consequently, in an asymptotic regime we obtain no beamforming gains under Rayleighfading channel with blocked direct channel.3.2.3 IRS Assisted System Under Perfect CSIFinally, we simplify the result in Theorem 3.1 for the scenario where perfect CSI is available atthe BS about all the IRS-assisted channels as well as the direct channels. The result is provided inthe following corollary.Corollary 3.3. Under the setting of Theorem 3.1, let perfect CSI be available. Then,γ◦k,MRT =pkK |tr(Ak)|2∑l,k tr( plKAlAk) +pkK∑Kk=1 tr(Ak )ρ, (3.18)where Ak = βd,kIM + β2,kH1HH1 .383.2. Deterministic Equivalence ResultsProof. The proof follows by letting ρtr →∞ in Theorem 3.1.Under this simplified setting where perfect CSI is known at the BS, IRS phase shifts alsoprovided no beamforming gain asymptotically. Notice that we have the direct channel in thissetting, but as shown in Theorem 3.1, phases only appear with the terms involving the MMSEcovariance matrix of direct channel error h˜d,k .In the following subsection, we will consider a simplifiedmodel of the channel to gain more insightson the benefits of IRS in system’s performance.3.2.4 Special CaseTo gain more explicit insights into the impact of IRS on the system performance, we considera simplified model from [HtD13] for the channel in (2.1). Before presenting the simplified model,note that (2.1) is statistically equivalent tohk =√βd,kzB,k +√β2,k(H1HH1 )1/2zI,k = A1/2k zk, (3.19)where Ak = β2,kH1HH1 + βd,kIM , zB,k ∼ CN(0, IM), zI,k ∼ CN(0, IM) and zk ∼ CN(0, IM) 2.We will focus on the perfect CSI scenario in this section for analytical tractability noting that itacts an an upper bound on how well the IRS-assisted system can perform under CSI errors. Thedeterministic equivalent for the channel in (2.2) under MRT precoding and perfect CSI has beenderived in corollary 3.3.Similar to [HtD13] which derives asymptotic performance limits of a conventional massiveMISO system, we consider a simplified channel model for hk given as,hk =√βd,kzB,k +√cβd,k(DDH)1/2zI,k, (3.20)whereD =√NU andU ∈ CM×N is composed of M ≤ N rows of an arbitrary N ×N unitary matrix.This model basically abstracts H1HH1 as DDH and since each diagonal entry of H1HH1 is the sumof N exponential terms of norm 1 scaled by β1, so we have introduced the normalization√N inD to ensure the variance of each element of IRS-assisted channel vector stays Nβ1. Moreover, we2The covariance matrix for IRS-assisted channel is obtained as E[H1Θh2,khH2,kΘHHH1 ] = β2,kH1HH1 , by notingthat ΘΘH = diag(α) = IN under the assumption α = [α1, . . . , αN ]T = 1N .393.2. Deterministic Equivalence Resultshave let β1β2,k = cβd,k , ∀k 3. The channel in (3.20) is equivalent to writing hk = D¯1/2k zk , wherezk ∼ CN(0, IM) and D¯k = βd,k(cDDH + IM). This is indeed a special case of (2.2).Under this model, the total energy of the channel grows linearly withM and N , since E[hHk hk] =tr (D¯k) = βd,k(cMN + M). The motivation behind this channel model is twofold. First, it assumesthe antenna aperture to increase with each additional antenna element and IRS, which is in contrastto existing works which assume more and more elements to be packed into a fixed volume. Aninsufficiency of this channel model is that the captured energy grows without bounds as M or Ngrow large. However, linear energy gains can be achieved up to very large numbers of antennas ifthe size of the antenna array is scaled accordingly as discussed in [HtD13]. Second, the model willallow us to explicitly study the impact of the IRS on the asymptotic system performance with theonly difference between (2.2) and (3.20) being that we have assumed rows of H1 to be orthogonalto each other. The LoS channel matrixH1 can be designed in practice to achieve this orthogonalityand therefore the model in (3.20) can actually be realized in practice. The performance obtainedusing (3.20) is an upper-bound to the actual performance under (2.2). However, the insights thatwe obtain using this model in the rest of this section will also hold under (2.2) and imperfect CSIas will be shown in simulations.The performance of the IRS-assisted system under perfect CSI in Corollary 3.3 can be given ina compact form closed form for the model in (3.20) as follows.Corollary 3.4. For the channel model (3.20) and pk = 1K , ∀k, γ◦k,MRT in Corollary 3.3 is given as,γ◦k,MRT =11M∑l,kβd,lβd,k︸ ︷︷ ︸Interference+∑Kl=1 βd,lMβ2d,k ρ(cN + 1)︸ ︷︷ ︸Noise. (3.21)The proof of Corollary 3.4 is provided in Appendix C.The resulting expression captures the interference caused by the system as well as the noiseexperienced by the system, where both effects are normalized by the desired signal power. Thiscorollary yields two very important insights. First it verifies the “massive MIMO effect" observedin [HtD13], which means that as M increases for fixed N and K , the SINR and therefore the userrates grows unboundedly.3This assumption is justified in scenarios where IRS is located very close to the BS. In such cases, c will accountfor the fixed extra loss that occurs in the IRS-assisted link.403.3. Reflect-Beamforming for Performance OptimizationSecond, the use of an IRS in Rayleigh fading environments is only useful when the averagereceived SNR is low, i.e. either the noise level is high or the path losses are very high (i.e. the valuesof β1 and β2,k are low). In such a noise-limited scenario, the second term in the denominator of(3.21) will dominate over the first and increasing N will produce a noticeable decrease in the noiseand increase in the SINR values. In an interference-limited scenario, i.e. where the first term in thedenominator of (3.21) dominates over the second, the use of an IRS yields no substantial benefit.This can be intuitively explained by noting that in Rayleigh fading channels, the phases of the IRSdo not matter and all the IRS does is provide an array gain of N . This array gain appears in boththe energy of the desired signal as well as the energy of the interfering signal and thereby the neteffect is negligible if the denominator of the SINR is dominated by the interference. Therefore, theuse of an IRS in Rayleigh fading environments will only be beneficial in noise-limited scenarios,for example, where users are located at cell edges such that the path loss is very high.It will be interesting to study the performance of the IRS-assisted system under Rician fadingchannels, there is a strong LoS component from IRS to user. Under such a setting, the values ofphase-shiftswillmatter and the IRSwill yield a reflect beamforming gainwhere reflect beamformingcan be designed to minimize the interference term. However, this is outside the scope of the currentwork as our focus is to gauge the practicality of using an IRS in the commonly studied Rayleighfading environments in existing works on IRS.3.3 Reflect-Beamforming for Performance OptimizationWe have already seen that the values of the phase shifts applied by the IRS elements do notmatter in Rayleigh fading environments under perfect CSI. Moreover they also do not appear inthe asymptotic expression of the SINR under imperfect CSI when the direct BS-to-user channelis absent. Under the proposed channel estimation protocol, the elements of v do appear in thedeterministic equivalent of the SINR in (3.6) in Theorem 3.1 in the terms related to the error inthe estimation of hd,k propagated to the estimation of h0,i,k , i = 1, . . . , N . Therefore, even thoughasymptotically the IRS elements do not need to be optimized under Rayleigh fading environments,but under the proposed channel estimation protocol we can minimize the channel estimation errorin hˆd,k propagated to the estimation of IRS-assisted channels by solving the following optimization413.3. Reflect-Beamforming for Performance Optimizationproblem.(P1) maxθ1,...,θNR◦sum =K∑k=1log(1 + γ◦k) (3.22a)s.t. |θn | = 1, ∀n. (3.22b)(P1) is a constrained maximization problem that can be solved using projected gradient ascentto increase the objective function by taking iterative steps proportional to the positive gradient. Ineach step we project the solution to the closest feasible point that satisfies the constraint in (3.22b).The algorithm is explained in Algorithm 1. To proceed, we need to the derivative of R◦sum withrespect to each θn, n = 1, . . . N which is stated in the following lemma.Lemma 3.4. The derivative of R◦sum defined in 3.6 with respect to θn is given as∂R◦sum∂θn=K∑k=11(1 + γok ) ln(2)2dk pkK |nk |tr(Ch˜d,k h˜d,kR0,n,kQn,k) −pkK |nk |2d′kd2k, (3.23)where,nk =N∑i=1θitr(Ch˜d,k h˜d,kR0,i,kQi,k) +β2d,kMβd,k +1ρtr+N∑i=1tr(R0,i,kR0,i,kQi,k), (3.24)dk =∑l,kplKtr(ClAk) + pkKρK∑k=1trCk, (3.25)d′k = 2N∑i=1θi(∑l,kplKtr(R0,i,lQi,lCh˜d,l h˜d,lQHn,lRH0,n,lAk)+K∑k=1pkKρtr(R0,i,kQi,kCh˜d,k h˜d,kQHn,kRH0,n,k)).(3.26)Proof. The poof of lemma 3.4 is provided in Appendix DWe denote the IRS phase-shifts vector as vs = [θs1, θs2, . . . , θsN ], where θsn is the nth IRS elementresponse at step s of the algorithm. The adopted gradient ascent direction is denoted as ps, where[ps]n = ∂Rosum∂θn|θsn . The next step updates the phase-shifts vector in a step proportional to the positivegradient as v˜s+1 = vs+µps, where the step size µ is obtained using backtracking line search [BV04].The solution v˜s+1 is projected onto the closest feasible point that satisfies the constraint (3.22b) as423.4. Conclusion and DiscussionAlgorithm 1: Projected Gradient Ascent Algorithm for the IRS Design1: procedure Design of IRS Phase-Shifts Vector(v∗)2: Initialize: v1 = exp( jpi/2)1N , R◦1sum = f (v1) given by (3.9),  > 0.3: set s=1;4: Repeat5: [ps]n = ∂R◦sum∂θn|θsn , n = 1, . . . , N , where ∂Rosum∂θnis given in (3.23);6: µ=backtrack line search( f (vs), ps, vs) [BV04];7: v˜s+1 = vs + µps;8: vs+1 = exp( jarg (v˜s+1));9: Ros+1sum = f (vs+1) ;10: Update s = s + 1;11: Until | |Ros+1sum − Rossum | |2 <  ; Obtain v∗ = vs+1;vs+1 = exp( jarg (v˜s+1)). The complete algorithm is outlined in Algorithm 1.As one might expect, optimizing the IRS phase-shifts provides a very small gain in the sum-rateperformance as will be seen in the simulations since the phase-shifts appear only in terms involvingthe error covariance matrix Ch˜d,k h˜d,k , which are almost negligible. Nevertheless, developing agradient ascent algorithm for the phase shifts that appear in the asymptotic expression in Theorem3.1 under the ON/OFF channel estimation protocol might be beneficial in scenarios where the errorin the estimation of direct channel is high.3.4 Conclusion and DiscussionIn this chapter, the asymptotic performance of an IRS-assisted systemwas studied under variouscases to gain an insight on the benefit of IRS-assisted communication. Specifically, we derived thedeterministic equivalence of the SINR and sum rate expressions, and drew conclusions on the effectof IRS phase shifts in the overall performance of the system in an asymptotic regime. We wereable to conclude that an IRS-assisted system operating in large dimensions under Rayleigh fadingenvironment, whose direct channel between the BS and user is blocked, does not benefit from433.4. Conclusion and Discussionbeamforming gains, since IRS phase shift values do not appear in the asymptotic equivalences.Same conclusion can be drawn for the case where perfect CSI is known while direct channelbetween BS and user exists. Nevertheless, in the case where imperfect CSI is considered anddirect channel is present, the IRS phase shifts do appear but only in terms related to the error ofthe direct channel estimation. These conclusions are important to asses the performance of largedimension wireless network in a Rayleigh block fading environment with an IRS deployed to assistin communication. To provide more visual results, the next chapter help support our conclusionsby providing numerical results and simulations that prove the tightness of the derived deterministicequivalences under large dimensions as well as analyze the IRS-assisted system performance underoptimized phase shifts with comparison to conventional MISO systems.44Chapter 4Numerical AnalysisThis chapter provides simulation results that (i) validate the derived deterministic approximationof the SINR for large dimensional systems, provide (ii) insights on the size of IRS needed to providecompeting achievable sum rate performance as compared to conventional MISO systems at a lowercost of implementation, and (iii) useful insights on the performance limit of IRS-assisted systemsunder varying cell radius.BS (0, 0, 25)y axisz axisx axisR 30o−30oIRS(x¯, y¯, 40)users(xi, yi, 1.5)Figure 4.1: Multiuser IRS-assisted system setupIn our simulations model we consider a 3D coordinate system with a linear array of antennasat the BS and passive reflecting elements at the IRS. The coordinates of the BS, IRS and userare denoted as (xBS, yBS, 25), (xIRS, yIRS, 40), (xU, yU, 1.5), respectively. We assume a full rankBS-to-IRS LoS channel matrix given as [NKDA15, NKC+19a][H1]m,n = exp(j2piλ((m − 1)dBS sin(θLoS1(n)) sin(φLoS1(n))+(n − 1)dIRS sin(θLoS2(n)) × sin(φLoS2(n)))),(4.1)454.1. Deterministic Equivalent Validationwhere dBS and dIRS represent the distance between the BS antennas and the IRS elements respec-tively set as 0.5λ, θLoS1(n) and φLoS1(n), n = 1 . . . N are generated uniformly between 0 to pi and0 to 2pi, respectively, while θLoS2(n) = pi − θLoS1(n) and φLoS2(n) = pi − φLoS1(n). The channelattenuation coefficients for the IRS-assisted channel and the direct channel are set as β2,k and βd,k ,respectively.The considered multi-user setup is shown in Fig. 2 with K cell-edge users placed along anarc of radius 150m that spans angles from −30o to 30o with respect to the x-axis. Using (x, y, z)coordinates (in meters), the BS and IRS are deployed at (0, 0, 25) and (x¯, y¯, 40) respectively wherex¯ is the mean of x coordinates of all users and y¯ is the mean of y coordinates of all users. Since thecell edge users have the most sensitive performance since they are more susceptible to blockages aswell as high path losses, we will focus on the performance of these users. Additionally, we definepk = 1/K , ∀k and Pmax = 1. The path loss for each link is calculated as β = C0dα , where C0 is thefixed loss set as 30dB for all links and α is the path loss exponent set as 2 for H1, 2.8 for h2,k and3.5 for hd,k as used in [WZ19b].4.1 Deterministic Equivalent ValidationWe will start with validating the tightness of the deterministic equivalence at large systemdimension. For the purpose of this result, we let β2,k = ck βd,k and abstractβ2,kσ2as ρ. Then, weplot in Fig. 4.2 against ρ, the sum-rate in (3.9) using the deterministic equivalent of the SINRin Theorem 3.1 as well as the Monte-Carlo simulated sum-rate in (2.23). The results are plottedagainst ρ for M = K = N = 48 and ρtr = {2, 8}dB.From Fig. 4.2 we can see that the deterministic equivalent of the sum rate in (3.9) providesa tight approximation to the Monte-Carlo simulated sum-rate for moderate values of M,K, N .As expected, the sum-rate performance of the system deteriorates as ρtr decreases. Also, weobserve that the approximation becomes less tight as ρ increases due to slower convergence tothe deterministic equivalent. However, the match is still quite good for the entire range of ρ andtherefore the deterministic equivalents derived in this thesis serve as useful tools to analyticallystudy the performance of IRS-assisted systems.464.2. IRS Performance with Respect to the Conventional System−30 −20 −10 0 10 200102030ρ [dB]SumRate(bps/Hz)Per. CSI, MCPer. CSI, Det.Imp. CSI, MC, ρtr=8dBImp. CSI, Det., ρtr=8dBImp. CSI, MC, ρtr=2dBImp. CSI, Det., ρtr=2dBFigure 4.2: Numerical validation of the deterministic (det.) equivalent of the SINR against theMonte-Carlo (MC) values, under perfect (per.) and imperfect (imper.) CSI., where ρ is abstractedas 1/σ2.4.2 IRS Performance with Respect to the Conventional SystemIn this section, we will compare the performance of an IRS-assisted system with conventionalMISO systems along with providing useful insights on the required size of IRS to achieve similarperformance as the conventional MISO system but at lower implementation cost.4.2.1 Performance ComparisonWe compare the performance of an IRS-assisted system with the conventional MISO systemand showcase that for the scenario considered in Fig. 4.3, the IRS yields notable gain over theconventional system wherein the gains promised by large MISO systems can be obtained by usingan IRS with many passive reflecting elements while using a smaller number of active antennas atthe BS. In the following simulation results, we introduce the model for ρtr as ρtr = pcτcσ2 , whichdepends on the noise variance σ2 set as 10−17 Joules as well as the pilot Tx power pc set as 1Wand length of each of the N + 1 channel estimation phases τs sec. The channel estimation periodτc is defined as 0.01τ, where τ is the length of the coherence interval set as 10ms. Thereforeτs =0.01τ(N+1) . Note that for the conventional MIMO system where no IRS is deployed, we will haveτs = 0.01τ. Under the block-fading channel model, the net achievable rate of user k in (2.6) is given474.2. IRS Performance with Respect to the Conventional Systemas Rk =(1 − τcτ)log2(1 + SINRk). The net achievable sum-rate is then given as Rsum =∑Kk=1 Rk .This is plotted in the subsequent figures.100 200 300 400 500812Red lines: IRS-asst. M = 64Dash-dot lines:IRS-asst. Imp. CSI, L=4NSumRate(bps/Hz)Conv., Imper. CSI, MC, M=80Conv., Per. CSI, MC, M=80IRS-asst., Imper. CSI, MC, M=48, G.A.O.phasesIRS-asst., Per. CSI, MC, M=48, θn = 1, ∀nIRS-asst., Imper. CSI, Det., M=48, G.A.O.phasesFigure 4.3: Sum-rate comparison of IRS-assisted (IRS-asst.) MISO system under gradient ascentalgorithm optimized (G.A.O) phases and exhaustive search optimized phases, and comparison withsum-rate of conventional (conv.) MISO systems.In Fig. 4.3, we show the effect of introducing an IRS on the sum-rate achieved by cell-edgeusers for varying number of BS antennas and IRS reflecting elements, i.e. M = {48, 64, 80} andN = {128, 256, 400, 512}. We consider K = 32 users in the system and compare the performancewith that of a conventional MISO systems with 80 active antennas at the BS. We plot IRS-assistedsystem sum rate while optimizing phase shifts using the gradient ascent algorithm proposed inAlgorithm 1 (square marked curve), as well as using an exhaustive search (dot-dashed curve) overan L-dimension set F defined as F = 0, 4θ, . . . , 4θ(L − 1), where 4θ = 2piL and L = 4. Notice thatthe curves almost overlap with negligible difference, which depicts the minor effect of phase shifton the overall performance of an asymptotic IRS-assisted system. Additionally, with increasingnumber of IRS elements N , the IRS-assisted system outperforms the conventional large MISOsystem, by using a fewer number of active antennas at the BS. The result shows that the IRS-assisted system with 170 passive reflecting elements at the IRS and 64 active antennas at the BS canachieve the same performance as a conventional large MISO system with 80 active antennas. Thesame performance can also be achieved with M = 48 antennas using N = 470 reflecting elementsat the IRS. Therefore, an IRS-assisted system can perform as well as a conventional large MIMOsystemwith a reduced number of active antennas at the BS, making it an energy-efficient alternative484.2. IRS Performance with Respect to the Conventional Systemto technologies like Massive MIMO and network densification.4.2.2 Impact of Channel Estimation Time10−14 10−13 10−12 10−11 10−10 10−9 10−8 10−7 10−6 10−5 10−4 10−3 10−2012345678910τc(sec)SumRate(bps/Hz)M=64, N=170No IRS, M=80M=48, N=470Figure 4.4: Sum-rate comparison of IRS-assisted (IRS-asst.) and conventional (conv.) MISOsystem under varying channel estimation time, where τc is the channel estimation time seconds.We plot in Fig. 4.4 the net achievable rate against τc for a conventional MISO system (no IRS)with 80 active antennas at the BS serving K = 32 users. We also plot the net rates for IRS-assistedsystems with a fewer number of active antennas at the BS. The value of N for each M is chosen toensure that the IRS-assisted system performs as well as the conventional system that has 80 activeantennas. We observe that the net rate is a unimodal function of τc. Specifically, τc = 10−4 isoptimal for all the considered IRS-assisted MISO systems, whereas the conventional MISO systemcould perform as well at even lower value of τc. This shows that the performance of IRS-assistedsystems is more sensitive to channel estimate quality as compared to conventional MISO systems,and that is due to the number of channels required to estimate in IRS-assisted system as comparedto the conventional MISO system.494.2. IRS Performance with Respect to the Conventional System4.2.3 Impact of Channel Noise Under an IRS-Simplified Channel Model−20 −15 −10 −5 0 5 10 15 20−20−10010ρ [dB]AverageSINR[dB]Conv. system, MCCorollary 2 from [HtD13]IRS-ass. system, MC, N = 32Corollary 3.21, N = 32IRS-ass. system, MC, N = 64Corollary 3.21, N = 64Figure 4.5: Performance of IRS-assisted system under the channel in (3.20), where ρ is abstractedas 1/σ2.Fig. 4.5 studies the performance of IRS-assisted system against ρ under the simplified channelmodel in (3.20) for perfect CSI. The result in (3.21) in Corollary 3.21 is also plotted and the matchbetween the Monte-Carlo simulated average SINR and the theoretical result is very good. Asdiscussed in Sec. III-C, IRS is only beneficial under large system dimensions and Rayleigh fadingwhen ρ takes small to moderate values, i.e. the system is noise-limited, which is often the casefor cell edge users and users in dense urban-macro settings. For interference-limited scenarios (i.e.high ρ), the performance of IRS-assisted system approaches that of the conventional MISO systemunder Rayleigh fading. Note that the conventional MISO system was studied under the model in(3.20) in [HtD13].504.2. IRS Performance with Respect to the Conventional System4.2.4 Impact of Cell Radius1 3 5 7 918222630343842Radius=70mRadius=85mRadius=100mlog2(N)SumRate(bps/Hz) IRS-asst., Imper. CSI, MC, Radius=55mConv., Imper. CSI, MC, Radius=55mFigure 4.6: Performance analysis of IRS-assisted system under varying cell radius with M = 50and K = 32In Fig. 4.6 we depict the performance limit of IRS-assisted systems over varying cell radius.The curves are plotted for M = 50, K = 32 and under N = [21, 23, 25, 27, 29]. It is clear that anIRS in small cell radius, i.e. radius < 55 meters drops the system’s performance while deployingIRS in larger cells, i.e 85 and 100 meters improves the signal received at the cell edge users. Thisobservation can be explained as follows: In the case of small cell radius, the IRS-assisted systemsuffers a more dominant interference effect in the SINR value at the user’s end as compared to theconventional MISO system. As the cell radius decreases, i.e. users become closer to BS, userswill receive better intended symbol power from the direct channel than when receiving the signalfrom both the direct channel and IRS-assisted channel. For this reason, as shown in the blackcurves in Fig.4.6, we find that the conventional MISO systems for small cell edge users performsbetter than an IRS-assisted system. Nevertheless, towards larger cell radius, as shown in red andblue curves, the IRS-assisted system out-performs the conventional MISO system. This is clearlydue to the large direct channel attenuation effect at the user’s end, which is when the IRS arraygain benefit would kick in. Deploying an IRS in such large cells will help the users leverage thegains that the IRS provides and improve the overall system’s performance. For this reason we find514.3. Discussionusers in conventional MISO systems in small cells suffer from channel attenuation at a greater dealcompared to users in IRS-assisted system.4.3 DiscussionThe IRS-assisted system was studied under a practical setting and compared to the conventionalMISO system performance in terms of edge users’ sum rate and number of BS active elements.IRS technology showed promising gains under large number of passive IRS elements while underrelatively less active BS elements, thus providing cost efficient solution. The performance alsoshowed sensitivity in the channel estimation time given the number of channels expected to estimate.IRS-assisted system performance limit was also investigated under varying cell radius and wasshown that edge cell users receive stronger intended signal power under the conventional MISOsystem as compared to the IRS-assisted system in small cells. This proves that IRS-assisted systemsare interference limited and are best deployed in large cells, where the high pathloss effect dropsthe performance of the direct channel and the IRS array gain benefit shows up. Finally, the derivedSINR deterministic equivalence we proven to be tight in moderate system dimensions and a strongtool to use for system analysis.52Chapter 5Conclusion and Future Direction5.1 ConclusionThe IRS technology was proven to hold great potential in improving communication by effec-tively controlling the reflection of incident signals in favor of user reception, thus manipulating thepropagation of radio waves in the environment and introducing a radically new way of communi-cation. In this thesis work, we consider a MISO system in a Rayleigh fading environment with anIRS deployed to assist in communication. Specifically, we study a system with a BS consistingof M antennas communicating to K single antenna users via an IRS, which is equipped with Npassive reflecting elements. We propose a novel MMSE based channel estimation protocol thatconsiders the radio limitations in IRS technology. MMSE estimates for IRS-assisted channel anddirect channels were derived under the estimation protocol and sum rate expressions under MRTprecoding were formulated. Asymptotic performance was also considered in which the asymptoticequivalences of SINR and sum rate were obtained using RMT tools and insights on the impactof IRS phase shifts were provided. The derived asymptotic performance limit showed to yield anarray gain but no beamforming gain. More specifically, IRS phases appear only in terms involvingthe estimation error of direct channel, which account for a negligible overall effect on performance.Our results also restrict the benefits achieved from an IRS assisted system under Rayleigh fadingenvironment to interference limited scenarios. Our conclusions and derivations were supported bysimulation results in Chapter 5.5.1.1 TakeawaysWith the above mentioned results, we can summarize the takeaways as follows:535.1. Conclusion− The derived asymptotic deterministic sum rate equivalences proved to be tight expressionseven in moderate system dimension, and can be considered as useful tools in performanceanalysis and optimization.− Asymptotic performance limit of an IRS-assisted system reveal that an IRS promises an arraygain in a Rayleigh fading environment but loses on the beamforming gains. This was shownin both the derived asymptotic expressions and simulation results.− IRS-assisted system performance is interference limited, were IRS’s high sum rate gainsappear to diminish in interference limited scenarios. This scenario can be shown in smallcells, where the conventional MISO system appears to outperform an IRS-assisted MISOsystem for cell radius < 55m.5.1.2 LimitationsThe limitations of this work include the following:− This work considers an ON-OFF channel estimation protocol that involves the N + 1 uplinksub-phases to obtain the final cascaded channel estimate for each user k. This obviouslyrequires a good deal of training overhead to successfully estimate all channels for K users.− This work considers MRT precoding for downlink transmission, which is not optimal whenit comes to mitigating the interference effect in our proposed IRS-assisted MISO system.To address the above limitations, the number of estimation sub-phases could perhaps be reduced,where the direct channel as well as IRS-assisted channels could be estimated in t sub-phases, suchthat 1 < t < N + 1 . This develops a trade-off between channel estimation accuracy and trainingoverhead, but is worth investigating in case of limited resources. To address the interferencedrawback due to MRT precoding, we can consider zero forcing (ZF) or regularized zero forcing(RZF) precoders instead. The use of such precoders will show significant improvement in system’sperformance especially in scenarios where interference is high.545.2. Future Direction5.2 Future DirectionIRS technology is a relatively new technology proposed to improve multiuser communicationsystems. Therefore, the potential research directions are numerous and promising to consider. Onedirectionwhich is now considered as an extension to this work, is studying the system’s performanceunder correlatedRician environment and usingRegularized zero forcing (RZF) precoder for transmitbeamforming. This setting depicts a more practical scenario and the results are expected to showbeamforming gain in the asymptotic limit given the line of sight (LOS) component of the Ricianchannel.Another potential direction in IRS technology is integrating IRS in some current leadingtechnologies including D2D communication as well as mmWave and THz communication. 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(A.3)Due to independence of h2,k(i) and h2,k( j) for i , j the above expression can be reduced to,Ak = βd,kIM + β2,kN∑i=1H1(:, i)θiθ∗iH1(:, i)H . (A.4)Representing (A.4) in a matrix form, we obtain,Ak = βd,kIM + β2,kH1ΘΘHHH1 (A.5)as given in (2.2).A.2 Proof of (2.13)Given the observed training signal, rtr1,k , in the first sub-interval of the channel estimation phasegiven in (2.12), we can write the MMSE estimate of hˆd,k ashˆd,k = Artr1,k, (A.6)64A.3. Proof of (2.16)whereA is found as the solution tominAE[|hˆd,k−hd,k |2] and turns out to beA = E[rtr1,khHd,k](E[rtr1,krtrH1,k ])−1.We now compute E[rtr1,khHd,k] by noting that ni,k and hd,k are independent random vectors, toobtain,E[rtr1,khHd,k] = E[(hd,k + n1,k)hHd,k], (A.7)= E[(hd,khHd,k + hHd,kn1,k)] (A.8)= E[hd,khHd,k] = βd,kIM . (A.9)Next we find E[rtr1,krtrH1,k ] as,E[rtr1,krtrH1,k ] = E[(hd,k + n1,k)(hd,k + n1,k), (A.10)= E[hd,khd,k] + E[n1,knH1,k], (A.11)= βd,kIM +1ρtrIM . (A.12)Putting these expression together, we obtain,hˆd,k = βd,kIM(βd,kIM +1ρtrIM)−1rtr1,k (A.13)which confirms (2.13)A.3 Proof of (2.16)The MMSE estimate of h0,i,k is obtained as,hˆ0,i,k = Ar˜tri,k (A.14)where A = E[r˜tri,khH0,i,k](E[r˜tri,k r˜trHi,k ])−1 and r˜tri,k is given in (2.15). We first find the expression ofE[r˜tri,khH0,i,k] by noting that that ni,k and h0,i,k are independent random vectors, and h˜d,k and hH0,i,k areindependent random vectors. Therefore,E[r˜tri,khH0,i,k] = E[(h˜d,k + h0,i,k + ni,k)hHo,i,k] (A.15)= E[(h˜d,khH0,i,k + h0,i,khH0,i,k + hH0,i,kni,k)], (A.16)= E[h0,i,khH0,i,k] = β2,kH1(:, i)H1(:, i)H . (A.17)65A.4. Proof of (2.17)Next we find E[r˜tri,k r˜trHi,k ] by using the independence of the following pair of vectors (ni,k , h0,i,k) and(h˜d,k ,hH0,i,k), to obtain,E[r˜tri,k r˜trHi,k ] = E[(h˜d,k + h0,i,k + ni,k)(h˜d,k + h0,i,k + ni,k)H], (A.18)= E[h˜d,k h˜d,k] + E[h0,i,khH0,i,k] + E[ni,knHi,k], (A.19)= Ch˜d,k h˜Hd,k + β2,kH1(:, i)H1(:, i)H +1ρtrIM . (A.20)To obtain CHh˜d,k h˜d,k , we recall that h˜d,k = hd,k − hˆd,k and therefore,Ch˜d,k h˜Hd,k = E[h˜d,k h˜Hd,k] = E[hd,khHd,k] − E[hˆd,k hˆHd,k], (A.21)= βd,kIM + E[Rd,kQd,k(hd,k + n1,k)(hHd,k + nH1,k)QHd,kRHd,k]. (A.22)Note that hd,k and n1,k are independent random vectors and therefore,Ch˜d,k h˜Hd,k = βd,kIM + Rd,kQd,kE[hd,khHd,k]QHd,kRHd,k+Rd,kQd,kE[n1,knH1,k]QHd,kRHd,k,(A.23)= βd,kIM + βd,kRd,kQd,kQHd,kRHd,k +1ρtrRd,kQd,kQHd,kRHd,k, (A.24)= βd,kIM + Rd,kQd,kRd,k . (A.25)Therefore, putting (A.25) back in (A.20) and using it along with (A.17) in (A.14) we obtain,hˆ0,i,k = β2,kH1(:, i)H1(:, i)H(Ch˜d,k h˜Hd,k + β2,kH1(:, i)H1(:, i)H +1ρtrIM)−1r˜tri,k(A.26)as given in (2.16).A.4 Proof of (2.17)Recall hˆk = ˆhd,k +∑Ni=1 hˆ0,i,kθi and that hˆ0,i,k and hˆd,k are independent, therefore:E[hˆk hˆHk ] = E[( ˆhd,k +N∑i=1hˆ0,i,kθi)( ˆhd,k +N∑j=1hˆ0, j,kθ j)∗], (A.27)= E[hˆd,k hˆHd,k] +N∑j=1N∑i=1θiE[hˆ0,i,k hˆH0, j,k]θ∗j, (A.28)66A.4. Proof of (2.17)where E[hˆd,k hˆHd,k] = Rd,kQd,kRd,k .We now compute∑Nj=1∑Ni=1 θiE[hˆ0,i,k hˆH0, j,k]θ∗j as,N∑j=1N∑i=1θiE[hˆ0,i,k hˆH0, j,k]θ∗j=N∑j=1N∑i=1θiE[R0,i,kQi,k(h˜d,k + h0,i,k + ni,k)(h˜Hd,k + hH0, j,k + nHj,k)QHj,kR0, j,k]θ∗j,=N∑j=1N∑i=1θi(R0,i,kQi,kE[h˜d,k h˜d,k]QHj,kRH0, j,k+R0,i,kQi,kE[h0,i,kh0,i,k]QHj,kRH0, j,k+R0,i,kQi,kE[ni,knHi,k]QHj,kRH0, j,k)θ∗j .(A.29)Considering the independence between h˜d,k , h0,i,k and ni,kN∑j=1N∑i=1θiE[hˆ0,i,k hˆH0, j,k]θ∗j =N∑j=1N∑i=1θiR0,i,kQi,kCh˜d,k h˜d,kQHj,kRH0, j,kθ∗j+N∑i=1R0,i,kQi,kR0,i,kQHi,kRH0,i,k+N∑i=1R0,i,kQi,k1ρtrIMQHi,kRH0,i,k(A.30)The expression can be simplified to obtain,N∑j=1N∑i=1θiE[hˆ0,i,k hˆH0, j,k]θ∗j =N∑j=1N∑i=1θiR0,i,kQi,kCh˜d,k h˜d,kQHj,kRH0, j,kθ∗j+N∑i=1R0,i,kQi,k(R0,i,k +1ρtrIM)QHi,kRH0,i,k (A.31)Therefore,E[hˆk hˆHk ] = Rd,kQd,kRd,k +N∑j=1N∑i=1θiR0,i,kQi,kCh˜d,k h˜d,kQHj,kRH0, j,kθ∗j+N∑i=1R0,i,kQi,k(R0,i,k +1ρtrIM)QHi,kRH0,i,k(A.32)as in (2.17).67Appendix BProof of Theorem 3.1In this section we derive the deterministic equivalence of SINR under the MRT precoding.The strategy is to look at the SINR expression in (2.22) as consisting of three separate terms: 1)the scaled signal power pkK |hHk hˆk |2, 2) the scaled interference power∑l,kplK |hHk hˆl |2, and 3) thepower normalization term Ψ. We will derive the deterministic equivalent for each of these termsseparately and then obtain the final expression γ◦k,MRT .Deterministic equivalence for 1KhHk hˆk :We have,1KhHk hˆk =1K(hd,k +N∑i=1h0,i,kθi)H(hˆd,k +N∑j=1hˆ0,i,kθ j), (B.1)(B.2)Noting the independence of hˆd,k and h0,i,k vectors and applying Lemma 3.2.1KhHk hˆk −1K©­«hHd,k hˆd,k + hHd,kN∑j=1hˆ0,i,kθ j +N∑j=1N∑i=1θHi hH0,i,k hˆ0,i,kθ jª®¬ a.s.−−−−−−−−→M,N,K→∞ 0, (B.3)Notice there are three separate terms in (B.3) so we find the deterministic equivalents for themseparately.Using (2.13), we obtain1KhHd,k hˆd,k =1KhHd,k(Rd,kQd,k(hd,k + nUL1 )) (B.4)Next we notice the independence between nUL1 and hHd,k and apply Lemma 3.2 to obtain,1KhHd,k hˆd,k −1KhHd,kRd,kQd,khd,ka.s.−−−−−−−−→M,N,K→∞ 0, (B.5)68Appendix B. Proof of Theorem 3.1Given hd,k ∼ CN(0, βd,kIM), we apply Lemma 3.1 to obtain,1KhHd,k hˆd,k −1Kβd,k tr(Rd,kQd,k) a.s.−−−−−−−−→M,N,K→∞ 0. (B.6)We can simplify the expression by using the definitions of Rd,k and Qd,k in (2.13) to obtain,1KhHd,k hˆd,k −1Kβ2d,kMβd,k +1ρtra.s.−−−−−−−−→M,N,K→∞ 0 (B.7)Let’s look at the second term in (B.3) and by applying (2.16).1KhHd,kN∑j=1hˆ0,i,kθ j =1KN∑j=1hHd,k(R0,i,kQ j,k r˜trj,k)θ j (B.8)=1KN∑j=1hHd,k(R0, j,kQ j,k( ˜hd,k + h0, j,k + nULj ))θ j (B.9)We note that hHd,k is independent of h0, j,k and nULj . Moreover, we recall that hd,k = h˜d,k + hˆd,kand notice that hˆd,k and h˜d,k are independent. Under these observations, we apply Lemma 3.2successively to obtain1KN∑j=1hHd,k hˆ0, j,kθ j −1KN∑j=1hHd,kR0, j,kQ j,k h˜d,kθ ja.s.−−−−−−−−→M,N,K→∞ 0, (B.10)1KN∑j=1hHd,kR0, j,kQ j,k h˜d,kθ j −1KN∑j=1h˜Hd,kR0, j,kQ j,k h˜d,kθ ja.s.−−−−−−−−→M,N,K→∞ 0. (B.11)Next we apply the trace lemma on the quadratic term in h˜d,k where h˜d,k ∼ CN(0,Ch˜d,k h˜d,k ), whereCh˜d,k h˜d,k is defined in Lemma 3.1. The resulting convergence is given as1KN∑j=1hHd,k hˆ0, j,kθ j −1KN∑j=1tr(Ch˜d,k h˜d,kR0, j,kQ j,k)θ ja.s.−−−−−−−−→M,N,K→∞ 0. (B.12)Next we look at the deterministic equivalence of third term by using the definitions in (2.15)and (2.16).1KN∑j=1N∑i=1θ∗i hH0,i,k hˆ0, j,kθ j =1KN∑j=1N∑i=1θ∗i hH0,i,k(R0, j,kQ j,k r˜trj,k)θ j (B.13)69Appendix B. Proof of Theorem 3.1=1KN∑j=1N∑i=1θ∗i hH0,i,k(R0, j,kQ j,k(h˜d,k + h0, j,k +nULjρtr))θ j (B.14)Notice that h0,i,k is independent of h˜d,k and nULj ∀i, j and independent of h0, j,k for j , i, and byapplying Lemma 3.2 we obtain:=1KN∑i=1θ∗i θihH0,i,kR0,i,kQi,kh0,i,k (B.15)Now as M,K →∞ and by applying the trace lemma Lemma 3.1 and noting that θ∗i θi = 1=1KN∑i=1hH0,i,kR0,i,kQi,kh0,i,k −1KN∑i=1tr(R0,i,kR0,i,kQi,k) a.s.−−−−−−−−→M,N,K→∞ 0 (B.16)Therefore, by (B.7), (B.12), and (B.16) we obtain:1KhHk hˆk−1K(β2d,kMβd,k +1ρtr+N∑i=1tr(Ch˜d,k h˜d,kR0,i,kQi,k)θi +N∑i=1tr(R0,i,kR0,i,kQi,k))a.s−−−−−−−−→M,N,K→∞ 0.(B.17)Deterministic equivalent of 1K∑l,k |hHk hˆl |2pl :Note that 1K∑l,k pl |hHk hˆl |2 = 1KhHk HˆH[k]P[k]Hˆ[k]hk , where H[k]= [h1, . . . , hk−1, hk+1, . . . , hK]H . Also, note HˆH[k]P[k]Hˆ[k] is independent of hk . Moreover hkdefined in (2.5) is distributed as CN ∼ (0,Ak) since Ak = E[hkhHk ] = H1ΘE[h2,khH2,k]ΘHHH1 +E[hd,khHd,k] = β2,kH1HH1 + βd,kIM . Using these observations, as M,K → ∞ and by applyingLemma 3.1 we obtain:1KhHk HˆH[k]P[k]Hˆ[k]hk −1Ktr(AkHˆH[k]P[k]Hˆ[k])a.s−−−−−−→M,K→∞ 0 (B.18)70Appendix B. Proof of Theorem 3.1Notice 1K tr(AkHˆH[k]P[k]Hˆ[k]) = 1K∑l,k pl hˆHl Ak hˆl . Applying Lemma 3.1 and as M,K →∞1K∑l,kpl hˆHl Ak hˆl −1K∑l,kpltrClAka.s−−−−−−→M,K→∞ 0 (B.19)where Ck = E[hˆk hˆHk ] proven in Appendix A (D)Therefore:1K∑l,k|hHk hˆl |2pl −1K∑l,kpltrClAka.s−−−−−−→M,K→∞ 0 (B.20)Deterministic equivalent of 1KΨ =1K tr(PHˆHˆH) :Note that,Ψ = tr(PHˆHˆH) =K∑k=1pk hˆHk hˆk (B.21)As M,K →∞ and by applying the Lemma 3.1 on the quadratic form in hˆl , we obtain1KK∑k=1pk hˆHk hˆk −1KK∑k=1pk trCka.s−−−−−−→M,K→∞ 0 (B.22)Therefore:1KΨ − 1KK∑k=1pk trCka.s−−−−−−→M,K→∞ 0 (B.23)Combining (B.17), (B.20), and (B.23) we obtain the deterministic equivalent of SINR as shownin Theorem 3.1.71Appendix CProof of Corollary 3.4The proof follows by writing γ◦k,MRT in Corollary 3.3 using the definition in (3.20) as,γ◦k,MRT =1K2 |tr(βd,kIM + cβd,kDDH)|21K2∑l,k tr((βd,kIM + cβd,kDDH)2) +1K2∑Kl=1 tr(βd,lIM+cβd,lDDH )ρ, (C.1)=β2d,k(cMN + M)2∑l,k βd,lβd,k(c2MN2 + 2cMN + M) +∑Kl=1 βd,l(cMN+M)ρ, (C.2)=βd,k(cMN + M)∑l,k βd,lM(cN+1)2M(cN+1) +∑Kl=1 βd,lβd,k ρ, (C.3)=βd,k(cMN + M)∑l,k βd,l(cN + 1) +∑Kl=1 βd,lβd,k ρ, (C.4)=11M∑l,kβd,lβd,k+∑Kl=1 βd,lMβ2d,kρ(cN+1), (C.5)where (C.2) is obtained by noting that tr(βd,kIM+cβd,kDDH) = βd,k(M+cMN) and tr(DDHDDH) =MN2. This completes the proof of Corollary (3.4).72Appendix DProof of Lemma 3.4In this section, we derive the derivative expression in Lemma 3.4. To begin, we can look at thedeterministic equivalent defined in (3.8) and use the linearity property of the derivative to find thederivative of the expression:∂Rosum∂θi=K∑k=1∂ log(1 + γok )∂θi(D.1)=K∑k=11(1 + γok ) ln(2)∂(γok )∂θi(D.2)Nowwe can look at the deterministic equivalent γok defined in (3.6) and notice that it is a quotientof two terms, i.e. the numerator pkK |nk |2 and the denominator d =∑l,kplK tr(ClAk) +pkK∑Kk=1 trCkρ .Following the quotient rule ,the derivative of the deterministic equivalence of SINR at user k withrespect to θi can then be defined as:∂γok∂θi=pkK∂ |nk |2∂θi(d) + pkK |nk |2 ∂d∂θid2(D.3)where the gradient can be expressed as: ∇γok = [∂γok∂θ1,∂γok∂θ2. . .∂γok∂θN]In the following two subsection, we will derive the expression for the following terms: ∂ |nk |2∂θiand ∂d∂θi , separately.D.1 Proof of Derivative Expression for ∂nk∂θnNotice that |nk |2 is nothing but the norm squared, in which we may use the power rule to findthe derivative as follows: ∂ |nk |2∂θi=2pkK |nk | ∂nkθi73D.2. Proof of (3.26)Recall:nk =∑Ni=1 vitr(Ch˜d,k h˜d,kR0,i,kQi,k) +β2d,kMβd,k+1ρtr+∑Ni=1 tr(R0,i,kR0,i,kQi,k)Notice that only the first term∑Ni=1 θitr(Ch˜d,k h˜d,kR0,i,kQi,k) includes the θi term. For this reason, itcan be easily shown that:∂nk∂θi= tr(Ch˜d,k h˜d,kR0,i,kQi,k). Therefore:∂ |nk |2∂θi=2pkK|nk |tr(Ch˜d,k h˜d,kR0,i,kQi,k) (D.4)D.2 Proof of (3.26)We will now derive the derivative d′ = ∂dk∂θn of the denominator dk =∑l,kplK tr(ClAk) +pkK∑Kk=1 trCkρ as,∂d∂θn=∑l,kplKtr(∂ClAk∂θn)+pkKρK∑k=1tr(∂Ck∂θn), (D.5)where Cl =β2d,lβd,l+1ρtr+∑Ni=1 R0,i,lQi,l(R0,i,l + IMρtr )QHi,lRH0,i,l+∑Nj=1∑Ni=1 θiR0,i,lQi,lCh˜d,l h˜d,lQHj,lRH0, j,lθ∗j .Noting that tr(ClAk) and tr(Ck) are real valued functions of the complex variable θn, for whichwe have the expression∂tr(ClAk)∂θn= 2∂tr(ClAk)∂θ∗n, (D.6)∂tr(Ck)∂θn= 2∂tr(Ck)∂θ∗n. (D.7)Only the third term in the expression of Cl given as74D.2. Proof of (3.26)∑Nj=1∑Ni=1 θiR0,i,lQi,lCh˜d,l h˜d,lQHj,lRH0, j,lθ∗j will contribute to the derivative. Using this we can obtain∂tr(ClAk)∂θ∗n=N∑i=1θitr(R0,i,lQi,lCh˜d,l h˜d,lQHn,lRH0,n,lAk), (D.8)∂tr(Ck)∂θ∗n=N∑i=1θitr(R0,i,kQi,kCh˜d,k h˜d,kQHn,kRH0,n,k). (D.9)Plugging (D.8) and (D.9) in (D.6) and (D.7) respectively and using the resulting expressions in(D.5), we obtain ∂d∂θn . Plugging (D.5) and (D.4) in (D.3) will complete the proof of Lemma 3.4.75
Thesis/Dissertation
eng
Electrical Engineering
Vancouver : University of British Columbia Library
Attribution-NonCommercial-NoDerivatives 4.0 International
Graduate
IRS-Assisted Rayleigh Fading MISO Systems : beamforming and asymptotic Analysis
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