Antenna Selection for Improving Energy Efficiency in XL-MIMO Systems
José Carlos Marinello, Taufik Abrão, Abolfazl Amiri, Elisabeth de Carvalho, Petar Popovski
aa r X i v : . [ ee ss . SP ] S e p Antenna Selection for Improving EnergyEfficiency in XL-MIMO Systems
José Carlos Marinello, Taufik Abrão, Abolfazl Amiri, Elisabeth de Carvalho,Petar Popovski
Abstract
We consider the recently proposed extra-large scale massive multiple-input multiple-output (XL-MIMO)systems, with some hundreds of antennas serving a smaller number of users. Since the array length is of thesame order as the distance to the users, the long-term fading coefficients of a given user vary with the differentantennas at the base station (BS). Thus, the signal transmitted by some antennas might reach the user with muchmore power than that transmitted by some others. From a green perspective, it is not effective to simultaneouslyactivate hundreds or even thousands of antennas, since the power-hungry radio frequency (RF) chains of theactive antennas increase significantly the total energy consumption. Besides, a larger number of selected antennasincreases the power required by linear processing, such as precoding matrix computation, and short-term channelestimation. In this paper, we propose four antenna selection (AS) approaches to be deployed in XL-MIMO systemsaiming at maximizing the total energy efficiency (EE). Besides, employing some simplifying assumptions, wederive a closed-form analytical expression for the EE of the XL-MIMO system, and propose a straightforwarditerative method to determine the optimal number of selected antennas able to maximize it. The proposed ASschemes are based solely on long-term fading parameters, thus, the selected antennas set remains valid for arelatively large time/frequency intervals. Comparing the results, we find that the genetic-algorithm based ASscheme usually achieves the best EE performance, although our proposed highest normalized received power ASscheme also achieves very promising EE performance in a simple and straightforward way.
Index Terms
Extra large-scale MIMO; Antenna selection; Energy efficiency; Spectral efficiency; Visibility region (VR);Non-stationary; Near-field.
I. I
NTRODUCTION
In the fifth-generation (5G) networks, massive multiple-input multiple-output (MIMO) is identified as a keytechnology for achieving large gains in spectral and energy efficiencies [1], [2]. Recently, a new type of verylarge antenna arrays, which can be integrated into large structures like stadiums, or shopping malls, has been
Copyright (c) 2015 IEEE. Personal use of this material is permitted.This work was supported in part by the Arrangement between the European Commission (ERC) and the Brazilian National Council of StateFunding Agencies (CONFAP), CONFAP-ERC Agreement H2020, by the National Council for Scientific and Technological Development(CNPq) of Brazil under grants 404079/2016-4 and 310681/2019-7.J. C. Marinello is with Electrical Engineering Department, Federal University of Technology PR, Cornélio Procópio, PR, Brazil; [email protected] .T. Abrão is with Electrical Engineering Department, State University of Londrina (UEL), Londrina, PR, Brazil; [email protected] .A. Amiri, E. de Carvalho and P. Popovski are with the Department of Electronic Systems, Technical Faculty of IT and Design; AalborgUniversity, Denmark; [email protected] . September 8, 2020 DRAFT conceived: the so called extra-large scale massive MIMO (XL-MIMO) [3]–[5]. XL-MIMO system is a verypromising and recent technology, pointed out as important candidate for sixth-generation (6G) and beyondtechnologies [6], [7], which is still in its inception, lacking for further elaborated techniques in order to maturethe technology. Indeed, due to the large dimension of the antenna array in XL-MIMO systems, different kinds ofspatial non-stationarities appear accross the array [3]–[5]; hence, admitting constant long-term fading coefficientsbetween a user and all the antennas of the array is not a valid assumption. This is the main difference betweenthe XL-MIMO scenario and the typical massive MIMO system model assumed in most part of massive MIMOliterature. In [8], it is shown through experimental measurements how different regions of an extremely largearray see different propagation paths, and in some cases, the terminals might see just a portion of the array,called visibility region (VR). Authors also discuss how the non-stationarity properties of this new scenariochange several important design aspects.In [3] authors seek for mapping users in terms of XL-MIMO array partition, such that the downlink (DL) sum-rate using a truncated zero-forcing (ZF) precoder is maximized. Numerical results show that a properly trainednetwork via deep learning approach solves the problem nearly as well as an optimal mapping algorithm. Hence,increasing the size of current massive MIMO arrays is promising in terms of boosting the spectral efficiency(SE) of the wireless systems.Since the centralized processing may present very high computational complexity in XL-MIMO arrays, auseful approach is to split the signal processing between subarrays. A subarray-based system architecture forXL-MIMO systems is proposed in [4], where closed-form uplink (UL) SE approximations with linear receiversare derived; the goal is to maximize the system sum achievable SE. Two statistical channel state information(CSI) based greedy user scheduling algorithms are developed, providing improved performance for XL-MIMOsystems.In [5], a simple non-stationary channel model is proposed for XL-MIMO systems, and the performance ofconjugate beamforming (CB) and ZF in the DL have been investigated considering such channel. The non-stationarities are modeled in a binary fashion, such that each antenna can be visible or not for a specific user,giving rise to the VRs: an area of the massive antenna array concentrating the most of the received user’senergy. However, the authors did not consider long-term fading variations between the visible antennas of agiven user.In [9] authors develop procedures for XL-MIMO receivers design. There are two important challenges indesigning receivers for XL-MIMO systems: increased computational cost of the multi-antenna processing, andhow to deal with the variations of user energy distribution over the antenna elements due to the spatial non-stationarities across huge distributed antenna-elements in the 2D or 3D array. Indeed, non-stationarities limit theXL-MIMO system performance. Hence, the authors propose a distributed receiver based on variational messagepassing that can address both challenges. In the proposed receiver structures, the processing is distributed intolocal processing units, that can perform most of the complex processing in parallel, before sharing their outcomewith a central processing unit. Such designs are specifically tailored to exploit the spatial non-stationarities andrequire lower computations than linear ZF or minimum mean square error (MMSE) receivers.In [10], the ZF and regularized ZF schemes operating in XL-MIMO scenarios with a fixed number of subarrayshave been emulated using the randomized Kaczmarz algorithm (rKA), deploying non-stationary properties
September 8, 2020 DRAFT through VRs. Numerical results have shown that, in general, the proposed rKA-based combiner applicable toXL-MIMO systems can considerably decrease computational complexity of the signal detector at the expenseof small performance losses. On the other hand, in [11], an expectation propagation detector for XL-MIMOsystems has been proposed. In order to reduce complexity, the subarray-based architecture employed distributesbaseband data from disjoint subsets of antennas into parallel processing procedures coordinated by a centralprocessing unit. Additionally, authors also propose strategies for further reducing the complexity and overheadof the information exchange between parallel subarrays and the central processing unit to facilitate the practicalimplementation of the proposed detector.Recently, to deal with subarrays and channel scatterers in non-stationary XL-MIMO environment, [12]proposed two channel estimation methods based on subarray-wise and scatterer-wise near-field non-stationarychannel properties. Authors model the multipath channel with the last-hop scatterers under a spherical wavefrontand divide the large aperture array into multiple subarrays. The proposed channel estimation methods positionthe scatterers and perform a mapping between subarrays and scatterers. Hence, the scatterer-wise methodsimultaneously positions each scatterer and detects its VR to further enhance the positioning accuracy. Moreover,the subarray-wise method can achieve low mean square error (MSE) performance under low-complexity, whereasthe scatterer-wise method can accurately arrange the scatterers and determine the non-stationary channel.In [13], authors propose and validate realistic channel models when employing physically-large arrays, inwhich non-stationarities and visibility regions are present, as in the XL-MIMO system. The statistical distri-bution of important channel parameters are found based on measurements. Such contributions are proposed asextensions to the COST 2100 channel model. Besides, key statistical properties of the proposed extensions, e.g.,autocorrelation functions, maximum likelihood estimators, and Cramer-Rao bounds, are derived and analyzed.Furthermore, the performance of a spatial modulation massive MIMO system is investigated in [14] under anon-stationary channel model. Authors show that spatial modulation can outperform typical employed spatialmultiplexing transmission in certain scenarios of low correlation among sub-channels, for example under a richscattering environment.A novel random access (RA) protocol for crowded XL-MIMO systems is proposed in [15]. Authors haveproposed a decentralized and uncoordinated decision rule, which can be evaluated at the users side, forretransmitting or not the RA pilots during the connection stage, taking advantage of the XL-MIMO propagationfeatures. The proposed protocol achieves significant performance improvements in terms of reducing theconnection delay and providing access for larger number of devices.
A. Motivation, Contributions and Novelties in Comparison with Existing Works
Current design approaches in telecommunication systems include a global effort in saving energy and reducingpollution [2], [16], [17]. We show in this paper that antenna selection (AS) methods in XL-MIMO systemsis a very important issue since the energy expenditure of such systems could be very high if activating theradio frequency (RF) chains of all antennas simultaneously. Besides, some antennas might contribute very littlewith the system performance due to the non-stationarities and visibility regions, in such a way that the powerrequired to activate their RF chain becomes a burden that severely penalizes the total energy efficiency (EE) of
September 8, 2020 DRAFT the system. Therefore, the very large number of antennas deployed in the XL-MIMO systems in conjunctionwith the spatial non-stationarities make the application of AS schemes very important.The main contributions of this work are threefold:( i ) Reformulating the signal to interference plus noise ratio (SINR) performance expressions of [5], consideringlong-term fading variations across the array and incorporating the maximum transmit power constraint intothe expressions for CB and ZF, and finding more compact and comprehensive results, readily applicablefor antenna selection procedures.( ii ) Based on the obtained expressions, and on a realistic power consumption model, we evaluate the total EEof the XL-MIMO system. Besides, we propose and compare four low-complexity AS procedures aimingto maximize the total EE of the system, different than [3], [4] which proposed SE-based AS schemes. Ourproposed schemes are based solely on the long-term fading parameters, and the obtained solutions remainvalid for larger time/frequency intervals.( iii ) Based on our proposed AS schemes, and some simplifying assumptions, we derive approximated closed-form EE expressions, and propose an iterative method for finding the optimal number of selected antennaswhich maximizes EE. Finally, numerical simulations have validated the proposed performance expressionsand compared the different XL-MIMO AS schemes.AS methods for typical spatially stationary massive MIMO systems [18], [19] is a well investigated topic.However, the XL-MIMO system is a different scenario. While the spatially stationary model applies for typicalcellular systems, where the BS antenna array dimension is much lower than the distance to the users and a singlelong-term fading coefficient holds for all antennas, significant power variations appear along the XL-MIMOarray, due to its large dimension and number of antennas, and proximity with users. The non-stationary XL-MIMO scenario just very recently was introduced in the literature. To the best of our knowledge, this contributionis the first evaluating the EE of the XL-MIMO scenario, showing that AS methods are especially importantto improve EE due to the spatial non-stationarities that naturally arise in XL-MIMO systems, proposing long-term fading based AS procedures, and deriving the optimal number of active antennas for this new wirelesscommunication context.With respect to the existing XL-MIMO literature, we can point out as the main novelties of our paper:although our system model and CB and ZF performance expressions are similar to that of [5], authors haveconsidered a binary visibility region model for the XL-MIMO scenario, in which no long-term fading variationoccurs for the visible antennas. Besides, performance expressions are dependent of power coefficients obtainedresolving a separated optimization problem for meeting power constraint, and no antenna selection is considered.Differently, we incorporated the power constraint into the performance expressions, arriving at more compactand comprehensive results, readily applicable for AS procedures, and considered long-term fading variationsalong the array. Besides, AS for XL-MIMO systems has been investigated only in [3], [4] at the moment ofwriting this paper; however, both works proposed SE-based AS schemes for XL-MIMO systems. Differently,based on only long-term fading coefficients, we propose AS schemes aiming to maximize the XL-MIMO totalEE, since this is a very important issue due to the very large number of antennas at the XL-MIMO array,and the non-stationarities and visibility regions which arise in this scenario. Furthermore, the long-term fading September 8, 2020 DRAFT approach has the advantages of being simpler than short-term ones, and of providing solutions which remainvalid for larger time periods and all subcarriers (if employing a wideband system), reducing the computationalcomplexity of the antenna selection approach and simplifying hardware due to switching and RF chain on-offrequirements.
Notations:
Boldface lower and upper case symbols represent vectors and matrices, respectively. I N denotes theidentity matrix of size N , while {·} T and {·} H denote the transpose and the Hermitian transpose operator,respectively. We use CN ( m, σ ) when referring to a circular symmetric complex Gaussian distribution withmean m and variance matrix σ . Besides, tr( · ) and diag( · ) are the trace and diagonal matrix operators,respectively, while [ A ] i,j holds to the element in the i th row and j th column of matrix A , and a i refersto its i th column vector. II. S YSTEM M ODEL
We consider a base station (BS) equipped with a linear XL-MIMO array with M antennas uniformlydistributed along a length of L meters, Fig. 1. In front of the extra-large array structure, K users are randomlydistributed in a rectangular area, of length L in the array parallel dimension, and with a distance to the arrayin the range [0 . · L, L ] . Since the distances of the users to the antennas is of the same order of the arraylength L the average received power varies along the XL-MIMO array, and therefore we cannot consider asingle long-term fading coefficient for a given user [3], [8]. Instead, we consider a long-term fading coefficient β m,k regarding the m -th antenna of the XL-MIMO array and the k -th user, similarly as in [3], [9], [10], [15],given by β m,k = q · d − κm,k , (1)in which q is a constant determining the path loss in a reference distance, d m,k is the distance between the m -th antenna of the XL-MIMO array and the k -th user, and κ is the path loss decay exponent. The channelmatrix H ∈ C M × K is thus formed by elements h m,k = p β m,k · h m,k , in which h m,k ∼ CN (0 , , assuming arich scattering environment as in [4], [5]. If we arrange the long-term fading coefficients of a user in a diagonalmatrix: R k = diag([ β ,k , β ,k , . . . , β M,k ]) ∈ R M × M , (2)and the elements h m,k in a vector h k ∈ C M × , we have that each column of H can be defined as h k = R k h k as in [5].In the DL, considering an average received signal-to-noise ratio (SNR) ρ at the users, an average long-termfading coefficient β avg (among all antennas and users’ positions), and a uniform power allocation policy forthe users, the total transmit power, P max , should satisfy [1] ρ = P max · β avg σ , (3)in which σ is the noise power. Since the channel gain β m,k varies significantly along the array, it is moreeffective to select just the stronger antennas to transmit signal to the k -th user, reducing the number of active In order to guarantee a minimum distance of the users to the XL-MIMO array, as in [10], [16].
September 8, 2020 DRAFT
Fig. 1. Illustration of the adopted system model. antennas, as well the power spent with power-hungry RF chains. We discuss in the next Section differentapproaches to obtain the set of antennas selected to serve the users, A . For simplicity, we considered β avg ≈ q · L − κ in our simulations. The signal for user k , s k , is precoded by g k ∈ C M × and scaled by p k ≥ , which adjuststhe signal power, before transmission. Considering a similar XL-MIMO system model than [5], the transmitvector x is the linear combination of the precoded and scaled signal of all the users, i.e. , x = K X k =1 √ p k · g k · s k . (4)Let G = [ g , g , . . . , g K ] ∈ C M × K be the combined precoding matrix, and P = diag([ p , p , . . . , p K ]) ∈ R K × K be the diagonal matrix of signal powers. The combined precoding matrix G is normalized to satisfythe power constraint E [ || x || ] = tr( PG H G ) = P max . (5)The signal received by the k -th user is y k = h Hk x + n k , k = 1 , , . . . K, (6)in which n k ∼ CN (0 , σ ) is an additive white Gaussian noise (AWGN) sample. Assuming independent Gaussiansignaling, i.e. , s k ∼ CN (0 , and E [ s i s ∗ j ] = 0 , i = j , the SINR γ k of the k -th user can be defined as [5]: γ k = p k | h Hk g k | P Kj =1 ,j = k p j | h Hk g j | + σ . (7)We selected the CB and ZF approaches as representative low-complexity linear precoding schemes. The CBprecoder matrix is simply defined as G CB = α CB H , (8)and the ZF precoding matrix is G ZF = α ZF H ( H H H ) − , (9)where the scaling factors α CB = p P max / tr( PH H H ) and α ZF = p P max / tr( P ( H H H ) − ) ensure that thepower constraint (5) is met. September 8, 2020 DRAFT
Using (8) in (7), the SINR of the k th user for CB is γ ( CB ) k = p k | h Hk h k | P Kj =1 ,j = k p j | h Hk h j | + σ P max tr( PH H H ) . (10)Similarly, using (9) in (7), the SINR of the k th user for ZF is γ ( ZF ) k = p k P max σ tr( P ( H H H ) − ) . (11)Given the system model presented in this Section in eq. (1)–(11), and the deterministic equivalent analysisof [20], it is presented in [5] the deterministic equivalent of γ ( CB ) k in (10) as γ ( CB ) k = p k (tr( R k )) P Kj =1 ,j = k p j tr( R k R j ) + σ P max P Kj =1 p j tr( R j ) , (12)and the deterministic equivalent of γ ( ZF ) k in (11) as γ ( ZF ) k = p k P max σ P Ki =1 p i (cid:16) tr( R i ) − P Kj =1 ,j = i tr( R i R j )tr( R j ) (cid:17) − . (13)where R i is defined as in (2).Having found the SINR of the k th user, the spectral efficiency is readily obtained as η sk = log (1 + γ k ) . Onthe other hand, the energy efficiency is [16], [17] η e = B P Kk =1 η sk P , (14)in which B is the system bandwidth, and P is the total power consumption, discussed in Section II-C. A. Further Advances in the Performance Expressions
We revisit the performance expressions for non-stationary XL-MIMO discussed in [5], while propose furtherelaborations to arrive at lean and more comprehensive results. Note that the results of (12) and (13) dependon the signal powers in both numerator and denominators, and such coefficients should be chosen in order tosatisfy the power constraint in (5). In the simulation code made available by the authors of [5], they apply theCVX solver of [21] to find a matrix P satisfying (5). This makes the performance expressions less intuitive,while limiting the application of AS schemes as proposed in Section III of this paper. Hence, in this subsection,we shed light on deriving self-contained closed-form SINR expressions recalling the channel hardening massiveMIMO properties. For that, we first rewrite (5) in the following form: E [ || x || ] = tr( PG H G ) = K X k =1 p k || g k || = P max . (15)If a uniform power allocation scheme is applied, the following equality holds p k || g k || = P max K , k = 1 , , . . . K. (16)Hence, when adopting CB, eq. (16) becomes p k α CB || h k || = P max K , k = 1 , , . . . K, (17) September 8, 2020 DRAFT and we have an undetermined system with K equations and K + 1 variables. By choosing α CB = 1 forsimplicity, the p k coefficients can be obtained for CB as p ( CB ) k = P max K || h k || , k = 1 , , . . . K. (18)Following similar assumptions as in [5], we have that || h k || = h Hk h k = h Hk R k h k M →∞ −−−−→ tr( R k ) , (19)and a deterministic equivalent of (18) is p ( CB ) k = P max K tr( R k ) , k = 1 , , . . . K. (20)Substituting (20) in (12), we arrive at γ ( CB ) k = tr( R k ) P Kj =1 ,j = k tr( R k R j )tr( R j ) + Kσ P max . (21)On the other hand, for the case of ZF, (5) becomes E [ || x || ] = tr (cid:0) PG H G (cid:1) = P max , = α ZF tr (cid:0) P ( H H H ) − H H H ( H H H ) − (cid:1) = P max , = α ZF tr (cid:0) P ( H H H ) − (cid:1) = P max , = α ZF tr ( PV ) = P max , (22)in which the matrix V is a diagonal matrix formed by the main diagonal elements of ( H H H ) − . We can thusrewrite (22) as α ZF K X k =1 p k [ V ] k,k = P max , (23)and if a uniform power allocation is employed α ZF p k [ V ] k,k = P max K , k = 1 , , . . . K. (24)Again, making α ZF = 1 , the p k coefficients can be obtained for the ZF precoding as p ( ZF ) k = P max K [ V ] k,k , k = 1 , , . . . K. (25)Following the analysis of [5, App. A], it can be shown that [ V ] k,k M →∞ −−−−→ tr( R k ) − K X j =1 ,j = k tr( R k R j )tr( R j ) − , (26)and a deterministic equivalent of (25) is p ( ZF ) k = P max K tr( R k ) − K X j =1 ,j = k tr( R k R j )tr( R j ) , k = 1 , . . . K. (27) September 8, 2020 DRAFT
Substituting (27) in (13), we arrive at γ ( ZF ) k = P max Kσ tr( R k ) − K X j =1 ,j = k tr( R k R j )tr( R j ) . (28)Equations (21) and (28) show the XL-MIMO DL system performance employing CB and ZF, respectively,as further extensions of eq. (12) and (13) from [5]. This is a first contribution of this manuscript, which servesas basis for the following EE and AS analysis. Remark 1:
Although we have considered α CB = α ZF = 1 in our analysis, any other choice for these parameterswould result in the same expressions, since would affect every numerator and denominator terms in the sameway. Remark 2:
The SINR performance expressions presented in [5, Table I] can be seen as particular cases of(21) and (28) when neglecting long-term fading and applying the normalization tr( R k ) = tr( Θ k ) = M or tr( Θ k ) = D , where Θ k and D are the matrix describing the VR of k th user and the number of visible antennasper user, respectively, as in [5]. B. Antenna Selection Model
Given our deterministic equivalent performance expressions for CB and ZF in eq. (21) and (28), respectively,we can rewrite these expressions considering the activation subset of antennas. Hence, denoting A as the setcontaining the indices of the active antennas, the deterministic equivalent SINR for the CB precoding results γ ( CB ) k = P m ∈A β m,k P Kj =1 ,j = k P m ∈A β m,k β m,j P m ∈A β m,j + Kσ P max , (29)while for the ZF: γ ( ZF ) k = P max Kσ X m ∈A β m,k − K X j =1 ,j = k P m ∈A β m,k β m,j P m ∈A β m,j . (30)It is worth to note that, in our formulation, the activation subset of antennas is the same for all users,differently from [3], in which each user has its own set of active antennas aiming to maximize the systemsum-rate. We justify our formulation since, when aiming to maximize the total energy efficiency, once thepower-hungry RF chain of an antenna is active, it is better to take full advantage of it, transmitting signal forall users. It has no significant benefit in defining the activation subset of antennas in a per-user fashion, since theZF approach is able to eliminate the inter-user interference, while the power increment necessary to computethe precoding vector with a slightly large number of antennas is small if compared to the power to activatethe RF chain of the additional antenna, as evinced in the next subsection. Besides, it would result in morecomplicated performance expressions, probably in terms of short-term fading coefficients, and the dimension ofthe search space of the AS algorithms would scale with K , becoming considerably more complex and powerconsuming. C. Power Consumption Model
We follow the same power consumption model of [16], which is very similar to that in [17], and is a veryrealistic model. However, as we focus on the DL transmission, we do not consider the UL data rates as well as
September 8, 2020 DRAFT0 the UL transmit powers. In the XL-MIMO scenario analysed herein, we consider the power expenditures of theirradiated DL data signal (with the amplifier efficiency), P DLTX , the UL training, P tr TX , the channel estimation, P CE ,the coding/decoding, P C / D , the backhaul, P BH , the linear processing computation, P PR , the transceiver chains, P TC , and a fixed quantity regarding the circuitry power consumption required for site-cooling, control signaling,and load-independent power of backhaul infrastructure and baseband processors, P FIX . Thus, the overall powerconsumption results P = P DLTX + P tr TX + P CE + P C / D + P BH + P PR + P TC + P FIX . (31)Our objective here is to investigate the dependence of the selected subset of antennas, A , with the totalenergy efficiency of the system. Note that the total energy efficiency of the system depends on A in differentways. First, the sum rate of the system depends on the SE of the users, which is a function of their SINRsdependent of A . Moreover, the sum rate impacts on the power expenditures of the coding/decoding, and thebackhaul. Besides, the power consumption of the transceiver chains is modeled as P TC = P SYN + |A| P BS + KP MT , (32)in which P SYN is the power of the local oscillator, P BS is the power required to each active BS antenna operate,while P MT is the power required to each single-antenna mobile terminal (MT) operate. Note that M is usuallyvery high in an XL-MIMO system , while P BS accounting for the power-hungry RF chains is considered in[16] as 1 W per antenna. Thus, activating the RF chains of all BS antennas would result in a very large powerexpenditure, in such a way that it is very important to perform a suitable antenna selection procedure.The power consumed with processing, P PR , corresponds to the power required to obtain the transmit signalin (4), to obtain the precoding matrix, and to obtain the AS set. Note that this power is also dependent on thenumber of active antennas |A| . Following the model in [16], but including the term of power related to the ASprocessing, we have P PR = B (cid:16) − τ S (cid:17) C ts L BS + B S C prec L BS + 1 T LT C as L BS , (33)in which τ is the length of the uplink pilot signals, S is the coherence block size, C ts is the computationalcomplexity for evaluating eq. (4). Besides, L BS is the computational efficiency of the BS (in W /f lop ), C prec isthe complexity of obtaining the precoding vectors for all users, T LT is the long-term fading coherence time, and C as is the complexity of obtaining the antenna selection set. The obtained AS set remains valid for a long-termcoherence interval, since our analysis is based only in long-term fading parameters. One can see from (33) thatthis approach results in a lower influence of the AS set computation in P PR , since it is multiplied by the factor /T LT , which is much lower than B/ S and B (cid:0) − τ S (cid:1) .Following the analysis in [16], [17], we consider 1 f lop as an arithmethic operation between two complexnumbers. Thus, the multiplication between a matrix A ∈ C m × n and a matrix B ∈ C n × p spends mnp flops. Therefore, we have C ts = 2 |A| K f lops from [17]. Besides, if using the CB precoder, C prec = C CB =3 |A| K f lops from [17], against C prec = C ZF = K / |A| K + |A| K f lops if adopting ZF. The complexity Typically hundreds or even thousands of antennas.
September 8, 2020 DRAFT1 C as is discussed in the next Section. Besides, the terms in (31) not discussed in this Section can be computedin the same way as in [16].Finally, we can rewrite (31) as P = P † + P CE + P C / D + P BH + P PR + |A| P BS , (34)in which we have gathered the power components that do not depend of A in the term: P † = P DLTX + P tr TX + P SYN + KP MT + P FIX . (35)The dependence of the terms in (34) with A can be justified as follows: P CE depends on A since the short-term channel estimates are obtained only for the active antennas, P C / D and P BH because they depend on thesystem sum-rate, which depends on A , and P PR because the processing complexity is dependent on the numberof active antennas. III. A NTENNA S ELECTION S CHEMES
In this section we propose different AS schemes for XL-MIMO aiming to obtain a suitable subset of antennas A selected to transmit the DL signal to the mobile users subject to channel non-stationarities. First we propose asimple, deterministic, greedy scheme based on the highest received normalized power (HRNP) criterion. Then,three heuristic schemes are proposed using the HRNP active antennas set as initial solution: local search (LS), genetic algorithm (GA), and particle swarm optimization (PSO). A. HRNP criterion
A first and greedy approach is to select just the M s antennas responsible for the major part of the powerreceived by the users. However, since closer users receive more power, this should be performed in a normalizedfashion in order to achieve a fair result for all users. In this case, we first compute the metric: ϕ m = K X k =1 β m,k P Mj =1 β j,k , m = 1 , , . . . , M. (36)Then, the selected subset of antennas A HRNP will be composed by the M s antennas with the highest values of ϕ m . A pseudo-code for the HRNP-AS procedure is presented in Algorithm 1, in which ϕ = [ ϕ , ϕ , . . . ϕ M ] .The complexity of the HRNP AS scheme is described by C HRNP as = 3 M K + M log( M ) [ f lops ] , (37)corresponding to the computation of (36) for all antennas, and a sorting algorithm to select the M s antennaswith highest ϕ m . It is noteworthy, however, that the HRNP EE performance is highly dependent on the M s choice, since the system would provide low sum-rates with few active antennas, or it would consume a highpower with many active antennas. Thus, we propose in Section IV an approximated closed-form analyticalexpression for the EE of the XL-MIMO system employing ZF and HRNP-AS as a function of M s . Then, we We evaluate the computational complexities of the investigated schemes in terms of floating point operations (flops) , defined as anaddition, subtraction, multiplication or division between two floating point numbers [22].
September 8, 2020 DRAFT2 propose an iterative method for obtaining the M s value which maximizes this expression. We do not considerthe complexity of this method in eq. (37) since it is not dependent on the channel parameters, but only controlledby the system parameters, such as the number of users, transmit power, dimensions of XL-MIMO array andcoverage area. Therefore, its computation can be performed over larger time periods. We discuss in SectionV-A the complexity of the proposed method for obtaining the optimal M s value. Algorithm 1
Proposed HRNP AS Scheme
Input: M s , β m,k , ∀ m, k . Initialize A as an empty set; for m = 1 , , . . . , M do Evaluate ϕ m as in (36); end for for n = 1 , , . . . , M s do Evaluate a = arg max m ϕ m ; Update A as A = A ∪ a ; Remove ϕ a from ϕ ; end for Output: A HRNP . B. LS-based Antenna Selection
A simple strategy for seeking a better active antennas set is to perform a local search (LS) in the neighborhoodof the HRNP solution. For this purpose, we first represent the set A as a binary vector a of length M , in whichif m ∈ A , a m = 1 ; otherwise a m = 0 . Then, we compute the total energy efficiency (14) of every candidatewithin a certain Hamming distance d Ham from it. If a better candidate is found, the solution is updated, andthe procedure is repeated on its neighborhood. This iterative procedure is repeated for a predefined number ofiterations or until the convergence. A pseudo-code representation of the LS-based AS scheme is provided inAlgorithm 2, in which N el as defined in step 3 is the number of elements within the Hamming distance d Ham from the current solution. For simplicity, we have limited our search with a unitary Hamming distance.The complexity of the LS-AS scheme is C LS as = C HRNP as + N it M C EE [ f lops ] , (38)in which N it is the average number of iterations until convergence, and C EE = 2 M K [ f lops ] is the complexityof computing the total energy efficiency cost function. An interesting point to observe in the LS algorithm isthat if a new solution is not found into an iteration, the search can be interrupted, since the algorithm hasconverged. This contributes to decrease the complexity of the algorithm, and, therefore, improve EE. C. GA-based Antenna Selection
The genetic algorithm is a widely-known bio-inspired heuristic optimization algorithm, which has been usedto solve optimization problems in different areas. In the context of massive MIMO antenna selection, GA hasbeen employed in the conventional stationary case in [18]. Herein, we employ a similar algorithm from [18], butadjusted to the non-stationary XL-MIMO configurations. The GA-AS uses the HRNP output as initial solution,
September 8, 2020 DRAFT3
Algorithm 2
Proposed LS-based AS Scheme
Input: d Ham , N max it , A HRNP , β m,k , ∀ m, k . Initialize a as the binary vector representation of A HRNP ; Initialize η best e as the total energy efficiency of a ; Evaluate N el = (cid:0) Md Ham (cid:1) ; for n = 1 , , . . . , N max it do Generate the search space matrix S of size M × N el with all vectors within the distance d Ham from a n − ; for ℓ = 1 , , . . . , N el do Evaluate η e as the total energy efficiency of s ℓ ; if η e > η best e then Update η best e = η e , and a n = s ℓ ; else Break; end if end for end for
Output: A LS as the set representation of a n .also, other random candidates forming an initial population of size p GA , which is evaluated in terms of thecost function in (14). A given number φ of the best candidates in this population is selected as parents, whichwill generate descendants in a new population. For this purpose, two parents are selected at random for eachdescendant, and the crossover operator is applied with a random crossover point. Then, the mutation operatoris also applied, which inverts the entries of each candidate with certain probability p mut . After a predefinednumber of iterations or until the convergence of the algorithm, it returns the best solution found so far. Apseudo-code representation for the GA-based AS scheme is provided in Algorithm 3.The complexity of our proposed GA-AS procedure is C GA as = C HRNP as + N it [ p GA C EE + p GA log( p GA )] [ f lops ] , (39)due to the cost function evaluation of each candidate in the population, and a sorting algorithm for selectingthe best candidates. D. PSO-based Antenna Selection
The particle swarm optimization algorithm is another bio-inspired optimization algorithm, similarly as GA.However, it is commonly recognized as a simpler algorithm, in terms of fewer mechanisms to escape fromlocal maxima, and reduced computational complexity per iteration. Therefore, we also suggest the use of aPSO-based AS scheme for the non-stationary XL-MIMO case, similarly as proposed in [19] for conventionalstationary massive MIMO scenario.The PSO-AS algorithm uses the HRNP output as initial solution, as well as other random candidates to forman initial swarm of p PSO particles. At each iteration, each particle updates its position in terms of its previousvelocity (inertial effect, with inertia weight ν ), its individual best solution found (cognitive information, withcognitive factor µ c ), and the best solution found by all particles (social information, with social factor µ s ).After a predefined number of iterations or the convergence of the algorithm, it returns the best solution found. September 8, 2020 DRAFT4
Algorithm 3
Proposed GA-based AS Scheme
Input: p GA , φ , p mut , A HRNP , N max it , β m,k , ∀ m, k . Initialize the population Θ GA with the binary vector representation of A HRNP and other p GA -1 random binaryvectors; Evaluate the total energy efficiency of each candidate in Θ GA , forming the vector η GA e ; Sort η GA e in descending order, reorganizing the columns of Θ GA accordingly; Initialize η best e = η GA e, , and a GA = θ GA ; for n = 2 , , . . . , N max it do for ℓ = 1 , , . . . , p GA do Generate two different random integers ∈ [1 , φ ] to be the parents of θ GA ℓ , applying the crossoveroperator in a random crossover point ∈ [2 , M ] ; Apply the mutation operator in θ GA ℓ with probability p mut ; Evaluate the total energy efficiency of θ GA ℓ , and assign it to η GA e,ℓ ; end for Sort η GA e in descending order, reorganizing the columns of Θ GA accordingly; if η GA e, > η best e then Update η best e = η GA e, , and a GA = θ GA ; end if end for Output: A GA as the set representation of a GA .A pseudo-code representation for the PSO-based AS scheme is provided in Algorithm 4, in which Γ ∈ R M × p PSO is a random matrix generated each time it is called with each element uniformly distributed in [0 , interval,and binround ( x ) is the binary round operator, which returns 1 if x > . , and 0 otherwise.The complexity of the proposed PSO-AS algorithm is C PSO as = C HRNP as + N it ( p PSO C EE + p PSO ) [ f lops ] , (40)due to the cost function evaluation (14) for all particles and finding the maximum EE particle, at each iteration.IV. O PTIMAL NUMBER OF SELECTED ANTENNAS : AN ITERATIVE - ANALYTICAL METHOD
In this Section we derive approximated performance analytical expressions for XL-MIMO systems employingthe ZF precoder and the HRNP-based AS method. Such expressions are compared with numerical resultsobtained via Monte-Carlo simulation method in Section V, confirming the tightness of the derivations proposedherein. Then, based on these analytical expressions, we devise an analytical iterative algorithm based on Newton-Raphson (NR) method to determine the optimal number of activated antennas for XL-MIMO systems, whichmaximizes the approximated EE expression.
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Algorithm 4
Proposed PSO-based AS Scheme
Input: p PSO , ν , µ c , µ s , A HRNP , N max it , β m,k , ∀ m, k . Initialize the positions Θ PSO with the binary vector representation of A HRNP and other p PSO -1 random binaryvectors; Evaluate the total energy efficiency of each candidate in Θ PSO , forming the vector η PSO e ; Initialize the social information η best e = η PSO e,φ , and a PSO = θ PSO φ , in which φ = arg max n η PSO e,n ; Initialize the cognitive information η ce = η PSO e , and Θ PSO c = Θ PSO ; Initialize the velocity matrix V ∈ R M × p PSO with random elements uniformly distributed in [ − , ; for n = 2 , , . . . , N max it do Update the velocity matrix V = ν V + µ c Γ (cid:2) Θ PSO c − Θ PSO (cid:3) + µ s Γ (cid:2) a PSO − Θ PSO (cid:3) ; Update the positions Θ PSO = binround (cid:0) Θ PSO + V (cid:1) ; for ℓ = 1 , , . . . p PSO do Evaluate the total energy efficiency of θ PSO ℓ , and assign it to η PSO e,ℓ ; if η PSO e,ℓ > η ce,ℓ then
Update η ce,ℓ = η PSO e,ℓ , and θ PSO c,ℓ = θ PSO ℓ ; if η PSO e,ℓ > η best e then Update η best e = η PSO e,ℓ , and a PSO = θ PSO ℓ ; end if end if end for end for Output: A PSO as the set representation of a PSO .In order to compute the average ZF SINR expression, one can directly evaluate from eq. (30): E h γ ( ZF ) k i = E P max Kσ X m ∈A β m,k − K X j =1 ,j = k P m ∈A β m,k β m,j P m ∈A β m,j = E P max Kσ X m ∈A β m,k − X m ∈A β m,k K X j =1 ,j = k β m,j P n ∈A β n,j = E P max Kσ X m ∈A β m,k − K X j =1 ,j = k β m,j P n ∈A β n,j = P max Kσ X m ∈A E [ β m,k ] − K X j =1 ,j = k E (cid:20) β m,j P n ∈A β n,j (cid:21) (41) in which the expectation is taken with respect to the random users’ positions.Instead of advancing with (41) seeking an exact solution, we approximate the average SINR by the SINR ofa user in the most expected position ( UMEP ). Given the uniform distribution of the users as illustrated in Fig.1, this most expected position would be as depicted in Fig. 2.Then, considering this position for the users, and noting that the HRNP AS activate in this case the M s closest antennas, the ZF SINR expression becomes E h γ ( ZF ) k i ≈ P max Kσ X m ∈A β m − ( K − P m ∈A β m P m ∈A β m ! , ≈ P max Kσ M s / X m =1 β m − ( K −
1) 2 P M s / m =1 β m P M s / m =1 β m , (42) September 8, 2020 DRAFT6
Fig. 2. Illustration of the most expected user’s position (
UMEP ). with β m = q · ( d m ) − κ , and d m = q y + [( m − ) dx ] = y q m − ) dxy ] ≈ y q m dxy ) isrepresented in Fig. 2 for m = 2 . Eq. (42) can thus be simplified as in the next page, in which from (48)to (49) we have used the binomial approximation (1 + x ) α ≈ αx for | αx | ≪ . In our scenario, thiscondition becomes κ M s dx y ≪ , (43)which usually holds for typical XL-MIMO systems. For example, the binomial approximation results in relativeerrors lower than for | αx | < . , which in our XL-MIMO scenario corresponds to M s < . Besides,with this approximated ZF HRNP-AS SINR expression, we can also approximate the EE expression as in eq.(44). Moreover, by expanding all the power terms in the denominator of (44), as discussed in Section II-C, andgrouping them according to their dependence with M s , we arrive at eq. (45), in which P BC = P COD + P DEC + P BT ,and T , T are defined in eq. (46) and (47), respectively. η e ≈ BK log (cid:16) E h γ ( ZF ) k i(cid:17) P DLTX + P tr TX + P CE + P C / D + P BH + P PR + P TC + P FIX , (44) ≈ BK log (cid:16) E h γ ( ZF ) k i(cid:17) P BC BK log (cid:16) E h γ ( ZF ) k i(cid:17) + T + T M s . (45) T = P † + 3 MK + M log( M ) T LT L BS + B K S L BS . (46) T = P BS + 5 B K S L BS + (cid:16) − τ S (cid:17) B K L BS + B K
S L BS . (47) A. Optimal Number of Activated Antennas
Considering our previous analytical results, we propose in this Section a method for obtaining the optimal M s value when employing ZF with HRNP AS, by taking the derivative of eq. (45), with the SINR given ineq. (50), with respect to M s , and equaling it to 0 when M s = M ∗ s . Following this procedure, and after somesimplifications, we arrive at f ( M ∗ s ) = 0 , with f ( M s ) defined as f ( M s ) = ∂ E h γ ( ZF ) k i ∂M s − T ln(2) (cid:16) E h γ ( ZF ) k i(cid:17) log (cid:16) E h γ ( ZF ) k i(cid:17) T + T M s , (54) where ∂ E h γ ( ZF ) k i ∂M s is given in (52). September 8, 2020 DRAFT7 E h γ ( ZF ) k i ≈ P max qKσ M s / X m =1 ( d m ) − κ − ( K −
1) 2 P M s / m =1 ( d m ) − κ P M s / m =1 ( d m ) − κ , ≈ P max qKσ M s / X m =1 y − κ " (cid:18) m dxy (cid:19) − κ − ( K −
1) 2 P M s / m =1 y − κ (cid:20) (cid:16) m dxy (cid:17) (cid:21) − κ P M s / m =1 y − κ (cid:20) (cid:16) m dxy (cid:17) (cid:21) − κ , ≈ P max q y − κ Kσ M s / X m =1 " (cid:18) m dxy (cid:19) − κ − ( K −
1) 2 P M s / m =1 (cid:20) (cid:16) m dxy (cid:17) (cid:21) − κ P M s / m =1 (cid:20) (cid:16) m dxy (cid:17) (cid:21) − κ ( ZF ME ) , (48) ≈ P max q y − κ Kσ M s / X m =1 " − κ (cid:18) m dxy (cid:19) − ( K −
1) 2 P M s / m =1 (cid:20) − κ (cid:16) m dxy (cid:17) (cid:21) P M s / m =1 (cid:20) − κ (cid:16) m dxy (cid:17) (cid:21) , (49) ≈ P max q y − κ Kσ "(cid:0) T , M s − T , M s − T , M s (cid:1) − ( K − (cid:0) T , M s − T , M s − T , M s (cid:1) ( T , M s − T , M s − T , M s ) ( ZF BA ) , (50) with T , = 1 − K dx y , T , = K dx y , T , = K dx y , T , = 1 − K dx y , T , = K dx y , T , = K dx y . ∂f ( M s ) ∂M s = ∂ E h γ ( ZF ) k i ∂M s − T ln(2) (cid:16) E h γ ( ZF ) k i(cid:17) log (cid:16) E h γ ( ZF ) k i(cid:17) − T ( T + T M s ) ∂ E (cid:20) γ ( ZF ) k (cid:21) ∂Ms (cid:16) (cid:16) E h γ ( ZF ) k i(cid:17)(cid:17) ( T + T M s ) , (51) with ∂ E h γ ( ZF ) k i ∂M s = P max q y − κ Kσ (cid:20) F ′ − ( K − F F ′ − F F ′ F (cid:21) , (52) and ∂ E h γ ( ZF ) k i ∂M s = P max q y − κ Kσ " F ′′ − ( K − F (cid:0) F F ′′ − F F ′′ (cid:1) − (cid:0) F F ′ − F F ′ (cid:1) F F ′ F , (53) in which F = T , M s − T , M s − T , M s , F = T , M s − T , M s − T , M s , F ′ = T , − T , M s − T , M s , F ′ = T , − T , M s − T , M s , F ′′ = − T , − T , M s , F ′′ = − T , − T , M s . Since E h γ ( ZF ) k i and its derivative are dependent of M s , we cannot arrive at a closed-form expression for M ∗ s .However, we can find the root of f ( M s ) by applying some iterative numerical method, like Newton-Raphson(NR) method, which obtains a sequence of M s values M s, , M s, , M s, , . . . M s,n converging to M ∗ s if thestarting point M s, is not too far from it. The values in the sequence obey M s,n = M s,n − − f ( M s,n − ) ∂f ( M s ) ∂M s (cid:12)(cid:12)(cid:12) M s,n − , (55)in which the derivative of f ( M s ) is given in (51).V. N UMERICAL R ESULTS AND D ISCUSSION
Our adopted simulation parameters are indicated in Table I. While we have chosen very similar powerconsumption parameters than that of [16], [17], the XL-MIMO system parameters are chosen similarly as [3]–[5], as well as in accordance with common XL-MIMO scenario applications. Considering M = 512 antennasat the XL-MIMO BS, Fig. 3 depicts the SINR, sum SE and the energy efficiency as a function of numberof users K (from 1 to M/ ), for both CB and ZF precoders. The sum SE is presented in units of bits perchannel use (bpcu). One can note that ZF precoding always achieve a higher total energy efficiency than CB in September 8, 2020 DRAFT8 the scenario investigated. The presented results were averaged among 1000 random realizations of the users’positions. It is also shown in the Figure the equivalence between the results of performance expressions from[5], eq. (12) and (13), and the expressions with our proposed simplifications, eq. (21) and (28).
TABLE IS
IMULATION P ARAMETERS . Parameter Value
Carrier frequency: f M [500; 512] XL-MIMO array length: L
30 mDistance of users to BS: [0 . · L, L ] Path loss decay exponent: κ q − . Transmission bandwidth: B
20 MHzChannel coherence bandwidth: B C
100 kHzChannel coherence time: T C T LT σ − dBmUL pilot transmit power: ρ p
20 mWDL radiated power: P max = ρσ qL − κ S
200 symbolsLength of the uplink pilot signals: τ K
Computational efficiency at BSs: L BS . (cid:2) GflopsW (cid:3)
Fraction of DL transmission: ξ d ξ u η d η u T P FIX
18 WPower for local oscillators at BSs: P SYN P BS P MT P COD . h WGbit / s i Power density for decoding data: P DEC . h WGbit / s i Power density for backhaul traffic: P BT . h WGbit / s i Now, considering M = 500 antennas at the XL-MIMO BS, and the same power consumption parameters,Fig. 4 shows the SINR, sum SE and the EE as a function of M s ∈ { M } , with K = 100 users, for bothCB and ZF precoders when employing the HRNP AS scheme. Notice that ZF precoding achieves a highertotal energy efficiency than CB in the scenario investigated. Besides, by activating a number of M s = 146 BS antennas, one can attain the maximum total energy efficiency for ZF precoder with K = 100 users (" M ∗ s by NR" point in Fig. 4.c), as found by our proposed NR method of Section IV-A. Fig. 4 also compares theperformance obtained by averaging eq. (30) with several random realizations for the users’ positions (denotedas ZF), with the approximated deterministic result from eq. (48), denoted as ZF ME , and with the binomialapproximation in eq. (50), denoted as ZF BA . It also shows the results in terms of sum SE and EE of theXL-MIMO system. One can conclude that both proposed approximations are tight, and that the M s values thatmaximize them are nearly the same.Next, in order to obtain the performance results of GA, LS, and PSO-based AS schemes, we have set themaximum number of iterations N max it = 60 for such schemes, and analysed their convergence for K = 100 users, as depicted in Fig. 5.a. One can see from the Figure that the LS-AS convergence presents a non-decreasing September 8, 2020 DRAFT9
50 100 150 200 250
Number of Users S I NR [ d B ] CBCB prop.ZFZF prop.
50 100 150 200 250
Number of Users S u m S pe c t r a l E ff. [ bp c u ]
50 100 150 200 250
Number of Users E ne r g y E ff. [ b / J ] Fig. 3. (a) SINR, (b) sum-SE, and (c) EE vs. K for M = 512 antennas, selecting all available antennas. Proposed eq. (21) and (28), arerepresented by dotted and solid line curves, respectively, while the performances from [5], eq. (12) and (13), are indicated by the curveswith ’ ♦ ’ and ’o’ markers.
200 300 400 500
Number of Active Antennas S I NR [ d B ]
200 300 400 500
Number of Active Antennas S u m S pe c t r a l E ff. [ bp c u ]
200 300 400 500
Number of Active Antennas E ne r g y E ff. [ b / J ] CBZFZF ME ZF BA M s* by NR (a) SINR (b) sum SE (c) EEFig. 4. HRNP-AS scheme under ZF and CB precoders: (a) SINR, (b) sum SE, and (c) EE as a function of M s for M = 500 antennasand K = 100 . September 8, 2020 DRAFT0 behavior, since when a new solution is not found in certain iteration, the algorithm interrupts its search, anddoes not spend more processing power. On the other hand, for GA and PSO-based AS for XL-MIMO systems,if the algorithms do not find new solutions and keep searching during additional iterations, the EE of thatsolution decreases due to the progressive processing power consumed in the subsequent iterations. Therefore, itis not efficient to predefine the number of iterations for these two schemes in the XL-MIMO antenna selectionproblem, since in this optimization problem it would be very difficult do adjust the number of iterations insuch a way to obtain a suitable EE solution for the algorithms. To circumvent while taking advantage of thisfeature, we implement an early-interruption criterion, in which if the GA or the PSO-based AS schemes donot find a new solution within 5 iterations, the search is interrupted, obtaining the convergences depicted inFigure 5.b. Besides, for the GA-based AS scheme, we have considered a population size of M/ , of which are selected as parents at each iteration, and a mutation probability of . For the PSO-based one, wehave considered a swarm of M/ particles, and an inertia weight, cognitive factor and social factor of 0.5.
10 20 30 40 50 60
Iterations E ne r g y E ff. [ b / J ]
20 40 60 80
Iterations E ne r g y E ff. [ b / J ] HRNP ASPSO ASGA ASLS AS (a) without early-interruption (b) with early-interruptionFig. 5. Convergence of the AS schemes: (a) without, and (b) with early-interruption stopping search criterion. K = 100 and M = 500 antennas. Fig. 6 depicts the SINR, sum SE, and EE as a function of K for the HRNP, GA, LS, and PSO-based ASschemes employing ZF precoding, with M = 500 antennas at the XL-MIMO array. While the achieved sumSE performance is nearly the same for all investigated schemes, the graphs reveal that SINR and EE gains canbe achieved in comparison with HRNP. The Figure also shows that, in terms of SINR and EE, the GA, LS,and PSO-based AS schemes achieve a similar performance, and their gains in comparison with HRNP AS aresmall, since the processing required for finding a suitable antennas subset in the XL-MIMO system increasesthe energy consumption; thus, the EE gains become marginal. Except for small number of users, the GA AS September 8, 2020 DRAFT1 scheme achieves one of the best EEs in most part of the investigated scenario, although for high number ofusers, its performance becomes very similar to HRNP AS scheme. Besides, due to its simplicity and celerityto return the results, one can point out that the HRNP criterion coupled to the NR procedure for M ∗ s selectionrepresents a very promising XL-MIMO AS scheme.
50 100 150
Number of Users S I NR [ d B ] HRNP ASPSO ASGA ASLS AS
50 100 150
Number of Users S u m S pe c t r a l E ff. [ bp c u ]
50 100 150
Number of Users E ne r g y E ff. [ b / J ] (a) SINR (b) sum SE (c) EEFig. 6. AS schemes for ZF precoding: (a) SINR, (b) sum SE, and (c) EE as a function of K for M = 500 antennas. A. Complexity of XL-MIMO AS Methods
Fig. 7.a depicts the average number of active antennas as a function of K for the investigated AS methods.One can see that the M ∗ s value obtained by our proposed NR method usually matches the number of antennasselected by LS, PSO, and GA-based AS schemes, corroborating the tightness of the approximations made andthe effectiveness of the method. The major advantage of our proposed NR method for obtaining M ∗ s is thatit can be evaluated for any system configuration satisfying eq. (43). In our numerical simulations, the methodhas converged in at most 3 iterations from the starting point M s, = 1 . K . Besides, the M ∗ s value is notdependent on the channel coefficients, but only on the system parameters, like number of users, transmit power,dimensions of XL-MIMO array and coverage area. Therefore, once found M ∗ s , the NR method just has to beevaluated again when one of these parameters change. The fixed complexity of evaluating M ∗ s under 3 NRiterations is about 380 flops , which is negligible in comparison with that of selecting the antennas subset, eq.(37), (38), (39), and (40), besides of remaining valid for larger time periods.Fig. 7.b depicts the average number of iterations required by each investigated AS scheme, recalling that thenumber of iterations are not fixed, since the LS interrupts when a new solution is not find in an iteration, andGA and PSO implement the early-interruption criterion. Besides, due to the non-decreasing behavior of the LS September 8, 2020 DRAFT2
50 100 150
Number of Users C o m p l e x i t y I n c r ea s e w . r .t. HRN P
50 100 150
Number of Users A v e r age N u m be r o f It e r a t i on s
50 100 150
Number of Users N u m be r o f A c t i v e A n t enna s HRNP ASPSO ASGA ASLS AS (a) Active Antennas (b) K for M = 500 antennas. convergence depicted in Fig. 5, the average number of iterations for this scheme in Fig. 7.b does not correspondto the point in which the LS convergence curve becomes horizontal. Besides, the advantage of HRNP criterionin selecting antennas within the XL-MIMO array can also be confirmed by the extra computational complexityrequired for the other analysed methods. Hence, considering the average number of iterations from Fig. 7.b, the relative complexity increment of the LS, GA and PSO AS schemes w.r.t. the HRNP AS method are depictedin Fig. 7.c. The relative complexity increment metric is defined as: ∆ C = C LS , GA , PSO as − C HRNP as C HRNP as considering typical XL-MIMO network configurations for K users and M BS antennas. One can confirm thevery large relative complexity increase of the AS methods for XL-MIMO, i.e. , this complexity increment is inthe order of , which make the benefits they would bring less significant in terms of energy efficiency.It is noteworthy that the computational complexity spent with the AS methods is included in the EE values,in terms of the processing power. In summary, the performance improvement of the AS scheme comes at theexpense of high complexity, which results in marginal EE gains. On the other hand, the HRNP-AS procedureis able to achieve an improved EE of 34.85 Mbit/J for K = 100 users, in comparison with 18.71 Mbit/J ofselecting all antennas, i.e. , not applying any AS procedure, corresponding in a 86.3% of EE increasing, as onecan infer from Fig. 4.Elaborating further regarding the dependence of the optimal number of selected antennas M ∗ s on the systemparameters, such as number of users, total transmit power available, dimensions of XL-MIMO array, and September 8, 2020 DRAFT3 coverage area, one can argue that such system parameters vary quite slowly with respect to the data symbolperiod. Therefore, it could be possible to evaluate the proposed AS scheme, and turning-on the optimal numberof RF chains M ∗ s , which are then switched to the best antenna subset according to our proposed HRNP criterion.Notice that only when the number of users changes significantly that it would be necessary to re-evaluate the(54)-(55), and then turning-on or turning-off some RF chains. Besides, the proposed method for finding theoptimal number of selected antennas can provide very useful information for XL-MIMO system designers.VI. C ONCLUSION
In this paper, we have investigated the XL-MIMO systems subject to channel non-stationarities. First, wehave revisited the performance expressions from [5], and proposed to incorporate the power constraint at theSINR expressions of CB and ZF to arrive at more lean and comprehensive results. Then, based on suchobtained expressions, we have proposed four XL-MIMO AS schemes aiming at maximizing the EE based onthe following criteria: HRNP, LS, GA, and PSO. Some simplifying assumptions allowed us to derive closed-form EE expressions, based on which we proposed a NR iterative method to obtain the optimal number ofactive antennas. Numerical results have shown that GA usually achieves one of the best EEs, although thegains were marginal in comparison with HRNP, since the processing required for achieving a suitable antennassubset increases the consumed energy, limiting the achieved EE gains. Thus, due to its simplicity and celerityin returning results, the proposed HRNP-AS scheme, with the NR method providing the optimal subarray sizevalue M ∗ s , can be seen as a very promising solution for AS XL-MIMO systems, achieving an EE gain of 86.3%in comparison with selecting all antennas strategy.R EFERENCES[1] T. Marzetta, E. Larsson, H. Yang, and H. Ngo,
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