Efficient Angle-Domain Processing for FDD-based Cell-free Massive MIMO Systems

01/21/2020 ∙ by Asmaa Abdallah, et al. ∙ American University of Beirut 0

Cell-free massive MIMO communications is an emerging network technology for 5G wireless communications wherein distributed multi-antenna access points (APs) serve many users simultaneously. Most prior work on cell-free massive MIMO systems assume time-division duplexing mode, although frequency-division duplexing (FDD) systems dominate current wireless standards. The key challenges in FDD massive MIMO systems are channel-state information (CSI) acquisition and feedback overhead. To address these challenges, we exploit the so-called angle reciprocity of multipath components in the uplink and downlink, so that the required CSI acquisition overhead scales only with the number of served users, and not the number of AP antennas nor APs. We propose a low complexity multipath component estimation technique and present linear angle-of-arrival (AoA)-based beamforming/combining schemes for FDD-based cell-free massive MIMO systems. We analyze the performance of these schemes by deriving closed-form expressions for the mean-square-error of the estimated multipath components, as well as expressions for the uplink and downlink spectral efficiency. Using semi-definite programming, we solve a max-min power allocation problem that maximizes the minimum user rate under per-user power constraints. Furthermore, we present a user-centric (UC) AP selection scheme in which each user chooses a subset of APs to improve the overall energy efficiency of the system. Simulation results demonstrate that the proposed multipath component estimation technique outperforms conventional subspace-based and gradient-descent based techniques. We also show that the proposed beamforming and combining techniques along with the proposed power control scheme substantially enhance the spectral and energy efficiencies with an adequate number of antennas at the APs.

READ FULL TEXT VIEW PDF
POST COMMENT

Comments

There are no comments yet.

Authors

page 1

This week in AI

Get the week's most popular data science and artificial intelligence research sent straight to your inbox every Saturday.

I Introduction

With the growing demand on high data-rate wireless communications, fifth generation (5G) cellular mobile communications 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. To achieve this aggressive goal, massive multiple-input multiple-output (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 path-loss and reusing spectrum [Ji2017OverviewMIMOLTE] efficiently.

Although massive MIMO and network densification bring forward several advantages, the performance of cellular networks is limited by inter-cell interference (ICI) and frequent handovers for fast moving users. In particular, users close to the cell edge suffer from strong interference.

Cell-free (CF) massive MIMO has recently been considered as a practical and useful embodiment of network MIMO that can potentially reduce such inter-cell interference through coherent cooperation between base stations [Nayebi2015CFmMIMO, Ngo2017CellFreevsSmallCell, Ngo2018EEofCF, interdonato2018ubiquitous]

. In cell-free massive MIMO, the serving antennas are distributed over a large area. Distributed systems can potentially provide higher coverage probability than co-located 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], “cell-free” massive MIMO implies that, from a user perspective during data transmission, all access points (APs) cooperate to jointly serve the end-users; hence there are no cell boundaries and no inter-cell 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 cell-free 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 time-division duplex (TDD) mode so that each AP only needs to estimate the uplink CSI.

An attractive alternative to consider is frequency-division duplexing (FDD) based cell-free 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 sub-6 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, FDD-based cell-free 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 cell-free 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 FDD-based cell-free massive MIMO systems. Therefore, a low complexity estimation approach that can efficiently estimate the angle information is required.

I-a Related Work

Much of the recent interest in cell-free massive MIMO systems has focused mainly on TDD-mode only [Ngo2017CellFreevsSmallCell, Ngo2018EEofCF, interdonato2018ubiquitous, Zhou2003Dist, Nayebi2015CFmMIMO, Alonzo2017CFandUCmmWave, Alonzo2019CF, maxminPCUL2019Bashar, Ngo2018CF]. In [Nayebi2015CFmMIMO], a cell-free system is considered and algorithms for power optimization and linear precoding are analyzed. Compared with the conventional small-cell scheme, cell-free massive MIMO can yield more than ten-fold improvement in terms of outage rate. While in [Ngo2017CellFreevsSmallCell], the APs perform multiplexing/de-multiplexing through conjugate beamforming in the downlink and matched filtering in the uplink.

In [Ngo2018EEofCF], a cell-free massive MIMO downlink is considered, wherein a large number of distributed multiple-antenna APs serve many single-antenna 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], cell-free and user-centric 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 non-convex problem of power allocation for downlink global energy efficiency maximization is addressed. In [maxminPCUL2019Bashar], an uplink TDD-based cell-free massive MIMO system is considered. Geometric programming GP is used to sub-optimally solve a quasi-linear max–min signal-to-interference-and-noise ratio (SINR) problem.

Angle estimation has been studied in other wireless networks without considering cell-free massive MIMO networks (see e.g.  [Shmidt1986MUSIC1, Roy1989ESPIRIT, Krim1996ASP, Wang2015DOAmMIMO, Gao2014ESPIRIT, Cheng2015DoA, Shafin2016DoAmmwave, Gao2017DoAestMmwave, Gao2017UnifiedTDDFDD, GAO2018AODest]). For instance, subspace-based 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 two-dimensional discrete Fourier transform (2D-DFT), 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 FDD-based cell-free massive MIMO, and employ it to estimate another channel multipath component, namely large-scale fading. Using these estimated components, we leverage from the angle coherence time and angle-reciprocity to propose low-complexity angle-based 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 FDD-based cell-free massive MIMO systems are presented. However, no closed-form expression of the mean-square-error (MSE) of the considered multipath estimation is presented.

I-B Contributions of the Paper

In this work, we consider a cell-free 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 time-frequency 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:

  1. We propose a multipath component estimation for the AoA and large-scale fading coefficients based on the DFT operation and log-likelihood function with reduced overhead. In particular, we leverage from the observation that the angle-of-departures (AoDs) and the large scale fading components vary more slowly than path gains [Heath2017Beamwidth], as well as from the property of angle-reciprocity. We further derive a closed-form 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 gradient-based [Kim2018CFreeFDD] and the classical subspace-based [Shmidt1986MUSIC1, Roy1989ESPIRIT] multipath component estimation in terms of MSE of the estimated AoA and large-scale fading coefficients since the MSE of the proposed DFT-based estimator coincides with that of the ML estimator.

  2. We propose linear angle-based beamforming/combining techniques for the downlink/uplink transmission that incorporate the estimated AoA and large-scale 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.

  3. We derive closed-form expressions for the spectral efficiencies for the FDD-based cell-free 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.

  4. We propose a solution to the max-min power control problem by formulating it as a standard semi-definite programming (SDP) approach. The proposed max-min power control maximizes the smallest rate of all users within the angle-coherence time-scale. In addition, we present a user-centric 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 FDD-based cell-free 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 pseudo-inverse . In addition, is used to indicate that is a positive semi-definite matrix. represents element of a vector .

refers to a circularly-symmetric complex Gaussian distribution with zero mean and variance

.

Figure 1: Cell-free massive MIMO system model

Ii System Model

As shown in Fig. 1, we consider an FDD-based cell-free 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), large-scale fading and small-scale fading coefficients are called the multipath components of the channel. Due to angle reciprocity in FDD systems  [Gao2017UnifiedTDDFDD], and frequency in-dependency, we assume that 1) the uplink AoA and downlink AoD are similar, and 2) the uplink and downlink large-scale fading coefficients (slow fading and distant-dependent path loss components) are similar [Marzetta2013ICILargescale, Larsson2018FDDperformance]. However, uplink and downlink small-scale 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 small-scale Rayleigh fading, and is the large-scale fading coefficient that accounts for path-loss 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 matrix-vector form as

(2)

where

(3)

As mentioned previously, the quantities are dependent on frequency; however and are constant with respect to frequency over an angle-coherence time interval (as discussed in subsection  III-D).

To model a realistic system where we have non-ideal 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].

Ii-a 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  (II-A) 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.

Ii-B 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 .

Ii-C 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
Table I: System Parameters

Iii Proposed Angle information aided channel estimation for FDD systems

In this section, we present the FDD-based cell-free 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 DFT-based AoA estimation, and then we propose the large-scale fading estimation based on the estimated angle information. Note that we need to estimate both components (AoA, and large-scale fading) for every angle coherence interval, in order to apply low complexity beamforming/combining techniques.

Iii-a 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 phase-shift 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 non-zero peak elements when the optimal phase shifter satisfies .

Therefore, the estimate can be expressed as , and the estimated AoA matrix is given by

(11)
1:Input: , , and
2:Output: ,
3:// AoA Estimation
4:for  do
5:     for  do
6:          Find the central point () of each bin in where
7:           where is the column of .
8:                
9:     end
10:     
11:end
12:// Large scale fading Estimation
13:, where
14:
15:
16:end
Algorithm 1 Extended DFT and Angle-Rotation-Based Multipath Component Estimation

Iii-B Large-Scale 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 log-likelihood 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 large-scale 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 large-scale 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].

Iii-C Performance Analysis

Using the same methodology as in [Gao2017DoAestMmwave, GAO2018AODest] in addition to estimating the large-scale fading parameter, we derive the theoretical MSE of the AoA estimates and the large-scale fading coefficients for the cell-free massive MIMO system. In general, a closed-form 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 single-path case, is the projection matrix onto the subspace spanned by , and represents the steering vector given in  (1). For the multi-path 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 path-loss parameter , the MSE (III-C) of the considered DFT estimator coincides with that of the ML estimator (20). Using Lemma 1 in [GAO2018AODest] while including the large-scale 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 first-order approximation of is given by

(24)

Substituting  (24) into  (19) and after collecting samples, we rewrite as

(25)

where .

Moreover,

(26)

Therefore, the MSE of can be obtained

(27)

Furthermore, the MSE expressions of the estimated AoA and large-scale fading components derived in (22) and (III-C) give important insights when assessing the impact of beamforming/combining techniques on the spectral efficiency of the proposed FDD-based cell-free massive MIMO system.

Iii-D 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 angle-based matched-filtering, angle-based zero-forcing and angle-based minimum-mean-square-error beamforming/combining that incorporate the estimated angle 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 transmitted to the end-users and receives soft-estimates of the received data-symbols from all the APs. Neither multipath estimates nor beamforming/combining vectors are transmitted through the backhaul network.

Iv-a Angle-Based Beamforming

The angle-based beamforming (or precoding) vector for the AP and the user is defined as

(28)

where is the column of defined below for the proposed angle-based 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 .

Iv-A1 Angle-Based Matched-Filtering Beamforming (A-MF)

The precoder matrix based on the angle information is given by

(30)

where and are the estimated AoA and large-scale fading matrices according to  (11) and  (16). Moreover, A-MF is a simple beamforming approach that only requires the channel multipath components (AoA and large-scale fading) of the direct link between the AP and the user. However, the inter user interference is ignored.

Iv-A2 Angle-Based Zero-Forcing Beamforming (A-ZF)

We use A-ZF 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 angle-based ZF beamforming used in this work is distinct from the conventional ZF beamforming in the sense that only the angle information and large-scale 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 angle-based ZF beamforming is that the beamforming vector is orthogonal to all other array steering vectors as given below:

(32)

The pseudo-inverse in A-ZF is more complex than A-MF, but the interference is suppressed.

Iv-A3 Angle-Based MMSE Beamforming (A-MMSE)

We use an angle-based MMSE beamforming design that can efficiently suppress interference, noise and channel estimation error. The A-MMSE strikes a balance between attaining the best signal amplification and reducing the interference. The proposed angle-based MMSE beamforming matrix is given by

(33)

where

such that and , where and account for non-ideal DL angle reciprocity, and , are the MSEs as defined in (22) and  (III-C), respectively.

Therefore, for A-ZF/A-MMSE, 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 A-ZF is suitable for high signal-to-noise ratio (SNR) conditions since it is expected that A-ZF and A-MMSE would have the same performance when the effect of noise is low.

Iv-B Angle-Based 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 UL-DL duality [bjornson2017massive], the combining vectors of the uplink case for A-MF combining, A-ZF combining and A-MMSE combining are also defined as

(35)

such that and . The corresponding combining matrices were defined in  (30),  (31) and  (IV-A3).

The benefits of relying on only the angle information and large-scale 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 closed-form 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 closed-form 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.

V-a Spectral Efficiency

The downlink spectral efficiency per user using the proposed beamforming schemes is given by

(36)

where