recon๏ฌgurable intelligent surface-assisted cell-free
TRANSCRIPT
1
Reconfigurable Intelligent Surface-Assisted Cell-Free MassiveMIMO Systems Over Spatially-Correlated Channels
Trinh Van Chien, Member, IEEE, Hien Quoc Ngo, Senior Member, IEEE, Symeon Chatzinotas, Senior Member, IEEE,Marco Di Renzo, Fellow, IEEE, and Bjorn Ottersten, Fellow, IEEE
AbstractโCell-Free Massive multiple-input multiple-output(MIMO) and reconfigurable intelligent surface (RIS) are twopromising technologies for application to beyond-5G networks.This paper considers Cell-Free Massive MIMO systems withthe assistance of an RIS for enhancing the system performanceunder the presence of spatial correlation among the engineeredscattering elements of the RIS. Distributed maximum-ratio pro-cessing is considered at the access points (APs). We introduce anaggregated channel estimation approach that provides sufficientinformation for data processing with the main benefit of reducingthe overhead required for channel estimation. The consideredsystem is studied by using asymptotic analysis which lets thenumber of APs and/or the number of RIS elements grow large.A lower bound for the channel capacity is obtained for a finitenumber of APs and engineered scattering elements of the RIS,and closed-form expressions for the uplink and downlink ergodicnet throughput are formulated in terms of only the channelstatistics. Based on the obtained analytical frameworks, we unveilthe impact of channel correlation, the number of RIS elements,and the pilot contamination on the net throughput of each user. Inaddition, a simple control scheme for optimizing the configurationof the engineered scattering elements of the RIS is proposed,which is shown to increase the channel estimation quality, and,hence, the system performance. Numerical results demonstratethe effectiveness of the proposed system design and performanceanalysis. In particular, the performance benefits of using RISsin Cell-Free Massive MIMO systems are confirmed, especially ifthe direct links between the APs and the users are of insufficientquality with high probability.
Index TermsโCell-free Massive MIMO, reconfigurable intelli-gent surface, maximum ratio processing, ergodic net throughput.
I. INTRODUCTION
In the last few decades, we have witnessed an exponentialgrowth of the demand for wireless communication systemsthat provide reliable communications and ensure ubiquitouscoverage, high spectral efficiency and low latency [2]. To meet
The work of T. V. Chien, S. Chatzinotas, and B. Ottersten was supported byRISOTTI - Reconfigurable Intelligent Surfaces for Smart Cities under projectFNR/C20/IS/14773976/RISOTTI. The work of H. Q. Ngo was supported bythe UK Research and Innovation Future Leaders Fellowships under GrantMR/S017666/1. The work of M. Di Renzo was supported in part by theEuropean Commission through the H2020 ARIADNE project under grantagreement number 871464 and through the H2020 RISE-6G project undergrant agreement number 101017011. Parts of this paper were presented atIEEE GLOBECOM 2021 [1]. The associate editor coordinating the reviewof this paper and approving it for publication was Z. Zhang. (Correspondingauthor: Trinh Van Chien.)
T. V. Chien, S. Chatzinotas, B. Ottersten are with the InterdisciplinaryCentre for Security, Reliability and Trust (SnT), University of Luxem-bourg, L-1855 Luxembourg, Luxembourg (email: [email protected],[email protected], and [email protected]).
H. Q. Ngo is with the School of Electronics, Electrical Engineering andComputer Science, Queenโs University Belfast, Belfast BT7 1NN, UnitedKingdom (email: [email protected]).
M. Di Renzo is with Universite Paris-Saclay, CNRS, CentraleSupelec,Laboratoire des Signaux et Systemes, 3 Rue Joliot-Curie, 91192 Gif-sur-Yvette, France (email: [email protected]).
these requirements, several new technologies have been incor-porated in 5G communication standards, which include Mas-sive multiple-input multiple-output (MIMO) [3], millimeter-wave communications [4], and network densification [5].Among them, Massive MIMO has gained significant attentionsince it can offer a good service to many users in the network.Moreover, the net throughput offered by a Massive MIMOsystem is close to the Shannon capacity, in many scenarios,by only employing simple linear processing techniques, suchas maximum ratio (MR) or zero forcing (ZF) processing.Since the net throughput can be computed in a closed-form expression that only depends on the channel statistics,the optimized designs are applicable for a long period oftime [6]. The colocated Massive MIMO architecture has theadvantage of low backhaul requirements since the base stationantennas are installed in a compact array. Conventional cellularnetworks, however, are impaired by intercell interference. Inparticular, the users at the cell boundaries are impaired byhigh intercell interference and path loss, and hence, they mayexperience insufficient performance. More advanced signalprocessing methods are necessary to overcome the inherentintercell interference that characterizes conventional cellularnetwork deployments.
Cell-Free Massive MIMO has recently been introduced toreduce the intercell interference that characterizes colocatedMassive MIMO architectures. Cell-Free Massive MIMO is anetwork deployment where a large number of access points(APs) are located in a given coverage area to serve a smallnumber of users [7]โ[10]. All APs collaborate with each othervia a backhaul network and serve all the users in the absence ofcell boundaries. The system performance is enhanced in Cell-Free Massive MIMO systems because they inherit the benefitsof the distributed MIMO and network MIMO architectures, butthe users are also close to the APs. When each AP is equippedwith a single antenna, MR processing results in a good netthroughput for every user, while ensuring a low computationalcomplexity and offering a distributed implementation that isconvenient for scalability purposes [11]. However, Cell-FreeMassive MIMO cannot guarantee a good quality of serviceunder harsh propagation conditions, such as in the presenceof poor scattering environments or high attenuation due to thepresence of large obstacles.
Reconfigurable intelligent surface (RIS) is an emergingtechnology that is capable of shaping the radio waves at theelectromagnetic level without applying digital signal process-ing methods and without requiring power amplifiers [12]โ[14]. Each element of the RIS scatters (e.g., reflects) theincident signal without using radio frequency chains and poweramplification [15]. Integrating an RIS into wireless networks
arX
iv:2
104.
0864
8v3
[cs
.IT
] 1
7 D
ec 2
021
2
introduces digitally controllable links that scale up with thenumber of engineered scattering elements of the RIS, whoseestimation is, however, challenged by the lack of digital signalprocessing units at the RIS [16]โ[20]. For simplicity, themain attention has so far been concentrated on designing thephase shifts under the assumption of perfect channel stateinformation (CSI) [16], [21] and the references therein. In [20],the authors have recently discussed the fundamental issuesof performing channel estimation in RIS-assisted wirelesssystems. The impact of the channel estimation overhead andreporting on the spectral efficiency, energy efficiency, andtheir tradeoff has recently been investigated in [17]. In [16]and [18], to reduce the impact of the channel estimationoverhead, the authors have investigated the design of RIS-assisted communications in the presence of statistical CSI. Asfar as the integration of Cell-Free Massive MIMO and RISis concerned, recent works have formulated and solved op-timization problems with different communication objectivesunder the assumption of perfect (and instantaneous) CSI [22]โ[26]. Recent results in the context of single-input single-output(SISO) and multi-user MIMO systems have, however, shownthat designs for the engineered scattering elements of the RISthat are based on statistical CSI may be of practical interestand provide good performance [18], [27]โ[29].
In the depicted context, no prior work has analyzed theperformance of RIS-assisted Cell-Free Massive MIMO sys-tems in the presence of spatially-correlated channels. In thiswork, motivated by these considerations, we introduce ananalytical framework for analyzing and optimizing the uplinkand downlink transmissions of RIS-assisted Cell-Free MassiveMIMO systems under spatially correlated channels and in thepresence of direct links subject to the presence of blockages.In particular, the main contributions made by this paper canbe summarized as follows:
โข We consider an RIS-assisted Cell-Free Massive MIMOunder spatially correlated channels. All APs estimate theinstantaneous channels in the uplink pilot training phase.We exploit a channel estimation scheme that estimates theaggregated channels including both the direct and indirectlinks, instead of every individual channel coefficient asin previous works [20], [24]. For generality, the pilotcontamination is assumed to originate from an arbitrarypilot reuse pattern.
โข We analytically show that, even by using a low complex-ity MR technique, the non-coherent interference, small-scale fading effects, and additive noise are averaged outwhen the number of APs and RIS elements increases.The received signal includes, hence, only the desiredsignal and the coherent interference. In addition, we showthat the indirect links become dominant if the number ofengineered scattering elements of the RIS increases.
โข We derive a closed-form expression of the net throughputfor both the uplink and downlink data transmissions. Theimpact of the array gain, coherent joint transmission,channel estimation errors, pilot contamination, spatialcorrelation, and phase shifts of the RIS, which determinethe system performance, are explicitly observable in the
Fig. 1. An RIS-assisted Cell-Free Massive MIMO system where ๐ APscollaborate with each other to serve ๐พ distant users.
obtained analytical expressions.โข With the aid of numerical simulations, we verify the
effectiveness of the proposed channel estimation schemeand the accuracy of the closed-form expressions of thenet throughput. The obtained numerical results show thatthe use of RISs significantly enhances the net throughputper user, especially when the direct links are blocked withhigh probability.
The rest of this paper is organized as follows: Section IIpresents the system model, the channel model, and the channelestimation protocol. The uplink data transmission protocol andthe asymptotic analysis by assuming a very large numberof APs and engineered scattering elements at the RIS arediscussed in Section III. A similar analysis for the downlinkdata transmission is reported in Section IV. Finally, Section Villustrates several numerical results, while the main conclu-sions are drawn in Section VI.
Notation: Upper and lower bold letters are used to denotematrices and vectors, respectively. The identity matrix of size๐ ร ๐ is denoted by I๐ . The imaginary unit of a complexnumber is denoted by ๐ with
โ๐ = โ1. The superscripts (ยท)โ,
(ยท)๐ , and (ยท)๐ป denote the complex conjugate, transpose, andHermitian transpose, respectively. E{ยท} and Var{ยท} denote theexpectation and variance of a random variable. The circularlysymmetric Gaussian distribution is denoted by CN(ยท, ยท) anddiag(x) is the diagonal matrix whose main diagonal is givenby x. tr(ยท) is the trace operator. The Euclidean norm ofvector x is โxโ, and โXโ2 is the spectral norm of matrix X.Finally, mod(ยท, ยท) is the modulus operation and bยทc denotes thetruncated argument.
II. SYSTEM MODEL, CHANNEL ESTIMATION, AND RISPHASE SHIFT CONTROL
We consider an RIS-assisted Cell-Free Massive MIMOsystem, where ๐ APs connected to a central processing unit(CPU) serve ๐พ users on the same time and frequency resource,as schematically illustrated in Fig. 1. All APs and users areequipped with a single antenna and they are randomly locatedin the coverage area. Since the considered users are far awayfrom the APs, the communication is assisted by an RIS thatcomprises ๐ engineered scattering elements that can modify
3
the phases of the incident signals.1 The matrix of phase shiftsof the RIS is denoted by ฮฆฮฆฮฆ = diag
([๐ ๐ \1 , . . . , ๐ ๐ \๐ ]๐
), where
\๐ โ [โ๐, ๐] is the phase shift applied by the ๐-th elementof the RIS. The phase shifts are adjusted by a controllerwhich exchanges information with the APs via a backhaullink (see Fig. 1). As a canonical form of Cell-Free MassiveMIMO systems, we assume that the system operates in time-division duplexing (TDD) mode. Thus, we assume that channelreciprocity holds in the consisted system model.A. Channel Model
We assume a quasi-static block fading model where thechannels are static and frequency flat in each coherenceinterval comprising ๐๐ symbols. We assume that the APsestimate the channel during the uplink pilot training phase.Thus, ๐๐ symbols (๐๐ < ๐๐) in each coherence interval arededicated to the channel estimation phase and the remaining(๐๐ โ ๐๐) symbols are utilized for the uplink and downlinkdata transmission phases.
The following notation is used: ๐๐๐ is the channel betweenthe user ๐ and the AP ๐, which is the direct link [12]; h๐ โC๐ is the channel between the AP ๐ and the RIS; and z๐ โC๐ is the channel between the RIS and the user ๐ . The pairh๐ and z๐ , which constitutes the cascaded channel, results inan indirect link (virtual line-of-sight link), which enhances thecommunication reliability between the AP ๐ and the user ๐[30]. The majority of existing works assume that the wirelesschannels undergo uncorrelated Rayleigh fading. In this paper,we consider a more realistic channel model by taking intoaccount the spatial correlation among the engineered scatteringelements of the RIS, which is due to their sub-wavelengthsize, sub-wavelength inter-distance, and geometric layout. Inan isotropic propagation environment, in particular, ๐๐๐ , h๐,and z๐ can be modeled as follows
๐๐๐ โผ CN(0, ๐ฝ๐๐ ), h๐ โผ CN(0,R๐), z๐ โผ CN(0, R๐ ), (1)
where ๐ฝ๐๐ is the large-scale fading coefficient; R๐ โ C๐ร๐
and R๐ โ C๐ร๐ are the covariance matrices that characterizethe spatial correlation among the channels of the RIS elements.The covariance matrices in (1) correspond to a general model,which can be further particularized for application to typicalRIS designs and propagation environments. For example, asimple exponential model was used to describe the spatial cor-relation among the engineered scattering elements of the RISin [31]. Another recent model that is applicable to isotropicscattering with uniformly distributed multipath components inthe half-space in front of the RIS was recently reported in[32], whose covariance matrices are2
R๐ = ๐ผ๐๐๐ป ๐๐R and R๐ = ๏ฟฝ๏ฟฝ๐๐๐ป ๐๐R, (2)1In general, a completed system should include many users randomly
located in the coverage area. There are some favorable users where their linksto some APs are strong. But there are also some unfavorable users where thetheir links to the APs are weak. This may come from the large path loss(long distances) or heavy shadowing. In our work, we consider the caseswhere RISs are deployed to improve the performance of these unfavorableusers, and hence, the coverage of the whole system can be increased.
2This paper considers a spatial correlation model between engineeredscattering elements in the far-field scenarios that is applicable and when thecovariance matrices among the users differ only in terms of large-scale channelcoefficients. Other scenarios where the users have covariance matrices withdifferent phases [33] are of interest and are left for future work.
where ๐ผ๐, ๏ฟฝ๏ฟฝ๐ โ C are the large-scale channel coefficients,which, for example, model the signal attenuation due to largeobjects and due to the transmission distance. The matricesin (2) assume that the size of each element of the RIS is๐๐ป ร ๐๐ , with ๐๐ป being the horizontal width and ๐๐ beingthe vertical height of each RIS element. In particular, the(๐โฒ, ๐โฒ)โth element of the spatial correlation matrix R โC๐ร๐ in (2) is [R]๐โฒ๐โฒ = sinc(2โu๐โฒ โ u๐โฒ โ/_), where _
is the wavelength and sinc(๐ฅ) = sin(๐๐ฅ)/(๐๐ฅ) is the sincfunction. The vector u๐ฅ , ๐ฅ โ {๐โฒ, ๐โฒ} is given by u๐ฅ = [0,mod (๐ฅ โ 1, ๐๐ป )๐๐ป , b(๐ฅ โ 1)/๐๐ป c๐๐ ]๐ , where ๐๐ป and ๐๐denote the total number of RIS elements in each row andcolumn, respectively. The channel model in (1) is significantlydistinct from related works since the small-scale fading andthe spatial correlation matrices are included in both links ofthe virtual line-of-sight link that comprises the RIS. In [31],by contrast, the channels between the transmitters and the RISare assumed to be deterministic, for analytical tractability.
B. Uplink Pilot Training PhaseThe channels are independently estimated from the ๐๐ pilot
sequences transmitted by the ๐พ users. All the users share thesame ๐๐ pilot sequences. In particular, ๐๐๐๐ โ C๐๐ with โ๐๐๐๐ โ2 =
1 is defined as the pilot sequence allocated to the user ๐ . Wedenote by P๐ the set of indices of the users (including theuser ๐) that share the same pilot sequence as the user ๐ . Thepilot sequences are assumed to be mutually orthogonal suchthat the pilot reuse pattern is
๐๐๐๐ป๐โฒ๐๐๐๐ =
{1, if ๐ โฒ โ P๐ ,0, if ๐ โฒ โ P๐ .
(3)
During the pilot training phase, all the ๐พ users transmit thepilot sequences to the ๐ APs simultaneously. In particular,the user ๐ transmits the pilot sequence โ
๐๐๐๐๐๐ . The receivedtraining signal at the AP ๐, y๐๐ โ C๐๐ , can be written as
y๐๐ =
๐พโ๐=1
โ๐๐๐๐๐๐๐๐๐๐ +
๐พโ๐=1
โ๐๐๐h๐ป๐ฮฆฮฆฮฆz๐๐๐๐๐ + w๐๐, (4)
where ๐ is the normalized signal-to-noise ratio (SNR) of eachpilot symbol, and w๐๐ โ C๐๐ is the additive noise at the AP ๐,which is distributed as w๐๐ โผ CN(0, I๐๐ ). In order for theAP ๐ to estimate the desired channels from the user ๐ , thereceived training signal in (4) is projected on ๐๐๐๐ป
๐as
๐ฆ๐๐๐ = ๐๐๐๐ป๐ y๐๐ =
โ๐๐๐
(๐๐๐ + h๐ป๐ฮฆฮฆฮฆz๐
)+โ
๐โฒโP๐\{๐ }
โ๐๐๐
(๐๐๐โฒ + h๐ป๐ฮฆฮฆฮฆz๐โฒ
)+ ๐ค๐๐๐ , (5)
where ๐ค๐๐๐ = ๐๐๐๐ป๐
w๐๐ โผ CN(0, 1). We emphasize that theco-existence of the direct and indirect channels due to thepresence of the RIS results in a complicated channel estimationprocess. In particular, the cascaded channel in (5) results in anontrivial procedure for applying the minimum mean-squareerror (MMSE) estimation method, as reported in previousworks, for processing the projected signals [8], [9]. Based
4
on the specific signal structure in (5), we denote the channelbetween the AP ๐ and the user ๐ through the RIS as
๐ข๐๐ = ๐๐๐ + h๐ป๐ฮฆฮฆฮฆz๐ , (6)
which is referred to as the aggregated channel that comprisesthe direct and indirect link between the user ๐ and the AP ๐.In contrast to previous works, where the matrix ฮฆฮฆฮฆ of the RISphase shifts is optimized based on instantaneous CSI, in thispaper, ฮฆฮฆฮฆ is optimized based on statistical CSI. This is detailedin Section II-C. By capitalizing on the definition of the aggre-gated channel in (6), the required channels can be estimatedin an effective manner even in the presence of the RIS. Inparticular, the aggregated channel in (6) is given by the productof weighted complex Gaussian and spatially correlated randomvariables, as given in (1). Despite the complex analytical form,the following lemma gives information on the statistics of theaggregated channels.
Lemma 1. The second and fourth moments of the aggregatedchannel ๐ข๐๐ can be formulated as follows
E{|๐ข๐๐ |2} = ๐ฟ๐๐ , (7)
E{|๐ข๐๐ |4} = 2๐ฟ2๐๐ + 2tr(ฮฮฮ2
๐๐ ), (8)
where ฮฮฮ๐๐ = ฮฆฮฆฮฆ๐ปR๐ฮฆฮฆฮฆR๐ and ๐ฟ๐๐ = ๐ฝ๐๐ + tr(ฮฮฮ๐๐ ).Moreover, the aggregated channels are mutually independentfor ๐ โ ๐โฒ and ๐ โ ๐ โฒ, i.e.,
E{๐ข๐๐๐ขโ๐โฒ๐โฒ} = E{๐ข๐๐ }E{๐ขโ๐โฒ๐โฒ} = 0. (9)
In addition, the aggregated channels ๐ข๐๐ and ๐ข๐โฒ๐ ,โ๐ โ ๐โฒ,and the aggregated channels ๐ข๐๐ and ๐ข๐๐โฒ ,โ๐ โ ๐ โฒ, aremutually uncorrelated, i.e.,
E{๐ข๐๐๐ขโ๐โฒ๐ } = 0 and E{๐ข๐๐โฒ๐ขโ๐๐ } = 0. (10)
Besides, the aggregated channels ๐ข๐๐ , ๐ข๐๐โฒ , ๐ข๐โฒ๐ , and ๐ข๐โฒ๐โฒ ,
fulfill the following conditions
E{|๐ข๐๐๐ขโ๐โฒ๐โฒ |2} = ๐ฟ๐๐๐ฟ๐โฒ๐โฒ , ๐ โ ๐โฒ, ๐ โ ๐ โฒ, (11)
E{๐ขโ๐๐๐ข๐๐โฒ๐ขโ๐โฒ๐โฒ๐ข๐โฒ๐ } = tr(ฮฮฮ๐๐โฒฮฮฮ๐โฒ๐ ), ๐ โ ๐โฒ, ๐ โ ๐ โฒ, (12)
E{|๐ข๐๐๐ขโ๐โฒ๐ |2} = ๐ฟ๐๐๐ฟ๐โฒ๐ + tr(ฮฮฮ๐๐ฮฮฮ๐โฒ๐ ), ๐ โ ๐โฒ, (13)
E{|๐ข๐๐๐ขโ๐๐โฒ |2} = ๐ฟ๐๐๐ฟ๐๐โฒ + tr(ฮฮฮ๐๐ฮฮฮ๐๐โฒ), ๐ โ ๐ โฒ. (14)
Proof. See Appendix B. ๏ฟฝ
The moments in Lemma 1 are employed next for analyzingthe channel estimation and the net throughput performance.We note, in addition, that the odd moments of ๐ข๐๐ , e.g., thefirst and third moments, are equal to zero. Conditioned onthe phase shifts, we employ the linear MMSE method forestimating ๐ข๐๐ at the AP. In spite of the complex structureof the RIS-assisted cascaded channel, Lemma 2 providesanalytical expressions of the estimated channels.
Lemma 2. By assuming that the AP ๐ employs the linearMMSE estimation method based on the signal observationin (5), the estimate of the aggregated channel ๐ข๐๐ can beformulated as
๏ฟฝ๏ฟฝ๐๐ =(E{๐ฆโ๐๐๐๐ข๐๐ }๐ฆ๐๐๐
)/E{|๐ฆ๐๐๐ |2} = ๐๐๐ ๐ฆ๐๐๐ , (15)
where ๐๐๐ = E{๐ฆโ๐๐๐
๐ข๐๐ }/E{|๐ฆ๐๐๐ |2} has the followingclosed-form expression
๐๐๐ =
โ๐๐๐๐ฟ๐๐
๐๐๐โ๐โฒโP๐ ๐ฟ๐๐โฒ + 1
. (16)
The estimated channel in (15) has zero mean and variance๐พ๐๐ equal to
๐พ๐๐ = E{|๏ฟฝ๏ฟฝ๐๐ |2} =โ๐๐๐๐ฟ๐๐๐๐๐ . (17)
Also, the channel estimation error ๐๐๐ = ๐ข๐๐ โ ๏ฟฝ๏ฟฝ๐๐ and thechannel estimate ๏ฟฝ๏ฟฝ๐๐ are uncorrelated. The channel estima-tion error has zero mean and variance equal to
E{|๐๐๐ |2
}= ๐ฟ๐๐ โ ๐พ๐๐ . (18)
Proof. It is similar to the proof in [34], and is obtained byapplying similar analytical steps to the received signal in (5)and by taking into account the structure of the RIS-assistedcascaded channel and the spatial correlation matrices in (1).
๏ฟฝ
Lemma 2 shows that, by assuming ฮฆฮฆฮฆ fixed, the aggregatedchannel in (6) can be estimated without increasing the pilottraining overhead, i.e., only ๐๐ symbols in each coherenceinterval are needed for channel estimation, which is the sameas for conventional Cell-Free Massive MIMO systems. If theuser ๐ โฒ uses the same pilot sequence as the user ๐ does, then๏ฟฝ๏ฟฝ๐๐โฒ = ๐๐๐โฒ๐ฆ๐๐๐ from (15). Consequently, we obtain therelation ๏ฟฝ๏ฟฝ๐๐โฒ =
๐๐๐โฒ๐๐๐
๏ฟฝ๏ฟฝ๐๐ , which implies that, because of pilotcontamination, it may be difficult to distinguish the signalsof two generic users. In that regard it is worth noting that,to get rid of pilot contamination, one can assign mutuallyorthogonal pilot signals to all the users in the network (ifthe coherence time is long enough so that ๐๐ โฅ ๐พ). Undermutually orthogonal pilot sequences, ๐๐๐ and ๐พ๐๐ simplify to๐๐๐๐
and ๐พ๐๐๐
, respectively, as follows
๐๐๐๐ =
โ๐๐๐๐ฟ๐๐
๐๐๐๐ฟ๐๐ + 1, ๐พ๐๐๐ =
โ๐๐๐๐ฟ๐๐๐
๐๐๐ . (19)
This implies that, in the absence of pilot contamination, wehave ๐พ๐
๐๐โ ๐ฟ๐๐ as ๐๐ โ โ, i.e., the variance of the
channel estimation error in (17) is equal to zero. The channelestimates given in Lemma 2 can be applied to an arbitraryset of phase shifts and covariance matrices. To facilitatethe performance analysis presented next, we introduce thefollowing corollary that characterizes the correlation betweenthe aggregated channels and their estimates.
Corollary 1. Let us consider the two aggregated channels ๐ข๐๐and ๐ข๐โฒ๐ with ๐ โ ๐โฒ, and let us denote
๐๐๐ =โ๐๐๐ ๏ฟฝ๏ฟฝ
โ๐๐๐ข๐๐ โ
โ๐๐๐E{๏ฟฝ๏ฟฝโ๐๐๐ข๐๐ }, (20)
๐๐โฒ๐ =โ๐๐โฒ๐ ๏ฟฝ๏ฟฝ
โ๐โฒ๐๐ข๐โฒ๐ โ
โ๐๐๐E{๏ฟฝ๏ฟฝโ๐๐๐ข๐๐ }, (21)
where ๐๐๐ and ๐๐โฒ๐ are two non-negative deterministicscalars. Then, the following holds
E{๐๐๐๐โ๐โฒ๐ } = ๐๐๐โ๐๐๐๐๐โฒ๐๐๐๐๐๐โฒ๐
โ๐โฒโP๐
tr(ฮฮฮ๐๐ฮฮฮ๐โฒ๐โฒ).
(22)
Proof. See Appendix C. ๏ฟฝ
5
Both Lemma 1 and Corollary 1 indicate that the presence ofan RIS makes the channel statistics more complex, comparedto a conventional Cell-Free Massive MIMO system, due tothe correlation among the propagation channels. In the nextsections, the analytical expression of the channel estimates inLemma 2 and the properties in Corollary 1 and Lemma 1are employed for signal detection in the uplink and forbeamforming in the downlink. Also, the same results are usedto optimize the phase shifts of the RIS in order to minimizethe channel estimation error and to evaluate the correspondingergodic net throughput.
C. RIS Phase Shift Control and OptimizationChannel estimation is a critical aspect in Cell-Free Massive
MIMO. As discussed in previous text, in many scenarios,non-orthogonal pilots have to be used. This causes pilotcontamination, which may significantly reduce the systemperformance. In this section, we design an RIS-assisted phaseshift control scheme that is aimed to improve the quality ofchannel estimation. To this end, we introduce the normalizedmean square error (NMSE) of the channel estimate of theuser ๐ at the AP ๐, as follows
NMSE๐๐ =E{|๐๐๐ |2}E{|๐ข๐๐ |2}
= 1 โ๐๐๐๐ฟ๐๐
๐๐๐โ๐โฒโP๐ ๐ฟ๐๐โฒ + 1
, (23)
where the last equality is obtained from (7) and (18). TheNMSE is a suitable metric to evaluate the channel estimationquality and to measure the relative channel estimation errorper AP. By definition, the NMSE lies in the range [0, 1]. Inparticular, the NMSE tends to zero if orthogonal pilot signalsare used for every user and the pilot power is sufficiently large.In general, however, the NMSE is greater than zero if the ๐พusers reuse the pilot signals, i.e., ๐๐ < ๐พ , since NMSE๐๐ โ1โ๐ฟ๐๐/(
โ๐โฒโP๐ ๐ฟ๐๐โฒ) as ๐ โ โ. We propose to optimize the
phase shift matrix ฮฆฮฆฮฆ of the RIS so as to minimize the totalNMSE obtained from all the users and all the APs, as follows
minimize{\๐ }
๐โ๐=1
๐พโ๐=1
NMSE๐๐
subject to โ ๐ โค \๐ โค ๐,โ๐.(24)
We emphasize that the optimal phase shifts obtained by solv-ing the problem in (24) are independent of the instantaneousCSI and depend only on the statistical CSI, i.e., the large-scale fading coefficients and the channel covariance matrices.Problem (24) is a fractional program, whose globally-optimalsolution is not simple to be obtained for an RIS with a largenumber of independently tunable elements. Nonetheless, in thespecial network setup where the direct links from the APs tothe users are weak enough to be negligible with respect tothe RIS-assisted links, the optimal solution to problem (24) isavailable in a closed-form expression as in Corollary 2.
Corollary 2. If the direct links are weak enough to be negli-gible and the RIS-assisted channels are spatially correlated asformulated in (2), the optimal minimizer of the optimizationproblem in (24) is \1 = . . . = \๐ , i.e., the equal phase shiftdesign is optimal.
Proof. See Appendix D. ๏ฟฝ
If the direct links are completely blocked and the spa-tial correlation model in (2) holds, Corollary 2 provides asimple but effective option to design the phase shifts of theRIS while ensuring the optimal estimation of the aggregatedchannels according to the sum-NMSE minimization criterion.Therefore, an efficient channel estimation protocol can bedesigned even in the presence of an RIS equipped with alarge number of engineered scattering elements. The numericalresults in Section V show that the phase shifts design obtainedin Corollary 2 offers good gains in terms of net throughputeven if the direct links cannot be completely ignored.
Remark 1. The proposed optimization method of the phaseshifts of the RIS is based on the minimization of the sum-NMSE, and it is, therefore, based on improving the channelestimation quality. This is a critical objective in MassiveMIMO systems, since improving the accuracy of channelestimation results in a noticeable enhancement of the uplinkand downlink net throughput [35], [36]. If the direct links arenot weak enough, the equal phase shift design is not optimalanymore, and the optimal solution to problem (24) may beobtained numerically. For example, one can compute the first-order derivative of the sum-NMSE in (24) with respect toeach reflecting element, and the gradient descent algorithmmay be utilized to obtain a locally-optimal solution of (24) inan iterative manner. Another option would be to optimize thephase shifts of the RIS based on the maximization of the uplinkor downlink ergodic net throughput. The solution of the cor-responding optimization problem is, however, challenging anddepends on whether the uplink or the downlink transmissionphases are considered. Due to space limitations, therefore, wepostpone this latter criterion for optimizing the phase shifts ofthe RIS to a future research work.
III. UPLINK DATA TRANSMISSION AND PERFORMANCEANALYSIS WITH MR COMBINING
In this section, we first introduce a procedure to detectthe uplink transmitted signals by capitalizing on the channelestimation method introduced in the previous section. Then,we derive an asymptotic closed-form expression of the ergodicnet throughput.
A. Uplink Data Transmission PhaseIn the uplink, all the ๐พ users transmit their data to the
๐ APs simultaneously. Specifically, the user ๐ transmits amodulated symbol ๐ ๐ with E{|๐ ๐ |2} = 1. This symbol isweighted by a power control factor
โ[๐ , 0 โค [๐ โค 1, which
enhances the spectral efficiency by, for example, compensatingthe near-far effects and mitigating the mutual interferenceamong the users. Then, the received baseband signal, ๐ฆ๐ข๐ โ C,at the AP ๐ is
๐ฆ๐ข๐ =โ๐๐ข
๐พโ๐=1
โ[๐
(๐๐๐ + h๐ป๐ฮฆฮฆฮฆz๐
)๐ ๐ + ๐ค๐ข๐
=โ๐๐ข
๐พโ๐=1
โ[๐๐ข๐๐ ๐ ๐ + ๐ค๐ข๐,
(25)
6
where ๐๐ข is the normalized uplink SNR of each data symbol,which is defined as the maximum transmit power divided bythe noise variance, and ๐๐ข[๐ is the corresponding SNR ofthe user ๐; and ๐ค๐ข๐ โผ CN(0, 1) is the normalized additivenoise. For data detection, the MR combining method is usedat the CPU, i.e., ๏ฟฝ๏ฟฝ๐๐ ,โ๐, ๐, in (15) is employed to detectthe data transmitted by the user ๐ . In mathematical terms, thecorresponding decision statistic is
๐๐ข๐ =
๐โ๐=1
๏ฟฝ๏ฟฝโ๐๐ ๐ฆ๐ข๐
=โ๐๐ข
๐โ๐=1
๐พโ๐โฒ=1
โ[๐โฒ ๏ฟฝ๏ฟฝ
โ๐๐๐ข๐๐โฒ๐ ๐โฒ +
๐โ๐=1
๏ฟฝ๏ฟฝโ๐๐๐ค๐ข๐.
(26)
Based on the observation ๐๐ข๐ , the uplink ergodic net through-put of the user ๐ is analyzed in the next subsection.
B. Asymptotic Analysis of the Uplink Received Signal
In the considered system model, the number of APs, ๐ , andthe number of tunable elements of the RIS, ๐ , can be large.Therefore, we analyze the performance of two case studies: (๐)๐ is fixed and ๐ is large; and (๐๐) both ๐ and ๐ are large.The asymptotic analysis is conditioned upon a given setupof the CSI, which includes the large-scale fading coefficients,the covariance matrices, and the power utilized for the pilotand data transmission phases. To this end, the uplink weightedsignal in (26) is split into three terms based on the pilot reuseset P๐ , as follows
๐๐ข๐ =โ๐๐ข
โ๐โฒโP๐
๐โ๐=1
โ[๐โฒ ๏ฟฝ๏ฟฝ
โ๐๐๐ข๐๐โฒ๐ ๐โฒ๏ธธ ๏ธท๏ธท ๏ธธ
T๐1
+
โ๐๐ข
โ๐โฒโP๐
๐โ๐=1
โ[๐โฒ ๏ฟฝ๏ฟฝ
โ๐๐๐ข๐๐โฒ๐ ๐โฒ๏ธธ ๏ธท๏ธท ๏ธธ
T๐2
+๐โ๐=1
๏ฟฝ๏ฟฝโ๐๐๐ค๐ข๐๏ธธ ๏ธท๏ธท ๏ธธT๐3
,
(27)
where T๐1 accounts for the signals received from all the usersin P๐ , and T๐2 accounts for the mutual interference from theusers that are assigned orthogonal pilot sequences. The impactof the additive noise obtained after applying MR combining isgiven by T๐3. From (5), (6), and (15), we obtain the followingidentity
๐โ๐=1
โ[๐โฒ ๏ฟฝ๏ฟฝ
โ๐๐๐ข๐๐โฒ
=
๐โ๐=1
โ[๐โฒ๐๐๐๐ข๐๐โฒ
( โ๐โฒโฒโP๐
โ๐๐๐๐ข
โ๐๐โฒโฒ + ๐ค
โ๐๐๐
)=
๐โ๐=1
โ๐๐๐[๐โฒ๐๐๐ |๐ข๐๐โฒ |2 +
โ๐โฒโฒโP๐\{๐โฒ }
๐โ๐=1
โ๐๐๐[๐โฒ
ร ๐๐๐๐ข๐๐โฒ๐ขโ๐๐โฒโฒ +๐โ๐=1
โ[๐โฒ๐๐๐๐ข๐๐โฒ๐ค
โ๐๐๐ .
(28)
1) Case I: ๐ is fixed and ๐ is large, i.e., ๐ โ โ. Inthis case, we divide both sides of (28) by ๐ and exploitsTchebyshevโs theorem [37]3 and (7) to obtain
1๐
๐โ๐=1
โ[๐โฒ ๏ฟฝ๏ฟฝ
โ๐๐๐ข๐๐โฒ
๐โโโโโโ๐โโ
1๐
๐โ๐=1
โ๐๐๐[๐โฒ๐๐๐๐ฟ๐๐โฒ , (29)
where๐โโ denotes the convergence in probability.4 Specifically,
the second and third terms in (28) converge to zero due tothe so-called favorable propagation conditions and since theaggregated channel and the noise are mutually independent[38]. By inserting (29) into the decision variable in (27), weobtain the following deterministic value
1๐๐๐ข๐
๐โโโโโโ๐โโ
1๐
โ๐โฒโP๐
๐โ๐=1
โ๐๐๐๐๐ข[๐โฒ๐๐๐๐ฟ๐๐โฒ๐ ๐โฒ , (30)
because T๐2/๐ โ 0 and T๐3/๐ โ 0 as ๐ โ โ. The resultin (30) unveils that, for a fixed ๐ , the channels become asymp-totically orthogonal. In particular, the small-scale fading, thenon-coherent interference, and the additive noise vanish. Theonly residual impairment is the pilot contamination caused bythe users that employ the same pilot sequence. This resultis the evidence that, due to pilot contamination, the systemperformance cannot be improved by adding more APs if MRcombining is used. The contributions of both the direct andRIS-assisted indirect channels explicitly appear in (30) throughthe terms ๐ฝ๐๐โฒ and tr(ฮฮฮ๐๐โฒ), respectively.
2) Case II: Both ๐ and ๐ are large, i.e., ๐ โ โ and ๐ โโ. To analyze this case study, we need some assumptions onthe covariance matrices R๐ and R๐ , as summarized as follows.
Assumption 1. For ๐ = 1, . . . , ๐ and ๐ = 1, . . . , ๐พ, thecovariance matrices R๐ and R๐ are assumed to fulfill thefollowing properties
lim sup๐
โR๐โ2 < โ, lim inf๐
1๐
tr(R๐) > 0, (31)
lim sup๐
โR๐ โ2 < โ, lim inf๐
1๐
tr(R๐ ) > 0. (32)
The assumptions in (31) and (32) imply that the largestsingular value and the sum of the eigenvalues (counted withtheir mutiplicity) of the ๐ ร ๐ covariance matrices thatcharacterize the spatial correlation among the channels of theRIS elements are finite and positive. Dividing both sides of(28) by ๐๐ and then applying Tchebyshevโs theorem and(7), we obtain
1๐๐
๐โ๐=1
โ[๐โฒ ๏ฟฝ๏ฟฝ
โ๐๐๐ข๐๐โฒ
๐โโโโโโ๐โโ๐โโ
1๐๐
๐โ๐=1
โ๐๐๐[๐โฒ๐๐๐ tr(ฮฮฮ๐๐โฒ).
(33)We first observe that ฮฮฮ๐๐โฒ is similar to R1/2
๐โฒ ฮฆฮฆฮฆR๐ฮฆฮฆฮฆ๐ป R1/2๐โฒ ,
which is a positive semi-definite matrix.5 Since similar matri-3Let ๐1, . . . , ๐๐ be independent random variables such that E{๐๐ } =
๏ฟฝ๏ฟฝ๐ and Var{๐๐ } โค ๐ < โ. Then, Tchebyshevโs theorem states1๐
โ๐๐โฒ=1 ๐๐โฒ
๐โโโโโโ๐โโ
1๐
โ๐โฒ ๏ฟฝ๏ฟฝ๐โฒ .
4A sequence {๐๐ } of random variables converges in probability to therandom variable ๐ if, for all ๐ > 0, it holds that lim๐โโ Pr( |๐๐ โ ๐ | >๐ ) = 0, where Pr( ยท) denotes the probability of an event.
5Two matrices A and B of size ๐ร๐ are similar if there exists an invertible๐ ร ๐ matrix U such that B = Uโ1AU.
7
ces have the same eigenvalues, it follows that tr(ฮฮฮ๐๐โฒ) > 0.Based on Assumption 1, we obtain the following inequalities
1๐
tr(ฮฮฮ๐๐โฒ)(๐)โค 1๐โฮฆฮฆฮฆโ2tr
(R๐ฮฆฮฆฮฆR๐โฒ
)(๐)=
1๐
tr(ฮฆฮฆฮฆR๐โฒR๐
) (๐)โค 1๐โR๐โฒ โ2tr(R๐),
(34)
where (๐) is obtained from Lemma 3 in Appendix A; (๐)follows because โฮฆฮฆฮฆโ2 = 1; and (๐) is obtained by applyingagain Lemma 3. Based on Assumption 1, the last inequality in(34) is bounded by a positive constant. From (33), therefore,the decision variable in (27) can be formulated as
1๐๐
๐๐ข๐๐โโโโโโ
๐โโ๐โโ
1๐๐
โ๐โฒโP๐
๐โ๐=1
โ๐๐๐๐๐ข[๐โฒ๐๐๐ tr(ฮฮฮ๐๐โฒ)๐ ๐โฒ ,
(35)which is bounded from above thanks to (34). The expressionobtained in (35) reveals that, as ๐, ๐ โ โ, the post-processedsignal at the CPU consists of the desired signal of the intendeduser ๐ and the interference from the other users in P๐ .Compared with (30), we observe that (35) is independent ofthe direct links and depends only on the RIS-assisted indirectlinks. This highlights the potentially promising contribution ofthe RIS, in the limiting regime ๐, ๐ โ โ, for enhancing thesystem performance.C. Uplink Ergodic Net Throughput with a Finite Number ofAPs and RIS Elements
In this section, we focus our attention on the practical setupin which ๐ and ๐ are both finite. By utilizing the user-and-then forget channel capacity bounding method [39], the uplinkergodic net throughput of the user ๐ can be written as follows
๐ ๐ข๐ = ๐ตa๐ข(1 โ ๐๐/๐๐
)log2 (1 + SINR๐ข๐ ) , [Mbps], (36)
where ๐ต is the system bandwidth measured in MHz and 0 โคa๐ข โค 1 is the portion of each coherence time interval that isdedicated to the uplink data transmission phase. The effectiveuplink signal-to-noise-plus-interference ratio (SINR), which isdenoted by SINR๐ข๐ , is defined as follows
SINR๐ข๐ =|DS๐ข๐ |2
E{|BU๐ข๐ |2} +โ๐พ๐โฒ=1,๐โฒโ ๐ E{|UI๐ข๐โฒ๐ |2} + E{|NO๐ข๐ |2}
, (37)
where the following definitions hold
DS๐ข๐ =โ๐๐ข[๐E
{๐โ๐=1
๏ฟฝ๏ฟฝโ๐๐๐ข๐๐
}, (38)
BU๐ข๐ =โ๐๐ข[๐
(๐โ๐=1
๏ฟฝ๏ฟฝโ๐๐๐ข๐๐ โ E{๐โ๐=1
๏ฟฝ๏ฟฝโ๐๐๐ข๐๐
}), (39)
UI๐ข๐โฒ๐ =โ๐๐ข[๐โฒ
๐โ๐=1
๏ฟฝ๏ฟฝโ๐๐๐ข๐๐โฒ , (40)
NO๐ข๐ =
๐โ๐=1
๏ฟฝ๏ฟฝโ๐๐๐ค๐ข๐. (41)
In particular, DS๐ข๐ denotes the (average) strength of thedesired signal, BU๐ข๐ denotes the beamforming uncertainty,
which represents the randomness of the effective channel gainfor a given beamforming method, UI๐ข๐โฒ๐ denotes the interfer-ence caused by the user ๐ โฒ to the user ๐ , and NO๐ข๐ denotes theadditive noise. We emphasize that the net throughput in (36)is achievable since it is a lower bound of the channel capacity.A closed-form expression for (36) is given in Theorem 1.
Theorem 1. If the CPU utilizes the MR combining method,a lower-bound closed-form expression for the uplink netthroughput of the user ๐ is given by (36), where the SINRis
SINR๐ข๐ =๐๐ข[๐
(โ๐๐=1 ๐พ๐๐
)2
MI๐ข๐ + NO๐ข๐
, (42)
where MI๐ข๐ is the mutual interference and NO๐ข๐ is the noise,which are formulated as follows
MI๐ข๐ = ๐๐ข๐พโ๐โฒ=1
๐โ๐=1
[๐โฒ๐พ๐๐๐ฟ๐๐โฒ
+ ๐๐๐๐๐ข๐พโ๐โฒ=1
โ๐โฒโฒโP๐
๐โ๐=1
๐โ๐โฒ=1
[๐โฒ๐๐๐๐๐โฒ๐ tr(ฮฮฮ๐๐โฒฮฮฮ๐โฒ๐โฒโฒ) (43)
+ ๐๐๐๐๐ขโ๐โฒโP๐
๐โ๐=1
[๐โฒ๐2๐๐ tr(ฮฮฮ
2๐๐โฒ)
+ ๐๐๐๐๐ขโ
๐โฒโP๐\{๐ }[๐โฒ
(๐โ๐=1
๐๐๐๐ฟ๐๐โฒ
)2
,
NO๐ข๐ =
๐โ๐=1
๐พ๐๐ , (44)
with ๐ฟ๐๐โฒ = ๐ฝ๐๐โฒ + tr(ฮฮฮ๐๐โฒ), ๐๐๐ given in (16), and ๐พ๐๐ givenin (17).
Proof. See Appendix E. ๏ฟฝ
By direct inspection of the SINR in (42), we observe thatthe numerator increases with the square of the sum of thevariances of the channel estimates, ๐พ๐๐ ,โ๐ thanks to thejoint signal processing, i.e., the received signals form the ๐APs are sent to the CPU for centralized data detection. Onthe other hand, the first term in (43) represents the powerof the mutual interference. The use of an RIS to supportmultiple users introduce the extra interference shown in thesecond and third terms in (43). Due to the limited and finitenumber of orthogonal pilot sequences being used, the fourthterm in (43) dominates the impact of pilot contamination.The second term in the denominator in (42) is the additivenoise. If the coherence time is sufficiently large that everyuser can utilize its own orthogonal pilot sequence, the uplinknet throughput of the user ๐ can still be obtained from (36),but the effective SINR simplifies to (45). The SINR in (42)is a multivariate function of the matrix of phase shifts of theRIS and of the channel statistics, i.e., the channel covariancematrices. Table I gives a comparison of the obtained uplinkSINR of the user ๐ with and without the presence of the RIS.By direct inspection of |DS๐ข๐ |2, we evince that the strengthof the desired signal increases thanks to the assistance of theRIS. However, the mutual interference becomes more severe as
8
SINR๐ข๐ =๐๐ข[๐
(๐โ๐=1
๐พ๐๐๐
)2
๐โ๐=1
๐พ๐๐๐
+ ๐๐ข๐พโ๐โฒ=1
๐โ๐=1
[๐โฒ๐พ๐๐๐๐ฟ๐๐โฒ + ๐๐๐๐๐ข
๐พโ๐โฒ=1
๐โ๐=1
๐โ๐โฒ=1
[๐โฒ๐๐๐๐๐๐๐โฒ๐ tr(ฮฮฮ๐๐โฒฮฮฮ๐โฒ๐ ) + ๐๐๐๐๐ข[๐
๐โ๐=1
๐2๐๐
tr(ฮฮฮ2๐๐
)(45)
well, due to the need of estimating both the direct and indirectlinks in the presence of the RIS. By assigning orthogonalpilot signals to all the ๐พ users, the coherent interference canbe completely suppressed. In Section V, the performance ofCell-Free Massive MIMO and RIS-assisted Cell-Free MassiveMIMO is compared with the aid of numerical simulations.IV. DOWNLINK DATA TRANSMISSION AND PERFORMANCE
ANALYSIS WITH MR PRECODINGIn this section, we consider the downlink data transmission
phase and analyze the received signal at the users when thenumber of APs is large or when the numbers of RIS elementsand APs are both large. A closed-form expression of thedownlink ergodic net throughput that is attainable with MRprecoding and for an arbitrary phase shift matrix of the RISelements is provided.
A. Downlink Data Transmission PhaseBy exploiting channel reciprocity, the AP ๐ treats the
channel estimates obtained in the uplink as the true channels inorder to construct the beamforming coefficients. Accordingly,the downlink signal transmitted from the AP ๐ is6
๐ฅ๐ =โ๐๐
๐พโ๐=1
โ[๐๐ ๏ฟฝ๏ฟฝ
โ๐๐๐๐ , (46)
where ๐๐ is the normalized SNR in the downlink; ๐๐ is thecomplex data symbol that is to be sent (cooperatively by virtueof the considered coherent joint transmission scheme) by allthe ๐ APs to the user ๐ , with E{|๐๐ |2} = 1; and [๐๐ isthe power control coefficient of the AP ๐, which satisfies thepower budget constraint as follows
E{|๐ฅ๐ |2} โค ๐๐ โ๐พโ๐=1
[๐๐๐พ๐๐ โค 1. (47)
The cooperation among the ๐ APs for jointly transmittingthe same data symbol to a particular user creates the majordistinction between the downlink and uplink data transmissionphases. Based on (46), the received signal at the user ๐ is thesuperposition of the signals transmitted by the ๐ APs as
๐๐๐ =
๐โ๐=1
๐ข๐๐๐ฅ๐ + ๐ค๐๐
=โ๐๐
๐โ๐=1
๐พโ๐โฒ=1
โ[๐๐โฒ๐ข๐๐ ๏ฟฝ๏ฟฝ
โ๐๐โฒ๐๐โฒ + ๐ค๐๐ .
(48)
where ๐ค๐๐ โผ CN(0, 1) is the additive noise at the user ๐ .The user ๐ decodes the desired data symbol based on theobservation in (48).
6In this paper, the downlink data transmission is conducted based on theuplink channel estimates, which depend on the RIS phase shift matrix. Sincethe MR processing is used for downlink data transmission based on the uplinkchannel estimates, closed-form analytical expressions for the downlink can beobtained if the same phase shift matrix is utilized in the uplink and in thedownlink.
B. Asymptotic Analysis of the Downlink Received SignalIn contrast with the uplink data processing where the CPU
needs only the channel estimate ๏ฟฝ๏ฟฝ๐๐ for detecting the dataof the user ๐ , as displayed in (27), the received signal in(48) depends on the channel estimates of the ๐พ users in thenetwork, since the channel estimates from the ๐พ users are usedfor MR precoding. Therefore, the analysis of the uplink anddownlink data transmission phases are different. First, we split(48) into three terms, by virtue of the pilot reuse pattern P๐ ,as follows
๐๐๐ =โ๐๐
โ๐โฒโP๐
๐โ๐=1
โ[๐๐โฒ๐ข๐๐ ๏ฟฝ๏ฟฝ
โ๐๐โฒ๐๐โฒ+
โ๐๐
โ๐โฒโP๐
๐โ๐=1
โ[๐๐โฒ๐ข๐๐ ๏ฟฝ๏ฟฝ
โ๐๐โฒ๐๐โฒ + ๐ค๐๐ . (49)
Then, we investigate the two asymptotic regimes for ๐ โ โand ๐, ๐ โ โ. In particular, the first term in (49) can berewritten as
๐โ๐=1
โ[๐๐โฒ๐ข๐๐ ๏ฟฝ๏ฟฝ
โ๐๐โฒ
(๐)=
๐โ๐=1
โ[๐๐โฒ๐๐๐โฒ๐ข๐๐
( โ๐โฒโฒโP๐
โ๐๐๐๐ข
โ๐๐โฒโฒ + ๐ค
โ๐๐๐โฒ
)(๐)=
๐โ๐=1
โ[๐๐โฒ ๐๐๐๐๐๐โฒ |๐ข๐๐ |2 +
โ๐โฒโฒโP๐\{๐ }
๐โ๐=1
โ[๐๐โฒ ๐๐๐
ร ๐๐๐โฒ๐ข๐๐๐ขโ๐๐โฒโฒ +๐โ๐=1
โ[๐๐โฒ๐๐๐โฒ๐ข๐๐๐ค
โ๐๐๐โฒ ,
(50)
where (๐) is obtained by utilizing the channel estimates in(15) and (๐) is obtained by extracting the aggregated channelof the user ๐ from the summation. By letting ๐ and/or ๐ belarge, similar to the uplink data transmission phase, we obtainthe following asymptotic results
1๐
๐โ๐=1
โ[๐๐โฒ๐ข๐๐ ๏ฟฝ๏ฟฝ
โ๐๐โฒ
๐โโโโโโ๐โโ
1๐
๐โ๐=1
โ[๐๐โฒ ๐๐๐๐๐๐โฒ๐ฟ๐๐ ,
(51)
1๐๐
๐โ๐=1
โ[๐๐โฒ๐ข๐๐ ๏ฟฝ๏ฟฝ
โ๐๐โฒ
๐โโโโโโ๐โโ๐โโ
1๐๐
๐โ๐=1
โ[๐๐โฒ ๐๐๐๐๐๐โฒ tr(ฮฮฮ๐๐ ), (52)
which are bounded from above based on Assumption 1.Consequently, the received signal at the user ๐ converges toa deterministic equivalent as the number of APs is large, i.e.,
9
TABLE ICOMPARISON OF THE UPLINK SINR BETWEEN CELL-FREE MASSIVE MIMO AND RIS-ASSISTED CELL-FREE MASSIVE MIMO
Uplink SINR Cell-Free Massive MIMO RIS-Assisted Cell-Free Massive MIMO
(42)
๐ฟ๐๐ ๐ฝ๐๐ ๐ฝ๐๐ + tr(ฮฆฮฆฮฆ๐ปR๐ฮฆฮฆฮฆR๐
)๐๐๐
โ๐๐๐๐ฝ๐๐
๐๐๐โ
๐โฒโP๐๐ฝ๐๐โฒ+1
โ๐๐๐ ๐ฟ๐๐
๐๐๐โ
๐โฒโP๐๐ฟ๐๐โฒ+1
๐พ๐๐โ๐๐๐๐ฝ๐๐๐๐๐
โ๐๐๐ ๐ฟ๐๐๐๐๐
|DS๐ข๐ |2 ๐๐ข[๐
(๐โ๐=1
๐พ๐๐
)2๐๐ข[๐
(๐โ๐=1
๐พ๐๐
)2
MI๐ข๐
๐๐ข๐พโ๐โฒ=1
๐โ๐=1
[๐โฒ๐พ๐๐ ๐ฟ๐๐โฒ+
๐๐๐๐๐ขโ
๐โฒโP๐ \{๐}[๐โฒ
(๐โ๐=1
๐๐๐ ๐ฟ๐๐โฒ
)2
๐๐ข๐พโ๐โฒ=1
๐โ๐=1
[๐โฒ๐พ๐๐ ๐ฟ๐๐โฒ+
๐๐๐๐๐ข๐พโ๐โฒ=1
โ๐โฒโฒโP๐
๐โ๐=1
๐โ๐โฒ=1
[๐โฒ๐๐๐๐๐โฒ๐ tr(ฮฮฮ๐๐โฒฮฮฮ๐โฒ๐โฒโฒ )+
๐๐๐๐๐ขโ
๐โฒโP๐
๐โ๐=1
[๐โฒ๐2๐๐
tr(ฮฮฮ2๐๐โฒ ) + ๐๐๐๐๐ข
โ๐โฒโP๐ \{๐} [๐โฒ
(๐โ๐=1
๐๐๐ ๐ฟ๐๐โฒ
)2
NO๐ข๐๐โ๐=1
๐พ๐๐๐โ๐=1
๐พ๐๐
(45)
๐๐๐๐
โ๐๐๐๐ฝ๐๐
๐๐๐๐ฝ๐๐+1
โ๐๐๐ ๐ฟ๐๐
๐๐๐ ๐ฟ๐๐+1๐พ๐๐๐
โ๐๐๐๐ฝ๐๐๐
๐๐๐
โ๐๐๐ ๐ฟ๐๐๐
๐๐๐
|DS๐ข๐ |2 ๐๐ข[๐
(๐โ๐=1
๐พ๐๐๐
)2๐๐ข[๐
(๐โ๐=1
๐พ๐๐๐
)2
MI๐ข๐ ๐๐ข๐พโ๐โฒ=1
๐โ๐=1
[๐โฒ๐พ๐๐๐๐ฟ๐๐โฒ +
๐โ๐=1
๐พ๐๐๐
๐๐ข๐พโ๐โฒ=1
๐โ๐=1
[๐โฒ๐พ๐๐๐๐ฟ๐๐โฒ + ๐๐๐๐๐ข
๐พโ๐โฒ=1
๐โ๐=1
๐โ๐โฒ=1
[๐โฒ๐๐๐๐๐๐๐โฒ๐ tr(ฮฮฮ๐๐โฒฮฮฮ๐โฒ๐ )+
๐๐๐๐๐ข[๐๐โ๐=1
๐2๐๐
tr(ฮฮฮ2๐๐
)
NO๐ข๐๐โ๐=1
๐พ๐๐๐
๐โ๐=1
๐พ๐๐๐
๐ โ โ, and as the number of APs and RIS elements arelarge, i.e., ๐, ๐ โ โ. More precisely, the received signalconverges (asymptotically) to
1๐๐๐๐
๐โโโโโโ๐โโ
1๐
โ๐โฒโP๐
๐โ๐=1
โ๐๐๐๐๐[๐๐โฒ๐๐๐โฒ๐ฟ๐๐ ๐ ๐โฒ , (53)
1๐๐
๐๐๐๐โโโโโโ
๐โโ
1๐๐
โ๐โฒโP๐
๐โ๐=1
โ๐๐๐๐๐[๐๐โฒ๐๐๐โฒ tr(ฮฮฮ๐๐ )๐ ๐โฒ ,
(54)
which indicates the inherent coexistence of the users in P๐ .The deterministic equivalents in (53) and (54) unveil that theimpact of the channel estimation accuracy and the channelstatistics is different between the uplink and the downlink.In particular, the asymptotic received signal in the uplinkonly depends on the channel estimation quality of each in-dividual user, which is manifested by the coefficient ๐๐๐ .The asymptotic received signal in the downlink depends, onthe other hand, on the channel estimation quality of all theusers that share the same orthogonal pilot sequences, i.e.,๐๐,๐โฒ ,โ๐ โฒ โ P๐ .
C. Downlink Ergodic Net Throughput with a Finite Numberof APs and RIS Elements
By utilizing the channel capacity bounding technique [39],similar to the analysis of the uplink data transmission phase,the downlink ergodic net throughput of the user ๐ can bewritten as follows
๐ ๐๐ = ๐ตa๐(1 โ ๐๐/๐๐
)log2 (1 + SINR๐๐ ) , [Mbps], (55)
where 0 โค a๐ โค 1 is the portion of each coherence timeinterval dedicated to the downlink data transmission phase,
with a๐ข + a๐ = 1, and the effective downlink SINR is definedas
SINR๐๐ =|DS๐๐ |2
E{|BU๐๐ |2} +โ๐พ๐โฒ=1,๐โฒโ ๐ E{|UI๐๐โฒ๐ |2} + 1
, (56)
where the following definitions hold
DS๐๐ =โ๐๐E
{๐โ๐=1
โ[๐๐๐ข๐๐ ๏ฟฝ๏ฟฝ
โ๐๐
}, (57)
UI๐๐โฒ๐ =โ๐๐
๐โ๐=1
โ[๐๐โฒ๐ข๐๐ ๏ฟฝ๏ฟฝ
โ๐๐โฒ , (58)
BU๐๐ =โ๐๐
(๐โ๐=1
โ[๐๐๐ข๐๐ ๏ฟฝ๏ฟฝ
โ๐๐ โ E
{๐โ๐=1
โ[๐๐๐ข๐๐ ๏ฟฝ๏ฟฝ
โ๐๐
}),
(59)
In particular, DS๐๐ denotes the (average) strength of thedesired signal received by the user ๐ , BU๐๐ denotes the beam-forming uncertainty, UI๐๐โฒ๐ denotes the interference causedto the user ๐ by the signal intended to the user ๐ โฒ. Thedownlink ergodic net throughput in (55) is achievable since itis a lower bound of the channel capacity, similar to the uplinkdata transmission phase. In contrast to the uplink ergodic netthroughput, which only depends on the combining coefficientsof each individual user, the downlink net throughput of theuser ๐ depends on the precoding coefficients of all the ๐พ users.A closed-form expression for (55) is given in Theorem 2.
Theorem 2. If the CPU utilizes the MR precoding method,a lower-bound closed-form expression for the downlink netthroughput of the user ๐ is given by (55), where the SINR is
SINR๐๐ =๐๐
( โ๐๐=1
โ[๐๐๐พ๐๐
)2
MI๐๐ + 1, (60)
10
SINR๐๐ =
๐๐
(๐โ๐=1
โ[๐๐๐พ๐๐
)2
1 + ๐๐๐พโ๐โฒ=1
๐โ๐=1
[๐๐โฒ๐พ๐๐๐โฒ๐ฟ๐๐ + ๐๐๐๐๐
๐พโ๐โฒ=1
๐โ๐=1
๐โ๐โฒ=1
โ[๐๐โฒ[๐โฒ๐โฒ๐๐๐โฒ๐๐โฒ๐โฒ tr(ฮฮฮ๐๐ฮฮฮ๐โฒ๐โฒ) + ๐๐๐๐๐
๐โ๐=1
[๐๐๐2๐๐
tr(ฮฮฮ2๐๐
)(62)
where MI๐๐ is the mutual interference, which is defined asfollows
MI๐๐ = ๐๐๐พโ๐โฒ=1
๐โ๐=1
[๐๐โฒ๐พ๐๐โฒ๐ฟ๐๐ + ๐๐๐๐๐ร
๐พโ๐โฒ=1
โ๐โฒโฒโP๐โฒ
๐โ๐=1
๐โ๐โฒ=1
โ[๐๐โฒ[๐โฒ๐โฒ๐๐๐โฒ๐๐โฒ๐โฒ tr(ฮฮฮ๐๐ฮฮฮ๐โฒ๐โฒโฒ)
+ ๐๐๐๐๐โ๐โฒโP๐
๐โ๐=1
[๐๐โฒ๐2๐๐โฒ tr(ฮฮฮ
2๐๐ )+
๐๐๐๐๐
โ๐โฒโP๐\{๐ }
(๐โ๐=1
โ[๐๐โฒ๐๐๐โฒ๐ฟ๐๐
)2
.
(61)
Proof. The main steps of the proof are similar to those of theproof of Theorem 1. However, there are also major differencesthat are due to the coherent joint data transmission schemeamong the APs. The details of the proof are available inAppendix F. ๏ฟฝ
From Theorem 2, we observe that the effective downlinkSINR has some similarities and differences as compared withits uplink counterpart in Theorem 1. Similar to the uplink, thenumerator of (60) is a quadratic function that depends on thechannel estimation quality and the coherent joint transmissionprocessing. Differently from the uplink, the transmit powercoefficients appear explicitly in the numerator of (60) as aresult of the cooperation among the APs. The impact of pilotcontamination in (61), in contrast to the uplink, depends on allthe transmit power coefficients. In addition, we observe thatthe impact of pilot contamination scales up with the numberof APs and with the number of elements of the RIS. Whenthe ๐พ users employ orthogonal pilot sequences, the downlinkSINR expression of the user ๐ simplifies to (62).
A comparison of the obtained analytical expressions of thedownlink SINR for Cell-Free Massive MIMO and RIS-assistedCell-Free Massive MIMO systems is given in Table II. Bycomparing Table I and Table II, the difference and similaritiesbetween the uplink and downlink transmission phases canbe identified as well. With the aid of numerical results, inSection V, we will illustrate the advantages of RIS-assistedCell-Free Massive MIMO especially if the direct links arenot sufficiently reliable (e.g., they are blocked) with highprobability.
Remark 2. We observe that the ergodic net throughputin the uplink (Theorem 1) and downlink (Theorem 2) datatransmission phases depend only on the large-scale fadingstatistics and on the channel covariance matrices, while theyare independent of the instantaneous CSI. This simplifiesthe deployment and optimization of RIS-assisted Cell-FreeMassive MIMO systems. As anticipated in Remark 1, in fact,
the phase shifts of the RIS can be optimized based on the (long-term) analytical expressions of the ergodic net throughputsin Theorem 1 and Theorem 2, which are independent of theinstantaneous CSI. In this paper, we have opted for optimizingthe phase shifts of the RIS in order to minimize the channelestimation error, which determines the performance of both theuplink and downlink transmission phases. The optimizationof the phase shifts of the RISs based on the closed-formexpressions in Theorem 1 and Theorem 2 is postponed to afuture research work.
V. NUMERICAL RESULTS
In this section, we report some numerical results in orderto illustrate the performance of the RIS-assisted Cell-FreeMassive MIMO system introduced in the previous sections.We consider a geographic area of size 1.5 ร 1.5 km2, wherethe locations of the APs and users are given in terms of(๐ฅ, ๐ฆ) coordinates. The four vertices of the considered regionare [โ0.75,โ0.75] km, [โ0.75, 0.75] km, [0.75, 0.75] km,[0.75,โ0.75] km. To simulate a harsh communication environ-ment, the ๐ APs are uniformly distributed in the sub-region๐ฅ, ๐ฆ โ [โ0.75,โ0.5] km, while the ๐พ users are uniformlydistributed in the sub-region ๐ฅ, ๐ฆ โ [0.375, 0.75] km. The RISis located at the origin, i.e., (๐ฅ, ๐ฆ) = (0, 0). The carrier fre-quency is 1.9 GHz and the system bandwidth is 20 MHz. Eachcoherence interval comprises ๐๐ = 200 symbols, which maycorrespond to a coherence bandwidth equal to ๐ต๐ = 200 KHzand a coherence time equal to ๐๐ = 1 ms, except in Fig. 12(๐)where ๐๐ = 5000 symbols. We assume ๐๐ = 5 orthonormalpilot sequences that are shared by all the users. The large-scalefading coefficients in dB are generated according to the three-slope propagation model in [8, (51)โ(53)], where the path lossexponent depends on the distance between the transmitter andthe receiver. The shadow fading has a log-normal distributionwith standard deviation equal to 8 dB. The distance thresholdsfor the three slopes are 10 m and 50 m. The height of the APs,RIS, and users is 15 m, 30 m, and 1.65 m, respectively. Thedirect links, ๐๐๐ ,โ๐, ๐ , are assumed to be unblocked witha given probability. More specifically, the large-scale fadingcoefficient ๐ฝ๐๐ is formulated as follows
๐ฝ๐๐ = ๐ฝ๐๐๐๐๐ , (63)
where ๐ฝ๐๐ accounts for the path loss due to the transmissiondistance and the shadow fading according to the three-slopepropagation model in [8]. The binary variables ๐๐๐ accountsfor the probability that the direct links are unblocked, and itis defined as
๐๐๐ =
{1, with a probability ๐,
0, with a probability 1 โ ๐,(64)
11
TABLE IICOMPARISON OF THE DOWNLINK SINR BETWEEN CELL-FREE MASSIVE MIMO AND RIS-ASSISTED CELL-FREE MASSIVE MIMO (SOME
PARAMETERS ARE DEFINED IN TABLE I)
Downlink SINR Cell-Free Massive MIMO RIS-Assisted Cell-Free Massive MIMO
(60)
|DS๐๐ |2 ๐๐
(๐โ๐=1
โ[๐๐๐พ๐๐
)2๐๐
(๐โ๐=1
โ[๐๐๐พ๐๐
)2
MI๐๐
๐๐๐พโ๐โฒ=1
๐โ๐=1
[๐๐โฒ๐พ๐๐โฒ ๐ฟ๐๐+
๐๐๐๐๐โ
๐โฒโP๐ \{๐}
(๐โ๐=1
โ[๐๐โฒ๐๐๐โฒ ๐ฟ๐๐
)2
๐๐๐พโ๐โฒ=1
๐โ๐=1
[๐๐โฒ๐พ๐๐โฒ ๐ฟ๐๐+
๐๐๐๐๐๐พโ๐โฒ=1
โ๐โฒโฒโP๐โฒ
๐โ๐=1
๐โ๐โฒ=1
โ[๐๐โฒ [๐โฒ๐โฒ๐๐๐โฒ๐๐โฒ๐โฒ tr(ฮฮฮ๐๐ฮฮฮ๐โฒ๐โฒโฒ )+
๐๐๐๐๐โ
๐โฒโP๐
๐โ๐=1
[๐๐โฒ๐2๐๐โฒ tr(ฮฮฮ
2๐๐
) + ๐๐๐๐๐โ
๐โฒโP๐ \{๐}
(๐โ๐=1
โ[๐๐โฒ๐๐๐โฒ ๐ฟ๐๐
)2
NO๐๐ 1 1
(62)
|DS๐๐ |2 ๐๐
(๐โ๐=1
โ[๐๐๐พ
๐๐๐
)2๐๐
(๐โ๐=1
โ[๐๐๐พ
๐๐๐
)2
MI๐๐ ๐๐๐พโ๐โฒ=1
๐โ๐=1
[๐๐โฒ๐พ๐๐๐โฒ ๐ฟ๐๐ + 1
๐๐๐พโ๐โฒ=1
๐โ๐=1
[๐๐โฒ๐พ๐๐๐โฒ ๐ฟ๐๐+
๐๐๐๐๐๐พโ๐โฒ=1
๐โ๐=1
๐โ๐โฒ=1
โ[๐๐โฒ [๐โฒ๐โฒ๐๐๐โฒ๐๐โฒ๐โฒ tr(ฮฮฮ๐๐ฮฮฮ๐โฒ๐โฒ )+
๐๐๐๐๐๐โ๐=1
[๐๐๐2๐๐
tr(ฮฮฮ2๐๐
)
NO๐๐ 1 1
where ๐ โ [0, 1] is the probability that the direct link is notblocked. The noise variance is โ92 dBm, which corresponds toa noise figure of 9 dB. The covariance matrices are generatedaccording to the spatial correlation model in (2). The power ofthe pilot sequences is 100 mW and the power budget of eachAP is 200 mW. The time intervals of the data transmissionphase, in each coherence time, that are allocated to the uplinkand downink transmissions are, a๐ข = a๐ = 0.5. The uplinkand downlink power control coefficients are [๐ = 1,โ๐, and[๐๐ = (โ๐พ
๐โฒ=1 ๐พ๐๐โฒ)โ1,โ๐, ๐ , which is directly obtained from(47) to satisfy the limited power budget per AP.7 As far asthe optimization of the phase shifts of the RIS elements areconcerned, we assume that they are optimized according tothe sum-NMSE minimization criterion in the absence of directlinks, according to Corollary 2 and Remark 1. Without lossof generality, in particular, the ๐ phase shifts in ฮฆฮฆฮฆ are all setequal to ๐/4, except in Figs. 8 and 9 where different phaseshifts are considered for comparison. In order to evaluate theadvantages and limitations of RIS-assisted Cell-Free MassiveMIMO systems, three system configurations are considered forcomparison:
๐) RIS-Assisted Cell-Free Massive MIMO: This is the pro-posed system model in which the direct links are blockedaccording to (64). This setup is denoted by โRIS-CellFreeโ.
๐๐) (Conventional) Cell-Free Massive MIMO: This setup isthe same as the previous one with the only exceptionthat the RIS is not deployed in the network. This setupis denoted by โCellFreeโ.
๐๐๐) RIS-Assisted Cell-Free Massive MIMO with blocked di-rect links: This is the worst case study in which thedirect links are blocked with unit probability and theuplink and downlink transmission phases are ensured only
7In the worst case, if all the direct links are blocked, we introduce a dampingconstant when Cell-Free Massive MIMO systems in the absence of the RISare considered, since in those cases we have
โ๐๐=1 ๐พ๐๐โฒ = 0.
through the RIS. This setup is denoted by โRIS-CellFree-NoLOSโ.
In Fig. 2, we illustrate the cumulative distribution function(CDF) of the net throughput by using Monte Carlo simulationsand the proposed analytical framework. The CDF is computedwith respect to the locations of the APs and users in the con-sidered area. The Monte Carlo simulation results are obtainedby using (36) and (55) with the SINRs given in (37) and (56),while the analytical results are obtained by using Theorem 1and Theorem 2. We observe a very good overlap between thenumerical simulations and the obtained analytical expressions.From Fig. 2, we evince that the downlink net throughput peruser is about 2.6ร better than the uplink net throughput. Thisis due to the higher transmission power of the APs and thegain of the joint processing of the APs. Since the Monte Carlosimulations are not simple to obtain for larger values of thesimulation parameters, the rest of the figures are obtainedby using the closed-form expressions of the net throughputderived in Theorem 1 and Theorem 2.
In Fig. 3, we illustrate the average sum net throughput as afunction of the probability ๐ in (64). In particular, the averageuplink sum net throughput is defined as
โ๐พ๐=1 E{๐ ๐ข๐ } and
the downlink sum net throughput is defined asโ๐พ๐=1 E{๐ ๐๐ },
where the uplink and downlink SINRs are obtained by usingTheorem 1 and Theorem 2. In particular, the expectation iscomputed with respect to the locations of the APs and users inthe considered area. From the obtained results, we evince thatCell-Free Massive MIMO provides the worst performance ifthe blocking probability is large (๐ is small). As expected, theaverage net throughput offered by Cell-Free Massive MIMOtends to zero if ๐ โ 0 (the direct links are unreliable). Forexample, at ๐ = 0.1, the average sum net throughput ofCell-Free Massive MIMO is approximately 2.0ร and 1.4รsmaller, in the uplink and downlink, respectively, than theaverage sum net throughput of the worst-case RIS-assistedCell-Free Massive MIMO setup (i.e., RIS-CellFree-NoLOS).In the considered case study, in addition, we note that the
12
Fig. 2. Monte Carlo simulations versus the analytical frameworks with ๐ =
20, ๐พ = 5, ๐ = 64, ๐๐ = 2, and ๐๐ป = ๐๐ = _/4. The unblockedprobability of the direct links is ๏ฟฝ๏ฟฝ = 1.0.
Fig. 3. Average sum net throughput [Mbps] versus the unblocked probabilityof the direct links ๏ฟฝ๏ฟฝ with ๐ = 100, ๐พ = 10, ๐ = 900, ๐๐ = 5, and๐๐ป = ๐๐ = _/4.
Fig. 4. CDF of the sum net throughput [Mbps] with ๐ = 100, ๐พ = 10,๐ = 900, ๐๐ = 5, and ๐๐ป = ๐๐ = _/4. The unblocked probability of thedirect links is ๏ฟฝ๏ฟฝ = 0.2.
Fig. 5. Average sum net throughput [Mbps] versus the number of APs with๐พ = 10, ๐ = 900, ๐๐ = 5, and ๐๐ป = ๐๐ = _/4. The unblocked probabilityof the direct links is ๏ฟฝ๏ฟฝ = 0.2.
proposed RIS-assisted Cell-Free Massive MIMO setup offersthe best average net throughput, since it can overcome theunreliability of the direct links thanks to the presence of theRIS. The presence of the RIS is particularly useful if ๐ issmall, i.e., ๐ < 0.2 in Fig. 3, since the direct links are not ableto support a high throughput. In this case, the combination ofCell-Free Massive MIMO and RIS is capable of providing ahigh throughput and signal reliability.
In Fig. 4, we compare the three considered systems in termsof average sum net throughput when ๐ = 0.2. We observe thenet advantage of the proposed RIS-assisted Cell-Free MassiveMIMO system, especially in the downlink. In the uplink, inaddition, even the worst-case RIS-assisted Cell-Free MassiveMIMO system setup (i.e., ๐ = 0) outperforms the Cell-FreeMassive MIMO setup in the absence of an RIS. In Figs. 5โ7,we show the average sum net throughput as a function ofthe number of APs, the number of users, and the numberof orthonormal pilot signals, respectively. We observe thatthe average sum net throughput increases with the numberof APs, with the number or users, and with the number ofpilot sequences, and that the RIS-assisted Cell-Free MassiveMIMO setup outperforms, especially in the downlink, theother benchmark schemes. Gains of the order of 1.7ร and2.6ร are obtained in the considered setups.
In the following figures, we focus our attention only on the
RIS-assisted Cell-Free Massive MIMO setup, since it providesthe best performance in the analyzed setups. In Figs. 8 and 9,we report the uplink and downlink average sum net throughputas a function of number of engineered scattering elementsof the RIS. In particular, we compare the average sum netthroughput when the phase shifts of the RIS are randomlychosen and are optimized according to Corollary 2 overspatially-independent and spatially-corrrelated fading channelsaccording to (2). In the presence of spatial correlation, thechannel correlation matrices are R๐ = ๐ผ๐๐๐ป ๐๐ I๐ and R๐๐ =๏ฟฝ๏ฟฝ๐๐๐๐ป ๐๐ I๐ ,โ๐, ๐ . We see different performance trends overspatially-independent and spatially-correlated fading channels.If the spatial correlation is not considered, we observe thatthere is no significant difference between the random anduniform phase shifts setup. In the presence of spatial uncor-relation, on the other hand, the uniform phase shift designobtained from Corollary 2 provides a much higher averagethroughput. This result highlights the relevance of using evensimple optimization designs for RIS-assisted communicationsover spatially-correlated fading channels.
In Fig. 10, we analyze the impact of the size of theengineered scattering elements of the RISs on the uplinkand downlink average net throughput, while keeping the totalnumber of RIS elements ๐ fixed. The size of the consideredRIS, which is a compact surface, is ๐๐๐ป ๐๐ , which implies
13
Fig. 6. Average sum net throughput [Mbps] versus the number of userswith ๐ = 100, ๐ = 900, ๐๐ = 5, and ๐๐ป = ๐๐ = _/4. The unblockedprobability of the direct links is ๏ฟฝ๏ฟฝ = 0.2.
Fig. 7. Average sum net throughput [Mbps] versus the number of pilotsequences with ๐ = 100, ๐พ = 10, ๐ = 900, and ๐๐ป = ๐๐ = _/4. Theunblocked probability of the direct links is ๏ฟฝ๏ฟฝ = 0.2.
Fig. 8. Average uplink sum net throughput [Mbps] versus the number ofengineered scattering elements with ๐ = 100, ๐พ = 10, ๐๐ = 5, and ๐๐ป =
๐๐ = _/4. The unblocked probability of the direct links is ๏ฟฝ๏ฟฝ = 0.2.
Number of Engineered Scattering Elements
Fig. 9. Average downlink sum net throughput [Mbps] versus the numberof engineered scattering elements with ๐ = 100, ๐พ = 10, ๐๐ = 5, and๐๐ป = ๐๐ = _/4. The unblocked probability of the direct links is ๏ฟฝ๏ฟฝ = 0.2.
that it increases as the size ๐๐ป ๐๐ of each element of theRIS increases. In this setup, we observe that the average netthroughput increases as the physical size of each element of theRIS increases. In Fig. 11, on the other hand, we analyze a setupin which the total size of the RIS is kept constant and equalto ๐๐๐ป ๐๐ = 10_ ร 10_ while the triplet (๐, ๐ = ๐๐ป = ๐๐ )is changed accordingly. With the considered fading spatialcorrelation model and for a size of the RIS elements nosmaller than _/3, we do not observe a significant differenceon the average net throughput. Further studies are, however,necessary for deep sub-wavelength RIS structures, for differentoptimization criteria of the phase shifts of the RIS, and in thepresence of mutual coupling in addition to the fading spatialcorrelation [40], [41].
In Fig. 12, we plot the ergodic net throughput per user byutilizing different channel capacity bounding techniques, byassuming ๐๐ = 200 symbols and ๐๐ = 5000 symbols. Themain benefit of the use-and-then-forget bound in (36) andthe hardening bound in (55) is the possibility of obtaining aclosed-form expression for the net throughput by capitalizingon the fundamentals properties of Massive MIMO communi-cations, as demonstrated in Theorems 1 and 2. However, thechannel hardening capability reduces in the presence of an RIS[38]. Thus, other channel capacity bounding techniques mayresult in a better estimate of the net throughput as compared
to the actual channel capacity. The statistical bound in [42],[43] was originally derived for application to the downlinkdata transmission when no instantaneous CSI is availableat the users and the channel vectors may be less hardened[10]. In order to apply the same bound to the uplink datatransmission, we assume that only the channel statistics areavailable at the CPU. Even though the statistical bound resultsin a better ergodic net throughput per user, some realizationsof user locations and shadow fading may lead to negativevalues of the ergodic net throughput. This may occur in highmobility scenarios, which correspond to small values of ๐๐ ,as illustrated in Fig. 12(๐). The statistical bound providesconsistent values of the ergodic net throughput in low mobilityscenarios when ๐๐ is large, as displayed in Fig. 12(๐). Thenumerical results unveils the need of developing differentand more accurate channel bounding methods for evaluatingthe net throughput of RIS-assisted Cell-Free Massive MIMOsystems.
VI. CONCLUSIONCell-Free Massive MIMO and RIS are two disruptive
technologies for boosting the system performance of futurewireless networks. These two technologies are not competingwith each other, but have complementary features that canbe integrated and leveraged for enhancing the system per-formance in harsh communication environments. Therefore,
14
Fig. 10. Average sum net throughput [Mbps] versus the size of engineeredscattering elements {๐๐ป , ๐๐ }, but for a different size of the RIS with ๐ =
100, ๐พ = 10, ๐ = 900, and ๐๐ = 5. The unblocked probability of the directlinks is ๏ฟฝ๏ฟฝ = 0.2.
Fig. 11. Average sum net throughput [Mbps] for a fixed total size of theRIS but for a different number of RIS elements with ๐ = 100, ๐พ = 10,๐๐ = 5, and ๐ = ๐๐ป = ๐๐ . The unblocked probability of the direct linksis ๏ฟฝ๏ฟฝ = 0.2.
(๐) (๐)Fig. 12. A comparison of different channel capacity bounding techniques with ๐ = 20, ๐พ = 10, ๐ = 64, ๐๐ = 10, ๐๐ป = ๐๐ = _/4, ๏ฟฝ๏ฟฝ = 1.0, and twocoherence intervals: (๐) ๐๐ = 200 symbols; (๐) ๐๐ = 5000 symbols.
we have considered an RIS-assisted Cell-Free Massive MIMOsystem that operates according to the TDD mode. An efficientchannel estimation scheme has been introduced to overcomethe high overhead that may be associated with the estimationof the individual channels of the RIS elements. Based on theproposed channel estimation scheme, an optimal design for thephase shifts of the RIS that minimizes the channel estimationerror has been devised and has been used for system analysis.Based on the proposed channel estimation method, closed-form expressions of the ergodic net throughput for the uplinkand downlink data transmission phases have been proposed.Based on them, the performance of RIS-assisted Cell-FreeMassive MIMO has been analyzed as a function of the fadingspatial correlation and the blocking probability of the directAP-user links. The numerical results have shown that thepresence of an RIS is particularly useful if the AP-user linksare unreliable with high probability.
Possible generalizations of the results illustrated in thispaper include the optimization of the phase shifts of the RISthat maximize the uplink or downlink throughput, the analysisof the impact of the fading spatial correlation for non-compactand deep sub-wavelength RIS structures, and the analysis andoptimization of RIS-assisted systems in the presence of mutualcoupling.
APPENDIX
A. Useful Lemmas
This section reports three useful lemmas that are utilizedfor asymptotic analysis.
Lemma 3. [44, Lemma B.7] For an arbitrary matrix X โC๐ร๐ and a positive semi-definite matrix Y โ C๐ร๐ , it holdsthat |tr(XY) | โค โXโ2tr(Y). If X is also a positive semi-definitematrix, then tr(XY) โค โXโ2tr(Y).
Lemma 4. [45, Lemma 9] For a random variable x โ C๐distributed as CN(0, R) with R โ C๐ร๐ and a givendeterministic matrix M โ C๐ร๐ , it holds that E
{|x๐ปMx|2
}=๏ฟฝ๏ฟฝtr(RM)
๏ฟฝ๏ฟฝ2 + tr(RMRM๐ป
).
Lemma 5. For a random vector x โ C๐ distributed as x โผCN(0, R) with R โ C๐ร๐ and two deterministic matricesM,N โ C๐ร๐ , it holds that
E{x๐ปMxx๐ปNx} = tr(RMRN) + tr(RM)tr(RN). (65)
Proof. Consider x = R1/2x with x โผ CN(0, I๐ ). Let us furtherdenote M = R1/2MR1/2 and N = R1/2NR1/2, where [M]๐๐and [N]๐๐ are the (๐, ๐)โth elements of matrix M and N,
15
respectively. Then, the expectation on the left-hand side of(65) is
E{x๐ปMxx๐ปNx}
= E
{(๐โ๐=1
๐โ๐=1
๐ฅโ๐ [M]๐๐๐ฅ๐
) (๐โ๐=1
๐โ๐=1
๐ฅโ๐ [N]๐๐๐ฅ๐
)}=
๐โ๐=1
๐โ๐=1
๐โ๐โฒ=1
๐โ๐โฒ=1
[M]๐๐ [N]๐โฒ๐โฒE{๐ฅโ๐๐ฅ๐๐ฅโ๐โฒ๐ฅ๐โฒ}.
(66)
where ๐ฅ๐ is the ๐โth element of vector x. By noting thatE{|๐ฅ๐ |4} = 2 from Lemma 4 and E{|๐ฅ๐ |2 |๐ฅ๐ |2} = 1 if ๐ โ ๐,we obtain the following
E{๐ฅโ๐๐ฅ๐๐ฅโ๐โฒ๐ฅ๐โฒ} =
2, if ๐ = ๐ = ๐โฒ = ๐โฒ,
1, if (๐ = ๐) โ (๐โฒ = ๐โฒ),1, if (๐ = ๐โฒ) โ (๐โฒ = ๐),0, otherwise.
(67)
Consequently, (66) can be further simplified as
E{x๐ปMxx๐ปNx} =๐โ๐=1
๐โ๐=1
[M]๐๐ [N]๐๐+๐โ๐=1
๐โ๐=1
[M]๐๐ [N]๐๐,
(68)which, with the aid of some algebraic manipulations, coincideswith (65). ๏ฟฝ
B. Proof of Lemma 1We first compute the second moment of the aggregated
channel ๐ข๐๐ by capitalizing on the statistical independenceof the direct and indirect channels, as follows
E{|๐ข๐๐ |2} = E{|๐๐๐ |2} + E{|h๐ป๐ฮฆฮฆฮฆz๐ |2
}(๐)= ๐ฝ๐๐ + E
{tr(ฮฆฮฆฮฆ๐ปh๐h๐ป๐ฮฆฮฆฮฆz๐z๐ป๐
)}(๐)= ๐ฝ๐๐ + tr
(ฮฆฮฆฮฆ๐ปE
{h๐h๐ป๐
}ฮฆฮฆฮฆE
{z๐z๐ป๐
})= ๐ฝ๐๐ + tr(ฮฮฮ๐๐ ) = ๐ฟ๐๐ ,
(69)
where (๐) follows by applying the trace of product propertytr(XY) = tr(YX) for some given size-matched matrices Xand Y; and (๐) is obtained thanks to the independence ofthe cascaded channels h๐ and z๐ . The fourth moment of theaggregated channel can be written, from (6), as follows
E{|๐ข๐๐ |4} =
E
{๏ฟฝ๏ฟฝ๏ฟฝ|๐๐๐ |2 + ๐โ๐๐h๐ป๐ฮฆฮฆฮฆz๐ + ๐๐๐z๐ป๐ ฮฆฮฆฮฆ๐ปh๐ +
๏ฟฝ๏ฟฝh๐ป๐ฮฆฮฆฮฆz๐๏ฟฝ๏ฟฝ2๏ฟฝ๏ฟฝ๏ฟฝ2} (70)
By setting ๐ = |๐๐๐ |2, ๐ = ๐โ๐๐
h๐ป๐ฮฆฮฆฮฆz๐ , ๐ = ๐๐๐z๐ป๐ ฮฆฮฆฮฆ๐ปh๐,
and ๐ =๏ฟฝ๏ฟฝh๐ป๐ฮฆฮฆฮฆz๐
๏ฟฝ๏ฟฝ2, (70) can be equivalently written as follows
E{|๐ข๐๐ |4} = E{|๐ |2} + E{|๐ |2} + E{|๐ |2} + 2E{๐๐} + E{|๐ |2}.(71)
By applying Lemma 4 with ๐๐๐ โผ CN(0, ๐ฝ๐๐ ), the firstexpectation on the right-hand side of (71) is equal to
E{|๐ |2} = 2๐ฝ2๐๐ . (72)
By exploiting the independence between the direct and RIS-assisted links, the next three expectations on the right-handside of (71) are equal to
E{|๐ |2} = E{|๐ |2} = E{๐๐} = ๐ฝ๐๐ tr(ฮฮฮ๐๐ ). (73)
By introducing the normalized variable ๐ง =
h๐ป๐ฮฆฮฆฮฆz๐/ R1/2
๐ ฮฆฮฆฮฆz๐ with ๐ง โผ CN(0, 1), the last expectation
on the right-hand side of (71) is equal to
E{|๐ |2} = E{ R1/2
๐ ฮฆฮฆฮฆz๐ 4
|๐ง |4}
(๐)= E
{ R1/2๐ ฮฆฮฆฮฆz๐
4}E{|๐ง |4} = 2 (tr(ฮฮฮ๐๐ ))2 + 2tr
(ฮฮฮ2๐๐
),
(74)
where (๐) follows because ๐ง is independent of the remainingrandom variables; and (๐) is obtained by virtue of Lemma 4.Inserting (72)โ(74) into (71), the proof follows with the aidof some algebraic manipulations.
Also, by exploiting the second moment in (7), the expec-tation of the two independent aggregated channels can beformulated as shown in (11). The correlation between the twoaggregated channels ๐ข๐๐ and ๐ข๐โฒ๐ , ๐ โ ๐โฒ, is, by definition,as followsE{๐ข๐๐๐ขโ๐โฒ๐ } = E{๐๐๐๐
โ๐โฒ๐ } + E{๐๐๐ (h
๐ป๐โฒฮฆฮฆฮฆz๐ )โ}
+ E{h๐ป๐ฮฆฮฆฮฆz๐๐โ๐โฒ๐ } + E{h๐ป๐ฮฆฮฆฮฆz๐ (h๐ป๐โฒฮฆฮฆฮฆz๐ )โ}
(๐)= 0,
(75)
where (๐) follows because the propagation channels are inde-pendent.
The expectation in (13) can be written as follows
E{|๐ข๐๐๐ขโ๐โฒ๐ |2} = E{|๐ข๐๐ |2 |๐ข๐โฒ๐ |2}
= E{|๐๐๐ |2 |๐๐โฒ๐ |2} + E{|๐๐๐ |2z๐ป๐ ฮฆฮฆฮฆ๐ปh๐โฒh๐ป๐โฒฮฆฮฆฮฆz๐ }
+ E{|๐๐โฒ๐ |2z๐ป๐ ฮฆฮฆฮฆ๐ปh๐h๐ป๐ฮฆฮฆฮฆz๐ }
+ E{z๐ป๐ ฮฆฮฆฮฆ๐ปh๐h๐ป๐ฮฆฮฆฮฆz๐z๐ป๐ ฮฆฮฆฮฆ
๐ปh๐โฒh๐ป๐โฒฮฆฮฆฮฆz๐ }= ๐ฝ๐๐ ๐ฝ๐โฒ๐ + ๐ฝ๐๐ tr(ฮฮฮ๐โฒ๐ ) + ๐ฝ๐โฒ๐ tr(ฮฮฮ๐๐ )+ E{z๐ป๐ ฮฆฮฆฮฆ
๐ปR๐ฮฆฮฆฮฆz๐z๐ป๐ ฮฆฮฆฮฆ๐ปR๐โฒฮฆฮฆฮฆz๐ }
(๐)=
(๐ฝ๐๐ + tr(ฮฮฮ๐๐ )
) (๐ฝ๐โฒ๐ + tr(ฮฮฮ๐โฒ๐ )
)+ tr(ฮฮฮ๐๐ฮฮฮ๐โฒ๐ ),
(76)
where (๐) is obtained by utilizing Lemma 5. Similar steps canbe applied to the two aggregated channels ๐ข๐๐ and ๐ข๐๐โฒ with๐ โ ๐ โฒ.
The expectation in (12) can be written as follows
E{๐ขโ๐๐๐ข๐๐โฒ๐ขโ๐โฒ๐โฒ๐ข๐โฒ๐ }
(๐)= E{z๐ป๐ ฮฆฮฆฮฆ
๐ปh๐h๐ป๐ฮฆฮฆฮฆz๐โฒz๐ป๐โฒฮฆฮฆฮฆ๐ปh๐โฒh๐ป๐โฒฮฆฮฆฮฆz๐ }
= tr(ฮฆฮฆฮฆ๐ปR๐ฮฆฮฆฮฆR๐โฒฮฆฮฆฮฆ๐ปR๐โฒฮฆฮฆฮฆR๐ ),(77)
where (๐) follows from the independence of the direct links.
C. Proof of Corollary 1By utilizing the identities E{(๐ โ E{๐})(๐ โ E{๐ })} =
E{๐๐ }โE{๐}E{๐ } and E{|๐โE{๐}|2} = E{|๐ |2}โ |E{๐}|2,the expectation in (22) can be formulated as follows
E{๐๐๐๐โ๐โฒ๐ } =โ๐ผ๐๐๐ผ๐โฒ๐ E{๏ฟฝ๏ฟฝโ๐๐๐ข๐๐ ๏ฟฝ๏ฟฝ๐โฒ๐๐ข
โ๐โฒ๐ }๏ธธ ๏ธท๏ธท ๏ธธ
=๐๐๐โฒ๐
โ โ๐ผ๐๐๐ผ๐โฒ๐๐พ๐๐๐พ๐โฒ๐ . (78)
16
By using the analytical expressions of the projected trainingsignal in (5) and the channel estimate in (15), ๐๐๐โฒ๐ defined in(78) can be formulated in a closed-form expression, as follows
๐๐๐โฒ๐(๐)= ๐๐๐๐๐โฒ๐E
{(โ๐๐๐
โ๐โฒโP๐
๐ขโ๐๐โฒ + ๐ค๐๐๐
)๐ข๐๐ร(
โ๐๐๐
โ๐โฒโP๐
๐ข๐โฒ๐โฒ + ๐ค๐๐โฒ๐
)๐ขโ๐โฒ๐
}(๐)= ๐๐๐๐๐โฒ๐ ๐๐๐E
{|๐ข๐๐ |2 |๐ข๐โฒ๐ |2
}+ ๐๐๐๐๐โฒ๐ ๐๐๐รโ
๐โฒโP๐\{๐ }E{๐ขโ๐๐โฒ๐ข๐๐๐ข
โ๐โฒ๐๐ข๐โฒ๐โฒ}
= ๐๐๐๐๐โฒ๐ ๐๐๐๐ฟ๐๐๐ฟ๐โฒ๐ + ๐๐๐๐๐โฒ๐ ๐๐๐
โ๐โฒโP๐
tr(ฮฮฮ๐๐ฮฮฮ๐โฒ๐โฒ),
(79)
where (๐) is obtained by retaining only the terms whoseexpectation is not zero based on (10) and (๐) follows byutilizing (13). From (17), finally, we obtain
๐๐๐โฒ๐ = ๐พ๐๐๐พ๐โฒ๐ + ๐๐๐๐๐โฒ๐ ๐๐๐
โ๐โฒโP๐
tr(ฮฮฮ๐๐ฮฮฮ๐โฒ๐โฒ). (80)
The proof follows by inserting (78) in (80).
D. Proof of Corollary 2We introduce the shorthand notation ๐๐๐ = ๐๐๐๐ผ๐๏ฟฝ๏ฟฝ๐๐2
๐ป๐2๐
and ๐๐๐ = ๐๐๐๐2๐ป๐2๐๐ผ๐
โ๐โฒโP๐ ๏ฟฝ๏ฟฝ๐โฒ . When the direct links
are weak enough to be negligible, the NMSE of the channelestimate of the user ๐ at the AP ๐ can be reformulated asfollows
NMSE๐๐ = 1 โ๐๐๐ tr
(ฮฆฮฆฮฆ๐ปRฮฆฮฆฮฆR
)1 + ๐๐๐ tr
(ฮฆฮฆฮฆ๐ปRฮฆฮฆฮฆR
) . (81)
Let us denote by ๐(tr(ฮฆฮฆฮฆ๐ปRฮฆฮฆฮฆR
) )=
โ๐๐=1
โ๐พ๐=1 NMSE๐๐
the objective function of the problem in (24), the first-orderderivative of NMSE๐๐ with respect to tr
(ฮฆฮฆฮฆ๐ปRฮฆฮฆฮฆR
)is
๐๐(tr(ฮฆฮฆฮฆ๐ปRฮฆฮฆฮฆR
) )๐tr
(ฮฆฮฆฮฆ๐ปRฮฆฮฆฮฆR
) = โ๐โ๐=1
๐พโ๐=1
๐๐๐(1 + ๐๐๐ tr
(ฮฆฮฆฮฆ๐ปRฮฆฮฆฮฆR
) )2 < 0,
(82)which implies that the objective function is a monotonicallydecreasing function of tr
(ฮฆฮฆฮฆ๐ปRฮฆฮฆฮฆR
)since ๐๐๐ โฅ 0,โ๐, ๐ .
Moreover, ฮฆฮฆฮฆ๐ปRฮฆฮฆฮฆR is similar to R1/2ฮฆฮฆฮฆ๐ปRฮฆฮฆฮฆR1/2, which isa positive semidefinite matrix. Thus, we obtain
tr(ฮฆฮฆฮฆ๐ปRฮฆฮฆฮฆR
)=
๏ฟฝ๏ฟฝtr(ฮฆฮฆฮฆ๐ปRฮฆฮฆฮฆR) ๏ฟฝ๏ฟฝ = ๏ฟฝ๏ฟฝ๏ฟฝ๏ฟฝ๏ฟฝ ๐โ
๐=1
๐โ๐โฒ=1
(๐๐๐โฒ)2๐ ๐ (\๐โ\๐โฒ )
๏ฟฝ๏ฟฝ๏ฟฝ๏ฟฝ๏ฟฝ .(83)
Let us introduce the two vectors a, b โ C๐ 2defined as follows
a = [๐11, . . . , ๐๐๐โฒ, . . . , ๐๐๐ ]๐ , (84)
b = [๐11๐ ๐ (\1โ\1) , . . . , ๐๐๐โฒ๐ ๐ (\๐โ\๐โฒ ) , . . . , ๐๐๐ ๐ ๐ (\๐โ\๐ ) ]๐ .
(85)
With the aid of Cauchy-Schwarzโs inequality, we obtain thefollowing upper bound for (83)
tr(ฮฆฮฆฮฆ๐ปRฮฆฮฆฮฆR
)= |a๐ปb| โค โaโโbโ =
๐โ๐=1
๐โ๐โฒ=1
(๐๐๐โฒ)2, (86)
which holds with equality if and only if the two vectors a andb in (84) and (85) are parallel. This implies \๐ = \๐โฒ ,โ๐, ๐โฒ.By combining (82) and (86), the proof is concluded.
E. Proof of Theorem 1To obtain the closed-form expression of the uplink SINR in
(37), we first compute |DS๐ |2 by using the definition of ๐ข๐๐in (6) as
|DS๐ข๐ |2(๐)= ๐๐ข[๐
๏ฟฝ๏ฟฝ๏ฟฝ๏ฟฝ๏ฟฝE{๐โ๐=1
๏ฟฝ๏ฟฝโ๐๐ (๏ฟฝ๏ฟฝ๐๐ + ๐๐๐ )}๏ฟฝ๏ฟฝ๏ฟฝ๏ฟฝ๏ฟฝ2
(๐)= ๐๐ข[๐
๏ฟฝ๏ฟฝ๏ฟฝ๏ฟฝ๏ฟฝ ๐โ๐=1E
{|๏ฟฝ๏ฟฝ๐๐ |2
}๏ฟฝ๏ฟฝ๏ฟฝ๏ฟฝ๏ฟฝ2 = ๐๐ข[๐
(๐โ๐=1
๐พ๐๐
)2
,
(87)
where (๐) is obtained by expressing the original channel ๐ข๐๐into the summation of its channel estimate and its estimationerror as stated in Lemma 2; and (๐) follows because the chan-nel estimate and the channel estimation error are uncorrelated.Since the aggregated channels sharing the same AP index arecorrelated, the first expectation in the denominator of (37) isequal to
E{|BU๐ข๐ |2} = ๐๐ข[๐E๏ฟฝ๏ฟฝ๏ฟฝ๏ฟฝ๏ฟฝ ๐โ๐=1
๐๐ข๐๐
๏ฟฝ๏ฟฝ๏ฟฝ๏ฟฝ๏ฟฝ2= ๐๐ข[๐
๐โ๐=1
๐โ๐โฒ=1,๐โฒโ ๐
E{๐๐ข๐๐๐โ๐ข๐โฒ๐ }๏ธธ ๏ธท๏ธท ๏ธธ=๏ฟฝ๏ฟฝ๐ข0
+ ๐๐ข[๐๐โ๐=1E{|๐๐ข๐๐ |2}๏ธธ ๏ธท๏ธท ๏ธธ=๏ฟฝ๏ฟฝ๐ข1
,
(88)
where ๐๐ข๐๐ = ๏ฟฝ๏ฟฝโ๐๐๐ข๐๐ โ E{๏ฟฝ๏ฟฝโ
๐๐๐ข๐๐ } with ๐๐๐ = 1. The
closed-form expression of ๐๐ข0 defined in (88) is as follows
๐๐ข0 = ๐๐๐๐๐ข[๐
โ๐โฒโP๐
๐โ๐=1
๐โ๐โฒ=1,๐โฒโ ๐
๐๐๐๐๐โฒ๐ tr(ฮฮฮ๐๐ฮฮฮ๐โฒ๐โฒ),
(89)thanks to (22) in Corollary 1. The expectation of ๐๐ข1 definedin (88) can be rewritten as follows
๐๐ข1 = ๐๐ข[๐
๐โ๐=1E
{๏ฟฝ๏ฟฝ๏ฟฝ๏ฟฝโ๐๐๐ข๐๐ โ E {๏ฟฝ๏ฟฝโ๐๐๐ข๐๐
}๏ฟฝ๏ฟฝ2}(๐)= ๐๐ข[๐
๐โ๐=1E
{๏ฟฝ๏ฟฝ๏ฟฝ๏ฟฝโ๐๐๐ข๐๐ ๏ฟฝ๏ฟฝ2} โ ๐๐ข[๐ ๐โ๐=1
๏ฟฝ๏ฟฝE {๏ฟฝ๏ฟฝโ๐๐๐ข๐๐
}๏ฟฝ๏ฟฝ2(๐)= ๐๐ข[๐
๐โ๐=1E
{๏ฟฝ๏ฟฝ๏ฟฝ๏ฟฝโ๐๐๐ข๐๐ ๏ฟฝ๏ฟฝ2} โ ๐๐ข[๐ ๐โ๐=1
๐พ2๐๐ ,
(90)
where (๐) is obtained by using the identity E{|๐ โE{๐}|2} =E{|๐ |2} โ |E{๐}|2; and (๐) is obtained from ๐ขโ
๐๐= ๏ฟฝ๏ฟฝโ
๐๐+
๐โ๐๐
, by taking into account that the channel estimate and the
17
channel estimation error are uncorrelated random variables asstated in Lemma 2. By replacing ๏ฟฝ๏ฟฝโ
๐๐= ๐๐๐ ๐ฆ
โ๐๐๐
in the firstexpectation of (88), we obtain
E{๏ฟฝ๏ฟฝ๏ฟฝ๏ฟฝโ๐๐๐ข๐๐ ๏ฟฝ๏ฟฝ2} = ๐2
๐๐E
{๏ฟฝ๏ฟฝ๏ฟฝ๐ฆโ๐๐๐๐ข๐๐ ๏ฟฝ๏ฟฝ๏ฟฝ2}= ๐2
๐๐E
๏ฟฝ๏ฟฝ๏ฟฝ๏ฟฝ๏ฟฝ( โ๐โฒโP๐
โ๐๐๐๐ข
โ๐๐โฒ + ๐ค
โ๐๐๐
)๐ข๐๐
๏ฟฝ๏ฟฝ๏ฟฝ๏ฟฝ๏ฟฝ2= ๐2
๐๐ ๐๐๐E{|๐ข๐๐ |4}๏ธธ ๏ธท๏ธท ๏ธธ
=๐11
+ ๐2๐๐ ๐๐๐
โ๐โฒโP๐\{๐ }
E{|๐ขโ๐๐โฒ๐ข๐๐ |2}๏ธธ ๏ธท๏ธท ๏ธธ
=๐12
+ ๐2๐๐E
{|๐คโ๐๐๐๐ข๐๐ |
2}๏ธธ ๏ธท๏ธท ๏ธธ=๐13
.
(91)
Let us analyze the three terms, ๐11, ๐12, and ๐13, in the lastequality of (91). The first term can be computed by exploitingthe forth moment given in (8), as follows
๐11 = 2๐2๐๐ ๐๐๐๐ฟ
2๐๐ + 2๐2
๐๐ ๐๐๐tr(ฮฮฮ2๐๐ )
(๐)= ๐พ2
๐๐ + ๐2๐๐ ๐๐๐๐ฟ
2๐๐ + 2๐2
๐๐ ๐๐๐tr(ฮฮฮ2๐๐ ),
(92)
where (๐) is obtained by using the variance of the channelestimate in (17), with ๐ฟ๐๐ and b๐๐ that are defined in thestatement of the theorem. The second term can be computedby exploiting the uncorrelation of the two cascaded channels,as follows
๐12 = ๐2๐๐ ๐๐๐
โ๐โฒโP๐\{๐ }
E{|๐ข๐๐โฒ |2 |๐ข๐๐ |2}
= ๐2๐๐ ๐๐๐
โ๐โฒโP๐\{๐ }
๐ฟ๐๐โฒ๐ฟ๐๐ + ๐2๐๐ ๐๐๐
โ๐โฒโP๐\{๐ }
tr(ฮฮฮ๐๐โฒฮฮฮ๐๐ ).
(93)
The last term can be computed by exploiting the independenceof the channel and noise, as follows
๐13 = ๐2๐๐E{|๐ค๐๐๐ |
2}E{|๐ข๐๐ |2}= ๐2
๐๐๐ฟ๐๐ .(94)
By inserting (92)โ(94) into (91) and with the aid of somealgebraic steps, we obtain
E{|๏ฟฝ๏ฟฝโ๐๐๐ข๐๐ |2} = ๐2
๐๐ ๐๐๐tr(ฮฮฮ2๐๐ ) + ๐พ
2๐๐
+ ๐๐๐โ๐๐๐๐ฟ
2๐๐ + ๐
2๐๐ ๐๐๐
โ๐โฒโP๐
tr(ฮฮฮ๐๐โฒฮฮฮ๐๐ )
= ๐2๐๐ ๐๐๐tr(ฮฮฮ2
๐๐ ) + ๐พ2๐๐ + ๐พ๐๐๐ฟ๐๐
+ ๐2๐๐ ๐๐๐
โ๐โฒโP๐
tr(ฮฮฮ๐๐โฒฮฮฮ๐๐ ),
(95)
where the final identity is obtained by using the relationshipbetween ๐พ๐๐ and ๐๐๐ in (17). Combining (88) and (95), thefirst term in the denominator of (37) simplifies to
E{|BU๐ข๐ |2} = ๐๐ข[๐๐โ๐=1
๐พ๐๐๐ฟ๐๐ + ๐๐๐๐๐ข[๐โ๐โฒโฒโP๐
๐โ๐=1
๐โ๐โฒ=1
๐๐๐
ร ๐๐โฒ๐ tr(ฮฮฮ๐๐ฮฮฮ๐โฒ๐โฒโฒ) + ๐๐๐๐๐ข[๐๐โ๐=1
๐2๐๐ tr(ฮฮฮ
2๐๐ ).
(96)
The second term in the denominator of (37) can be split intothe two terms based on the pilot reuse pattern defined in P๐ ,as follows
๐พโ๐โฒ=1,๐โฒโ ๐
E{|UI๐ข๐โฒ๐ |2} =โ๐โฒโP๐
E{|UI๐ข๐โฒ๐ |2}
+โ
๐โฒโP๐\{๐ }E{|UI๐ข๐โฒ๐ |2}. (97)
The first term on the right-hand side of (97) is the non-coherentinterference, which is equal toโ
๐โฒโP๐E{|UI๐ข๐โฒ๐ |2} = ๐๐ข
โ๐โฒโP๐
[๐โฒE
๏ฟฝ๏ฟฝ๏ฟฝ๏ฟฝ๏ฟฝ ๐โ๐=1
๏ฟฝ๏ฟฝโ๐๐๐ข๐๐โฒ
๏ฟฝ๏ฟฝ๏ฟฝ๏ฟฝ๏ฟฝ2= ๐๐ข
โ๐โฒโP๐
๐โ๐=1
๐โ๐โฒ=1
[๐โฒ E{๏ฟฝ๏ฟฝโ๐๐๐ข๐๐โฒ ๏ฟฝ๏ฟฝ๐โฒ๐๐ข
โ๐โฒ๐โฒ
}๏ธธ ๏ธท๏ธท ๏ธธ=๐๐๐โฒ๐โฒ๐
.
(98)
We compute each expectation ๐๐๐โฒ๐โฒ๐ by utilizing the channelestimate in (15) with the aid of some algebraic manipulations,as follows
๐๐๐โฒ๐โฒ๐ =๐๐๐๐๐๐๐๐โฒ๐
โ๐โฒโฒโP๐
E{๐ขโ๐๐โฒโฒ๐ข๐๐โฒ๐ข๐โฒ๐โฒโฒ๐ขโ๐โฒ๐โฒ}
+ ๐๐๐๐๐โฒ๐E{๐คโ๐๐๐๐ข๐๐โฒ๐ค๐๐โฒ๐๐ข
โ๐โฒ๐โฒ}
=
๐๐๐๐๐๐๐๐โฒ๐
โ๐โฒโฒโP๐
๐๐๐โฒ๐โฒ๐โฒโฒ , if ๐ โ ๐โฒ,
๐๐๐โฒ๐๐ , if ๐ = ๐โฒ.
(99)
where the following definitions hold
๐๐๐โฒ๐โฒ๐โฒโฒ = E{๐ขโ๐๐โฒโฒ๐ข๐๐โฒ๐ข๐โฒ๐โฒโฒ๐ขโ๐โฒ๐โฒ}, (100)
๐๐๐โฒ๐๐ = ๐๐๐๐2๐๐
โ๐โฒโฒโP๐
E{|๐ขโ๐๐โฒโฒ๐ข๐๐โฒ |2} + ๐2
๐๐E{|๐ข๐๐โฒ |2}.
(101)
The closed-form expression of ๐๐๐โฒ๐โฒ๐โฒโฒ is obtained by uti-lizing the uncorrelated property of the quadruple aggregatedchannels in (12), as follows
๐๐๐โฒ๐โฒ๐โฒโฒ = tr(ฮฮฮ๐๐โฒฮฮฮ๐โฒ๐โฒโฒ). (102)
In addition, the closed-form expression of ๐๐๐โฒ๐๐ can becomputed by utilizing the results in Lemma 1, as follows
๐๐๐โฒ๐๐ = ๐2๐๐ ๐๐๐
โ๐โฒโฒโP๐
๐ฟ๐๐โฒโฒ๐ฟ๐๐โฒ + ๐2๐๐ ๐๐๐รโ
๐โฒโฒโP๐tr(ฮฮฮ๐๐โฒโฒฮฮฮ๐๐โฒ) + ๐2
๐๐๐ฟ๐๐โฒ
= ๐พ๐๐๐ฟ๐๐โฒ + ๐2๐๐ ๐๐๐
โ๐โฒโฒโP๐
tr(ฮฮฮ๐๐โฒฮฮฮ๐๐โฒโฒ).
(103)
Inserting (102) and (103) into (99), and with the aid of somealgebraic manipulations, (98) can be equivalently formulatedas followsโ
๐โฒโP๐E{|UI๐ข๐โฒ๐ |2} = ๐๐ข
โ๐โฒโP๐
๐โ๐=1
[๐โฒ๐พ๐๐๐ฟ๐๐โฒ + ๐๐๐๐๐ขร
โ๐โฒโP๐
โ๐โฒโฒโP๐
๐โ๐=1
๐โ๐โฒ=1
[๐โฒ๐๐๐๐๐โฒ๐ tr(ฮฮฮ๐๐โฒฮฮฮ๐โฒ๐โฒโฒ). (104)
18
The second term on the right-hand side of (97) is the coherentinterference. By using (5) and (15), it simplifies as follows
E{|UI๐ข๐โฒ๐ |2} = E๏ฟฝ๏ฟฝ๏ฟฝ๏ฟฝ๏ฟฝ ๐โ๐=1
๐๐๐ ๐ฆโ๐๐๐๐ข๐๐โฒ
๏ฟฝ๏ฟฝ๏ฟฝ๏ฟฝ๏ฟฝ2= E
๏ฟฝ๏ฟฝ๏ฟฝ๏ฟฝ๏ฟฝ ๐โ๐=1
๐๐๐
( โ๐โฒโฒโP๐
โ๐๐๐๐ข
โ๐๐โฒโฒ + ๐ค
โ๐๐๐
)๐ข๐๐โฒ
๏ฟฝ๏ฟฝ๏ฟฝ๏ฟฝ๏ฟฝ2= ๐๐ข[๐โฒE
๏ฟฝ๏ฟฝ๏ฟฝ๏ฟฝ๏ฟฝ ๐โ๐=1
๐๐๐๐คโ๐๐๐๐ข๐๐โฒ
๏ฟฝ๏ฟฝ๏ฟฝ๏ฟฝ๏ฟฝ2+ ๐๐ข[๐โฒ ๐๐๐ E
๏ฟฝ๏ฟฝ๏ฟฝ๏ฟฝ๏ฟฝ๏ฟฝ ๐โ๐=1
๐๐๐ยฉยญยซ
โ๐โฒโฒโP๐\{๐โฒ }
๐ขโ๐๐โฒโฒยชยฎยฌ ๐ข๐๐โฒ
๏ฟฝ๏ฟฝ๏ฟฝ๏ฟฝ๏ฟฝ๏ฟฝ2๏ธธ ๏ธท๏ธท ๏ธธ
=๐๐ข21
+ ๐๐ข[๐โฒ ๐๐๐ E๏ฟฝ๏ฟฝ๏ฟฝ๏ฟฝ๏ฟฝ ๐โ๐=1
๐๐๐ |๐ข๐๐โฒ |2๏ฟฝ๏ฟฝ๏ฟฝ๏ฟฝ๏ฟฝ2๏ธธ ๏ธท๏ธท ๏ธธ
=๐๐ข22
= ๐๐ข[๐โฒ
๐โ๐=1
๐2๐๐๐ฟ๐๐โฒ + ๐๐ข[๐โฒ ๐๐๐ (๐๐ข21 + ๐๐ข22),
(105)
which is obtained by using the identity E{|๐+๐ |2} = E{|๐ |2}+E{|๐ |2} that holds true for zero-mean and uncorrelated randomvariables. The expectation ๐๐ข21 can be simplified as follows
๐๐ข21 =โ
๐โฒโฒโP๐\{๐โฒ }
๐โ๐=1
๐2๐๐E{|๐ข๐๐โฒโฒ |
2 |๐ข๐๐โฒ |2}+
โ๐โฒโฒโP๐\{๐โฒ }
๐โ๐=1
๐โ๐โฒ=1,๐โฒโ ๐
๐๐๐๐๐โฒ๐E{๐ขโ๐๐โฒโฒ๐ข๐๐โฒ๐ข๐โฒ๐โฒโฒ๐ขโ๐โฒ๐โฒ}
=โ
๐โฒโฒโP๐\{๐โฒ }
๐โ๐=1
๐2๐๐๐ฟ๐๐โฒโฒ๐ฟ๐๐โฒ+
โ๐โฒโฒโP๐\{๐โฒ }
๐โ๐=1
๐โ๐โฒ=1
๐๐๐๐๐โฒ๐ tr(ฮฮฮ๐๐โฒฮฮฮ๐โฒ๐โฒโฒ).
(106)
Similarly, the expectation ๐๐ข22 in (105) can be simplified asfollows
๐๐ข22 =
๐โ๐=1
๐โ๐โฒ=1
๐๐๐๐๐โฒ๐E{|๐ข๐๐โฒ |2 |๐ข๐โฒ๐โฒ |2}
=
๐โ๐=1
๐2๐๐E{|๐ข๐๐โฒ |
4} +๐โ๐=1
๐โ๐โฒ=1,๐โฒโ ๐
๐๐๐๐๐โฒ๐E{|๐ข๐๐โฒ |2 |๐ข๐โฒ๐โฒ |2}
= 2๐โ๐=1
๐2๐๐๐ฟ
2๐๐โฒ + 2
๐โ๐=1
๐2๐๐ tr(ฮฮฮ
2๐๐โฒ) +
๐โ๐=1
๐โ๐โฒ=1,๐โฒโ ๐
๐๐๐ร
๐๐โฒ๐๐ฟ๐๐โฒ๐ฟ๐โฒ๐โฒ +๐โ๐=1
๐โ๐โฒ=1,๐โฒโ ๐
๐๐๐๐๐โฒ๐ tr(ฮฮฮ๐๐โฒฮฮฮ๐โฒ๐โฒ)
=
๐โ๐=1
๐2๐๐๐ฟ
2๐๐โฒ +
๐โ๐=1
๐2๐๐ tr(ฮฮฮ
2๐๐โฒ) +
(๐โ๐=1
๐๐๐๐ฟ๐๐โฒ
)2
+๐โ๐=1
๐โ๐โฒ=1
๐๐๐๐๐โฒ๐ tr(ฮฮฮ๐๐โฒฮฮฮ๐โฒ๐โฒ).
(107)
By inserting (106) and (107) into (105), and with the aid ofsome algebraic manipulations, we obtain
๐๐ข[๐โฒ
๐โ๐=1
๐2๐๐๐ฟ๐๐โฒ + ๐๐ข[๐โฒ ๐๐๐
โ๐โฒโฒโP๐
๐โ๐=1
๐2๐๐๐ฟ๐๐โฒโฒ๐ฟ๐๐โฒ
= ๐๐ข[๐โฒ
๐โ๐=1
๐2๐๐๐ฟ๐๐โฒ
(1 + ๐๐๐
โ๐โฒโฒโP๐
๐ฟ๐โฒโฒ๐
)(๐)= ๐๐ข[๐โฒ
๐โ๐=1
๐๐๐โ๐๐๐๐ฟ๐๐๐ฟ๐๐โฒ
(๐)= ๐๐ข[๐โฒ
๐โ๐=1
๐พ๐๐๐ฟ๐๐โฒ ,
(108)
where (๐) is obtained by using (16) and (๐) by using (17).Therefore, the total mutual interference between the user ๐ โฒ
and the user ๐ who share the same pilot sequence can bewritten as followsโ๐โฒโP๐\{๐ }
E{|UI๐ข๐โฒ๐ |2} = ๐๐ขโ
๐โฒโP๐\{๐ }
๐โ๐=1
[๐โฒ๐พ๐๐๐ฟ๐๐โฒ
+ ๐๐๐๐๐ขโ
๐โฒโP๐\{๐ }
๐โ๐=1
[๐โฒ๐2๐๐ tr(ฮฮฮ
2๐๐โฒ)
+ ๐๐๐๐๐ขโ
๐โฒโP๐\{๐ }
โ๐โฒโฒโP๐
๐โ๐=1
๐โ๐โฒ=1
[๐โฒ๐๐๐๐๐โฒ๐ร
tr(ฮฮฮ๐๐โฒฮฮฮ๐โฒ๐โฒโฒ) + ๐๐๐๐๐ขโ
๐โฒโP๐\{๐ }[๐โฒ
(๐โ๐=1
๐๐๐๐ฟ๐๐โฒ
)2
.
(109)
Let us denote I๐ข๐ =โ๐พ๐โฒ=1,๐โฒโ ๐ E{|UI๐ข๐โฒ๐ |2}. By combing
(104) and (109), the mutual interference at the user ๐ canbe formulated in a closed-form expression as follows
I๐ข๐ = ๐๐ข๐พโ
๐โฒ=1,๐โฒโ ๐
๐โ๐=1
[๐โฒ๐พ๐๐๐ฟ๐๐โฒ + ๐๐๐๐๐ขร
๐พโ๐โฒ=1,๐โฒโ ๐
โ๐โฒโฒโP๐
๐โ๐=1
๐โ๐โฒ=1
[๐โฒ๐๐๐๐๐โฒ๐ tr(ฮฮฮ๐๐โฒฮฮฮ๐โฒ๐โฒโฒ)+
+ ๐๐๐๐๐ขโ
๐โฒโP๐\{๐ }
๐โ๐=1
[๐โฒ๐2๐๐ tr(ฮฮฮ
2๐๐โฒ) + ๐๐๐๐๐ขร
โ๐โฒโP๐\{๐ }
[๐โฒ
(๐โ๐=1
๐๐๐๐ฟ๐๐โฒ
)2
.
(110)
Finally, the expectation of the additive noise after MR pro-cessing can be written as follows
E{|NO๐ข๐ |2} =๐โ๐=1E{|๏ฟฝ๏ฟฝโ๐๐๐ค๐ข๐ |
2}
=
๐โ๐=1E{|๏ฟฝ๏ฟฝ๐๐ |2}E{|๐ค๐ข๐ |2} =
๐โ๐=1
๐พ๐๐ ,
(111)
19
thanks to the independence between the channel estimate andthe noise. The proof follows by inserting (87), (91), (109), and(111) into the SINR in (37) with the aid of some algebraicmanipulations.
F. Proof of Theorem 2
Consider the downlink SINR in (56). Thanks to the uncorre-lation between the channel estimate and the channel estimationerror, the numerator simplifies to
|DS๐๐ |2 = ๐๐
๏ฟฝ๏ฟฝ๏ฟฝ๏ฟฝ๏ฟฝ ๐โ๐=1
โ[๐๐E{|๏ฟฝ๏ฟฝ๐๐ |2}
๏ฟฝ๏ฟฝ๏ฟฝ๏ฟฝ๏ฟฝ2 = ๐๐
(๐โ๐=1
โ[๐๐๐พ๐๐
)2
.
(112)The beamforming uncertainty term in the denominator of (56)can be simplified by using (15), as follows
E{|BU๐๐ |2} = ๐๐E๏ฟฝ๏ฟฝ๏ฟฝ๏ฟฝ๏ฟฝ ๐โ๐=1
๐๐๐๐
๏ฟฝ๏ฟฝ๏ฟฝ๏ฟฝ๏ฟฝ2= ๐๐
๐โ๐=1
๐โ๐โฒ=1,๐โฒโ ๐
E{๐๐๐๐๐โ๐๐โฒ๐ }๏ธธ ๏ธท๏ธท ๏ธธ=๏ฟฝ๏ฟฝ๐0
+ ๐๐๐โ๐=1E{|๐๐๐๐ |2}๏ธธ ๏ธท๏ธท ๏ธธ=๏ฟฝ๏ฟฝ๐1
,
(113)
where ๐๐๐๐ =โ[๐๐๐ข๐๐๐ข
โ๐๐
โ โ[๐๐E{๐ข๐๐๐ขโ๐๐ } with ๐๐๐ =
[๐๐ (see Corollary 1 for ๐๐๐ ). The closed-form expressionof ๐๐0 in (113) can be formulated as follows
๐๐0 =
๐๐๐๐๐
โ๐โฒโฒโP๐
๐โ๐=1
๐โ๐โฒ=1,๐โฒโ ๐
โ[๐๐[๐โฒ๐๐๐๐๐๐โฒ๐ tr(ฮฮฮ๐๐ฮฮฮ๐โฒ๐โฒโฒ),
(114)
by utilizing (22) in Corollary 1. The expectation ๐๐1 in (114)can be rewritten as follows
๐๐1 = ๐๐
๐โ๐=1
[๐๐E{|๐ข๐๐ ๏ฟฝ๏ฟฝโ๐๐ |
2}๏ธธ ๏ธท๏ธท ๏ธธ
=๐๐1
โ๐๐๐โ๐=1
[๐๐๏ฟฝ๏ฟฝE{|๏ฟฝ๏ฟฝ๐๐ |2}๏ฟฝ๏ฟฝ2
= ๐๐1 โ ๐๐๐โ๐=1
[๐๐๐พ2๐๐ ,
(115)
where we have used the identities E{|๐โE{๐}|2} = E{|๐ |2}โ|E{๐}|2 and E{|๐ + ๐ |2} = E{|๐ |2} + E{๐ |2} for zero-meanuncorrelated random variables. By using the identities in (91)and (95), ๐๐1 in (115) can be formulated as follows
๐๐1 = ๐๐๐๐๐
๐โ๐=1
[๐๐๐2๐๐ tr(ฮฮฮ
2๐๐ ) + ๐๐
๐โ๐=1
[๐๐๐พ2๐๐ + ๐๐ร
๐โ๐=1
[๐๐๐พ๐๐๐ฟ๐๐ + ๐๐๐๐๐โ๐โฒโฒโP๐
๐โ๐=1
[๐๐๐2๐๐ tr(ฮฮฮ๐๐ฮฮฮ๐๐โฒโฒ),
(116)
by using (5). Inserting (114) and (116) into (113), we obtain
E{|BU๐๐ |2} = ๐๐๐โ๐=1
[๐๐๐พ๐๐๐ฟ๐๐ + ๐๐๐๐๐รโ๐โฒโฒโP๐
๐โ๐=1
๐โ๐โฒ=1
โ[๐๐[๐โฒ๐๐๐๐๐๐โฒ๐ tr(ฮฮฮ๐๐ฮฮฮ๐โฒ๐โฒโฒ)
+ ๐๐๐๐๐๐โ๐=1
[๐๐๐2๐๐ tr(ฮฮฮ
2๐๐ ).
(117)
The mutual interference term in the denominator of (56) canbe rewritten as follows
๐พโ๐โฒ=1,๐โฒโ ๐
E{|UI๐๐โฒ๐ |2} =โ
๐โฒโP๐\{๐ }๐๐E{|UI๐๐โฒ๐ |2}๏ธธ ๏ธท๏ธท ๏ธธ
=๐๐
+
๐๐
โ๐โฒโP๐
๐โ๐=1
[๐๐โฒ๐พ๐๐โฒ๐ฟ๐๐ + ๐๐๐๐๐ร
โ๐โฒโP๐
โ๐โฒโฒโP๐โฒ
๐โ๐=1
๐โ๐โฒ=1
โ[๐๐โฒ[๐โฒ๐โฒ๐๐๐โฒ๐๐โฒ๐โฒ tr(ฮฮฮ๐๐ฮฮฮ๐โฒ๐โฒโฒ),
(118)
where the first term on the right-hand side of (118) is obtainedby exploiting the orthogonality of the pilot sequences. Inthe second summation of (118), ๐๐ = E{|UI๐๐๐โฒ |2} can berewritten as follows
๐๐ = ๐๐E
๏ฟฝ๏ฟฝ๏ฟฝ๏ฟฝ๏ฟฝ ๐โ๐=1
โ[๐๐โฒ๐ข๐๐ ๏ฟฝ๏ฟฝ
โ๐๐โฒ
๏ฟฝ๏ฟฝ๏ฟฝ๏ฟฝ๏ฟฝ2= ๐๐E
๏ฟฝ๏ฟฝ๏ฟฝ๏ฟฝ๏ฟฝ ๐โ๐=1
โ[๐๐โฒ๐๐๐โฒ๐ข๐๐ ๐ฆ
โ๐๐๐โฒ
๏ฟฝ๏ฟฝ๏ฟฝ๏ฟฝ๏ฟฝ2= ๐๐E
๏ฟฝ๏ฟฝ๏ฟฝ๏ฟฝ๏ฟฝ๏ฟฝ ๐โ๐=1
โ[๐๐โฒ๐๐๐โฒ๐ข๐๐
ยฉยญยซโ๐โฒโฒโP๐โฒ
โ๐๐๐๐ข
โ๐๐โฒโฒ + ๐ค
โ๐๐๐โฒ
ยชยฎยฌ๏ฟฝ๏ฟฝ๏ฟฝ๏ฟฝ๏ฟฝ๏ฟฝ2
= ๐๐
๐โ๐=1
[๐๐โฒ๐2๐๐โฒE{|๐ข๐๐๐ค
โ๐๐๐โฒ |
2}
+ ๐๐ ๐๐๐ E๏ฟฝ๏ฟฝ๏ฟฝ๏ฟฝ๏ฟฝ๏ฟฝ ๐โ๐=1
โ[๐๐โฒ๐๐๐โฒ๐ข๐๐
ยฉยญยซโ
๐โฒโฒโP๐โฒ\{๐ }๐ขโ๐๐โฒโฒ
ยชยฎยฌ๏ฟฝ๏ฟฝ๏ฟฝ๏ฟฝ๏ฟฝ๏ฟฝ2๏ธธ ๏ธท๏ธท ๏ธธ
=๐๐21
+
๐๐ ๐๐๐ E
๏ฟฝ๏ฟฝ๏ฟฝ๏ฟฝ๏ฟฝ ๐โ๐=1
โ[๐๐โฒ๐๐๐โฒ |๐ข๐๐ |2
๏ฟฝ๏ฟฝ๏ฟฝ๏ฟฝ๏ฟฝ2๏ธธ ๏ธท๏ธท ๏ธธ=๐๐22
= ๐๐
๐โ๐=1
[๐๐โฒ๐2๐๐โฒ๐ฟ๐๐ + ๐๐๐๐๐ (๐๐21 + ๐๐22),
(119)
which is obtained by taking into account that the noise iscircularly symmetric. The term ๐๐21 depends on the non-
20
coherent interference and can be simplified as follows
๐๐21 =โ
๐โฒโฒโP๐โฒ\{๐ }
๐โ๐=1
[๐๐โฒ๐2๐๐โฒE{|๐ข๐๐๐ข
โ๐๐โฒโฒ |
2}
+โ
๐โฒโฒโP๐โฒ\{๐ }
๐โ๐=1
๐โ๐โฒ=1,๐โฒโ ๐
โ[๐๐โฒ[๐โฒ๐โฒ๐๐๐โฒ๐๐โฒ๐โฒ
ร E{๐ขโ๐๐โฒโฒ๐ข๐๐๐ขโ๐โฒ๐๐ข๐โฒ๐โฒโฒ}
=โ
๐โฒโฒโP๐โฒ\{๐ }
๐โ๐=1
[๐๐โฒ๐2๐๐โฒ๐ฟ๐๐๐ฟ๐๐โฒโฒ+
โ๐โฒโฒโP๐โฒ\{๐ }
๐โ๐=1
๐โ๐โฒ=1
โ[๐๐[๐โฒ๐โฒ๐๐๐โฒ๐๐โฒ๐โฒ tr(ฮฮฮ๐๐ฮฮฮ๐โฒ๐โฒโฒ),
(120)
by using the second moment in (7) and the uncorrelationamong the aggregated channels. The last term ๐๐22 in (119)can be simplified as follows
๐๐22 =
๐โ๐=1
๐โ๐โฒ=1
โ[๐๐โฒ[๐โฒ๐โฒ๐๐๐โฒ๐๐โฒ๐โฒE
{|๐ข๐๐ |2 |๐ข๐โฒ๐ |2
}=
๐โ๐=1
[๐๐โฒ๐2๐๐โฒE
{|๐ข๐๐ |4
}+
๐โ๐=1
๐โ๐โฒ=1,๐โฒโ ๐
โ[๐๐โฒ[๐โฒ๐โฒ๐๐๐โฒ๐๐โฒ๐โฒE{|๐ข๐๐ |2 |๐ข๐โฒ๐ |2}
=
๐โ๐=1
[๐๐โฒ๐2๐๐โฒ๐ฟ
2๐๐ +
๐โ๐=1
[๐๐โฒ๐2๐๐โฒ tr(ฮฮฮ
2๐๐ )
+(๐โ๐=1
โ[๐๐โฒ๐๐๐โฒ๐ฟ๐๐
)2
+๐โ๐=1
๐โ๐โฒ=1
โ[๐๐โฒ[๐โฒ๐โฒ๐๐๐โฒ๐๐โฒ๐โฒ tr(ฮฮฮ๐๐ฮฮฮ๐โฒ๐ )
(121)
where the last equality follows from Lemma 1. By insert-ing (120) and (121) into (119), and by denoting I๐๐ =โ๐พ๐โฒ=1,๐โฒโ ๐ E{|UI๐๐โฒ๐ |2}, (118) can be rewritten as follows
I๐๐ = ๐๐๐พโ
๐โฒ=1,๐โฒโ ๐
๐โ๐=1
[๐๐โฒ๐พ๐๐โฒ๐ฟ๐๐ + ๐๐๐๐๐ร
๐พโ๐โฒ=1,๐โฒโ ๐
โ๐โฒโฒโP๐โฒ
๐โ๐=1
๐โ๐โฒ=1
โ[๐๐โฒ[๐โฒ๐โฒ๐๐๐โฒ๐๐โฒ๐โฒ tr(ฮฮฮ๐๐ฮฮฮ๐๐โฒโฒ)
+ ๐๐๐๐๐โ
๐โฒโP๐\{๐ }
๐โ๐=1
[๐๐โฒ๐2๐๐โฒ tr(ฮฮฮ
2๐๐ )+
๐๐๐๐๐
โ๐โฒโP๐\{๐ }
(๐โ๐=1
โ[๐๐โฒ๐๐๐โฒ๐ฟ๐๐
)2
,
(122)
The proof follows, by inserting (112), (113), and (122) into(56) and by using some algebraic manipulations.
REFERENCES
[1] T. V. Chien, H. Q. Ngo, S. Chatzinotas, M. D. Renzo, and B. Ottersten,โRIS and Cell-Free Massive MIMO: A marriage for harsh propagationenvironments,โ in Proc. IEEE GLOBECOM, 2021.
[2] M. Giordani, M. Polese, M. Mezzavilla, S. Rangan, and M. Zorzi,โToward 6G networks: Use cases and technologies,โ IEEE Commun.Mag., vol. 58, no. 3, pp. 55โ61, 2020.
[3] T. V. Chien and E. Bjornson, Massive MIMO Communications. SpringerInternational Publishing, 2017, pp. 77โ116.
[4] T. S. Rappaport, Y. Xing, G. R. MacCartney, A. F. Molisch, E. Mel-lios, and J. Zhang, โOverview of millimeter wave communications forfifth-generation (5G) wireless networksโWith a focus on propagationmodels,โ IEEE Trans. Antennas Propag., vol. 65, no. 12, pp. 6213โ6230,2017.
[5] E. Bjornson, L. Sanguinetti, and M. Kountouris, โDeploying densenetworks for maximal energy efficiency: Small cells meet massiveMIMO,โ IEEE J. Sel. Areas Commun., vol. 34, no. 4, pp. 832โ847,2016.
[6] T. Van Chien, T. N. Canh, E. Bjornson, and E. G. Larsson, โPowercontrol in cellular massive MIMO with varying user activity: A deeplearning solution,โ IEEE Trans. Wireless Commun., vol. 19, no. 9, pp.5732 โ 5748, 2020.
[7] G. Interdonato, E. Bjornson, H. Q. Ngo, P. Frenger, and E. G. Lars-son, โUbiquitous cell-free massive MIMO communications,โ EURASIPJournal on Wireless Communications and Networking, vol. 2019, no. 1,pp. 1โ13, 2019.
[8] H. Q. Ngo, A. Ashikhmin, H. Yang, E. G. Larsson, and T. L. Marzetta,โCell-free massive MIMO versus small cells,โ IEEE Trans. WirelessCommun., vol. 16, no. 3, pp. 1834โ1850, 2017.
[9] T. Van Chien, E. Bjornson, and E. G. Larsson, โJoint power allocationand load balancing optimization for energy-efficient cell-free MassiveMIMO networks,โ IEEE Trans. Wireless Commun., vol. 19, no. 10, pp.6798โ6812, 2020.
[10] G. Interdonato, H. Q. Ngo, and E. G. Larsson, โEnhanced normalizedconjugate beamforming for Cell-Free Massive MIMO,โ IEEE Trans.Commun., vol. 69, no. 5, pp. 2863โ2877, 2021.
[11] E. Bjornson and L. Sanguinetti, โScalable cell-free massive MIMOsystems,โ IEEE Trans. Commun., vol. 68, no. 7, pp. 4247โ4261, 2020.
[12] Q. Wu and R. Zhang, โIntelligent reflecting surface enhanced wirelessnetwork via joint active and passive beamforming,โ IEEE Trans. WirelessCommun., vol. 18, no. 11, pp. 5394โ5409, 2019.
[13] T. A. Le, T. Van Chien, and M. Di Renzo, โRobust probabilistic-constrained optimization for IRS-aided MISO communication systems,โIEEE Wireless Commun. Lett., vol. 10, no. 1, pp. 1โ5, 2021.
[14] M. Di Renzo, A. Zappone, M. Debbah, M. S. Alouini, C. Yuen, J. deRosny, and S. Tretyakov, โSmart radio environments empowered byreconfigurable intelligent surfaces: How it works, state of research, andthe road ahead,โ IEEE J. Sel. Areas Commun., vol. 38, no. 11, pp. 2450โ2525, 2020.
[15] Q. Wu, S. Zhang, B. Zheng, C. You, and R. Zhang, โIntelligentreflecting surface aided wireless communications: A tutorial,โ IEEETrans. Commun., vol. 69, no. 5, pp. 3313โ3351, 2021.
[16] M. M. Zhao, Q. Wu, M. J. Zhao, and R. Zhang, โIntelligent reflectingsurface enhanced wireless networks: Two-timescale beamforming op-timization,โ IEEE Trans. Wireless Commun., vol. 20, no. 1, pp. 2โ17,2021.
[17] A. Zappone, M. Di Renzo, F. Shams, X. Qian, and M. Debbah,โOverhead-aware design of reconfigurable intelligent surfaces in smartradio environments,โ IEEE Trans. Wireless Commun., vol. 20, no. 1, pp.126โ141, 2021.
[18] A. Abrardo, D. Dardari, and M. Di Renzo, โIntelligent reflecting sur-faces: Sum-rate optimization based on statistical position information,โIEEE Trans. Commun., vol. 69, no. 10, pp. 7121โ7136, 2021.
[19] B. Zheng and R. Zhang, โIntelligent reflecting surface-enhanced OFDM:Channel estimation and reflection optimization,โ IEEE Wireless Com-mun. Lett., vol. 9, no. 4, pp. 518โ522, 2020.
[20] X. Wei, D. Shen, and L. Dai, โChannel estimation for RIS assistedwireless communications: Part I-Fundamentals, solutions, and futureopportunities,โ IEEE Commun. Lett., vol. 25, no. 5, pp. 1398โ1402,2021.
[21] N. S. Perovic, L.-N. Tran, M. Di Renzo, and M. F. Flanagan, โAchievablerate optimization for MIMO systems with reconfigurable intelligentsurfaces,โ IEEE Trans. Wireless Commun., vol. 20, no. 6, pp. 3865 โ3882, 2021.
[22] T. Zhou, K. Xu, X. Xia, W. Xie, and J. Xu, โAchievable rate optimizationfor aerial intelligent reflecting surface-aided Cell-Free Massive MIMOsystem,โ IEEE Access, vol. 9, pp. 3828 โ 3837, 2020.
[23] Z. Zhang and L. Dai, โCapacity improvement in wideband recon-figurable intelligent surface-aided cell-free network,โ in Proc. IEEESPAWC, 2020, pp. 1โ5.
21
[24] M. Bashar, K. Cumanan, A. G. Burr, P. Xiao, and M. Di Renzo, โOn theperformance of reconfigurable intelligent surface-aided cell-free massiveMIMO uplink,โ in Proc. IEEE GLOBECOM, 2020, pp. 1โ6.
[25] Y. Zhang, B. Di, H. Zhang, J. Lin, Y. Li, and L. Song, โReconfigurableintelligent surface aided cell-free MIMO communications,โ IEEE Wire-less Commun. Lett., vol. 10, no. 4, pp. 775 โ 779, 2021.
[26] Y. Zhang, B. Di, H. Zhang, J. Lin, C. Xu, D. Zhang, Y. Li, and L. Song,โBeyond cell-free MIMO: Energy efficient reconfigurable intelligentsurface aided cell-free MIMO communications,โ IEEE Trans. on Cog.Comm. and Net., vol. 7, no. 2, pp. 412โ426, 2021.
[27] T. Van Chien, L. T. Tu, S. Chatzinotas, and B. Ottersten, โCoverage prob-ability and ergodic capacity of intelligent reflecting surface-enhancedcommunication systems,โ IEEE Commun. Lett., vol. 25, no. 1, pp. 69โ73, 2021.
[28] T. Hou, Y. Liu, Z. Song, X. Sun, Y. Chen, and L. Hanzo, โReconfig-urable intelligent surface aided NOMA networks,โ IEEE J. Sel. AreasCommun., vol. 38, no. 11, pp. 2575โ2588, 2020.
[29] T. Van Chien, A. K. Papazafeiropoulos, L. T. Tu, R. Chopra, S. Chatzino-tas, and B. Ottersten, โOutage probability analysis of IRS-assistedsystems under spatially correlated channels,โ IEEE Wireless Commun.Lett., vol. 10, no. 8, pp. 1815 โ 1819, 2021.
[30] M. Di Renzo, M. Debbah, D.-T. Phan-Huy, A. Zappone, M.-S.Alouini, C. Yuen, V. Sciancalepore, G. C. Alexandropoulos, J. Hoydis,H. Gacanin et al., โSmart radio environments empowered by recon-figurable AI meta-surfaces: An idea whose time has come,โ EURASIPJournal on Wireless Communications and Networking, vol. 2019, no. 1,pp. 1โ20, 2019.
[31] H. Alwazani, A. Kammoun, A. Chaaban, M. Debbah, and M.-S. Alouini,โIntelligent reflecting surface-assisted multi-user MISO communication:Channel estimation and beamforming design,โ IEEE Open Jour. of theCommun. Soc., vol. 1, pp. 661โ680, 2020.
[32] E. Bjornson and L. Sanguinetti, โRayleigh fading modeling and channelhardening for reconfigurable intelligent surfaces,โ IEEE Wireless Com-mun. Lett., vol. 10, no. 4, pp. 830 โ 834, 2021.
[33] B. Clerckx, G. Kim, and S. Kim, โCorrelated fading in broadcast MIMOchannels: Curse or blessing?โ in Proc. IEEE GLOBECOM, 2008.
[34] S. Kay, Fundamentals of Statistical Signal Processing: EstimationTheory. Prentice Hall, 1993.
[35] T. C. Mai, H. Q. Ngo, M. Egan, and T. Q. Duong, โPilot power controlfor Cell-Free Massive MIMO,โ IEEE Trans. Veh. Technol., vol. 67,no. 11, pp. 11 264โ11 268, 2018.
[36] T. Van Chien, E. Bjornson, and E. G. Larsson, โJoint pilot design anduplink power allocation in multi-cell Massive MIMO systems,โ IEEETrans. Wireless Commun., vol. 17, no. 3, pp. 2000โ2015, 2018.
[37] H. Cramer, Random variables and probability distributions. CambridgeUniversity Press, 2004, vol. 36.
[38] T. Van Chien, H. Q. Ngo, S. Chatzinotas, and B. Ottersten,โReconfigurable intelligent surface-assisted Massive MIMO: Favorablepropagation, channel hardening, and rank deficiency,โ IEEE SignalProcess. Mag., 2021, accepted for publication. [Online]. Available:arXivpreprintarXiv:2107.03434
[39] T. L. Marzetta, E. G. Larsson, H. Yang, and H. Q. Ngo, Fundamentalsof Massive MIMO. Cambridge University Press, 2016.
[40] G. Gradoni and M. Di Renzo, โEnd-to-end mutual coupling awarecommunication model for reconfigurable intelligent surfaces: Anelectromagnetic-compliant approach based on mutual impedances,โIEEE Wireless Commun. Lett., vol. 10, no. 5, pp. 938 โ 942, 2021.
[41] X. Qian and M. Di Renzo, โMutual coupling and unit cell awareoptimization for reconfigurable intelligent surfaces,โ IEEE WirelessCommun. Lett., vol. 10, no. 6, pp. 1183 โ 1187, 2021.
[42] G. Caire, โOn the ergodic rate lower bounds with applications to MassiveMIMO,โ IEEE Trans. Wireless Commun., vol. 17, no. 5, pp. 3258โ3268,2018.
[43] G. Interdonato, H. Q. Ngo, P. Frenger, and E. G. Larsson, โDownlinktraining in cell-free Massive MIMO: A blessing in disguise,โ IEEETrans. Wireless Commun., vol. 18, no. 11, pp. 5153โ5169, 2019.
[44] E. Bjornson, J. Hoydis, and L. Sanguinetti, โMassive MIMO networks:Spectral, energy, and hardware efficiency,โ Foundations and Trendsยฎ inSignal Processing, vol. 11, no. 3-4, pp. 154โ655, 2017.
[45] T. V. Chien and H. Q. Ngo, Massive MIMO Channels. IET Publishers,2020, ch. 11.
Trinh Van Chien (Sโ16-Mโ20) received the B.S. degree in Electronics
and Telecommunications from Hanoi University of Science and Technology(HUST), Vietnam, in 2012. He then received the M.S. degree in Electricaland Computer Enginneering from Sungkyunkwan University (SKKU), Korea,in 2014 and the Ph.D. degree in Communication Systems from LinkopingUniversity (LiU), Sweden, in 2020. He is now a research associate atUniversity of Luxembourg. His interest lies in convex optimization problemsand machine learning applications for wireless communications and image &video processing. He was an IEEE wireless communications letters exemplaryreviewer for 2016 and 2017. He also received the award of scientific excellencein the first year of the 5Gwireless project funded by European Union Horizonโs2020.
Hien Quoc Ngo received the B.S. degree in electrical engineering from the HoChi Minh City University of Technology, Vietnam, in 2007, the M.S. degree inelectronics and radio engineering from Kyung Hee University, South Korea,in 2010, and the Ph.D. degree in communication systems from LinkopingUniversity (LiU), Sweden, in 2015. In 2014, he visited the Nokia Bell Labs,Murray Hill, New Jersey, USA. From January 2016 to April 2017, Hien QuocNgo was a VR researcher at the Department of Electrical Engineering (ISY),LiU. He was also a Visiting Research Fellow at the School of Electronics,Electrical Engineering and Computer Science, Queenโs University Belfast,UK, funded by the Swedish Research Council.
Hien Quoc Ngo is currently a Reader (Associate Professor) at QueenโsUniversity Belfast, UK. His main research interests include massive (large-scale) MIMO systems, cell-free massive MIMO, physical layer security, andcooperative communications. He has co-authored many research papers inwireless communications and co-authored the Cambridge University Presstextbook Fundamentals of Massive MIMO (2016).
Dr. Hien Quoc Ngo received the IEEE ComSoc Stephen O. Rice Prize inCommunications Theory in 2015, the IEEE ComSoc Leonard G. AbrahamPrize in 2017, and the Best PhD Award from EURASIP in 2018. He alsoreceived the IEEE Sweden VT-COM-IT Joint Chapter Best Student JournalPaper Award in 2015. He was an IEEE Communications Letters exemplaryreviewer for 2014, an IEEE Transactions on Communications exemplaryreviewer for 2015, and an IEEE Wireless Communications Letters exemplaryreviewer for 2016. He was awarded the UKRI Future Leaders Fellowship in2019. Dr. Hien Quoc Ngo currently serves as an Editor for the IEEE Transac-tions on Wireless Communications, IEEE Wireless Communications Letters,Digital Signal Processing, Elsevier Physical Communication (PHYCOM), andIEICE Transactions on Fundamentals of Electronics, Communications andComputer Sciences. He was a Guest Editor of IET Communications, specialissue on โRecent Advances on 5G Communicationsโ and a Guest Editor ofIEEE Access, special issue on โModelling, Analysis, and Design of 5G Ultra-Dense Networksโ, in 2017. He has been a member of Technical ProgramCommittees for several IEEE conferences such as ICC, GLOBECOM, WCNC,and VTC.
Symeon Chatzinotas is Full Professor and Head of the SIGCOM ResearchGroup at SnT, University of Luxembourg. He is coordinating the researchactivities on communications and networking, acting as a PI for more than20 projects and main representative for 3GPP, ETSI, DVB. He is currentlyserving in the editorial board of the IEEE Transactions on Communications,IEEE Open Journal of Vehicular Technology and the International Journal ofSatellite Communications and Networking.
In the past, he has been a Visiting Professor at the University of Parma,Italy and was involved in numerous R&D projects for NCSR Demokritos,CERTH Hellas and CCSR, University of Surrey.
He was the co-recipient of the 2014 IEEE Distinguished Contributions toSatellite Communications Award and Best Paper Awards at EURASIP JWCN,CROWNCOM, ICSSC. He has (co-)authored more than 500 technical papersin refereed international journals, conferences and scientific books.
22
Marco Di Renzo (Fellow, IEEE) received the Laurea (cum laude) and Ph.D.degrees in electrical engineering from the University of LโAquila, Italy, in2003 and 2007, respectively, and the Habilitation a Diriger des Recherches(Doctor of Science) degree from University Paris-Sud (now Paris-SaclayUniversity), France, in 2013. Since 2010, he has been with the French NationalCenter for Scientific Research (CNRS), where he is a CNRS Research Director(Professor) with the Laboratory of Signals and Systems (L2S) of Paris-Saclay University โ CNRS and CentraleSupelec, Paris, France. In Paris-Saclay University, he serves as the Coordinator of the Communications andNetworks Research Area of the Laboratory of Excellence DigiCosme, and asa Member of the Admission and Evaluation Committee of the Ph.D. Schoolon Information and Communication Technologies. He is the Editor-in-Chiefof IEEE Communications Letters and a Distinguished Speaker of the IEEEVehicular Technology Society. In 2017-2020, he was a Distinguished Lecturerof the IEEE Vehicular Technology Society and IEEE Communications Society.He has received several research distinctions, which include the SEE-IEEEAlain Glavieux Award, the IEEE Jack Neubauer Memorial Best Systems PaperAward, the Royal Academy of Engineering Distinguished Visiting Fellowship,the Nokia Foundation Visiting Professorship, the Fulbright Fellowship, andthe 2021 EURASIP Journal on Wireless Communications and NetworkingBest Paper Award. He is a Fellow of the UK Institution of Engineeringand Technology (IET), a Fellow of the Asia-Pacific Artificial IntelligenceAssociation (AAIA), an Ordinary Member of the European Academy ofSciences and Arts (EASA), and an Ordinary Member of the AcademiaEuropaea (AE). Also, he is a Highly Cited Researcher.
Bjorn Ottersten (Sโ87โMโ89โSMโ99โFโ04) received the M.S. degree in elec-trical engineering and applied physics from Linkoping University, Linkoping,Sweden, in 1986, and the Ph.D. degree in electrical engineering from StanfordUniversity, Stanford, CA, USA, in 1990. He has held research positions withthe Department of Electrical Engineering, Linkoping University, the Infor-mation Systems Laboratory, Stanford University, the Katholieke UniversiteitLeuven, Leuven, Belgium, and the University of Luxembourg, Luxembourg.From 1996 to 1997, he was the Director of Research with ArrayComm,Inc., a start-up in San Jose, CA, USA, based on his patented technology.In 1991, he was appointed Professor of signal processing with the RoyalInstitute of Technology (KTH), Stockholm, Sweden. Dr. Ottersten has beenHead of the Department for Signals, Sensors, and Systems, KTH, and Deanof the School of Electrical Engineering, KTH. He is currently the Directorfor the Interdisciplinary Centre for Security, Reliability and Trust, Universityof Luxembourg. He is a recipient of the IEEE Signal Processing SocietyTechnical Achievement Award, the EURASIP Group Technical AchievementAward, and the European Research Council advanced research grant twice.He has co-authored journal papers that received the IEEE Signal ProcessingSociety Best Paper Award in 1993, 2001, 2006, 2013, and 2019, and 8 IEEEconference papers best paper awards. He has been a board member of IEEESignal Processing Society, the Swedish Research Council and currently servesof the boards of EURASIP and the Swedish Foundation for Strategic Research.Dr. Ottersten has served as Editor in Chief of EURASIP Signal Processing,and acted on the editorial boards of IEEE Transactions on Signal Processing,IEEE Signal Processing Magazine, IEEE Open Journal for Signal Processing,EURASIP Journal of Advances in Signal Processing and Foundations andTrends in Signal Processing. He is a fellow of EURASIP.