I Introduction
With the growing demand on high datarate wireless communications, fifth generation (5G) cellular mobile communications has emerged as the latest generation to offer 1000fold capacity enhancement over current fourth generation (4G) LongTerm Evolution (LTE) systems with reduced latency. To achieve this aggressive goal, massive multipleinput multipleoutput (MIMO) and network densification are promising 5G wireless technologies that improve the capacity of cellular systems by 1) scaling up the number of antennas in a conventional MIMO system by orders of magnitude [Marzetta2010MIMO, Marzetta2014mimo], and 2) reducing pathloss and reusing spectrum [Ji2017OverviewMIMOLTE] efficiently.
Although massive MIMO and network densification bring forward several advantages, the performance of cellular networks is limited by intercell interference (ICI) and frequent handovers for fast moving users. In particular, users close to the cell edge suffer from strong interference.
Cellfree (CF) massive MIMO has recently been considered as a practical and useful embodiment of network MIMO that can potentially reduce such intercell interference through coherent cooperation between base stations [Nayebi2015CFmMIMO, Ngo2017CellFreevsSmallCell, Ngo2018EEofCF, interdonato2018ubiquitous]
. In cellfree massive MIMO, the serving antennas are distributed over a large area. Distributed systems can potentially provide higher coverage probability than colocated massive MIMO due to their ability to efficiently exploit diversity against shadow fading effects, at the cost of increased backhaul requirements
[Zhou2003Dist].According to [interdonato2018ubiquitous], “cellfree” massive MIMO implies that, from a user perspective during data transmission, all access points (APs) cooperate to jointly serve the endusers; hence there are no cell boundaries and no intercell interference in the data transmission. The APs are connected to a central processing unit (CPU) via a backhaul link. This approach, with simple signal processing, can effectively control ICI, leading to significant improvements in spectral and energy efficiency over the cellular systems [Ngo2017CellFreevsSmallCell, Ngo2018EEofCF, interdonato2018ubiquitous, Zhou2003Dist, Nayebi2015CFmMIMO].
The main challenge in deploying cellfree networks lies mainly in acquiring sufficiently accurate channel state information (CSI) so that the APs can simultaneously transmit (receive) signals to (from) all user equipments (UEs) and cancel interference in the spatial domain. The conventional approach of sending downlink (DL) pilots and letting the UEs feed back channel estimates is unscalable since the feedback load is proportional to the number of APs. Therefore, to reduce the signaling overhead [Emil2010coop, marzetta2016fundamentals], channel reciprocity can be exploited in timedivision duplex (TDD) mode so that each AP only needs to estimate the uplink CSI.
An attractive alternative to consider is frequencydivision duplexing (FDD) based cellfree massive MIMO systems for the following reasons: 1) channel reciprocity in TDD mode might not be accurate due to calibration errors in radio frequency (RF) chains [Vieira2017calibration], 2) with the lack of downlink training symbols in TDD systems, users may not be able to acquire instantaneous CSI, and thus system performance will deteriorate in detecting and decoding the intended signals, 3) while TDD operation is preferable at sub6 GHz massive MIMO, in millimeter wave (mmWave) bands FDD may be equally good since the angular parameters of the channel are reciprocal over a wide bandwidth [bjornson2018massive], and 4) FDD systems dominate current wireless communications and have many benefits such as lower cost and greater coverage than TDD [qualcommFDDTDD].
On the other hand, FDDbased cellfree massive MIMO systems still suffer from CSI acquisition and feedback overhead since the amount of downlink CSI feedback scales linearly with the number of antennas [Lee2015FDDmMIMO] and the number of APs in cellfree massive MIMO system. However, we can still benefit from 1) angle reciprocity, which holds true for FDD systems as long as the uplink and downlink carrier frequencies are not too far from each other (less than several GHz [Gao2017UnifiedTDDFDD]), and 2) angle coherence time which is much longer than the conventional channel coherence time [Heath2017Beamwidth] where the channel angle information can be regarded as unchanged. Hence, angle information is essential in FDDbased cellfree massive MIMO systems. Therefore, a low complexity estimation approach that can efficiently estimate the angle information is required.
Ia Related Work
Much of the recent interest in cellfree massive MIMO systems has focused mainly on TDDmode only [Ngo2017CellFreevsSmallCell, Ngo2018EEofCF, interdonato2018ubiquitous, Zhou2003Dist, Nayebi2015CFmMIMO, Alonzo2017CFandUCmmWave, Alonzo2019CF, maxminPCUL2019Bashar, Ngo2018CF]. In [Nayebi2015CFmMIMO], a cellfree system is considered and algorithms for power optimization and linear precoding are analyzed. Compared with the conventional smallcell scheme, cellfree massive MIMO can yield more than tenfold improvement in terms of outage rate. While in [Ngo2017CellFreevsSmallCell], the APs perform multiplexing/demultiplexing through conjugate beamforming in the downlink and matched filtering in the uplink.
In [Ngo2018EEofCF], a cellfree massive MIMO downlink is considered, wherein a large number of distributed multipleantenna APs serve many singleantenna users. A distributed conjugate beamforming scheme is applied at each AP via the use of local CSI. Spectral efficiency and energy efficiency are studied while considering channel estimation error and power control.
In [Alonzo2017CFandUCmmWave, Alonzo2019CF], cellfree and usercentric architectures at mmWave frequencies are considered. A multiuser clustered channel model is introduced, and an uplink multiuser channel estimation scheme is described along with hybrid analog/digital beamforming architectures. Moreover, in [Alonzo2019CF], the nonconvex problem of power allocation for downlink global energy efficiency maximization is addressed. In [maxminPCUL2019Bashar], an uplink TDDbased cellfree massive MIMO system is considered. Geometric programming GP is used to suboptimally solve a quasilinear max–min signaltointerferenceandnoise ratio (SINR) problem.
Angle estimation has been studied in other wireless networks without considering cellfree massive MIMO networks (see e.g. [Shmidt1986MUSIC1, Roy1989ESPIRIT, Krim1996ASP, Wang2015DOAmMIMO, Gao2014ESPIRIT, Cheng2015DoA, Shafin2016DoAmmwave, Gao2017DoAestMmwave, Gao2017UnifiedTDDFDD, GAO2018AODest]). For instance, subspacebased angle estimation algorithms, such as multiple signal classification (MUSIC), estimation of signal parameters via rotational invariance technique (ESPRIT) and their extensions have gained interest in the array processing community due to their high resolution angle estimation capability [Shmidt1986MUSIC1, Roy1989ESPIRIT, Krim1996ASP]. Their applications in massive MIMO systems and MIMO systems for angle estimation have been presented in [Wang2015DOAmMIMO, Gao2014ESPIRIT, Cheng2015DoA, Shafin2016DoAmmwave]
. Unfortunately, the classical MUSIC and ESPRIT schemes are not suitable for mmWave communications due to the following main reasons: 1) They have high computational complexity mainly due to the singular value decomposition (SVD) operation on channels with massive number of antennas; 2) They are considered as blind estimation techniques originally targeted for radar applications, and do not make full use of training sequences in wireless communication systems.
In [Gao2017DoAestMmwave, Gao2017UnifiedTDDFDD, GAO2018AODest]
, an AoA estimation scheme for a conventional mmWave massive MIMO system with a uniform planar array at the base station is presented. The initial AoAs of each uplink path are estimated through the twodimensional discrete Fourier transform (2DDFT), and then the estimation accuracy is further enhanced via an angle rotation technique. In the present work, we extend the AoA estimation technique of
[Gao2017DoAestMmwave, Gao2017UnifiedTDDFDD, GAO2018AODest], adapt it to the context of FDDbased cellfree massive MIMO, and employ it to estimate another channel multipath component, namely largescale fading. Using these estimated components, we leverage from the angle coherence time and anglereciprocity to propose lowcomplexity anglebased beamforming/combining schemes and power control algorithms for downlink and uplink directions.In [Kim2018CFreeFDD], a multipath component estimation technique and base station cooperation scheme based on the multipath components for the FDDbased cellfree massive MIMO systems are presented. However, no closedform expression of the meansquareerror (MSE) of the considered multipath estimation is presented.
IB Contributions of the Paper
In this work, we consider a cellfree massive MIMO system with multiple antennas at each AP operating in FDD mode that do not require any feedback from the user. All APs cooperate via a backhaul network to jointly transmit signals to all users in the same timefrequency resources. By exploiting angle reciprocity, APs can acquire multipath component information from the uplink pilot signals using array signal processing techniques. The contributions of this paper are:

We propose a multipath component estimation for the AoA and largescale fading coefficients based on the DFT operation and loglikelihood function with reduced overhead. In particular, we leverage from the observation that the angleofdepartures (AoDs) and the large scale fading components vary more slowly than path gains [Heath2017Beamwidth], as well as from the property of anglereciprocity. We further derive a closedform expression for the MSE of the estimated channel multipath components. Both theoretical and numerical results are provided to verify the effectiveness of the proposed methods. These schemes are shown to provide a substantial enhancement over the gradientbased [Kim2018CFreeFDD] and the classical subspacebased [Shmidt1986MUSIC1, Roy1989ESPIRIT] multipath component estimation in terms of MSE of the estimated AoA and largescale fading coefficients since the MSE of the proposed DFTbased estimator coincides with that of the ML estimator.

We propose linear anglebased beamforming/combining techniques for the downlink/uplink transmission that incorporate the estimated AoA and largescale fading components. Interestingly, the proposed schemes scale only with the number of served users rather than the total number of serving antennas, and need to be updated every angle coherence time. Therefore, the impact of signaling overhead is substantially reduced with the proposed schemes.

We derive closedform expressions for the spectral efficiencies for the FDDbased cellfree massive MIMO downlink and uplink with finite numbers of APs and users. Our analysis takes into account the proposed beamforming/combining techniques and the effect of multipath estimation errors.

We propose a solution to the maxmin power control problem by formulating it as a standard semidefinite programming (SDP) approach. The proposed maxmin power control maximizes the smallest rate of all users within the anglecoherence timescale. In addition, we present a usercentric AP selection scheme to further enhance the energy efficiency of the system.
The rest of the paper is organized as follows. The system model for the FDDbased cellfree massive MIMO network is described in Section II. In Section III, the proposed multipath components estimation is introduced. In Section IV, the proposed beamforming and combining techniques are presented. Moreover, spectral efficiency analysis is introduced in Section V. Case studies with numerical results are simulated and analyzed based on the proposed schemes in Section VII. Section VIII concludes the paper.
Notation
: Bold upper case, bold lower case, and lower case letters correspond to matrices, vectors, and scalars, respectively. Scalar norms, vector
norms, and Frobenius norms, are denoted by , , and , respectively. , , , , , and stand for expected value, transpose, complex conjugate, Hermitian, orthogonal projection matrix, and the trace of a matrix. stands for the pseudoinverse . In addition, is used to indicate that is a positive semidefinite matrix. represents element of a vector .refers to a circularlysymmetric complex Gaussian distribution with zero mean and variance
.Ii System Model
As shown in Fig. 1, we consider an FDDbased cellfree massive MIMO system having APs, each equipped with a uniform linear array (ULA) of antennas, serving users with single antennas. We assume a geometric channel model with propagation paths [Gao2017UnifiedTDDFDD, Kim2018CFreeFDD]. Moreover, AoAs (or AoDs), largescale fading and smallscale fading coefficients are called the multipath components of the channel. Due to angle reciprocity in FDD systems [Gao2017UnifiedTDDFDD], and frequency independency, we assume that 1) the uplink AoA and downlink AoD are similar, and 2) the uplink and downlink largescale fading coefficients (slow fading and distantdependent path loss components) are similar [Marzetta2013ICILargescale, Larsson2018FDDperformance]. However, uplink and downlink smallscale fading coefficients in FDD systems are distinct since they are frequency dependent [Marzetta2013ICILargescale, Larsson2018FDDperformance]. Therefore, the channel vectors can be expressed as [Gao2017UnifiedTDDFDD, Kim2018CFreeFDD]
(1) 
where is the complex gain of the path that represents the smallscale Rayleigh fading, and is the largescale fading coefficient that accounts for pathloss and shadowing effects. The variable is the angle of arrival of the path. The array steering vector is defined as where , is the antenna spacing, and is the channel wavelength (Note that we also define ). Equivalently, the channel vector in (1) can be expressed in matrixvector form as
(2) 
where
(3) 
As mentioned previously, the quantities are dependent on frequency; however and are constant with respect to frequency over an anglecoherence time interval (as discussed in subsection IIID).
To model a realistic system where we have nonideal angle reciprocity, we assume that the differences between uplink and downlink multipath components, and
, are i.i.d. random variables with zero mean and variance
, [hugl2002spatial].Iia Uplink Training
Let be the uplink (UL) pilot signal sent by the user composed of symbols with unit norm. All pilot sequences used by different users are assumed to be pairwise orthogonal, since the angle coherence time is much longer than the conventional channel coherence time [Heath2017Beamwidth]. Therefore, we can assign a sufficiently large number to such that holds true.
Therefore, the received signal at the AP sent by the user is given by
(4) 
where is the uplink transmit power and the entries of the additive white Gaussian noise matrix are independent and identically distributed (i.i.d.) random variables. Multiplying (4) by and collecting samples, we have
(5) 
where and . Then, the samples of (IIA) are collected in a matrix form as
(6) 
where , , , and .
The multipath components estimation is performed in a distributed fashion, in which each AP independently estimates the multipath components to the users. The APs do not cooperate on the multipath components estimation, and no estimates need to be shared among the APs.
IiB Downlink Payload Data Transmission
The APs, based on the estimated multipath components, independently apply beamforming vector to transmit signals to the users. Moreover, APs do not cooperate on the beamforming vectors. The transmit DL signal from the AP is given by
(7) 
where is the data symbol for the user satisfying , and is the maximum transmit power satisfying, . It can be noted here that the multiplexing order is equal to 1.
Then, the received downlink signal at the user is given by
(8) 
where is the additive noise at the user. Note that the received signal can be decomposed into three parts: 1) desired signal part (), 2) interference part (), and 3) noise . Moreover, the user can detect signal from .
IiC Uplink Payload Data Transmission
In the uplink, all users simultaneously send their data symbols , where , to the APs. It can be noted here that the multiplexing order is equal to 1. The received UL signal at the AP is given by
(9) 
where is the uplink transmit power and is additive noise at the AP. The noise entries () are modeled as i.i.d. . The received signal is multiplied by the combiner at each AP where the resulting signal is sent to the CPU through a backhaul to detect the signal. The CPU will receive
(10) 
Then, is detected from .
The main system parameters are summarized in Table I.
Number of APs, and number of antennas per AP  
Total number of users  
Number of paths  
Channel gain for the AP and user  
Angular steering vector for the path  
Angular steering matrix for the AP and user  
Large scale fading matrix  
Small scale fading vector  
DFT matrix 
Iii Proposed Angle information aided channel estimation for FDD systems
In this section, we present the FDDbased cellfree massive MIMO systems that directly acquire multipath components from the uplink pilot signal and use them for the AP cooperation. Using array signal processing, we first present the low complexity DFTbased AoA estimation, and then we propose the largescale fading estimation based on the estimated angle information. Note that we need to estimate both components (AoA, and largescale fading) for every angle coherence interval, in order to apply low complexity beamforming/combining techniques.
Iiia AoA Estimation Algorithm
Based on our previous work [Asmaa2019cellfree], we apply AoA estimation step that relies on the classical DFT estimation and angle rotation. DFT is used to estimate the AoA wherein the peak of the DFT magnitude spectrum can select the column whose steering angle best matches the true AoA.
Moreover, the normalized DFT of the channel matrix is defined as where is an DFT matrix whose element is given by . Most of the channel power is concentrated around largest peaks determined by the elements where (for ) and [GAO2018AODest]. Therefore, the initial AoA estimate for the user is .
Furthermore, the accuracy of the AoA estimation is improved through an angle rotation operation [GAO2018AODest] by incorporating a phaseshift to the initial estimation to obtain more accurate peaks. The angle rotation of the original channel matrix is expressed as , where with is the angle rotation parameter. It is shown in [GAO2018AODest] that the entries of have only nonzero peak elements when the optimal phase shifter satisfies .
Therefore, the estimate can be expressed as , and the estimated AoA matrix is given by
(11) 
IiiB LargeScale Fading Estimation
Based on the AoA estimate and given that in (6
), the probability density function of
for given and over all can be expressed as(12) 
The loglikelihood function can be applied to (12) to give
(13) 
Knowing that is a concave function of and , the optimal estimates and can be obtained by taking a partial derivative with respect to and . Hence, and
(14) 
where is the estimate of which is obtained using array signal processing (DFT operation with angle rotation). Once is obtained, we next estimate the largescale fading coefficients . From (14), we can estimate and the covariance matrix . Note that the original covariance matrix is given by
(15) 
Hence, we can obtain the estimates of the largescale fading coefficients as
(16) 
The proposed multipath component estimation is shown in Algorithm 1, where is the search grid within needed for angle estimation.
Note that the search grid parameter determines the complexity and accuracy of the algorithm. The complexity of the whole algorithm is of the order where the factor comes from the DFT operation and comes from rotation operation over a search grid for all paths over antennas. Moreover, the complexity of the proposed algorithm is less than that of the classical subspace ESPRIT algorithm of complexity , with being the number of snapshots required during blind estimation [Stoeckle2015espiritCom].
IiiC Performance Analysis
Using the same methodology as in [Gao2017DoAestMmwave, GAO2018AODest] in addition to estimating the largescale fading parameter, we derive the theoretical MSE of the AoA estimates and the largescale fading coefficients for the cellfree massive MIMO system. In general, a closedform solution of the MSE for multiple AoA estimations is hard to obtain [Gao2017DoAestMmwave]. An alternative approach is to consider the single user and single propagation path and derive corresponding MSE of and as benchmark [Gao2017DoAestMmwave].
For a single propagation path according to (6), the received training signal at the AP transmitted by the user is given by
(17) 
where is the steering vector with its entry given by .
For brevity, we henceforth omit the subscript representing the link between the AP and the user. The proposed angle estimator can be expressed as
(18) 
where , is the column of , and is the nearest integer to .
Moreover, using (14), the ML estimate of is obtained as
(19) 
The joint ML estimates of and can be obtained from
(20) 
where are the optimizing variables.
Therefore, using (19), the ML estimate of is given by
(21) 
where is the cost function of . For the singlepath case, is the projection matrix onto the subspace spanned by , and represents the steering vector given in (1). For the multipath case, represents the projection matrix onto the subspace spanned by , and is the steering matrix given in (II). As shown in [GAO2018AODest] while including the large scale pathloss parameter , the MSE (IIIC) of the considered DFT estimator coincides with that of the ML estimator (20). Using Lemma 1 in [GAO2018AODest] while including the largescale fading parameter and , the MSE of is expressed as
(22) 
where , is the projection matrix onto the orthogonal space spanned by and is the diagonal matrix given by Based on the fact that and , we further examine the MSE of
(23) 
Using Taylor series expansion, a of firstorder approximation of is given by
(24) 
Moreover,
(26) 
Therefore, the MSE of can be obtained
(27) 
IiiD Angle Coherence Time
Different from the conventional channel coherence time, the angle coherence time is defined as typically an order of magnitude longer, during which the AoDs can be regarded as static [Heath2017Beamwidth]. Specifically, the path AoD in (1) mainly depends on the surrounding obstacles around the BS, which may not physically change their positions often. On the contrary, the path gain of the user depends on a number of unresolvable paths, each of which is generated by a scatter surrounding the user. Therefore, path gains vary much faster than the path AoDs [Heath2017Beamwidth]. Accordingly, the angle coherence time is much longer than the conventional channel coherence time. Therefore, we can leverage from this fact and perform multipath estimation in every angle coherence time instead of the much shorter channel coherence time as the impact of the overhead is substantially reduced.
Iv Proposed Beamforming and Combining Techniques
We next propose the anglebased matchedfiltering, anglebased zeroforcing and anglebased minimummeansquareerror beamforming/combining that incorporate the estimated angle information, and the largescale fading components.
The APs are connected via a backhaul network to a CPU, which sends to the APs the datasymbols to be transmitted to the endusers and receives softestimates of the received datasymbols from all the APs. Neither multipath estimates nor beamforming/combining vectors are transmitted through the backhaul network.
Iva AngleBased Beamforming
The anglebased beamforming (or precoding) vector for the AP and the user is defined as
(28) 
where is the column of defined below for the proposed anglebased beamforming techniques. In addition, is the normalized complex weight for the propagation path that satisfies and . Moreover, using (7),
(29) 
will satisfy the maximum transmit power .
IvA1 AngleBased MatchedFiltering Beamforming (AMF)
The precoder matrix based on the angle information is given by
(30) 
where and are the estimated AoA and largescale fading matrices according to (11) and (16). Moreover, AMF is a simple beamforming approach that only requires the channel multipath components (AoA and largescale fading) of the direct link between the AP and the user. However, the inter user interference is ignored.
IvA2 AngleBased ZeroForcing Beamforming (AZF)
We use AZF beamforming as a means to efficiently suppress interference. To do so, the conventional ZF beamforming employs all the downlink CSI from the users. However, the anglebased ZF beamforming used in this work is distinct from the conventional ZF beamforming in the sense that only the angle information and largescale fading coefficients of the channel are required in the beamforming design. We collect the corresponding array steering vectors into and similarly for . Then, the precoder matrix is given by
(31) 
where beamforming vector is defined as the column of .
A key property of the anglebased ZF beamforming is that the beamforming vector is orthogonal to all other array steering vectors as given below:
(32) 
The pseudoinverse in AZF is more complex than AMF, but the interference is suppressed.
IvA3 AngleBased MMSE Beamforming (AMMSE)
We use an anglebased MMSE beamforming design that can efficiently suppress interference, noise and channel estimation error. The AMMSE strikes a balance between attaining the best signal amplification and reducing the interference. The proposed anglebased MMSE beamforming matrix is given by
(33) 
where
such that and , where and account for nonideal DL angle reciprocity, and , are the MSEs as defined in (22) and (IIIC), respectively.
Therefore, for AZF/AMMSE, the only overhead for DL channel acquisition at each AP comes from UL training, which only scales with the number of served users. In addition, one can note that AZF is suitable for high signaltonoise ratio (SNR) conditions since it is expected that AZF and AMMSE would have the same performance when the effect of noise is low.
IvB AngleBased Combining
Similarly, the combining vector for the AP and the user is defined as
(34) 
where is the column of which corresponds to , and and .
Using ULDL duality [bjornson2017massive], the combining vectors of the uplink case for AMF combining, AZF combining and AMMSE combining are also defined as
(35) 
such that and . The corresponding combining matrices were defined in (30), (31) and (IVA3).
The benefits of relying on only the angle information and largescale fading are: (i) the need for downlink training is avoided; (ii) the beamforming/combining matrices can be updated every angle coherence time, and (iii) a simple closedform expression for the spectral efficiency can be derived which enables us to obtain important insights.
V Spectral and Energy efficiency Analysis
In this section, we derive closedform expressions for the spectral efficiencies per user for DL and UL transmissions using the analysis technique from [Ngo2017CellFreevsSmallCell, Ngo2018EEofCF, Kim2018CFreeFDD]. Then, we define the total energy efficiency of the system.
Va Spectral Efficiency
The downlink spectral efficiency per user using the proposed beamforming schemes is given by
(36) 
where
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