Efficient Angle-Domain Processing for FDD-based Cell-free Massive MIMO Systems
11 Efficient Angle-Domain Processing for FDD-basedCell-free Massive MIMO Systems
Asmaa Abdallah,
Student Member, IEEE, and Mohammad M. Mansour,
Senior Member, IEEE
Abstract —Cell-free massive MIMO communications is anemerging network technology for 5G wireless communicationswherein distributed multi-antenna access points (APs) servemany users simultaneously. Most prior work on cell-free massiveMIMO systems assume time-division duplexing mode, althoughfrequency-division duplexing (FDD) systems dominate currentwireless standards. The key challenges in FDD massive MIMOsystems are channel-state information (CSI) acquisition andfeedback overhead. To address these challenges, we exploit the so-called angle reciprocity of multipath components in the uplinkand downlink, so that the required CSI acquisition overheadscales only with the number of served users, and not the numberof AP antennas nor APs. We propose a low complexity multipathcomponent estimation technique and present linear angle-of-arrival (AoA)-based beamforming/combining schemes for FDD-based cell-free massive MIMO systems. We analyze the perfor-mance of these schemes by deriving closed-form expressions forthe mean-square-error of the estimated multipath components,as well as expressions for the uplink and downlink spectralefficiency. Using semi-definite programming, we solve a max-min power allocation problem that maximizes the minimum userrate under per-user power constraints. Furthermore, we present auser-centric (UC) AP selection scheme in which each user choosesa subset of APs to improve the overall energy efficiency of thesystem. Simulation results demonstrate that the proposed multi-path component estimation technique outperforms conventionalsubspace-based and gradient-descent based techniques. We alsoshow that the proposed beamforming and combining techniquesalong with the proposed power control scheme substantiallyenhance the spectral and energy efficiencies with an adequatenumber of antennas at the APs.
Index Terms —FDD mode, cell-free massive MIMO, multi-path component estimation, array signal processing, angle-basedbeamforming/combining, power control.
I. I
NTRODUCTION
With the growing demand on high data-rate wireless com-munications, fifth generation (5G) cellular mobile communi-cations has emerged as the latest generation to offer 1000-fold capacity enhancement over current fourth generation (4G)Long-Term Evolution (LTE) systems with reduced latency. Toachieve this aggressive goal, massive multiple-input multiple-output (MIMO) and network densification are promising 5Gwireless technologies that improve the capacity of cellular sys-tems by 1) scaling up the number of antennas in a conventional
Part of this work has been presented at the 2019 IEEE InternationalWorkshop on Signal Processing Advances in Wireless Communications(SPAWC) [1].This work is supported by the National Council for Scientific Researchof the Lebanese Republic (CNRS-L) and the American University of Beirut(AUB) fellowship program, as well as by the University Research Board(URB) at AUB.A. Abdallah and M. M. Mansour are with the Department of Electrical andComputer Engineering, American University of Beirut, Beirut 1107 2020,Lebanon (e-mail: [email protected]; [email protected]).
MIMO system by orders of magnitude [2], [3], and 2) reducingpath-loss and reusing spectrum [4] efficiently.Although massive MIMO and network densification bringforward several advantages, the performance of cellular net-works is limited by inter-cell interference (ICI) and frequenthandovers for fast moving users. In particular, users close tothe cell edge suffer from strong interference.Cell-free (CF) massive MIMO has recently been consideredas a practical and useful embodiment of network MIMOthat can potentially reduce such inter-cell interference throughcoherent cooperation between base stations [5]–[8]. In cell-free massive MIMO, the serving antennas are distributed overa large area. Distributed systems can potentially provide highercoverage probability than co-located massive MIMO due totheir ability to efficiently exploit diversity against shadow fad-ing effects, at the cost of increased backhaul requirements [9].According to [8], “cell-free” massive MIMO implies that,from a user perspective during data transmission, all accesspoints (APs) cooperate to jointly serve the end-users; hencethere are no cell boundaries and no inter-cell interference inthe data transmission. The APs are connected to a centralprocessing unit (CPU) via a backhaul link. This approach, withsimple signal processing, can effectively control ICI, leadingto significant improvements in spectral and energy efficiencyover the cellular systems [5]–[9].The main challenge in deploying cell-free networks liesmainly in acquiring sufficiently accurate channel state infor-mation (CSI) so that the APs can simultaneously transmit(receive) signals to (from) all user equipments (UEs) andcancel interference in the spatial domain. The conventionalapproach of sending downlink (DL) pilots and letting the UEsfeed back channel estimates is unscalable since the feedbackload is proportional to the number of APs. Therefore, to reducethe signaling overhead [10], [11], channel reciprocity can beexploited in time-division duplex (TDD) mode so that eachAP only needs to estimate the uplink CSI.An attractive alternative to consider is frequency-divisionduplexing (FDD) based cell-free massive MIMO systems forthe following reasons: 1) channel reciprocity in TDD modemight not be accurate due to calibration errors in radiofrequency (RF) chains [12], 2) with the lack of downlinktraining symbols in TDD systems, users may not be ableto acquire instantaneous CSI, and thus system performancewill deteriorate in detecting and decoding the intended signals,3) while TDD operation is preferable at sub-6 GHz massiveMIMO, in millimeter wave (mmWave) bands FDD may beequally good since the angular parameters of the channelare reciprocal over a wide bandwidth [13], and 4) FDDsystems dominate current wireless communications and have a r X i v : . [ c s . I T ] J a n many benefits such as lower cost and greater coverage thanTDD [14].On the other hand, FDD-based cell-free massive MIMO sys-tems still suffer from CSI acquisition and feedback overheadsince the amount of downlink CSI feedback scales linearlywith the number of antennas [15] and the number of APs incell-free massive MIMO system. However, we can still benefitfrom 1) angle reciprocity, which holds true for FDD systemsas long as the uplink and downlink carrier frequencies arenot too far from each other (less than several GHz [16]),and 2) angle coherence time which is much longer than theconventional channel coherence time [17] where the channelangle information can be regarded as unchanged. Hence, angleinformation is essential in FDD-based cell-free massive MIMOsystems. Therefore, a low complexity estimation approach thatcan efficiently estimate the angle information is required. A. Related Work
Much of the recent interest in cell-free massive MIMOsystems has focused mainly on TDD-mode only [5]–[9], [18]–[21]. In [5], a cell-free system is considered and algorithmsfor power optimization and linear precoding are analyzed.Compared with the conventional small-cell scheme, cell-freemassive MIMO can yield more than ten-fold improvementin terms of outage rate. While in [6], the APs performmultiplexing/de-multiplexing through conjugate beamformingin the downlink and matched filtering in the uplink.In [7], a cell-free massive MIMO downlink is considered,wherein a large number of distributed multiple-antenna APsserve many single-antenna users. A distributed conjugatebeamforming scheme is applied at each AP via the use oflocal CSI. Spectral efficiency and energy efficiency are studiedwhile considering channel estimation error and power control.In [18], [19], cell-free and user-centric architectures atmmWave frequencies are considered. A multiuser clusteredchannel model is introduced, and an uplink multiuser chan-nel estimation scheme is described along with hybrid ana-log/digital beamforming architectures. Moreover, in [19], thenon-convex problem of power allocation for downlink globalenergy efficiency maximization is addressed. In [20], anuplink TDD-based cell-free massive MIMO system is con-sidered. Geometric programming GP is used to sub-optimallysolve a quasi-linear max–min signal-to-interference-and-noiseratio (SINR) problem.Angle estimation has been studied in other wireless net-works without considering cell-free massive MIMO networks(see e.g. [16], [22]–[30]). For instance, subspace-based angleestimation algorithms, such as multiple signal classification(MUSIC), estimation of signal parameters via rotational in-variance technique (ESPRIT) and their extensions have gainedinterest in the array processing community due to their highresolution angle estimation capability [22]–[24]. Their applica-tions in massive MIMO systems and MIMO systems for angleestimation have been presented in [25]–[28]. Unfortunately,the classical MUSIC and ESPRIT schemes are not suitable formmWave communications due to the following main reasons:1) They have high computational complexity mainly due to the singular value decomposition (SVD) operation on channelswith massive number of antennas; 2) They are consideredas blind estimation techniques originally targeted for radarapplications, and do not make full use of training sequencesin wireless communication systems.In [16], [29], [30], an AoA estimation scheme for a conven-tional mmWave massive MIMO system with a uniform planararray at the base station is presented. The initial AoAs ofeach uplink path are estimated through the two-dimensionaldiscrete Fourier transform (2D-DFT), and then the estimationaccuracy is further enhanced via an angle rotation technique.In the present work, we extend the AoA estimation techniqueof [16], [29], [30], adapt it to the context of FDD-based cell-free massive MIMO, and employ it to estimate another channelmultipath component, namely large-scale fading. Using theseestimated components, we leverage from the angle coherencetime and angle-reciprocity to propose low-complexity angle-based beamforming/combining schemes and power controlalgorithms for downlink and uplink directions.In [31], a multipath component estimation technique andbase station cooperation scheme based on the multipath com-ponents for the FDD-based cell-free massive MIMO systemsare presented. However, no closed-form expression of themean-square-error (MSE) of the considered multipath estima-tion is presented.
B. Contributions of the Paper
In this work, we consider a cell-free massive MIMO systemwith multiple antennas at each AP operating in FDD mode thatdo not require any feedback from the user. All APs cooperatevia a backhaul network to jointly transmit signals to all usersin the same time-frequency resources. By exploiting anglereciprocity, APs can acquire multipath component informationfrom the uplink pilot signals using array signal processingtechniques. The contributions of this paper are:1) We propose a multipath component estimation for the AoAand large-scale fading coefficients based on the DFT oper-ation and log-likelihood function with reduced overhead. Inparticular, we leverage from the observation that the angle-of-departures (AoDs) and the large scale fading componentsvary more slowly than path gains [17], as well as from theproperty of angle-reciprocity. We further derive a closed-form expression for the MSE of the estimated channel mul-tipath components. Both theoretical and numerical resultsare provided to verify the effectiveness of the proposedmethods. These schemes are shown to provide a substantialenhancement over the gradient-based [31] and the classicalsubspace-based [22], [23] multipath component estimationin terms of MSE of the estimated AoA and large-scalefading coefficients since the MSE of the proposed DFT-based estimator coincides with that of the ML estimator.2) We propose linear angle-based beamforming/combiningtechniques for the downlink/uplink transmission that incor-porate the estimated AoA and large-scale fading compo-nents. Interestingly, the proposed schemes scale only withthe number of served users rather than the total numberof serving antennas, and need to be updated every angle
CPU ! " AP $ user % AP AP ' user user ( ! " AP $ user % st path ) * + ,- path ) . / ,- path Figure 1. Cell-free massive MIMO system model coherence time. Therefore, the impact of signaling overheadis substantially reduced with the proposed schemes.3) We derive closed-form expressions for the spectral efficien-cies for the FDD-based cell-free massive MIMO downlinkand uplink with finite numbers of APs and users. Our analy-sis takes into account the proposed beamforming/combiningtechniques and the effect of multipath estimation errors.4) We propose a solution to the max-min power controlproblem by formulating it as a standard semi-definite pro-gramming (SDP) approach. The proposed max-min powercontrol maximizes the smallest rate of all users within theangle-coherence time-scale. In addition, we present a user-centric AP selection scheme to further enhance the energyefficiency of the system.The rest of the paper is organized as follows. The systemmodel for the FDD-based cell-free massive MIMO networkis described in Section II. In Section III, the proposed mul-tipath components estimation is introduced. In Section IV,the proposed beamforming and combining techniques arepresented. Moreover, spectral efficiency analysis is introducedin Section V. Case studies with numerical results are simulatedand analyzed based on the proposed schemes in Section VII.Section VIII concludes the paper.
Notation : Bold upper case, bold lower case, and lower caseletters correspond to matrices, vectors, and scalars, respec-tively. Scalar norms, vector L norms, and Frobenius norms,are denoted by |·| , (cid:107)·(cid:107) , and (cid:107)·(cid:107) F , respectively. E [ · ] , ( · ) T , ( · ) ∗ , ( · ) H , P ⊥ , and tr ( · ) stand for expected value, transpose,complex conjugate, Hermitian, orthogonal projection matrix,and the trace of a matrix. X † stands for the pseudo-inverse ( X H X ) − X H . In addition, X (cid:23) is used to indicate that X isa positive semi-definite matrix. [ x ] i represents i th element of avector x . CN (0 , σ n ) refers to a circularly-symmetric complexGaussian distribution with zero mean and variance σ n .II. S YSTEM M ODEL
As shown in Fig. 1, we consider an FDD-based cell-freemassive MIMO system having M APs, each equipped with auniform linear array (ULA) of N antennas, serving K userswith single antennas. We assume a geometric channel modelwith L propagation paths [16], [31]. Moreover, AoAs (orAoDs), large-scale fading and small-scale fading coefficients are called the multipath components of the channel. Due toangle reciprocity in FDD systems [16], and frequency in-dependency, we assume that 1) the uplink AoA and downlinkAoD are similar, and 2) the uplink and downlink large-scale fading coefficients (slow fading and distant-dependentpath loss components) are similar [32], [33]. However, uplinkand downlink small-scale fading coefficients in FDD systemsare distinct since they are frequency dependent [32], [33].Therefore, the N × channel vectors can be expressed as[16], [31] h = (cid:114) L L (cid:88) l =1 (cid:112) β l α l a ( φ l ) , (1)where α l ∼ CN (0 , is the complex gain of the l th paththat represents the small-scale Rayleigh fading, and β l is thelarge-scale fading coefficient that accounts for path-loss andshadowing effects. The variable φ l ∈ [0 , π ] is the angleof arrival of the l th path. The array steering vector a ( φ l ) is defined as a ( φ l ) = √ N (cid:2) , e jη sin( φ l ) ,. . . ,e j ( N − η sin( φ l ) (cid:3) T , where η = πuλ , u is the antenna spacing, and λ is thechannel wavelength (Note that we also define υ l = η sin ( φ l ) ).Equivalently, the channel vector in (1) can be expressed inmatrix-vector form as h = (cid:114) L AB α , (2)where A N × L = [ a ( φ ) , . . . , a ( φ L )] , B L × L = diag ( (cid:112) β , . . . , (cid:112) β L ) , and α L × = [ α , . . . , α L ] T . (3)As mentioned previously, the quantities α are dependenton frequency; however B and A are constant with respect tofrequency over an angle-coherence time interval (as discussedin subsection III-D).To model a realistic system where we have non-ideal anglereciprocity, we assume that the differences between uplinkand downlink multipath components, ˜ υ u / d l and ˜ β u / d l , are i.i.d.random variables with zero mean and variance σ υ , σ β (cid:28) [34]. A. Uplink Training
Let p k ∈ C × τ be the uplink (UL) pilot signal sent by the k th user composed of τ symbols with unit norm. All pilotsequences used by different users are assumed to be pairwiseorthogonal, since the angle coherence time is much longer thanthe conventional channel coherence time [17]. Therefore, wecan assign a sufficiently large number to τ such that τ ≥ K holds true.Therefore, the received signal Y mk ∈ C N × τ at the m th APsent by the k th user is given by Y mk = √ ρ h mk p k + N mk , (4)where ρ is the uplink transmit power and the entries [ N mk ] n,i of the additive white Gaussian noise matrix N mk ∈ C N × τ are independent and identically distributed (i.i.d.) CN (0 , σ n ) random variables. Multiplying (4) by p H k and collecting T samples, we have Y mk ( t ) p H k = (cid:114) ρL A mk B mk α mk ( t ) + N mk p H k = √ ρ A mk d mk ( t ) + ¯ n mk , t = 1 , . . . , T, (5)where d mk = √ L B mk α mk and ¯ n mk = N mk p H k ∼CN ( N × , σ n I N ) . Then, the T samples of (5) are collectedin a matrix form as ¯Y mk = √ ρ H mk + ¯N mk = √ ρ A mk D mk + ¯N mk , (6)where ¯Y mk = [ Y mk (1) p H k , . . . , Y mk ( T ) p H k ] , H mk =[ h mk (1) , . . . , h mk ( T )] , D mk = [ d mk (1) , . . . , d mk ( T )] , and ¯N mk = [¯ n mk (1) , . . . , ¯ n mk ( T )] .The multipath components estimation is performed in adistributed fashion, in which each AP independently estimatesthe multipath components to the K users. The APs do notcooperate on the multipath components estimation, and noestimates need to be shared among the APs. B. Downlink Payload Data Transmission
The APs, based on the estimated multipath components,independently apply N × beamforming vector ˆ w mk to trans-mit signals to the K users. Moreover, APs do not cooperateon the beamforming vectors. The transmit DL signal from the m th AP is given by x m = (cid:112) ρ d K (cid:88) k =1 ˆ w mk s d k , (7)where s d k is the data symbol for the k th user satisfying E [ | s d k | ] = 1 , and ρ d is the maximum transmit power satisfy-ing, E [ || x m || ] ≤ ρ d . It can be noted here that the multiplexingorder is equal to 1.Then, the received downlink signal at the k th user is givenby r d k = M (cid:88) m =1 h H mk x m + n d k = (cid:112) ρ d M (cid:88) m =1 h H mk ˆ w mk s d k (cid:124) (cid:123)(cid:122) (cid:125) S + (cid:112) ρ d K (cid:88) j (cid:54) = k M (cid:88) m =1 h H mk ˆ w mj s d j (cid:124) (cid:123)(cid:122) (cid:125) I + n d k , (8)where n d k ∼ CN (0 , is the additive noise at the k th user.Note that the received signal can be decomposed into threeparts: 1) desired signal part ( S ), 2) interference part ( I ), and3) noise n d k . Moreover, the k th user can detect signal s d k from r d k . C. Uplink Payload Data Transmission
In the uplink, all K users simultaneously send their datasymbols s u k , where E (cid:8) | s u k | (cid:9) = 1 , to the APs. It can be notedhere that the multiplexing order is equal to 1. The receivedUL signal at the m th AP is given by y u m = √ ρ u K (cid:88) k =1 h mk s u k + n u m , (9)where ρ u is the uplink transmit power and n u m is additive noiseat the m th AP. The noise entries ( [ n u m ] i ) are modeled as i.i.d. CN (0 , σ n ) . The received signal is multiplied by the N × Table IS
YSTEM P ARAMETERS
Number of APs, and number of antennas per AP
M, N
Total number of users K Number of paths L Channel gain for the m th AP and k th user h mk Angular steering vector for the l th path a ( φ l ) Angular steering matrix for the m th AP and k th user A mk Large scale fading matrix B mk Small scale fading vector α mk N × N DFT matrix F N combiner ˆ v mk at each AP where the resulting signal is sentto the CPU through a backhaul to detect the signal. The CPUwill receive r u k = M (cid:88) m =1 ˆ v H mk y u m = K (cid:88) k (cid:48) =1 M (cid:88) m =1 √ ρ u ˆ v H mk h mk (cid:48) s u k (cid:48) + M (cid:88) m =1 ˆ v H mk n u m . (10)Then, s k is detected from r u k .The main system parameters are summarized in Table I.III. P ROPOSED A NGLE INFORMATION AIDED CHANNELESTIMATION FOR
FDD
SYSTEMS
In this section, we present the FDD-based cell-free massiveMIMO systems that directly acquire multipath componentsfrom the uplink pilot signal and use them for the AP co-operation. Using array signal processing, we first presentthe low complexity DFT-based AoA estimation, and thenwe propose the large-scale fading estimation based on theestimated angle information. Note that we need to estimateboth components (AoA, and large-scale fading) for everyangle coherence interval, in order to apply low complexitybeamforming/combining techniques.
A. AoA Estimation Algorithm
Based on our previous work [1], we apply AoA estimationstep that relies on the classical DFT estimation and anglerotation. DFT is used to estimate the AoA wherein the peakof the DFT magnitude spectrum can select the column whosesteering angle best matches the true AoA.Moreover, the normalized DFT of the channel matrix isdefined as h DFT mk = F N h mk where F N is an N × N DFT matrixwhose ( n, q ) th element is given by [ F N ] nq = √ N e − j πnqN .Most of the channel power is concentrated around L largestpeaks determined by the ( (cid:98) q l (cid:101) th ) elements where q l = Nυ l,mk π (for l = 1 , · · · , L ) and υ l,mk = η sin ( φ l,mk ) [30]. There-fore, the initial AoA estimate for the k th user is ˆ φ ini l,mk =sin − (cid:16) λq ini l Nd (cid:17) .Furthermore, the accuracy of the AoA estimation is im-proved through an angle rotation operation [30] by incorpo-rating a phase-shift to the initial estimation to obtain moreaccurate peaks. The angle rotation of the original channelmatrix is expressed as h rDFT l,mk = Φ N ( (cid:52) φ l,mk ) h DFT mk , where Φ ( (cid:52) φ l,mk ) = diag (cid:8)(cid:2) , e j (cid:52) φ l,mk , . . . , e j ( N − (cid:52) φ l,mk (cid:3)(cid:9) with (cid:52) φ l ∈ [ − ( π/N ) , π/N ] is the angle rotation parameter. It isshown in [30] that the entries of [ h rDFT mk ] have only L non-zero peak elements when the optimal phase shifter satisfies (cid:52) φ l,mk = 2 πq l /N − υ l,mk = 2 πq l /N − η sin ( φ l,mk ) . Therefore, the estimate ˆ φ l,mk can be expressed as ˆ φ l,mk =sin − (cid:16) πq l Nη − (cid:52) φ l,mk η (cid:17) , and the estimated AoA matrix isgiven by ˆ A mk = (cid:104) a (cid:16) ˆ φ ,mk (cid:17) , . . . , a (cid:16) ˆ φ L,mk (cid:17)(cid:105) . (11) B. Large-Scale Fading Estimation
Based on the AoA estimate and given that ¯ n mk ∼CN ( N × , σ n I N ) in (6), the probability density function of ¯Y mk for given φ l,mk and β l,mk over all l = 1 , · · · , L can beexpressed as f ( ¯Y mk | φ l,mk , β l,mk ) = exp {− σ n || ¯Y mk − √ ρ A mk D mk || F } ( πσ n ) N . (12)The log-likelihood function can be applied to (12) to give L ( D mk , σ n ) = − N ln π − N ln σ n − || ¯Y mk −√ ρ A mk D mk || F σ n . (13)Knowing that L is a concave function of σ n and D mk , theoptimal estimates ˆ σ n and ˆ D mk can be obtained by taking apartial derivative with respect to σ n and D mk . Hence, ˆ σ n = N || ¯Y mk − √ ρ ˆ A mk ˆ D mk || F , and ˆ D mk = √ ρ ˆ A † mk ¯Y mk , (14)where ˆ A mk = [ a ( ˆ φ ,mk ) , . . . , a ( ˆ φ L,mk )] is the estimateof A mk which is obtained using array signal processing(DFT operation with angle rotation). Once ˆ A mk is obtained,we next estimate the large-scale fading coefficients β l,mk .From (14), we can estimate D mk and the covariance matrix ˆ R mk,d = LT E [ ˆ D mk ˆ D H mk ] . Note that the original covariancematrix R mk,d is given by R mk,d = L × E [ d mk d mk H ] = B mk E [ α mk α mk H ] B H mk = diag ( β ,mk , . . . , β L,mk ) . (15)Hence, we can obtain the estimates of the large-scale fadingcoefficients as ˆ β mk = [ ˆ β ,mk , . . . , ˆ β L,mk ] T = diag ( ˆ R mk,d ) . (16)The proposed multipath component estimation is shown inAlgorithm 1, where G is the search grid within [ − πN , πN ] needed for angle estimation.Note that the search grid parameter G determines thecomplexity and accuracy of the algorithm. The complexityof the whole algorithm is of the order O ( N log N + G N L ) where the factor N log N comes from the DFT operation and G N L comes from rotation operation over a search grid G for all paths L over N antennas. Moreover, the complexityof the proposed algorithm is less than that of the classicalsubspace ESPRIT algorithm of complexity O ( N + U N ) ,with U (cid:29) G being the number of snapshots required duringblind estimation [35]. C. Performance Analysis
Using the same methodology as in [29], [30] in additionto estimating the large-scale fading parameter, we derive thetheoretical MSE of the AoA estimates and the large-scalefading coefficients for the cell-free massive MIMO system. Ingeneral, a closed-form solution of the MSE for multiple AoAestimations is hard to obtain [29]. An alternative approach is to consider the single user and single propagation path andderive corresponding MSE of φ and β as benchmark [29].For a single propagation path according to (6), the receivedtraining signal at the m th AP transmitted by the k th user isgiven by ¯ y mk = Y mk p H k = √ ρ h mk + ¯ n mk = √ ρ a ( φ ) d mk + ¯ n mk = √ ρ (cid:112) β mk α mk a ( φ ) + ¯ n mk , (17)where a ( φ ) is the N × steering vector with its q th entrygiven by [ a ( φ )] q = √ N e ( q − υ mk .For brevity, we henceforth omit the subscript mk repre-senting the link between the m th AP and the k th user. Theproposed angle estimator can be expressed as ˆ υ = arg max υ (cid:107) (cid:107) a ( φ ) (cid:107) a ( φ ) H ¯y (cid:107) = arg max υ (cid:107) a ( φ ) H ¯y (cid:107) = arg max υ ¯y H a ( φ ) a ( φ ) H ¯y , (18)where a ( φ ) = Φ ( (cid:52) φ ) f N q , (cid:107) a ( φ ) (cid:107) = 1 , f Nq is the q thcolumn of F N , and q is the nearest integer to Nυ π .Moreover, using (14), the ML estimate of d is obtained as ˆ d ML = √ ρ ( a ( ˆ φ ) H a ( ˆ φ )) − a ( ˆ φ ) H ¯ y = √ ρ (cid:107) a ( ˆ φ ) (cid:107) a ( ˆ φ ) H ¯ y = √ ρ (cid:107) a ( ˆ φ ) (cid:107) a ( ˆ φ ) H a ( φ ) d + √ ρ (cid:107) a ( ˆ φ ) (cid:107) a ( ˆ φ ) H ¯ n = √ ρ (cid:107) a ( ˆ φ ) (cid:107) a ( ˆ φ ) H a ( φ ) (cid:112) βα + √ ρ (cid:107) a ( ˆ φ ) (cid:107) a ( ˆ φ ) H ¯ n . (19)The joint ML estimates of υ and d can be obtained from [ˆ υ ML ˆ d ML ] = arg min υ,d (cid:107) ¯ y − a ( φ ) d (cid:107) , (20)where ˆ υ ML , ˆ d ML are the optimizing variables.Therefore, using (19), the ML estimate of υ is given by ˆ υ ML = arg max υ ¯ y H P a ¯ y = arg max υ g ( υ ) , (21)where g ( υ ) is the cost function of υ . For the single-path case, P a = a ( φ ) a ( φ ) H is the projection matrix onto the subspacespanned by a ( φ ) , and a ( φ ) represents the steering vector givenin (1). For the multi-path case, P A = AA † = A ( A H A ) − A H represents the projection matrix onto the subspace spannedby A , and A is the steering matrix given in (3). As shownin [30] while including the large scale path-loss parameter β ,the MSE (18) of the considered DFT estimator coincides withthat of the ML estimator (20). Using Lemma 1 in [30] whileincluding the large-scale fading parameter and p k p H k = 1 , theMSE of υ is expressed as E (cid:2) (cid:52) υ (cid:3) = E [(ˆ υ − υ )(ˆ υ − υ ) H ] = σ n ρβ a ( ˆ φ ) H EP ⊥ a Ea ( ˆ φ ) , (22)where E [ (cid:52) υ ] = 0 , P ⊥ a = I − P a is the projection matrix ontothe orthogonal space spanned by a ( φ ) and E is the diagonalmatrix given by E = diag { , · · · , ( N − } . Based on the factthat υ = η sin φ and φ = sin − ( υη ) , we further examine theMSE of φ E (cid:2) (cid:52) φ (cid:3) = ( η ) − ( υη ) × σ n β a ( ˆ φ ) H EP a ⊥ Ea ( ˆ φ ) . (23)Using Taylor series expansion, a of first-order approxima-tion of a ( φ ) is given by a ( φ ) = a ( ˆ φ ) + j Ea ( ˆ φ ) (cid:52) υ. (24)Substituting (24) into (19) and after collecting T samples, Algorithm 1
Extended DFT and Angle-Rotation-Based Multipath Component Estimation Input: ¯Y ∈ C N × T , L , G and λ Output: ˆ φ ∈ R L × , ˆ β ∈ C L × // AoA Estimation for l = 1 : L do for t = 1 : T do Find the central point ( q ini l ) of each bin in ˆ h DFT mk = F N ¯y p mk ( t ) where ( q ini l ) = arg max ( q ) ∈ bin ( l ) (cid:107) [ˆ h DFT mk ] q (cid:107) , l = 1 , · · · L. ( ˆ (cid:52) φ l ) = arg max (cid:52) φ ∈G (cid:107) f Nq ini l Φ ( (cid:52) φ l ) ¯y p mk ( t ) (cid:107) , where f Nq ini l is the q ini l th column of F N . ˆ θ l ( t ) = ˆ θ l ( t −
1) + sin − (cid:16) πq ini l Nη − (cid:52) φ l η (cid:17) end ˆ φ l,mk = T ˆ θ l ( T ) end // Large scale fading Estimation ˆ D mk = √ ρ (cid:16) ˆ A H mk ˆ A mk (cid:17) − ˆ A H mk ¯Y mk , where ˆ A mk = [ a (cid:16) ˆ φ ,mk (cid:17) , . . . , a (cid:16) ˆ φ L,mk (cid:17) ] ˆ R d = LT [ ˆ D mk ˆ D H mk ] ˆ β mk = [ ˆ β ,mk , . . . , ˆ β L,mk ] T = diag ( ˆ R d ) end we rewrite ˆ d as ˆ d = [ ˆ d , · · · , ˆ d T ]= d + j (cid:107) a ( ˆ φ ) (cid:107) a ( ˆ φ ) H Ea ( ˆ φ ) (cid:52) υ d + √ ρ (cid:107) a ( ˆ φ ) (cid:107) a ( ˆ φ ) H ¯ N , (25)where ¯ N = [¯ n , · · · , ¯ n T ] .Moreover, ˆ β = LT E [ ˆdˆd H ] = β + β E (cid:2) ( (cid:52) υ ) (cid:3) | (cid:107) a ( ˆ φ ) (cid:107) a ( ˆ φ ) H Ea ( ˆ φ ) | + √ ρ (cid:107) a ( ˆ φ ) (cid:107) a ( ˆ φ ) H E (cid:2) nn H (cid:3) ( √ ρ (cid:107) a ( ˆ φ ) (cid:107) a ( ˆ φ ) H ) H = β + σ n | a ( ˆ φ ) H Ea ( ˆ φ ) | ρ a ( ˆ φ ) H EP a ⊥ Ea ( ˆ φ ) + σ n ρ . (26)Therefore, the MSE of β can be obtained E (cid:2) (cid:52) β (cid:3) = E (cid:104) ( ˆ β − β )( ˆ β − β ) H (cid:105) = (cid:32) σ n | a ( ˆ φ ) H Ea ( ˆ φ ) | ρ a ( ˆ φ ) H EP a ⊥ Ea ( ˆ φ ) + σ n ρ (cid:33) . (27)Furthermore, the MSE expressions of the estimated AoAand large-scale fading components derived in (22) and (27)give important insights when assessing the impact of beam-forming/combining techniques on the spectral efficiency of theproposed FDD-based cell-free massive MIMO system. D. Angle Coherence Time
Different from the conventional channel coherence time,the angle coherence time is defined as typically an order ofmagnitude longer, during which the AoDs can be regarded asstatic [17]. Specifically, the path AoD in (1) mainly dependson the surrounding obstacles around the BS, which may notphysically change their positions often. On the contrary, thepath gain of the k th user depends on a number of unresolvablepaths, each of which is generated by a scatter surrounding theuser. Therefore, path gains vary much faster than the pathAoDs [17]. Accordingly, the angle coherence time is muchlonger than the conventional channel coherence time. There-fore, we can leverage from this fact and perform multipathestimation in every angle coherence time instead of the much shorter channel coherence time as the impact of the overheadis substantially reduced.IV. P ROPOSED B EAMFORMING AND C OMBINING T ECHNIQUES
We next propose the angle-based matched-filtering, angle-based zero-forcing and angle-based minimum-mean-square-error beamforming/combining that incorporate the estimatedangle information, and the large-scale fading components.The APs are connected via a backhaul network to a CPU,which sends to the APs the data-symbols to be transmittedto the end-users and receives soft-estimates of the receiveddata-symbols from all the APs. Neither multipath estimatesnor beamforming/combining vectors are transmitted throughthe backhaul network.
A. Angle-Based Beamforming
The angle-based beamforming (or precoding) vector ˆ w mk for the m th AP and the k th user is defined as ˆ w mk = L (cid:88) l =1 γ mk,l ˆ g mk,l = ˆ G mk || ˆ G mk || γ mk , (28)where ˆ g mk,l is the l th column of ˆ G mk = [ˆ g mk, , . . . , ˆ g mk,L ] defined below for the proposed angle-based beamforming tech-niques. In addition, γ mk,l is the normalized complex weightfor the l th propagation path that satisfies (cid:80) Ll =1 | γ mk,l | = 1 and γ mk = [ γ mk, , . . . , γ mk,L ] T . Moreover, using (7), E [ || x m || ] = ρ d K (cid:88) k =1 || ˆ G mk γ mk || || ˆ G mk || ≤ ρ d (29)will satisfy the maximum transmit power ρ d .
1) Angle-Based Matched-Filtering Beamforming (A-MF):
The precoder matrix based on the angle information is givenby ˆ G A-MF mk = ˆ A mk ˆ B mk , (30)where ˆ A mk = (cid:104) a (cid:16) ˆ φ ,mk (cid:17) , . . . , a (cid:16) ˆ φ L,mk (cid:17)(cid:105) and ˆ B mk = diag (cid:18)(cid:113) ˆ β ,mk , . . . , (cid:113) ˆ β L,mk (cid:19) are the estimated AoA and large-scale fading matrices according to (11) and (16). More-over, A-MF is a simple beamforming approach that onlyrequires the channel multipath components (AoA and large-scale fading) of the direct link between the m th AP and the k th user. However, the inter user interference is ignored.
2) Angle-Based Zero-Forcing Beamforming (A-ZF):
Weuse A-ZF beamforming as a means to efficiently suppressinterference. To do so, the conventional ZF beamforming em-ploys all the downlink CSI from the users. However, the angle-based ZF beamforming used in this work is distinct from theconventional ZF beamforming in the sense that only the angleinformation and large-scale fading coefficients of the channelare required in the beamforming design. We collect the corre-sponding array steering vectors into ˆ A m = [ ˆ A m , . . . , ˆ A mK ] and similarly for ˆ B m = diag (cid:16) [ ˆ B m , . . . , ˆ B mK ] T (cid:17) . Then, theprecoder matrix is given by ˆ G A-ZF m = ˆ A m ˆ B m (cid:16) ˆ B H m ˆ A H m ˆ A m ˆ B m (cid:17) − , (31)where beamforming vector is ˆ g mk,l defined as the (( k − L + l ) th column of ˆ G A-ZF m .A key property of the angle-based ZF beamforming isthat the beamforming vector is orthogonal to all other arraysteering vectors as given below: ˆ h H mk ˆ w A-ZF mi = (cid:26) s Tmk γ mk if i = k ;0 if i (cid:54) = k. (32)The pseudo-inverse in A-ZF is more complex than A-MF,but the interference is suppressed.
3) Angle-Based MMSE Beamforming (A-MMSE):
We usean angle-based MMSE beamforming design that can efficientlysuppress interference, noise and channel estimation error. TheA-MMSE strikes a balance between attaining the best signalamplification and reducing the interference. The proposedangle-based MMSE beamforming matrix is given by G A-MMSE mk = (cid:32) K (cid:88) k =1 (( ˆ A mk ˆ B mk ˆ B H mk ˆ A H mk + Υ m,k ) + σ n I N (cid:33) − ˆ A mk ˆ B mk , (33)where Υ m,k = ˜ σ υ ( E ˆ A mk ˆ B mk )( E ˆ A mk ˆ B mk ) H +˜ σ υ ˜ σ β ( E ˆ A mk )( E ˆ A mk ) H + ˜ σ β ˆ A mk ˆ A H mk , such that ˜ σ υ = σ υ + E (cid:2) (cid:52) υ (cid:3) and ˜ σ β = σ β + E (cid:2) (cid:52) β (cid:3) , where σ υ and σ β account for non-ideal DL angle reciprocity, and E (cid:2) (cid:52) υ (cid:3) , E (cid:2) (cid:52) β (cid:3) are the MSEs as defined in (22) and (27),respectively.Therefore, for A-ZF/A-MMSE, the only overhead for DLchannel acquisition at each AP comes from UL training, whichonly scales with the number of served users. In addition, onecan note that A-ZF is suitable for high signal-to-noise ratio(SNR) conditions since it is expected that A-ZF and A-MMSEwould have the same performance when the effect of noise islow. B. Angle-Based Combining
Similarly, the combining vector ˆ v mk for the m th AP andthe k th user is defined as ˆ v mk = L (cid:88) l =1 γ mk,l ˆ c mk,l = ˆ C mk γ mk , (34)where ˆ c mk,l is the (( k − L + l ) th column of ˆ C m whichcorresponds to ˆ C mk = [ˆ c mk, , . . . , ˆ c mk,L ] , and γ mk,l = L and γ mk = [ γ mk, , . . . , γ mk,L ] T .Using UL-DL duality [36], the combining vectors of theuplink case for A-MF combining, A-ZF combining and A-MMSE combining are also defined as ˆ C mk = G A-MF mk for A-MF combining ; G A-ZF mk for A-ZF combining ; G A-MMSE mk for A-MMSE combining . (35)such that ˜ σ υ = E (cid:2) (cid:52) υ (cid:3) and ˜ σ β = E (cid:2) (cid:52) β (cid:3) . The correspond-ing combining matrices were defined in (30), (31) and (33).The benefits of relying on only the angle information andlarge-scale fading are: (i) the need for downlink trainingis avoided; (ii) the beamforming/combining matrices can beupdated every angle coherence time, and (iii) a simple closed-form expression for the spectral efficiency can be derivedwhich enables us to obtain important insights.V. S PECTRAL AND E NERGY EFFICIENCY A NALYSIS
In this section, we derive closed-form expressions for thespectral efficiencies per user for DL and UL transmissionsusing the analysis technique from [6], [7], [31]. Then, wedefine the total energy efficiency of the system.
A. Spectral Efficiency
The downlink spectral efficiency per user using the proposedbeamforming schemes is given by R d k = log (cid:0) dk (cid:1) (cid:39) log (cid:32) ρ d S d k ρ d I d jk + ρ d BU d k + σ n (cid:33) , (36)where S d k = M (cid:88) m =1 E (cid:104) || ˆ h H mk ˆ w mk || (cid:105) ,I d jk = K (cid:88) j (cid:54) = k M (cid:88) m =1 E (cid:104) || ˆ h H mk ˆ w mj || (cid:105) , and BU d k = K (cid:88) j =1 M (cid:88) m =1 E (cid:104) || ˜ h H mk ˆ w mj || (cid:105) , represent the strength of the desired signal of the k th user( S d k ), the interference generated by the j th user ( I d jk ), andthe beamforming gain uncertainty ( BU d k ), respectively. Theelements inside the norm of S d k , I d jk and BU d k are uncorrelatedzero mean random variables. In addition, ˆ h mk = h mk − ˜ h mk = ˆ A mk ˆ B mk s mk and the channel uncertainty is ˜ h mk = (cid:52) ˜ υ ( E ˆ A mk ˆ B mk ) s mk + (cid:52) ˜ β ˆ A mk s mk + (cid:52) ˜ β (cid:52) ˜ υ E ˆ A mk s mk ,where (cid:52) ˜ υ and (cid:52) ˜ β differ in the DL and UL directions due toun-ideal angle reciprocity such that (cid:52) ˜ υ d = υ u − ˆ υ u − ˜ υ u / d , (cid:52) ˜ β d = β u − ˆ β u − ˜ β u / d , (cid:52) ˜ υ u = υ u − ˆ υ u , and (cid:52) ˜ β u = β u − ˆ β u . Similarly for the uplink case, the uplink spectral efficiencyper user using the proposed combining schemes is given by R u k (cid:39) log (cid:32) ρ u S u k ρ u I u jk + ρ u BU u k + σ n (cid:80) Mm =1 || ˆ v mk || (cid:33) , (37)where uplink desired signal power ( S uk ), the interferencecaused by the j th user ( I ujk ), and the combining gain uncer-tainty ( BU uk ) are defined similarly as the downlink case butby substituting ˆ w mj with the combining vector ˆ v mj .Using the fact that α l ∼ CN (0 , as well as the fact thatangle of arrival and the large-scale fading remain unchangedduring the angle coherence time, we can further reduce theDL and UL spectral efficiencies into closed forms as shownin (38) and (39) at the top of the next page. B. Energy Efficiency
The total energy efficiency (bit/Joule) is defined as the sumthroughput (bit/s) divided by the total power consumption(Watt) in the network: EE (cid:44) B · (cid:80) Kk =1 κR k P total , (40)where R k is the spectral efficiency (expressed in bit/s/Hz) forthe k th user, B is defined as the system bandwidth, P total isthe total power consumption, κ = (cid:16) − ττ c (cid:17) , and τ = K islength of pilot training sequence in samples, τ c is the anglecoherence interval in samples. Furthermore, we consider thepower consumption model defined in [7] P total = M (cid:88) m =1 P m + M (cid:88) m =1 P bh ,m , (41)where P m is the power consumed at the m th AP whichincludes the amplifier and the circuit power consumption andthe power consumption of the transceiver chains and thepower consumed for signal processing, and P bh ,m representsthe power consumed by the backhaul link that transfers databetween the CPU and the m th AP. The power consumptionterm P m can be defined as P m = 1 ϑ m ρ d σ n (cid:32) N K (cid:88) k =1 || ˆ w mk || (cid:33) + N P tc ,m , (42)where < ϑ m ≤ is the power amplifier efficiency, ρ d is thedownlink SNR, σ n is the noise power, ˆ w mk is the angle basedbeamforming vector for the m th AP and the k th user (definedin (28)), N is the number of antennas at the AP, and P tc ,m isthe internal power required to operate the circuit components(e.g., converters, mixers, and filters) per antenna at the m thAP.Moreover, the power consumption of the backhaul is pro-portional to the sum spectral efficiency and can be modeledas, P bh ,m = P ,m + B · K (cid:88) k =1 κR k · P bt ,m , (43)where P ,m is defined as a fixed power consumption of eachbackhaul (traffic-independent power) which may depend onthe distances between the APs and the CPU and the systemtopology, and P bt ,m is defined as the traffic-dependent power(in Watt per bit/s). VI. P ROPOSED M AX -M IN P OWER CONTROL
To obtain good system performance, the available powerresources must be efficiently managed. In this section, wepropose a solution to the max-min user-fairness problem inthe proposed cell-free Massive MIMO system, where the min-imum uplink rates of all users are maximized while satisfyinga per-user power constraint. We show that the FDD-basedcell-free massive MIMO system can provide uniformly goodservice to all users, regardless of their geographical location,by adopting a max-min power/weight control strategy. Theproposed power control algorithm is done at the CPU, andimportantly, is carried only at the angle-coherence time-scale.Hence the impact of the signaling overhead is substantiallyreduced. Moreover, we present a user centric AP selectionapproach to further enhance the energy efficiency of the CFmassive MIMO system.
A. Downlink Power Control
In the downlink, given realizations of the large-scale fadingand the array steering vectors, we find the power control coef-ficients γ mk , m = 1 , . . . , M , k = 1 , . . . , K, that maximize theminimum of the downlink rates of all users, under the powerconstraint (29). At the optimum point, all users attain the samerate. Mathematically, this is formulated as: max { γ mk,l } min k =1 , ··· ,K R d k subject to K (cid:88) k =1 || ˆ G mk γ mk || || ˆ G mk || ≤ , m = 1 , . . . , Mγ mk,l ≥ , ∀ k, ∀ m, ∀ l. (44)Then, using (38), we can reformulate (44) into a max-minSINR problem as follows: max { γ mk,l } min k =1 , ··· ,Kρ d (cid:80) Mm =1 || ˆB H mk ˆA H mk ˆ w mk || ρ d (cid:80) Kj (cid:54) = k (cid:80) Mm =1 || ˆB H mk ˆA H mk ˆ w mj || + ρ d (cid:80) Kj =1 (cid:80) Mm =1 Ω m,j + σ n s.t. K (cid:88) k =1 || ˆ G mk γ mk || || ˆ G mk || ≤ , ∀ m, ˆ w mk = ˆ G mk || ˆ G mk || γ mk , ∀ k, ∀ m , and γ mk,l ≥ , ∀ k, ∀ m, ∀ l. (45)One can note that (45) is a non-convex separable quadratically-constrained quadratic program (QCQP) in terms of powerallocation γ mk , for all k, m . Therefore, this problem cannotbe directly solved in an efficient manner using existing convexoptimization schemes. While the non-convex QCQP is NP-hard, it can be relaxed into a convex semi-definite program(SDP) using semi-definite relaxation (SDR) [37], in whichthe following property of a scalar is utilized: γ H mk Q γ mk = tr ( γ H mk Q γ mk ) = tr ( Q γ mk γ H mk ) , for any Q ∈ C L × L . There-fore, by introducing a new variable Γ mk = γ mk γ H mk , which isa rank-one symmetric positive semi-definite (PSD) matrix, thequadratic constraints can be transformed into linear constraints R d k (cid:39) log (cid:32) ρ d (cid:80) Mm =1 || ˆB H mk ˆA H mk ˆ w mk || ρ d (cid:80) Kj (cid:54) = k (cid:80) Mm =1 || ˆB H mk ˆA H mk ˆ w mj || + ρ d (cid:80) Kj =1 (cid:80) Mm =1 Ω m,j + σ n (cid:33) , (38)where Ω m,j = ˜ σ υ (cid:107) ( ˆ B H mk ˆ A H mk E ) ˆ w mj (cid:107) + ˜ σ β (cid:107) ( ˆ A H mk ˆ w mj (cid:107) + ˜ σ β ˜ σ υ (cid:107) ( ˆ A H mk E ˆ w mj (cid:107) .R u k (cid:39) log (cid:32) ρ u (cid:80) Mm =1 || ˆB H mk ˆA H mk ˆ v mk || ρ u (cid:80) Kj (cid:54) = k (cid:80) Mm =1 || ˆB H mk ˆA H mk ˆ v mj || + ρ u (cid:80) Kj =1 (cid:80) Mm =1 Λ m,j + σ n (cid:80) Mm =1 || ˆ v mk || (cid:33) , (39)where Λ m,j = ˜ σ υ (cid:107) ( ˆ B H mk ˆ A H mk E )ˆ v mj (cid:107) + ˜ σ β (cid:107) ( ˆ A H mk ˆ v mj (cid:107) + ˜ σ β ˜ σ υ (cid:107) ( ˆ A H mk E ˆ v mj (cid:107) . in the set of all real symmetric L × L matrices S L . Using SDP,problem (45) can be equivalently reformulated as max { Γ mk } min k =1 , ··· ,Kρ d (cid:80) Mm =1 tr ( Ξ mkk Ξ H mkk Γ mk ) ρ d (cid:80) Kj (cid:54) = k (cid:80) Mm =1 tr ( Ξ mkj Ξ H mkj Γ mj ) + ρ d (cid:80) Kj =1 (cid:80) Mm =1 Ω m,j + σ n s.t. K (cid:88) k =1 tr ( ˆ G H mk ˆ G mk Γ mk ) || ˆ G mk || ≤ , ∀ m, Γ mk (cid:23) , ∀ k, ∀ m, rank ( Γ mk ) = 1 , ∀ k, ∀ m, (46)where Ξ mkj = ˆB H mk ˆA H mk ˆ G mj || ˆ G mj || .Since the rank constraint of Γ mk is non-convex, we relaxit to obtain the feasible SDP formulation of (46) as max { Γ mk } µ s.t. ρ d (cid:80) Mm =1 tr ( Ξ mkk Ξ H mkk Γ mk ) ρ d (cid:80) Kj (cid:54) = k (cid:80) Mm =1 tr ( Ξ mkj Ξ H mkj Γ mj ) + ρ d (cid:80) Kj =1 (cid:80) Mm =1 Ω m,j + σ n ≥ µ, K (cid:88) k =1 tr ( ˆ G H mk ˆ G mk Γ mk ) || ˆ G mk || ≤ , ∀ m, and Γ mk (cid:23) , ∀ k, ∀ m. (47)The relaxed problem (47) is a convex SDP and can be solvedby standard convex optimization tools such as CVX [38]. Oncethe optimal variables ˆ Γ mk ( ∀ m, ∀ k ) are obtained, we canfind the rank-one approximations of ˆ Γ mk which are feasiblefor the original problem (45) by applying eigen-value decom-position (EVD) on ˆ Γ mk , and extracting the largest eigen-value and the corresponding eigen-vector to construct ˆ γ mk .Consequently, (47) can be solved efficiently via a bisectionsearch, in which each step involves solving a sequence ofconvex SDP feasibility subproblems [39]. The proposed max-min power control algorithm is summarized in Algorithm 2. Complexity Analysis:
Here, we provide the computationalcomplexity analysis for the proposed Algorithm 2, which usesiterative bisection search to solve the convex optimizationproblem (47) at each iteration. The complexity of (47) is O (( M K ) L / ) in each iteration [40]. Note that the totalnumber of iterations to solve the SDR Problem via a bisectionsearch method is given by log( µ max − µ min (cid:15) ) , where (cid:15) refers to apredetermined threshold [39]. Hence, the total complexity ofsolving (47) is O (( M K ) L / ) log( µ max − µ min (cid:15) ) . B. Uplink Weight Control
Similarly in the uplink, given realizations of the large-scalefading and the array steering vectors, we find the weight control coefficients γ mk , m = 1 , . . . , M , k = 1 , . . . , K, thatmaximize the minimum of the uplink rates of all users, underthe weight constraint. At the optimum point, all users attainthe same rate. So, max { γ mk,l } min k =1 , ··· ,K R u k subject to K (cid:88) k =1 || ˆ C mk γ mk || || ˆ C mk || ≤ , m = 1 , . . . , M,γ mk,l ≥ , ∀ k, ∀ m, ∀ l. (49)Moreover, (49) can be solved following the same steps asshown in subsection (VI-A) in the DL case. C. User-Centric (UC) AP Selection Method
As noted from the last term in (41) that represents the totalpower consumption of the backhaul, cell-free massive MIMOsystems require more backhaul connections to transfer databetween the APs and the CPU when compared to the co-located massive MIMO. Moreover, the second term of (43) hasa significant effect on the energy efficiency, especially when M increases in (41). To improve the total energy efficiency, wecan further decrease the denominator of the energy efficiencyin (40). We present an AP selection for the user-centric casewhich can reduce the backhaul power consumption, and hence,increase the energy efficiency. The AP selection scheme isbased on choosing for each user k a subset of APs M k that forms ( δ % ) of the total channel power. For a particularuser, there are many APs which are located very far away.These APs will not impact the overall spatial diversity gains.Hence, not all APs actually contribute in serving this user.Furthermore, M k is chosen based on the following: M k (cid:88) m || A (cid:63)mk B (cid:63)mk || (cid:80) Mm || A mk B mk || ≥ δ % (50)where {|| A (cid:63) k B (cid:63) k || , · · · , || A (cid:63)Mk B (cid:63)Mk ||} representsthe sorted (in descending order) set of the set {|| A k B k || , · · · , || A Mk B Mk ||} . Therefore, by applyingthe presented AP selection scheme, each access point m serves a subset K m of K users. Hence, the power allocationschemes proposed in the preceding subsections will allocatepower γ (cid:63)mk = γ mk if k, m ∈ K m , M k , respectively, and γ (cid:63)mk = L × otherwise. Therefore, Algorithm 2 can bedirectly applied where Γ mk is replaced by L × L when m / ∈ M k for k ∈ K m .VII. S IMULATION R ESULTS
In this section, we study the performance of the proposedmultipath components estimation compared to conventional Algorithm 2
SDR-based Bisection Algorithm for Solving (47) Initialization:
Define the initial values µ max , µ min that represent the range of relevant values of the objective function in (47), andChoose a tolerance (cid:15) > Set: µ = µ max + µ min , Solve the following convex SDP feasibility program: ρ d (cid:80) Mm =1 tr (cid:0) Ξ mkk Ξ H mkk Γ mk (cid:1) ≥ µ (cid:16) ρ d (cid:80) Kj (cid:54) = k (cid:80) Mm =1 tr (cid:0) Ξ mkj Ξ H mkj Γ mj (cid:1) + ρ d (cid:80) Kj =1 (cid:80) Mm =1 Ω m,j + σ n (cid:17) , ∀ k, (cid:80) Kk =1 tr ( ˆ G H mk ˆ G mk Γ mk ) || ˆ G mk || ≤ , ∀ m, and Γ mk (cid:23) , ∀ k, ∀ m, (48) if If problem (48) is feasible, then set µ min = µ else set µ max = µ . end if Stop if µ max − µ min < (cid:15) . Otherwise , go to Step 2. [ U mk , V mk ] = EVD ( Γ mk ) , ∀ k, ∀ m, where V L × L is the diagonal matrix of eigenvalues, and U L × L is a full matrix whose columnsare the corresponding eigenvectors ( u ). γ mk = (cid:112) max( V mk ) u max m,k , ∀ k, ∀ m, where u max is the corresponding eigenvector to the maximum eigenvalue in V . end schemes, and we provide numerical results to quantitativelystudy the performance of FDD cell-free massive MIMO interms of downlink and uplink spectral efficiency for all theproposed beamforming and combining techniques. A. Experimental Setup and Parameters
The APs and the users are located within a square of × km . The square is wrapped around at the edges toavoid boundary effects. Furthermore, for simplicity, randompilot assignment is used. With random pilot assignment, eachuser randomly chooses a pilot sequence from a predefined setof orthogonal pilot sequences of length τ = K . The large-scale fading coefficient β l,mk is modeled as the product ofpath loss and shadow fading as in [31]:
10 log ( β l,mk ) = (cid:26) P − . ( u mk ) + z mk,l −
15 log ( u ) , if u mk > u ; P −
35 log ( u mk ) + z mk,l , if u mk ≤ u . where u mk is the distance between the m th AP and k th userin kilometers, z mk,l ∼ N (0 , σ z ) is the shadow fading variablewith σ z = 8 dB , u = 0 .
05 km and P = −
148 dB for line-of-sight (LOS) and P = −
158 dB for non-line-of-sight (NLOS)propagation.Moreover, for the AP selection schemes, we choose δ =95% . The system parameters used throughout the experimentalsimulations are summarized in Table II. B. Results and Discussions1) Performance of Multipath Component Estimation:
InFig. 2, the root mean-square error (RMSE) of the presentedmultipath component estimation technique is evaluated for N = 32 and T = 16 . We compare the performance ofthe presented method with that of MUSIC and ESPRITalgorithms, which are subspace-based multipath componentestimation techniques that depend on the correlation matrixof the received data [22], [23] and the gradient-descent-basedalgorithm [31]. The plots demonstrate that the proposed DFT-based technique outperforms the conventional approaches in[22], [23] and [31]. Also, the normalized RMSE performance Table IIS
IMULATION P ARAMETERS
Parameter
ValueCell radius ( D ) System Bandwidth ( B )
100 MHz
Uplink/Downlink Frequencies . /
50 GHz
Uplink pilot training transmit power ρ
200 mW
Uplink transmit power ρ u
200 mW
Downlink transmit power ρ d Power amplifier parameter ϑ . Internal power consumption/each backhaul, P tc ,m ∀ m [7] . Fixed power consumption/each backhaul, P ,m ∀ m [7] .
825 W
Traffic dependent backhaul power, P bt ,m ∀ m [7] .
25 W / (Gbits / s) User Centric threshold ( δ ) Angle coherence interval ( τ c ) samplesMonte-Carlo Simulations 1000 SNR(dB) -40-35-30-25-20-15-10-505 R M SE ( d B ) Theor RMSE( , L=1)DFT-based ( , L=2)DFT-based ( , L=3)GD-based ( , L=2)ESPRIT ( , L=2)MUSIC ( , L=2)Theor MSE( , L=1)DFT-based ( , L=2)DFT-based ( , L=3)GD-based ( , L=2)ESPRIT ( , L=2)MUSIC ( , L=2)
Figure 2. RMSE performance of the multipath component estimation versusSNR for N = 32 and T = 16 compared with the gradient-descent basedestimation and subspace-based estimation. of the proposed large-scale fading coefficient estimation out-performs that of conventional subspace-based estimation [22],[23] and gradient-descent-based estimation [31]. The largescale fading estimation in [22], [23], [31] cannot work wellwhen number of samples (snapshots) T is small.Moreover, it can be seen that the presented AoA estimationand the large-scale fading estimation method performs slightlyworse than that of theoretical bound in (22) since the searchgrid is large enough ( G = 100 ). -10 -5 0 5 10 15 20 25 SNR(dB) s pe c t r a l e ff i c i en cy ( b i t/ s e c / H z ) A-MMSE AnalyticalA-MMSE SimulationA-ZF AnalyticalA-ZF SimulationA-MF AnalyticalA-MF SimulationMMSEZFMF (a) -10 -5 0 5 10 15 20 25
SNR(dB) s pe c t r a l e ff i c i en cy ( b i t/ s e c / H z ) A-MMSE AnalyticalA-MMSE SimulationA-ZF AnalyticalA-ZF SimulationA-MF AnalyticalA-MF SimulationMMSEZFMF (b)Figure 3. Spectral efficiency of the proposed beamforming schemes versus SNR for M = 10 APs with N = 32 antennas and K = 20 users under imperfectchannel estimation: (a) for DL, and (b) UL. Number of APs S pe c t r a l e ff i c i en cy ( b i t/ s e c / H z ) A-MMSE K=10A-ZF K=10A-MMSE K=20A-ZF K=20A-MMSE K=40A-ZF K=40 (a)
Num. of antennas N S pe c t r a l e ff i c i en cy ( b i t/ s e c / H z ) A-MMSE SNR=30dBA-ZF SNR=30dBA-MMSE SNR=10dBA-ZF SNR=10dBA-MMSE SNR=0dBA-MMSE SNR=0dB (b)Figure 4. DL sum-rate of the proposed combining schemes versus (a) number of APs at SNR=
10 dB for MN = 320 and K = { , , } users, and (b)versus number of antennas N at various SNR values for M = 10 APs and K = 20 users. -10 -5 0 5 10 15 20 25 SNR(dB) s pe c t r a l e ff i c i en cy ( b i t/ s e c / H z ) A-MMSE with equal power alloc.A-ZF with equal power alloc.A-MMSE with water-filling power alloc.A-ZF with water-filling power alloc.A-MMSE with proposed Max-Min power alloc.A-ZF with proposed Max-Min power alloc. (a) -10 -5 0 5 10 15 20 25
SNR(dB) s pe c t r a l e ff i c i en cy ( b i t/ s e c / H z ) A-MMSE with equal power alloc.A-ZF with equal power alloc.A-MMSE with proposed Max-Min power alloc.A-ZF with proposed Max-Min power alloc. (b)Figure 5. Spectral efficiency of the proposed combining schemes with equal power control, water-filling power control and the proposed max-min powercontrol versus SNR for M = 10 APs, and K = 20 users for the Cell-Free (CF) massive MIMO (AP selection is not applied): (a) DL and (b) UL.
2) Performance of Spectral Efficiency:
We compare theperformance of the proposed angle-based beamforming andcombining schemes (A-MF/A-ZF and A-MMSE) for the FDD- based cell-free massive MIMO with the conventional idealbeamforming and combining schemes (MF/ZF and MMSE)in terms of spectral efficiency for the case of M = 10 APs -10 -5 0 5 10 15 20 25 SNR(dB) s pe c t r a l e ff i c i en cy ( b i t/ s e c / H z ) A-MMSE with equal power alloc.A-ZF with equal power alloc.A-MMSE with water-filling power alloc.A-ZF with water-filling power alloc.A-MMSE with proposed Max-Min power alloc.A-ZF with proposed Max-Min power alloc. (a) -10 -5 0 5 10 15 20 25
SNR(dB) s pe c t r a l e ff i c i en cy ( b i t/ s e c / H z ) A-MMSE with equal power alloc.A-ZF with equal power alloc.A-MMSE with proposed Max-Min power alloc.A-ZF with proposed Max-Min power alloc. (b)Figure 6. Same as Fig. 5 but applying the user centric (UC) AP selection scheme: (a) DL and (b) UL.
Spectral Efficiency bps/Hz CD F CF with equal power alloc.CF with water-filling power alloc.CF with Max-Min power alloc.UC with equal power alloc.UC with water-filling power alloc.UC with Max-Min power alloc. (a)
Num. of APs E ne r g y E ff i c i en cy ( M b i t/ J ou l e ) CF A-MMSE with equal power alloc.CF A-ZF with equal power alloc.CF A-MMSE with proposed Max-Min power alloc.CF A-ZF with proposed Max-Min power alloc.UC A-MMSE with equal power alloc.UC A-ZF with equal power alloc.UC A-MMSE with proposed Max-Min power alloc.UC A-ZF with proposed Max-Min power alloc. (b)Figure 7. (a) Cumulative distribution of the spectral efficiency for all power control schemes with/without applying the proposed AP selection (CF/UC), and(b) DL energy efficiency of the proposed combining schemes with equal power control and max-min power control versus number of APs. Here, SNR=
10 dB for M = 10 , N = 32 , and K = 20 users. No. of antennas at UE S pe c t r a l e ff i c i en cy ( b i t/ s e c / H z ) K=10K=20K=40
Figure 8. DL spectral efficiency versus multiple antenna configurations at theusers for K = { , , } , M = 10 , and N = 32 . with N = 32 antennas and K = 20 users. We consider theconventional full-channel-based beamforming and combiningschemes (MF/ZF and MMSE) as benchmarks, but they are in-applicable in a realistic FDD cell-free massive MIMO system since complete channel knowledge requires large amount ofsignaling overhead and feedback.For the downlink scenario in Fig. 3(a), and for the uplinkscenario in Fig. 3(b), the spectral efficiency of the proposedbeamforming/combining schemes with imperfect multipathcomponent estimation is shown. As shown in the figures,the A-MMSE beamforming/combining outperforms A-ZF andA-MF beamforming/combining, due to their ability to sup-press interference and noise. In addition, at high SNR (lownoise) the A-ZF matches A-MMSE in performance as bothof the schemes are able to suppress interference. Moreoverat moderate to high SNR values, A-MMSE, A-ZF, and A-MF lead to about − sum rate loss compared to theconventional ideal beamforming/combining schemes (MF/ZFand MMSE). However, with the proposed angle-based beam-forming schemes, the DL CSI signaling overhead is avoided.Finally, we evaluate the validity of our closed-form expres-sion for the downlink achievable rate for the proposed anglebased beamformers given in (38) with imperfect multipathcomponent estimation. In Fig. 3(a), we show the accuracyof the proposed closed form of the proposed angle based beamformers (38) with the simulated form (51) ˜ R d = K (cid:88) k =1 E (cid:34) log (cid:32) ρ d (cid:80) Mm =1 || h H mk ˆ w mk || ρ d (cid:80) Kj (cid:54) = k (cid:80) Mm =1 || h H mk ˆ w mj || + σ n (cid:33)(cid:35) . (51)Moreover, (51) represents the achievable rate for genie-aidedusers that know the instantaneous channel gain [6].In Fig. 3(b), we also validate the closed-form expressionfor the uplink achievable rate for the proposed angle basedcombining given in (39) for imperfect multipath componentestimation with simulated form (52) ˜ R u = K (cid:88) k =1 E (cid:34) log (cid:32) ρ u (cid:80) Mm =1 || h H mk ˆ v mk || ρ u (cid:80) Kj (cid:54) = k (cid:80) Mm =1 || h H mk ˆ v mj || + Υ σ (cid:33)(cid:35) . (52)where Υ σ = σ n (cid:80) Mm =1 || ˆ v mk || .One can notice that the closed form achievable rate perfectlymatches with Monte Carlo simulated rates. This indicatesthat our derived expressions (38) and (39) are valid perfor-mance predictors of the proposed FDD-based cell-free massiveMIMO system.
3) Effect of the Number of APs M for a Fixed Total Numberof Service Antennas ( N M ) : Furthermore, we examine theperformance of the proposed FDD-based cell-free massiveMIMO system with different numbers of APs for the downlinkcase. For fair comparison, the total transmit power in thenetwork is the same, and the number of total service antennasis fixed, i.e.
N M = 320 . Figure 4(a) shows the averagespectral efficiency ( κ × (cid:80) Kk R d k where κ = 1 − ττ c , τ = K corresponds to the length of pilot training sequence in samples,and τ c corresponds to the angle coherence interval in samples)as a function of the number of APs. We are able to comparethe spectral efficiency of cell-free massive MIMO and co-located massive MIMO where the co-located massive MIMOcorresponds to the case M = 1 . It can be seen that thespectral efficiency of the cell-free massive MIMO (for M = 10 and N = 32 ) is better than that of the co-located massiveMIMO ( M = 1 and N = 320 ) due to spatial diversity gains.However, as the number of APs increases while decreasingthe number of antennas per AP, the performance of the cell-free massive MIMO starts to decay. The main reasons for thisdecay are: 1) for a particular user, there are many APs whichare located very far away. These APs will not add significantlyto the overall spatial diversity gains which implies that not allAPs really participate in serving this user; and 2) angle-basedbeamforming performs better for higher number of antennas.
4) Effect of the Number of Antennas per AP:
Finally, tosupport our findings in Fig. 4(a), we study the performanceof FDD-based cell-free massive MIMO system with differentnumbers of antennas per AP for a fixed number of APs ( M =10 ) in Fig. 4(b). As the number of antennas increases, thespectral efficiency increases due to the increased array gainin addition to the applied angle-based beamforming. It can beseen that the spectral efficiency saturates for N ≥ as nofurther gains are attained.
5) Performance of the Proposed Power/Weight Control onDL/UL Spectral Efficiency:
We compare the DL/UL spectralefficiency performance of the proposed angle-based beam-forming and combining schemes (A-ZF and A-MMSE) for the FDD-based cell-free massive MIMO with equal power alloca-tion, water-filling power allocation and the proposed max-minpower/weight control for the CF case (AP selection is not ap-plied) and the UC case (AP selection is applied). One can notethat the water-filling PC approach is based only on the angleand large-scale fading parameters in which the allocated poweris ρ mk = max { K m (cid:0) ρ tot + (cid:80) k ∈K m σ n ( || A mk B mk || ) − (cid:1) − σ n ( || A mk B mk || ) − , } , where ρ tot = K m ρ d is the totalpower, and K m = K only if the UC AP selection is notapplied. Moreover, the water-filling PC approach is applicablein the DL direction, since only the APs have the knowledgeof the angle and large scale fading parameters, whereas forthe UL direction the users cannot have this information dueto the incurred high signaling overhead.For the downlink scenario in Figs. 5(a) and 6(a), and for theuplink scenario in Figs. 5(b) and 6(b), the spectral efficiencyusing the proposed max-min power/weight control schemes issignificantly enhanced compared to the case of equal powercontrol and water-filling power control, especially at high SNRvalues. In particular, as shown in Fig. 5(a), the DL sum-rateof the proposed A-MMSE and A-ZF beamforming using max-min power control is increased by - compared to theequal power allocation case. While, in Fig. 5(b), the UL sumrate of the proposed A-MMSE and A-ZF combining usingmax-min weight control is increased by - due to thefact that the downlink uses more power (since ρ d > ρ u )and has more power control coefficients to choose than theuplink does, hence the DL performance is better than the ULperformance. Moreover, as shown in Figs. 5, and 6, the UCapproach has better performance than that of the CF case sincethe UEs obtain very noisy signals from the far APs, and notall APs actually participate in serving the users.In addition, the cumulative distribution function (CDF)curve for the proposed max-min power control scheme isplotted in Fig. 7(a), and compared with the equal PC and thewater-filling PC schemes at SNR = 10 dB . As expected, themax-min PC scheme was able to outperform the rest of thePC schemes and improve the system fairness for both casesCF and UC, respectively.
6) Energy Efficiency versus Number of APs M and aFixed Total Number of Service Antennas ( N M ) : Figure 7(b)examines the energy efficiency (40) as a function of thenumber of AP for a fixed total number of service antennas,when the number of AP increases, the number of antennas perAP decreases. As shown, the energy efficiency while applyingthe proposed max-min power control significantly outperformsthat of equal power control by - , especially when theUC AP selection scheme is applied. Furthermore, we are ableto compare the energy efficiency of cell-free massive MIMOand co-located massive MIMO where the co-located massiveMIMO corresponds to the case M = 1 . It can be seen that theenergy efficiency of the cell-free massive MIMO (for M = 10 and N = 32 ) is better than that of the co-located massiveMIMO ( M = 1 and N = 320 ) due to spatial diversity gains,and better spectral efficiency as shown in Fig. 4(a). Moreover,the number of APs will affect the level of backhaul powerconsumption; therefore, as the number of APs increases whiledecreasing the number of antennas per AP, the performance of the cell-free massive MIMO starts to decay due to theincreased backhaul power consumption as shown in (43).
7) Multi-antenna Users extension:
In this subsection, wefinally study the effect of having multi-antenna users on theproposed FDD cell-free massive MIMO system where eachuser is equipped with N (cid:48) antennas. First, the updated channelmodel is given by H N × N (cid:48) = (cid:114) L A AP BΛ α ( A UE ) H , (53)where A AP N × L = [ a (cid:0) φ AP1 (cid:1) , . . . , a (cid:0) φ AP L (cid:1) ] , B L × L = diag ( √ β , . . . , √ β L ) , ( Λ α ) L × L = diag ( α , . . . , α L ) , and A UE N (cid:48) × L = [ a (cid:0) φ UE1 (cid:1) , . . . , a (cid:0) φ UE L (cid:1) ] . Moreover, the DL spectralefficiency per user is given by κ × ˜ R d = (cid:16) − ττ c (cid:17) × K (cid:88) k =1 E (cid:20) log (cid:18) ρ d (cid:80) Mm =1 || ˆ v H m(cid:63)k H H mk ˆ w mk || ρ d (cid:80) Kj (cid:54) = k (cid:80) Mm =1 || ˆ v H m(cid:63)k H H mk ˆ w mj || + σ n (cid:19)(cid:21) , (54)where ˆ v m (cid:63) k corresponds to the combining vector at the multi-antenna k th user that is based on the estimated AoA of theuser from the strongest AP m (cid:63) . Moreover, the combiningvector ˆ v m (cid:63) k follows the same definition as the combiningvector defined in Section IV-B eq. (34), but in this case ˆ C m (cid:63) = ˆ A UE m (cid:63) (cid:16) ( ˆ A UE m (cid:63) ) H ˆ A UE m (cid:63) (cid:17) − , and the beamforming vector ˆ w mk follows the same definition as the A-ZF combiningvector defined in Section IV-A. The strongest AP m (cid:63) is theAP that has the best channel quality with k th user. Onecan note that only the m (cid:63) th AP will need to feed back thecombining vector ˆ v m (cid:63) k to the k th user; hence, no extensivesignaling overhead is needed from all the APs to feed backthe estimated multipath components to the k th user. Finally,note that τ = KN (cid:48) depends on the number of users K andscales linearly with the number of antennas at the users N (cid:48) .Therefore, the factor (1 − ττ c ) is an important limiting factorwhen determining the achievable rates for multi-antenna users.In Fig. 8, the performance of the simulated DL spectralefficiency is studied assuming that RMSE ˆ φ AP = RMSE ˆ φ UE = RMSE ˆ β = −
18 dB . As shown, the DL spectral efficiency firstincreases when the number of antennas per user increases.However, this spectral efficiency will reach a peak value andthen decrease when the number of antennas per user increases.This is due to the fact that although the spatial diversityper user increases, the multipath channel estimation overhead(the training duration relative to the angle coherence interval)also increases. This channel estimation overhead becomesdominant when N (cid:48) and K are large.VIII. C ONCLUSION
In this paper, an FDD-based cell-free massive MIMO sys-tem that directly acquires multipath components from theuplink pilot signal and processes them for AP cooperation hasbeen considered. It has been shown that an FDD-based cell-free massive MIMO system is a viable alternative comparedto a TDD-based system in which angle reciprocity can beexploited to avoid DL CSI feedback and overhead. A lowcomplexity multipath component (AoA and large-scale fading)estimation technique based on DFT operation, along with angle rotation with very small amount of training overheadand feedback cost, has been presented. To evaluate the benefitsof the proposed methods, theoretical bounds on the MSEhave been derived and validated. In addition, angle-basedbeamformers and combiners, which incur CSI overhead thatscales only with the number of served users rather than thetotal number of serving antennas, have been proposed. Finally,a new max-min power/weight control algorithm and associatedAP selection scheme that significantly improve the downlinkand uplink sum-rate and energy efficiency compared to equal-power allocation and water-filling power control have beenproposed.The spectral efficiency of the presented FDD-based cell-freemassive MIMO system has been shown to outperform thatof cell-based systems for an adequate number of antennas atthe APs and a small number of APs. Furthermore, when thenumber of active users in the system is small, the spectralefficiency also improves upon equipping the users with anadequate number of antennas.R
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