Israa Khaled

Charlotte Langlais

Ammar El Falou

Bachar Elhassan

Michel Jezequel

Low-Complexity Angle-Domain MIMO NOMA System with partial channel state information for MmWave Communications *

In millimeter-wave communication, digital beamsteering (DBS), only based on the user direction, is a promising angle-domain multi-antenna technique to mitigate the severe path loss and multi-user interference, with low-complexity and partial channel state information (CSI). In this paper, we design a power-domain non-orthogonal multiple access (NOMA) scheme that enhances the DBS performance trading-off complexity, energy-consumption and capacity performance. In particular, we propose a user-clustering algorithm to pair users, based on a geometric-interference metric, so that the inter-user interference is reduced. Afterward, based on a fixed inter-cluster power allocation, we derive analytically a sub-optimal intra-cluster power allocation optimization problem to maximize the network throughput. To address the issue of partial CSI, we rewrite the aforementioned optimization problem, by relying only on the user direction. Performance evaluation of the proposed schemes is developed in rural environment, based on the New York University millimeter-wave simulator. The obtained results demonstrate that the proposed low-complexity NOMA-DBS schemes with either full-or partial-CSI achieve significant performance improvement over the classical DBS, in terms of spectral-and energy-efficiencies (up to 26.8% bps/Hz rate gain for 45 users using the proposed scheme with partial CSI).

Introduction

To meet the exponential growth of data traffic, the new generation of cellular systems needs to achieve higher network capacity and better energy efficiency. This is done by exploiting new and innovative technologies such as massive multiple-input multiple-output (m-MIMO) system, non-orthogonal multiple access (NOMA) technique and millimeterwave (mmWave) bands [START_REF] Busari | Millimeter-wave massive MIMO communication for future wireless systems: A survey[END_REF][START_REF] Araújo | Massive MIMO: survey and future research topics[END_REF][START_REF] Dai | Non-orthogonal multiple access for 5G: solutions, challenges, opportunities, and future research trends[END_REF]. These technologies can be integrated together to further enhance the network capacity [START_REF] Zhang | Capacity analysis of NOMA with mmWave massive MIMO systems[END_REF]. Indeed, the m-MIMO beamforming (BF), which enables high directional array, overcomes the tremendous path loss in mmWave wireless communications where enormous bandwidths are available. In addition, using superposition coding (SC) at the base station (BS) and successive interference cancellation (SIC) at the receiver side, power-domain NOMA (PD-NOMA) effectively improves the connectivity density by multiplexing user equipments (UEs) into transmission power domain [START_REF] Dai | Non-orthogonal multiple access for 5G: solutions, challenges, opportunities, and future research trends[END_REF].

Recently, numerous research studies on NOMA-based multiuser MIMO (NOMA-MIMO) system have been initiated to enhance the spectrum efficiency. For instance, system-level simulations are presented in [START_REF] Benjebbovu | Systemlevel performance of downlink NOMA for future LTE enhancements[END_REF] and show a clear superiority of NOMA over orthogonal multiple access (OMA) in terms of system throughput. Moreover, the authors in [START_REF] Zeng | Capacity comparison between MIMO-NOMA and MIMO-OMA with multiple users in a cluster[END_REF] analytically compares the performance of NOMA-MIMO with OMA-MIMO in terms of the sum-channel-and ergodic-sum-capacities, when multiple users exist in a cluster. The analytical proof also indicates the superiority of NOMA-MIMO.

The most important challenges of NOMA-MIMO system include overall system overhead, user clustering, power allocation and beamforming techniques. For multicast NOMA-MIMO, an iterative algorithm is proposed in [START_REF] Choi | Minimum power multicast beamforming with superposition coding for multiresolution broadcast and application to NOMA systems[END_REF] to find the BF vector and the power for each UE that solves the power minimization problem. In [START_REF] Xiao | Joint power allocation and beamforming for non-orthogonal multiple access (NOMA) in 5G millimeter wave communications[END_REF], a joint beamforming and power allocation scheme was proposed to maximize the sum-rate of a 2-user mmWave NOMA-MIMO system using an analog BF structure with a phased array. A spectrum and energy efficient mmWave transmission scheme that integrates NOMA with beamspace MIMO was first proposed in [START_REF] Wang | Spectrum and energy-efficient beamspace MIMO-NOMA for millimeter-wave communications using lens antenna array[END_REF], in order to reduce the hardware complexity and energy consumption by using lens antennas. In [START_REF] Wang | Spectrum and energy-efficient beamspace MIMO-NOMA for millimeter-wave communications using lens antenna array[END_REF], the authors designed a precoding scheme based on the principle of zero forcing (ZF) to reduce the inter-cluster interferences.

However, [START_REF] Choi | Minimum power multicast beamforming with superposition coding for multiresolution broadcast and application to NOMA systems[END_REF][START_REF] Wang | Spectrum and energy-efficient beamspace MIMO-NOMA for millimeter-wave communications using lens antenna array[END_REF][START_REF] Xiao | Joint power allocation and beamforming for non-orthogonal multiple access (NOMA) in 5G millimeter wave communications[END_REF][START_REF] Zeng | Capacity comparison between MIMO-NOMA and MIMO-OMA with multiple users in a cluster[END_REF] require full channel state information (CSI), which is difficult to perform at BS and brings high feedback overhead in case of a large array system. To address this issue, the authors in [START_REF] Ding | Random beamforming in millimeter-wave NOMA networks[END_REF] implemented a random beamforming for a 2-user NOMA-MIMO cluster to avoid the requirement of full CSI at BS. The authors exploit the key feature of mmWave systems, i.e., the highly directional transmission. Thereby, they proposed a low-feedback mmWave NOMA-MIMO schemes with a fixed power allocation where the two UEs in the NOMA cluster located in the same sector are classified based on their distances from BS instead of their effective channel gains. In [START_REF] Sun | On the ergodic capacity of MIMO NOMA systems[END_REF], the power allocation for a Rayleigh fading 2-user NOMA-MIMO system without beamforming was investigated as an ergodic capacity maximization under statistical CSI. A m-MIMO NOMA system with limited feedback is designed in [START_REF] Ding | Design of massive-MIMO-NOMA with limited feedback[END_REF], where the m-MIMO NOMA channel is splited into multiple single input single output (SISO) NOMA channels, by exploiting the spatial correlation matrices of users' channels.

To reduce the BF implementation complexity and the channel overhead in m-MIMO systems, the authors in [START_REF] Rozé | Millimeter-wave digital beamsteering in highly line-of-sight environments for massive MIMO systems[END_REF] have been investigated the digital beamsteering (DBS). This geometric beamformer, based on the UEs' direction, steers a single beam toward each UE, using just digital phase shifters. Our previous works [START_REF] Khaled | Performance evaluation of linear precoding mmwave multi-user MIMO systems with NYUSIM channel simulator[END_REF][START_REF] Khaled | Multi-user digital beamforming based on path angle information for mm-Wave MIMO systems[END_REF][START_REF] Khaled | Joint SDMA and power-domain NOMA system for multi-user mm-Wave communications[END_REF] show that DBS is an appealing beamformer in the sparse mmWave environment trading-off complexity, performance and channel feedback. This is done using the stochastic mmWave channel model, called NYUSIM [START_REF] Samimi | 3-D millimeter-wave statistical channel model for 5G wireless system design[END_REF] built on real measurements and developed by New York University. The spatial resolution to discriminate users depends on the beamwidth, i.e. the number of antennas. However, the implementation of a large number of antennas is difficult due to high power consumption and transceiver complexity. We propose to tackle the problem of congested cell where the number of users is close to the number of antennas. To do so, we boost the DBS performance, without adding any additional antenna, by implementing PD-NOMA, as an alternative and innovative solution. Moreover, we aim to maintain a low complexity, low overhead solution, by assuming partial CSI at the BS.

Specifically, the major contributions of this paper are summarized as follows:

• We design a NOMA-DBS scheme that enhances the DBS performance without adding additional antennas, and then offers affordable hardware complexity and energy consumption. This is performed by integrating DBS with the PD-NOMA transmission, by which more than one UE can be served by each beam. The aim of the proposed NOMA-DBS scheme is to boost the DBS performance by reducing the spatial inter-beam interference (IBI).

• We leverage the spatial behavior of DBS in mono-and multi-path environments, and then we define a geometric interference metric based on the UE's direction. Accordingly, we propose a user clustering algorithm for 2-UE NOMA-DBS system based on the aforementioned geometric metric instead of the effective channel gain or the channel correlation as adopted in the literature, so that reduces spatial IBI based only on the UEs' directions.

• Given the set of clusters, we aim at maximizing the system throughput under transmission power and SIC constraints. Accordingly, the corresponding optimal intrabeam power allocation method is achieved by assuming a fixed inter-beam power allocation. However, the obtained scheme requires full channel feedback.

• To tackle the system overhead issue, we define a partial CSI-based geometric function relying on the overall system throughput. Thereby, we reformulate a suboptimal problem that maximizes the new geometric function, instead of the system throughput.

• By simulations, we verify the performance in terms of both spectrum and energy efficiencies of the proposed NOMA-DBS scheme with either full or partial channel feedback in rural environment, using NYUSIM.

The rest of this paper is organized as follows: Section 2 presents the model of the proposed NOMA-DBS scheme. In Section 3, we investigate the spatial behavior of DBS and propose a user clustering method. Section 4 and Section 5, respectively, discuss the proposed full and partial CSI-based intra-cluster power allocation schemes. Section 6 numerically evaluates the performance of the proposed schemes, while Section 7 concludes the paper. Symbol Notations: A, a and a denote matrix, vector and scalar, respectively. N (µ, σ 2 ) is a Gaussian random variable with mean µ and variance σ 2 . (.) T , (.) H and Tr(•) stand for the transpose, the conjugate transpose and the trace, respectively.

Description of NOMA-DBS System

This study is conducted on a downlink 3 dimensions (3D) MU-MIMO system consisting of a BS equipped with a 2 dimensions (2D) array of M = M H × M V antennas, where M H and M V are the number of horizontal and vertical antennas, respectively. The BS concurrently communicates with K single-antenna UEs (K < M ), which are randomly distributed in the cell. Denote K = {1, • • • , K} as the set of UEs. Based on the angle information of the UE's direction and using digital phase shifters, classical DBS generates K beams. Each beam only serves a single UE. Thus, with space division multiple access (SDMA) strategy implemented via DBS, all UEs exploit the whole system bandwidth B for high data rate communication.

In a congested cell, where the number of users is close to the number of antennas, the main problem of DBS is IBI, that impacts drastically the system performance in terms of sum-rate. To tackle this issue, we design a NOMA-DBS system, as illustrated in Fig. 1. In the proposed scheme, UEs located at the same direction are allowed to be served by one beam and the BS communicates with them by exploiting the PD-NOMA transmission protocol. Consequently, two different types of cluster exist, namely DBS-cluster and NOMA-DBS-cluster as depicted in Fig. 1. In the former cluster, only one UE is served using classical DBS. However, in the latter, multiple UEs are served by the same beam and share the same time-frequency resources using PD-NOMA. Now, the K UEs are served simultaneously by C (C ≤ K < M ) beams (or clusters).

System model

Let C d and C n be the number of DBS-and NOMA-DBS-clusters, respectively and K c be the number of UEs in the cth cluster. Hence, Based on available CSI and user clustering, detailed in section 3, the BS multiplexes UEs in the power domain using SC, so as to transmit the signals of UEs grouped in the same cluster within the same beam. Thus, BS constructs the superimposed signal s ∈ C C×1 , given by:

K c = 1, ∀c ∈ C d . Denote C n = {1, • • • , C n } and C d = {C n + 1, • • • , C}
s =    K 1 l=1 √ γ l,1 p 1 s l,1 . . . K C l=1 √ γ l,C p C s l,C    =    s 1 . . . s C    , (1)
where s l,c is the modulated complex symbol corresponding to the lth UE in the cth cluster, denoted as UE l,c (l ∈ K c , c ∈ C), p c is the power allocated to the cth cluster, and γ l,c is the power allocation coefficient for s l,c , such that Kc l=1 γ l,c = 1. Moreover, according to DBS, the BS steers a beam toward each cluster in both azimuth and elevation domains. This implies that the transmit BF matrix is defined as

W = [w 1 • • • w C ] ∈ C M ×C with w c = a c ∈ C M ×1
, the BF weight vector corresponding to the cth cluster. a c = a( Θ c ) ∈ C M ×1 is the array steering vector corresponding to the direction Θ c = (θ c , φ c ) of the cth cluster, with θ the azimuth angle and φ the elevation angle. In this study, we conduct a 3D MIMO channel with uniform planar array (UPA) located in xoz plane, as shown in Fig. 1. Therefore, the antenna array steering vector a( Θ) = a(θ, φ) = a az (θ, φ) ⊗ a el (φ) is a function of θ and φ with:

a az (θ, φ) = 1, e j2π d λ cos(θ) sin(φ) , • • • , e j2π(M H -1) d λ cos(θ) sin(φ) T , (2)
a el (φ) = 1, e j2π d λ cos(φ) , • • • , e j2π(M V -1) d λ cos(φ) T , (3)
where d is the antenna spacing and λ is the wavelength. Then, the signal x ∈ C M ×1 transmitted from BS can be expressed as:

x = √ ηWs, (4)
where

η = 1 Tr(W H W) = 1 M C
is the normalization factor that eliminates the beamforming effect on the transmission power. Thus, the received signal is expressed as:

y l,c = √ ηh l,c w c s c + b∈C,b =c √ ηh l,c w b s b + n l,c , (5)
where n l,c ∼ N (0, σ 2 n ) is the additive white Gaussian noise, and h l,c ∈ C 1×M is the multipath 3D channel vector between BS and UE l,c , which is given by:

h l,c = N l,c n=1 α n,l,c a H ( Θ n,l,c ), (6)
where N l,c is the number of multi-path components in UE l,c 's channel, α n,l,c is the complex gain and Θ n,l,c = (θ n,l,c , φ n,l,c ) is the departure angle of the nth path in UE l,c 's channel, respectively. Denote n = 1 as the index of the line-of-sight (LOS) path, i.e., the strongest path generated by NYUSIM.

System model with SIC

In the rest of the study, we consider only 2 UEs per NOMA-DBS cluster, such that K c = 2, to limit the complexity of SIC at the receiver. The number K c of UEs per DBS-cluster is equal to 1. For DBS-cluster and 2-UE NOMA-DBS-cluster, the direction Θ c of the cth cluster used to form the beams is calculated as follows:

Θ c = (θ c , φ c ) = (θ 1,1,c , φ 1,1,c ) if K c = 1, ( θ 1,1,c +θ 1,2,c 2 , φ 1,1,c +φ 1,2,c 2
) if K c = 2. (7)
Thus, only the departure angle of the LOS path (n = 1) is considered.

Based on the user clustering method and the set of instantaneous received power

|h l,c w c | 2 (l ∈ K c , c ∈ C n ) at
all UEs, the two UEs regrouped in each NOMA-DBS cluster are classified between strong and weak UEs according to the descending order of their received power, i.e., |h 1,c w c | 2 > |h 2,c w c | 2 . Moreover, SIC applied at strong UE, denoted as UE 1,c will work properly if the power allocation coefficients satisfy the following condition as mentioned in [START_REF] Ali | Dynamic user clustering and power allocation for uplink and downlink non-orthogonal multiple access (NOMA) systems[END_REF]:

ηγ 2,c p c |h 1,c w c | 2 σ 2 n - ηγ 1,c p c |h 1,c w c | 2 σ 2 n ≥ P min ∀c ∈ C n , (8)
where P min is the minimum power difference. In fact, (8) maintains the power difference between strong and weak UEs' signals, so that SIC at strong UE's receiver can decode first the data of the weak UE, and then subtract it from the superimposed signal to decode his own signal. However, NOMA-DBS system also suffers from the inter-cluster interference unlike the classical PD-NOMA in [START_REF] Ali | Dynamic user clustering and power allocation for uplink and downlink non-orthogonal multiple access (NOMA) systems[END_REF]. Therefore, we propose to take into account the interference from the other clusters and (8) will be given by:

ηγ 2,c p c |h 1,c w c | 2 b∈C,b =c ηp b |h 1,c w b | 2 I :→1,c +σ 2 n - ηγ 1,c p c |h 1,c w c | 2 b∈C,b =c ηp b |h 1,c w b | 2 + σ 2 n ≥ P min , (9)
where I :→l,c is the interference at UE l,c from all other clusters. From (9), we find that the transmit power of UE 2,c must be larger than that of UE 1,c , thus γ 1,c < γ 2,c satisfying the PD-NOMA transmission protocol.

According to PD-NOMA principle, UE 1,c in NOMA-DBS cluster conducts SIC to eliminate the intra-cluster interference from UE 2,c . We assume a perfect SIC at the receiver side of UE 1,c . In other words, the interference at UE 1,c from UE 2,c , denoted as I 2→1,c , is totally eliminated, i.e., I 2→1,c = 0. Therefore, the received signal y 1,c at UE 1,c after SIC can be expressed as:

y 1,c = √ η √ γ 1,c p c h 1,c w c s 1,c + b∈C,b =c √ η √ p b h 1,c w b s b + n 1,c . (10)
And, the signal-to-interference-plus-noise ratio (SINR) SINR NOMA-DBS

1,c

of UE 1,c in NOMA-DBS cluster is given by:

SINR NOMA-DBS 1,c = ηγ 1,c p c |h 1,c w c | 2 b∈C,b =c ηp b |h 1,c w b | 2 + σ 2 n . (11)
However, UE 2,c of NOMA-DBS cluster treats the UE 1,c signal as an interference and the received signal y 2,c at UE 2,c can be expressed as:

y 2,c = √ η √ γ 2,c p c h 2,c w c s 2,c
Desired signal

+ √ η √ γ 1,c p c h 2,c w c s 1,c Intra-beam interference + b∈C,b =c √ η √ p b h 2,c w b s b Inter-beam interference +n 2,c . (12)
Therefore, the SINR SINR NOMA-DBS

2,c

of UE 2,c is given by:

SINR NOMA-DBS 2,c = ηγ 2,c p c |h 2,c w c | 2 ηγ 1,c p c |h 2,c w c | 2 + b∈C,b =c ηp b |h 2,c w b | 2 + σ 2 n . (13)
For DBS-cluster, the received signal y 1,c and SINR DBS

1,c

are also calculated as in ( 10) and [START_REF] Sun | On the ergodic capacity of MIMO NOMA systems[END_REF], respectively, with γ 1,c = 1.

The SINR achieved at UE l,c in DBS and NOMA-DBS clusters can be rewritten as:

SINR DBS 1,c = ψ 1,c ν 1,c = ζ 1,c , (14)
SINR NOMA-DBS 1,c = ψ 1,c γ 1,c ν 1,c = ζ 1,c γ 1,c , (15)
SINR NOMA-DBS 2,c = ψ 2,c (1 -γ 1,c ) ν 2,c + ψ 2,c γ 1,c = ζ 2,c (1 -γ 1,c ) 1 + ζ 2,c γ 1,c , (16)
where

ψ l,c = ηp c |h H l,c w c | 2 , l ∈ K c , c ∈ C n , (17)
ν l,c = b∈C,b =c ηp b |h H l,c w b | 2 + σ 2 n , l ∈ K c , c ∈ C n , (18)
ζ l,c = ψ l,c ν l,c , l ∈ K c , c ∈ C n . (19)
As seen in ( 5), ψ l,c represents the received power at UE l,c of the superimposed signal, that is transmitted in the cth cluster. And, ν l,c represents the interference I :→l,c at UE l,c coming from the other clusters plus the noise power. This implies that ζ l,c is the superimposedsignal to other-clusters interference plus noise ratio at UE l,c , which is different than SINR l,c for NOMA-DBS cluster and represents SINR 1,c for DBS-cluster. Therefore, (9) can be simplified as follows:

γ 2,c -γ 1,c ≥ P min ζ 1,c . (20)

3D Geometric Interference Metric β and User Clustering Method (β-UC)

In this section, we discuss the factors affecting the interference level at each UE in case of classical DBS, in both mono-and multi-path environments, to define later a 3D geometric interference metric. Subsequently, we propose a user clustering strategy based on the aforementioned metric.

Classical DBS Performance

In case of classical DBS, we use the subscript k (∀k ∈ K) instead of (l, c).

Mono-path environment

In mono-path environment, the SINR SINR DBS k achieved using DBS at UE k (∀k ∈ K) is given by [START_REF] Khaled | Multi-user digital beamforming based on path angle information for mm-Wave MIMO systems[END_REF]:

SINR DBS k = |a H 1,k a 1,k | 2 u∈K,u =k |a H 1,k a 1,u | 2 I k + σ 2 n η DBS |α 1,k | 2 , ( 21
)
where a 1,k = a( Θ 1,k ) is the steering vector corresponding to the LOS path angle of UE k , denoted as

Θ 1,k = (θ 1,k , φ 1,k ), α 1,
= |a H 1,k a 1,u | (∀u = k), i.e.
, the spatial interference causing by the beam generated for UE u with the LOS path in the UE k 's channel. In other words, if UE k and UE u are located at the same direction, i.e., Θ 1,k is closed to Θ 1,u , then DBS system suffers from high IBI, and thus the DBS performance degrades.

Multi-path environment

Similarly, in multi-path environment, DBS also generates a single beam in the LOS path direction. Therefore, SINR DBS k is given by [START_REF] Khaled | Multi-user digital beamforming based on path angle information for mm-Wave MIMO systems[END_REF]:

SINR DBS k = a H 1,k a 1,k + N k n=2 α n,k α 1,k a H n,k a 1,k 2 u∈K,u =k N k n=1 α n,k α 1,k a H n,k a 1,u 2
I k + σ 2 n η DBS |α 1,k | 2 , ( 22
)
where α n,k , Θ n,k and a n,k = a( Θ n,k ) are complex gain, angle and steering vector of the nth path in UE k 's channel, respectively. As seen in ( 22), the interference I k at UE k is caused by the LOS beam of UE u (∀u = k), represented by a 1,u , that interferes with all paths of UE k , represented by a H n,k . Thus, the interference results from both non-LOS (NLOS) and LOS paths. However, DBS is a geometric beamformer, that doesn't exploit the NLOS paths and generates a single beam in the LOS path direction, i.e., the directions of the UEs. This simplifying assumption makes possible partial CSI and feedback solely based on the UEs' directions but restricts the sum rate, since all the path diversity is not exploited.

Interference Metric

To enhance the DBS performance, our proposed NOMA-DBS scheme divides UEs into clusters based on their spatial direction, which directly reflects in their spatial interferences.

Based on the simplifying assumption and the previous analysis in Section 3.1, we define

β k,u = 1 M µ k,u = 1 M |a H 1,k a 1,u | = 1 M |a H 1,u a 1,k | = 1 M µ u,k = β u,k
as a 3D geometric metric representing the normalized spatial interference at UE k from the beam generated for UE u , and vice versa. Applying (2) and (3), using UPA, β k,u is given by:

β k,u = 1 M M H m H =1 M V m V =1 e j2π d λ {(m H -1)(cos(θ 1,k ) cos(φ 1,k )-cos(θ 1,u ) cos(φ 1,u ))+(m V -1)(sin(φ 1,k )-sin(φ 1,u ))} (23a) = sin M H πd λ (cos(θ k ) cos(φ k ) -cos(θ u ) cos(φ u )) M H sin πd λ (cos(θ k ) cos(φ 1,k ) -cos(θ 1,u ) cos(φ 1,u )) sin M V πd λ (sin(φ 1,k ) -sin(φ 1,u ) M V sin πd λ (sin(φ 1,k ) -sin(φ 1,u ) (23b)
. According to (23), β k,u ∈ (0, 1), and β k,u represents the value of the normalized array pattern

AF n (θ 1,u ,φ 1,u ) (θ, φ) [19] of the beam pointed at UE u direction (θ 1,u , φ 1,u ) for (θ, φ) = (θ 1,k , φ 1,k ), i.e., β k,u = AF n (θ 1,u ,φ 1,u ) (θ 1,k , φ 1,k ). ( 24
)
Remark 1 Based on (24), we can see that if the angular distance (∆θ, ∆φ)

def = (|θ 1,k - θ 1,u |, |φ 1,k -φ 1,u |) → (0, 0) , i.e., (θ 1,u , φ 1,u ) → (θ 1,k , φ 1,k ), then β k,u → 1.
Thus, the smaller ∆θ and ∆φ are, i.e., UE k and UE u are closely at the same direction in azimuth and elevation domains, the larger β k,u is.

Remark 2 AF n (θ 1,u ,φ 1,u ) (θ, φ) = 1
that satisfy AF n (θ 1,u ,φ 1,u ) (θ, φ) = 1 2
. This implies that β k,u implicitly provides information regarding the beamwidth Ω 3dB u .

Therefore, we define the interference threshold β 0 , such that β k,u ≥ β 0 means that the LOS path of UE k lies in the UE u beam in both azimuth and elevation domains. Particularly, the β 0 -beamwidth Ω β 0 u defines the azimuth and elevation angular distances satisfying |AF n (θ 1,u ,φ 1,u ) (θ, φ)| = β 0 . Thus, β is an important metric to determine the level of spatial interference based on a geometric partial channel knowledge, namely the UE direction, i.e., (θ 1,k , φ 1,k ) (∀k ∈ K).

User clustering algorithm

We would like to apply NOMA to UEs that produce high spatial interference to each other, whereas the others are treated via DBS. To perform user clustering, we take profit of the 3D geometric interference based on the angles of the LOS path of each user. To the best of our knowledge, it is the first time that such a geometric metric is used in user clustering. The main idea of our proposed UC, denoted as β-UC, is to regroup two-by-two UEs (UE u and UE k ) with large inter-beam interference, i.e., β k,u > β 0 , into the same cluster, whose beam angle is defined in 7. Subsequently, we assign a single cluster to the remaining UEs with cluster angle equal to their LOS path angle, as mentioned in [START_REF] Choi | Minimum power multicast beamforming with superposition coding for multiresolution broadcast and application to NOMA systems[END_REF]. The proposed β-UC algorithm can be given as in Algorithm 1.

Full CSI-based Power Allocation Scheme

In this section, we design a power allocation (PA) scheme based on full CSI, that maximizes the throughput of the NOMA-DBS system. However, DBS is a geometric beamformer that requires only the UE's direction. For that, it's more significant to design a PA scheme based on partial CSI, which is presented in Section 5.

Problem Formulation

The system throughput, which is defined as the maximum quantity of the transmitted information over the NOMA-DBS system, is given by a sum of the UEs' data rates: (l, m) = max

R T = k∈K R k = c∈C l∈Kc R l,c (25a)
(k,u)∈D {β k,u }, G = G ∪ [l, m].
3: Remove from D the spatial interference of UEs selected in Step 2 with each other and with any other UEs to prevent their existence in another cluster;

D ← D -(l, m) -(l, w) -(v, l) -(m, q) -(t, m) (∀w > l, v < l, q > m, t < m ∈ K). 4: Repeat Step 2 and
Step 3 until D = ∅. 5: Assign for each one of the remaining UEs a beam in the corresponding UE's direction;

s ∈ K, s / ∈ G, G = G ∪ [s]. = c∈C d R 1,c DBS clusters + c∈Cn (R 1,c + R 2,c ) NOMA-DBS clusters , (25b)
where R l,c [bps] is the achievable data rate for UE l,c and can be expressed, according to Shannon formula, as:

R l,c = B log 2 (1 + SINR l,c ), l ∈ K c , c ∈ C d ∪ C n . (26)
Therefore, R T can be defined as a function of the inter-cluster PA p = {p c , c ∈ C} and the intra-cluster PA γ = {γ l,c , l ∈ {1, 2}, c ∈ C n }, since γ 1,c = 1 (∀c ∈ C d ), i.e., R T = F (γ, p):

R T = F DBS (γ, p) + F NOMA-DBS 1 (γ, p) + F NOMA-DBS 2 (γ, p), (27)
where F DBS is the throughput of UEs grouped in DBS-clusters, F NOMA-DBS 1 is the throughput of strong UEs grouped in NOMA-DBS-clusters and F NOMA-DBS 2 is the throughput of weak UEs grouped in NOMA-DBS-clusters. Using ( 14), ( 15) and ( 16), F DBS , F NOMA-DBS 1 and F NOMA-DBS 2 are given by ( 28), ( 29) and (30), respectively:

F NOMA-DBS 1 (γ, p) = B c∈Cn log 2 (1 + ζ 1,c (p)), (28)
F NOMA-DBS 1 (γ, p) = B c∈Cn log 2 (1 + ζ 1,c (p)γ 1,c ), (29)
F NOMA-DBS 1 (γ, p) = B c∈Cn log 2 1 + ζ 2,c (p)(1 -γ 1,c ) 1 + ζ 2,c (p)γ 1,c . (30)
The aim of the proposed PA scheme is to maximize the sum of the UEs' data rates over the NOMA-DBS system. Therefore, the corresponding optimization problem denoted by P 1 is formulated as follows:

P 1 : { γ, p} =max γ,p F (γ, p), s.t. : C 1 : γ 1,c + γ 2,c = 1 ∀c ∈ C n , C 2 : γ 1,c ≤ γ 2,c ∀c ∈ C n , C 3 : γ 2,c -γ 1,c ≥ P min ζ 1,c ∀c ∈ C n . (31)
P 1 must be done under the power allocation constraints C 1 and C 2 and the SIC constraint C 3 obtained in [START_REF] Yang | Performance of conjugate and zero-forcing beamforming in large-scale antenna systems[END_REF]. Problem P 1 maximizes the system throughput by jointly optimizing intra-and intercluster PA. However, because the coupling of the power allocation factors from different UEs, then it is difficult to obtain the optimal solutions. To this end, we add a constraint that addresses this issue and then simplifies P 1 by applying a fixed inter-cluster PA. Thus once β-UC is applied, the power allocated to each cluster p will be well known and equal to constant values as detailed in Section 4.2. Hence, P 1 can be reformulated as follows:

P 1 : γ=max γ F (γ), s.t. : C 1 : γ 1,c + γ 2,c = 1 ∀c ∈ C n , C 2 : γ 1,c ≤ γ 2,c ∀c ∈ C n , C 3 : γ 2,c -γ 1,c ≥ P min ζ 1,c ∀c ∈ C n . (32)
For DBS cluster, as seen in ( 28), the throughput of UEs grouped in DBS-cluster is the same regardless of the intra-cluster PA method, i.e., F DBS (γ) = constant (∀γ). However, for NOMA-DBS cluster, as seen in ( 29) and (30), the data rates of UE 1,c and UE 2,c only depend on their PA coefficients γ 1,c and γ 2,c = 1 -γ 1,c . To this end, based on β-UC, the PA scheme maximizing the system throughput is equivalent to that maximizing the throughput of each NOMA-DBS cluster separately. Accordingly, the maximization problem P 1 can be seen as an equivalent problem of C n independent optimization problems denoted by P 2 c for the c th NOMA-DBS cluster, and can be equivalently formulated as follows:

P 2 c : γ 1,c =max γ 1,c R c = max γ 1,c R 1,c + R 2,c ∀c ∈ C n , s.t : C 1 : γ 1,c + γ 2,c = 1 ∀c ∈ C n , C 2 : γ 1,c ≤ 1 2 , C 3 : γ 2,c -γ 1,c ≥ P min ζ 1,c . (33)
C 1 and C 2 in problem (31) can be reformulated as C 2 . Moreover, C 1 , C 2 and C 3 specify the feasible interval Γ c of γ 1,c , that is Γ c = [0, γc ] with:

γc = 1 2 1 - P min ζ 1,c . ( 34
)

Fixed Inter-Cluster Power Allocation

In our previous work [START_REF] Khaled | Joint SDMA and power-domain NOMA system for multi-user mm-Wave communications[END_REF], we applied a uniform power allocation per cluster, i.e., p c = constant (∀c). We define P c the emitted power toward the cth cluster as follows:

P c = η w c 2 Kc l=1 γ l,1 p c = η w c 2 p c . (35)
Moreover, C c=1 P c = P e where P e is the total transmit power. This implies that the uniform power allocation per cluster satisfies P c = Pe C , using η w c 2 = 1 C . However, in this study, we allocate a power p c to the cth cluster proportional to the number K c of UEs served in the cth cluster. Accordingly, based on (35), P c is also proportional to K c . This is supposed to guarantee the power fairness among UEs. Therefore, we calculate P c as follows:

P c = K c P e K . ( 36
)
From ( 35) and ( 36) we obtain that:

p c = K c C P e K . ( 37
)

Intra-Cluster Power Allocation Solution with Full CSI

In this subsection, we propose an intra-cluster PA scheme, for the problem P 2 c for each NOMA-DBS cluster, denoted as OPA, which is summarized in Algorithm 2.

Algorithm 2: Optimal Power Allocation (OPA)

initialization: c = C n ; while c ≤ C do if log 2 (1+ζ 1,c )-log 2 (1+ζ 2,c ) log 2 (1+ζ 1,c ) < then γ 1,c = γ1,c else if ζ 2,c == min(ζ 1,c , ζ 2,c ) then γ 1,c = γc else γ 1,c = 0 c=c+1; Lemma 1 R c is a real monotone function of γ 1,c ,

and its variation depends on the sign of

ζ 1,c -ζ 2,c . Proof 1 See Appendix 1.
From Lemma 1, R c is a real monotone function of γ 1,c , and its variation is as follows:

     ζ 1,c > ζ 2,c =⇒ R c is an increasing function of γ 1,c ζ 1,c < ζ 2,c =⇒ R c is an decreasing function of γ 1,c ζ 1,c = ζ 2,c =⇒ R c is constant ∀γ 1,c (38)
For the three cases, and since UE 1,c is the strong UE, i.e., |h

1,c w c | 2 > |h 2,c w c | 2 , this implies that ψ 1,c > ψ 2,c . If ζ 1,c < ζ 2,c
, then I :→1,c > I :→2,c . And, UE 1,c taken as a strong UE suffers from high other-clusters interference compared to UE 2,c . This implies that even though UE 1,c has the highest received superimposed signal, the level of interference I :→l,c at each UE from other clusters determines the variation of the function R c .

From Lemma 1 and based on the derivation and analysis in Section 4.3, the optimal solution γ 1,c of Problem P 2 c is achieved at the end point of the feasible interval Γ c of γ 1,c . Indeed, if ζ 1,c > ζ 2,c , i.e., R c is an increasing function, then γ 1,c = γc . The authors in [START_REF] Ali | Dynamic user clustering and power allocation for uplink and downlink non-orthogonal multiple access (NOMA) systems[END_REF] propose an optimal PA policy that maximizes the sum-throughput of a classical NOMA system (NOMA-SISO) by maximizing the sum-throughput per m-user NOMA cluster (2 ≤ m ≤ K). Using Karush-Kuhn-Tucker optimality conditions, they derived the optimal PA. We find that for 2-UE NOMA-SISO cluster (see Table 1 [START_REF] Ali | Dynamic user clustering and power allocation for uplink and downlink non-orthogonal multiple access (NOMA) systems[END_REF]), the optimal PA can be similar to our solution obtained without using Lagrange, by changing variables. Otherwise, if ζ 1,c < ζ 2,c , i.e., R c is a decreasing function, then γ 1,c = 0. This implies that OPA deactivates UE 1,c , which is classified as strong UE, but actually suffers from high inter-cluster interference, by allocating all the power p c to UE 2,c . In addition, when |ζ 1,c -ζ 2,c | < , each value belonging to the interval Γ c of γ 1,c can be taken as an optimal solution. But, we choose γ1,c that achieves fairness between UEs in the cth cluster, i.e.,

R 2,c (γ 1,c ) = R 1,c (γ 1,c ), thus γ 1,c = γ1,c with γ1,c = -(ζ 1,c + ζ 2,c ) + (ζ 1,c + ζ 2,c ) 2 + 4ζ 1,c ζ 2 2,c 2ζ 1,c ζ 2,c . (39)
Lemma 2 When R c is a constant function, then γ1,c chosen as the optimal solution of Problem P 2 satisfies the power allocation constraint C 1 .

Proof 2 As seen in (38

), R c is a constant function if ζ 1,c = ζ 2,c . Therefore, for ζ = ζ 1,c → ζ 2,c
, γ1,c can be expressed as:

γ1,c = -1 + √ 1 + ζ ζ . ( 40
)
Applying the first-order derivative of (40), we obtain that:

dγ 1,c dζ = -ζ -2 + 2 √ ζ + 1 2ζ 2 √ ζ + 1 = -( √ ζ + 1 -1) 2 2ζ 2 √ ζ + 1 . ( 41
)
From ( 41), we find that dγ 1,c dζ < 0 for ζ > 0, and thus γ1,c in ( 40) is a decreasing function.

This implies that if 0 < ζ min ≤ ζ ≤ ζ max , then γ1,c (ζ max ) ≤ γ1,c ≤ γ1,c (ζ min ). γ1,c → -1 + (1 + ζ) ζ =⇒        lim ζ→0 γ1,c = lim ζ→0 -1 + (1 + 1 2 ζ) ζ = 1 2 lim ζ→∞ γ1,c = lim ζ→∞ 1 1 + √ 1 + ζ = 0        =⇒ 0 < γ 1,c < 1 2 (42)
Accordingly, based on (42), γ1,c lies on (0, 1 2 ) for ζ > 0. Here, Lemma 2 is proved.

Remark 3 When R c is a slowly-increasing or slowly-decreasing function, the absolute difference between R c at γ 1,c = 0 and R c at γ 1,c = 1 is very small, i.e., |R c (1

) -R c (0)| = | log 2 (1 + ζ 1,c ) -log 2 (1 + ζ 2,c )| < . However, since γ 1,c ∈ Γ c = [0, γc ] ⊂ [0, 1], then | log 2 (1 + ζ 1,c ) - log 2 (1 + ζ 2,c )| < is also valid in the feasible interval Γ c of γ 1,c , i.e., |R c (γ c ) -R c (0)| < . There- fore, we use | log 2 (1 + ζ 1,c ) -log 2 (1 + ζ 2,c )| < instead of |R c (γ c ) -R c (0)| < ,
which is more complicated to compute. Moreover, the value of depends on the value of log 2 (1+ζ l,c ). To this end, we use the relative difference of the corresponding data rates

log 2 (1+ζ 1,c )-log 2 (1+ζ 2,c ) log 2 (1+ζ 1,c ) < instead of |ζ 1,c -ζ 2,c | < .

CSI-based Power Allocation Scheme

Using classical DBS, UEs only feedback their estimated angles, i.e., Θ 1,k = (θ 1,k , φ 1,k ) (∀k ∈ K). Accordingly, we design a NOMA-DBS scheme that uses these angles to schedule UEs between DBS and NOMA-DBS clusters. Subsequently, in Section 4, we propose an intra-cluster PA scheme for NOMA-DBS, denoted as OPA, that maximizes the system throughput under full CSI. This implies that NOMA-DBS misses the key features of DBS, namely partial CSI and low channel overhead. To this end, we will formulate an alternative problem from P 2 c , by defining a partial CSI-based function, to guarantee low feedback overhead. First, we will try to rewrite SINR NOMA-DBS l,c of UE l,c served in NOMA-DBS cluster, by exploiting the impacts of the NLOS and LOS paths on both received power and interference at UE l,c . Then, we will rewrite the system throughput to conclude a partial CSI-based geometric function.

Based on [START_REF] Zeng | Capacity comparison between MIMO-NOMA and MIMO-OMA with multiple users in a cluster[END_REF], |h H l,c a c | 2 can be expressed as:

|h H l,c a c | 2 = N l,c n=1 α n,l,c a H n,l,c a c 2 (43a) = α 1,l,c a H 1,l,c a c + N l,c n=2 α n,l,c α 1,l,c a H n,l,c a c 2 . (43b)
In (43b), the first term represents the spatial interference of the cth cluster with the LOS path in UE l,c 's channel. And, the second term represents the spatial interference of the cth cluster, with all the NLOS paths in UE l,c 's channel.

In high signal-to-noise ratio (SNR) regime and using (43), R 1,c in (11) and R 2,c in (13) are given by ( 44) and (45), respectively:

R 1,c high SNR -→ B log 2 1 + γ 1,c p c |a H 1,1,c a c + N 1,c n=2 α n,1,c α 1,1,c a H n,1,c a c | 2 b∈C,b =c p b |a H 1,1,c a b + N 1,c n=2 α n,1,c α 1,1,c a H n,1,c a b | 2 (44) R 2,c high SNR -→ B log 2 1+ γ 2,c p c a H 1,2,c a c + N 2,c n=2 α n,2,c α 1,2,c a H n,2,c a c 2 γ 1,c p c a H 1,2,c a c + N 2,c n=2 α n,2,c α 1,2,c a H n,2,c a c 2 + b∈C,b =c p b a H 1,2,c a b + N 2,c n=2 α n,2,c α 1,2,c a H n,2,c a b 2
(45) As seen in ( 44) and (45), the NLOS paths affect either the received power or the interference. Indeed, the interference I :→l,c , (l = 1, 2) is due to the beams of other UEs that interferes with LOS and NLOS paths. In addition, the received power at UE l,c is that transmitted by the beam interfered with the LOS path plus the effect of paths in the vicinity of the beam direction.

Assuming that this NLOS paths' impact is negligible, i.e.,

α n,l,c α 1,l,c is negligible, thus the data rates Ȓl,c at UE l,c (l = 1, 2) can be approximated as follows:

Ȓ1,c = B log 2 1 + γ 1,c p c |a H 1,1,c a c | 2 b∈C,b =c p b |a H 1,1,c a b | 2 , ( 46
) Ȓ2,c = B log 2 1 + γ 2,c p c |a H 1,2,c a c | 2 γ 1,c p c |a H 1,2,c a c | 2 + b∈C,b =c p b |a H 1,2,c a b | 2 . ( 47
)
As seen in ( 46) and (47), Ȓl,c depends on the LOS path angle information, i.e., Θ 1,l,c , the β-UC, i.e., Θ c , and the applied PA method, i.e., γ 1,c and γ 2,c . The obtained partial CSI-based function Ȓl,c corresponds to the data rate at UE l,c in case of mono-path environment, where the received power just comes from the LOS path and the interference comes from the other-clusters with the LOS path. However, DBS is a LOS path angle-based beamformer that generates a single beam toward each UE. Thus, in multi-path environment, DBS loses the potentiality of the NLOS paths and some of the radiated energy, as obtained in [START_REF] Khaled | Multi-user digital beamforming based on path angle information for mm-Wave MIMO systems[END_REF]. To guarantee the key feature of DBS, we formulate a partial CSI-based function that misses the potentiality of the NLOS paths. Accordingly, the problem P 3 c from P 2 c consists to find the PA coefficients that maximize the partial CSI function Ȓc instead of the NOMA-DBS cluster's throughput R c , as follows:

P 3 c : γ 1,c =max γ 1,c Ȓc = max γ 1,c Ȓ1,c + Ȓ2,c ; ∀c ∈ C n , s.t : C 1 : γ 1,c + γ 2,c = 1 ∀c ∈ C n , C 2 : γ 1,c ≤ 1 2 , C 3 : γ 2,c -γ 1,c ≥ P min ζ1,c . (48)
This Problem P 3 c will be solved in the same manner as Problem P 2 c , but with ζl,c = ψl,c νl,c , where:

ψl,c = ηp c |a H 1,l,c a c | 2 , l ∈ K c , c ∈ C n , (49) νl
,c = b∈C,b =c ηp b |a H 1,l,c a b | 2 , l ∈ K c , c ∈ C n . (50)

Performance Evaluation

In this section, numerical results are presented to verify the performance of the proposed NOMA-DBS scheme. All results are obtained by averaging over 5000 random trials. Perfect estimation of angles at the receivers and perfect instantaneous feedback of the estimated angles are assumed in the analysis of such systems. We set β 0 = 0.5 as the spatial interference threshold. Other system and channel parameters used for performance assessment are listed in Table 1.

{1, 2}

The proposed schemes with full and partial CSI are labeled, respectively, as "β-UC NOMA-DBS-FCSI" and "β-UC NOMA-DBS-PCSI". For comparison, the orthogonal multiple access (OMA) with DBS (OMA-DBS) is adopted as the baseline, in which the two UEs in NOMA-DBS cluster obtained using β-UC algorithm are served using OMA with equal degrees of freedom for each UE, which is labeled as "β-UC OMA-DBS". Moreover, a widely used beamformer, namely maximum rate ratio, also known as conjugate beamforming (CB) [START_REF] Yang | Performance of conjugate and zero-forcing beamforming in large-scale antenna systems[END_REF] is adopted to make a comparison. The number of real coefficients needed per channel estimation for the different systems is summarized in Table 2.

DBS 2 [θ 1,k , φ 1,k ] MRT 2 × M [H] β-UC NOMA-DBS-FCSI 2 × (M + 1) [H + (θ 1,k , φ 1,k )] β-UC NOMA-DBS-PCSI 3 [|h l,c w c | 2 + (θ 1,k , φ 1,k )] β-UC OMA-DB 2 × M [H]

Spectral Efficiency

Fig. 2 depicts the spectral efficiency (or the achievable sum-rate per the normalized bandwidth) versus the total number of users, in rural environment, when BS is equipped using UPA with M H = 32. It can be seen that the proposed NOMA-DBS scheme with full and partial CSI significantly improves the performance of classical DBS. Fortunately, attractive results are achieved, where NOMA-DBS-PCSI can almost achieve the same performance as NOMA-DBS-FCSI, with much lower channel feedback; 43.3 times less as indicated in Table 2. Indeed, one or two paths exist in rural environment (see Table 1), and thus the partial CSI-based function in NOMA-DBS-PCSI scheme is approximately the same as the throughput. For instance, NOMA-DBS-PCSI achieves higher spectral efficiency with respect to DBS up to 26.8% for 45 users in the cell using UPA.

As seen in Fig. 2, it is obvious that classical DBS outperforms CB in rural environment, thanks to a few paths as explained in our previous work [START_REF] Khaled | Multi-user digital beamforming based on path angle information for mm-Wave MIMO systems[END_REF]. Besides, NOMA-DBS improves the performance of DBS and thus surpasses CB. For instance, the performance gain in spectral efficiency for NOMA-DBS-PCSI against CB is approximately 29.4% for 45 users in the cell using UPA, with lower complexity and much lower channel overhead (42.6 times less as shown in Table 2).

It can be seen from Fig. 2 that NOMA-DBS surpasses the conventional OMA-DBS schemes in rural environment, and using UPA. This is also obtained in the prior work comparing NOMA-MIMO with OMA-MIMO, which concludes that OMA-MIMO can not support more users, while NOMA-MIMO enables a higher number of successfully connected users. Indeed, NOMA with the aid of SIC and SC enables more UEs in each beam to be supported at the same spectrum efficiency, while OMA serves only one UE in each beam.

Energy Efficiency

The energy efficiency (EE) is also a key performance metric used for future cellular network, because of environmental and economic interests. EE is defined as the ratio of the spectral efficiency [bps] to the total power consumed at BS:

EE = B K k=1 R k ρ K k=1 w k 2 + M P a + P 0 [bps/J]. (51)
where ρ ≥ 1 is a constant modeling the inefficiency of the power amplifier, P a is the constant power consumption per antenna independent of the transmitted power, and P 0 is the basic power consumed at the base station independent of the number of antennas. Assume ρ = 10, P a = 1W and P 0 = 200mW . In rural environment, using 96 less of antennas, our proposed schemes NOMA-DBS-FCSI and NOMA-DBS-PCSI with M H = 32 and M V = 2 have almost the same performance of the classical DBS with M H = 32 and M V = 5 in terms of spectrum efficiency, as shown in Fig. 3. It's obvious therefore that the energy efficiency using NOMA-DBS is 2.2 times greater than that using classical DBS with M H = 32 and M V = 5, as seen in Fig. 4. Accordingly, NOMA-DBS is an alternative solution to enhance the DBS performance without increasing the number of antennas.

Conclusion

In this paper, we integrate DBS with PD-NOMA transmission for mmWave massive MIMO systems in order to boost the DBS performance without increasing the number of antennas. We first define a geometric interference metric, so that the proposed NOMA-DBS scheme regrouped the users based on their directions. Subsequently, by applying a fixed inter-cluster PA, we derive an intra-cluster PA for 2-user NOMA-DBS system that maximizes the sum-throughput. More importantly, a new partial CSI-based geometric function is defined by exploiting the spatial behavior of DBS, to design a PA scheme with low-feedback rate. Simulation results using NYUSIM indicates that our proposed lowcomplex NOMA-DBS system with limited feedback yields a significant spectral efficiency improvements with respect to DBS, and can be an alternative solution to the use of a large number of antennas. However, our proposed scheme is limited to 2-user in NOMA-DBS cluster. It is in our interest to extend 2-user NOMA-DBS to support multiple users, to further enhance the NOMA-DBS scheme.

Appendix

Appendix 1

The first-order derivative of R 1,c , R 2,c and R c can be expressed as: Here, Lemma 1 is proved.

dR 1,c dγ 1,c = ln(2) ψ 1,c ν 1,c + ψ 1,c γ 1,c > 0, (52)

Figure 1 :

 1 Figure 1: NOMA-DBS beamforming system.

  as the set of DBS-and NOMA-DBS-clusters, respectively, C = C n ∪ C d as the set of all clusters and K c = {1, • • • , K c } as the set of UEs in the cth cluster.

2

 2 gives a measure of the 3dB-beamwidth Ω 3dB u = (Ω az 3dB u , Ω el 3dB u ) expressed in terms of the azimuth Ω azu and elevation Ω elu widths of the beam generated at UE u , which depend on the beam direction Θ u and the number of horizontal and vertical antennas. Ω az 3dB u = |θ 1,u -θ| and Ω el 3dB u = |φ 1,u -φ| are defined as the angular distances

Algorithm 1 : 1 :

 11 β-based User Clustering Algorithm Require: Θ 1,k = (θ 1,k , φ 1,k ), β k,u ∀k, u < k ∈ K, β 0 .Output: G = {g c }: set of clusters with g c the array of UEs belonging to the cth cluster. Select the groups of 2-UEs who have a spatial interference greater than β 0 ; D = {(k, u), β k,u ≥ β 0 , k < u ∈ K}. 2: Select from D the subset of 2-UEs having the largest spatial interference;

Figure 2 :

 2 Figure 2: Spectral efficiency versus number of users, for NOMA-DBS, OMA-DBS, classical DBS and CB in rural environment, when BS is equipped with UPA.

Figure 3 :Figure 4 :

 34 Figure 3: Spectral efficiency versus number of users, for NOMA-DBS using UPA with M H = 32 and M V = 2, and for classical DBS using UPA with M H = 32 and different values of M V (M V = {2, 3, 4, 5}) in rural environment.

Fig. 3

 3 Fig.3 and 4depict, respectively, the spectral efficiency and energy efficiency versus number of users, for NOMA-DBS using UPA with M H = 32 and M V = 2, and for classical DBS using UPA with M H = 32 and different values of M V (M V = {2, 3, 4, 5}) in rural environment.In rural environment, using 96 less of antennas, our proposed schemes NOMA-DBS-FCSI and NOMA-DBS-PCSI with M H = 32 and M V = 2 have almost the same performance of the classical DBS with M H = 32 and M V = 5 in terms of spectrum efficiency, as shown in Fig.3. It's obvious therefore that the energy efficiency using NOMA-DBS is 2.2 times greater than that using classical DBS with M H = 32 and M V = 5, as seen in Fig.4. Accordingly, NOMA-DBS is an alternative solution to enhance the DBS performance without increasing the number of antennas.

  dR 2,c dγ 1,c = -ln(2) ψ 2,c ν 2,c + ψ 2,c γ 1,c < 0, c -ζ 2,c (1 + ζ 1,c γ 1,c )(1 + ζ 2,c γ 1,c ) .(54b)(54) indicates that the sign of the first-order derivative of R c is the same of that of ζ 1,c -ζ 2,c .

Table 1 :

 1 Simulation parameters

	System and Channel Parameters	Values
	Number of transmit antennas	64
	Carrier frequency	28 GHz
	Channel bandwidth	20 MHz
	Cell edge radius	100 m
	Transmission power	30 dBm
	Noise power	-100.9178
		dBm
	Minimum power difference P min	1 mW
	Number of paths per time cluster in rural sce-
	nario

Table 2 : Channel Overhead System Number of real coefficients per chan- nel estimation for each UE Classical

 2

This work has been funded by IMT Atlantique, Lebanese University research fund program and the AZM association.


Fabien Tsin

Anne Pensel

Jacky Vigneron

Anne-Marie Gonçalvès

Sophie Peulon
email: sophie.peulon@univ-evry.fr

Innovative wastewater treatment based on electrodeposited thin film: Systematic studies of interfacial processes between birnessite and Mn(II) for a better efficiency

 Interests of electrodeposited birnessite thin films as eco-friendly materials for depollution  Systematic bulk and surface characterizations after interaction by XRD, SEM and XPS  No modification of the material after interaction even in unfavourable chemical conditions  High robustness of electrodeposited birnessite thin films compared to powder  Promising materials for efficient wastewater treatments at low costs GRAPHICAL

imply a possible inhibition of the degradation process, a depreciated condition for future applications. This paper focuses on the determination of the real robustness of electrodeposited birnessite thin films and their quality for innovative wastewater treatment. Systematical studies were performed with solutions containing different Mn(II) concentrations in presence and in absence of oxygen to have unfavorable conditions. Samples were characterized at macroscopic scales (SEM and XRD) in addition to a focused study on the chemical modification at the extreme near surface by XPS. Open circuit potential (OCP) was also followed to complete the study.

Similar characterizations were performed after interaction with organic pollutants solutions. As a result, nature and structure of the bulk and the near surface of birnessite stay unchanged after long times even in absence of oxygen or in presence of organic pollutants at high concentrations, proving their very good robustness and real interests for future applications. KEYWORDS: Electrodeposition; Manganese oxides; Depollution; Glyphosate; AMPA; Pesticide.

INTRODUCTION

Birnessite, a layered manganese oxide, is closely studied since many years in different fields such as energy production and/or storage [START_REF] Nakayama | Electrochemical synthesis of birnessitetype layered manganese oxides for rechargeable lithium batteries[END_REF][START_REF] Hu | Synthesis of layered birnessite-type manganese oxide thin films on plastic substrates by Chemical Bath Deposition for flexible transparent supercapacitors[END_REF][START_REF] Nam | The high performance of crystal water containing manganese birnessite cathodes for magnesium batteries[END_REF], or pollutants treatments [START_REF] Islam | Manganese oxides and their application to metal ion and contaminant removal from wastewater[END_REF][START_REF] Islam | Removal of dye from polluted water using novel nano manganese oxide-based materials[END_REF][START_REF] Lib | Synthesis of manganese dioxide with different morphologies for thallium removal from wastewater[END_REF][START_REF] Islam | Macroscopic and modeling evidence for nickel (II) adsorption onto selected manganese oxides and boehmite[END_REF][START_REF] Choumane | Electrodeposited birnessite thin film: An efficient eco-friendly sorbent for removing heavy metals from water[END_REF][START_REF] Barrett | Oxidative degradation of glyphosate and Aminomethylphosphonate by manganese oxide[END_REF]. Majority of studies used birnessite powder, however electrodeposited birnessite thin films appear as very interesting tools for wastewater treatment. For example, previous works have demonstrated the high efficiency of such samples for eliminating heavy metals by sorption [START_REF] Choumane | Electrodeposited birnessite thin film: An efficient eco-friendly sorbent for removing heavy metals from water[END_REF] or degrading and mineralizing organic pollutants as dyes [START_REF] Zaied | Spontaneous oxidative degradation of indigo carmine by thin films of birnessite electrodeposited onto SnO2[END_REF][START_REF] Zaied | Studies of N-Demethylation oxidative and degradation of methylene blue by thin layers of birnessite electrodeposited onto SnO2[END_REF] and pesticides (e.g. glyphosate and AMPA, its main by-product) [START_REF] Ndjeri | Degradation of glyphosate and AMPA (AminoMethylPhosphonic Acid) solutions by thin films of birnessite electrodeposited: A new design of material for remediation processes? Colloids Surf[END_REF] by spontaneous electronic transfers. These remarkable results were based on interactions between solution and birnessite thin film in very soft conditions by a simple contact between the film and the solution without causing damage [START_REF] Choumane | Electrodeposited birnessite thin film: An efficient eco-friendly sorbent for removing heavy metals from water[END_REF][START_REF] Zaied | Spontaneous oxidative degradation of indigo carmine by thin films of birnessite electrodeposited onto SnO2[END_REF][START_REF] Zaied | Studies of N-Demethylation oxidative and degradation of methylene blue by thin layers of birnessite electrodeposited onto SnO2[END_REF][START_REF] Ndjeri | Degradation of glyphosate and AMPA (AminoMethylPhosphonic Acid) solutions by thin films of birnessite electrodeposited: A new design of material for remediation processes? Colloids Surf[END_REF]. Moreover, the yield and/or the kinetic of the process degradation of organic pollutants can be largely even enhanced by coupling with electrochemistry but with very low electrical input [START_REF] Pensel | Efficient electrochemical treatment based on electrodeposited thin films of birnessite for mineralisation of AMPA (Aminomethylphosphonic Acid) in very soft conditions[END_REF].

Birnessite thin films are synthesized by electrodeposition, which is a well-adapted method to form adherent, crystallized and very homogeneous thin films in very reproducible conditions on conductive substrates with a good control of the amount and the thickness [START_REF] Peulon | Mechanistic study of cathodic electrodeposition of zinc oxide and zinc hydroxychloride films from oxygenated aqueous zinc chloride solutions[END_REF][START_REF] Nakayama | A novel electrochemical method for preparation of thin films of layered manganese oxides[END_REF][START_REF] Larabi-Gruet | Studies of electrodeposition from Mn(II) species of thin layers of birnessite onto transparent semiconductor[END_REF][START_REF] Jacob | Microstructure and properties of manganese dioxide films prepared by electrodeposition[END_REF]. The synthesis is fast, performed in aqueous solution at room temperature, and at low costs. Moreover, the use of material as thin films has many advantages for applications such as no filtration step after synthesis and uses, and the possibility of regeneration and reuse of the material [START_REF] Choumane | Electrodeposited birnessite thin film: An efficient eco-friendly sorbent for removing heavy metals from water[END_REF][START_REF] Zaied | Spontaneous oxidative degradation of indigo carmine by thin films of birnessite electrodeposited onto SnO2[END_REF][START_REF] Zaied | Studies of N-Demethylation oxidative and degradation of methylene blue by thin layers of birnessite electrodeposited onto SnO2[END_REF][START_REF] Ndjeri | Degradation of glyphosate and AMPA (AminoMethylPhosphonic Acid) solutions by thin films of birnessite electrodeposited: A new design of material for remediation processes? Colloids Surf[END_REF][START_REF] Pensel | Efficient electrochemical treatment based on electrodeposited thin films of birnessite for mineralisation of AMPA (Aminomethylphosphonic Acid) in very soft conditions[END_REF]. That is why, these non-toxic material as thin films appear as very promising for developing efficient and eco-friendly water depollution processes.

The high efficiency of this degradation treatment is due to birnessite, which belongs to manganese oxides. Indeed, these non-toxic compounds present natural sorption and oxidant properties. They exist in soils and play key role in environment by geochemistry [START_REF] Post | Manganese oxide minerals: Crystal structures and economic and environmental significance[END_REF][START_REF] Tebo | Biogenic manganese oxides: Properties and mechanisms of formation[END_REF]. Among manganese oxides, birnessite presents the most important redox and sorption properties due to its specific structure. It is a non-stoichiometric oxide with a very open and malleable layered structure formed by MnO6 octahedra sharing edges due to the substitution of Mn(IV) by Mn(III) or Mn(II), the negative charges being compensated by the presence of hydrated cations inserted inside the layered structure [START_REF] Post | Manganese oxide minerals: Crystal structures and economic and environmental significance[END_REF][START_REF] Tebo | Biogenic manganese oxides: Properties and mechanisms of formation[END_REF].

Nevertheless, the redox mechanism, which takes place at the interface between birnessite and solution containing pollutants is complex [START_REF] Barrett | Oxidative degradation of glyphosate and Aminomethylphosphonate by manganese oxide[END_REF][START_REF] Zaied | Spontaneous oxidative degradation of indigo carmine by thin films of birnessite electrodeposited onto SnO2[END_REF][START_REF] Zaied | Studies of N-Demethylation oxidative and degradation of methylene blue by thin layers of birnessite electrodeposited onto SnO2[END_REF][START_REF] Ndjeri | Degradation of glyphosate and AMPA (AminoMethylPhosphonic Acid) solutions by thin films of birnessite electrodeposited: A new design of material for remediation processes? Colloids Surf[END_REF][START_REF] Nesbitt | XPS study of reductive dissolution of 7Åbirnessite by H3AsO3, with constraints on reaction mechanism[END_REF][START_REF] Banerjee | XPS study of reductive dissolution of birnessite by oxalate: Rates and mechanistic aspects of dissolution and redox processes[END_REF][START_REF] Banerjee | Oxidation of aqueous Cr (III) at birnessite surfaces: Constraints on reaction mechanism[END_REF][START_REF] Ndjeri | Studies on the reduction of birnessite thin layers: Influence of medium[END_REF][START_REF] Stone | Reductive dissolution of manganese (III/IV) oxides by substituted phenols[END_REF][START_REF] Banerjee | XPS study of dissolution of birnessite by humate with constraints on reaction mechanism[END_REF]. The organic pollutants are oxidized and degraded until mineralization, while Mn(IV) and/or Mn(III) present in the birnessite structure are reduced into soluble Mn(II) species. Previous works have admitted the hypothesis that these Mn(II) species generated by this reduction may be adsorbed onto the birnessite surface [START_REF] Barrett | Oxidative degradation of glyphosate and Aminomethylphosphonate by manganese oxide[END_REF][START_REF] Zaied | Spontaneous oxidative degradation of indigo carmine by thin films of birnessite electrodeposited onto SnO2[END_REF][START_REF] Zaied | Studies of N-Demethylation oxidative and degradation of methylene blue by thin layers of birnessite electrodeposited onto SnO2[END_REF][START_REF] Ndjeri | Degradation of glyphosate and AMPA (AminoMethylPhosphonic Acid) solutions by thin films of birnessite electrodeposited: A new design of material for remediation processes? Colloids Surf[END_REF][START_REF] Stone | Reductive dissolution of manganese (III/IV) oxides by substituted phenols[END_REF].

Two pathways are possible: (i) dissolved oxygen in the solution may oxidize adsorbed Mn(II) to regenerate oxide, or (ii) adsorbed Mn(II) species may react by reduction reaction with birnessite leading to its possible transformation. Some studies performed with birnessite powder have described many possible evolutions of birnessite structure after interaction with solutions containing different Mn(II) concentrations at fixed pH during a couple of days. As a result, it has been observed spontaneous redox transformations of birnessite powder into new materials such as feitknechtite (-MnOOH), hausmannite (Mn3O4) or manganite ((-MnOOH) [START_REF] Elzinga | Reductive transformation of birnessite by aqueous Mn(II)[END_REF][START_REF] Zhao | Redox reactions between Mn (II) and hexagonal birnessite change its layer symmetry[END_REF]. Another study focused on reduction of birnessite thin films induced by electrochemical methods confirmed also the possible transformation of birnessite in similar compounds in presence of Mn(II) in solution [START_REF] Ndjeri | Studies on the reduction of birnessite thin layers: Influence of medium[END_REF]. The major problem is that these materials are passivating and inhibit the surface reactivity [START_REF] Ndjeri | Studies on the reduction of birnessite thin layers: Influence of medium[END_REF][START_REF] Zhao | Redox reactions between Mn (II) and hexagonal birnessite change its layer symmetry[END_REF], which could put the brakes on pollutants degradation.

In the present work, thin films synthesized by electrodeposition on conductive substrate (≈2 microns) were used rather than birnessite powder in order to study systematically the behavior of the material after spontaneous interactions with solution containing various concentrations of Mn(II) during long durations (e.g. from few days to several months). Oxygen influence was also studied in order to highlight the material regeneration step. After interaction, materials were investigated at macroscopic scale by Scanning Electron Microscopy (SEM) and X-Ray Diffraction (XRD), and at microscopic scale by X-Ray Photoelectron Spectroscopy (XPS) to focus on any chemical transformation on the extreme surface of the solid. Indeed, XPS is a very well characterization method for determining the oxidation states at the near surface of materials (10 to 20 nanometers). This is a key point for understanding precisely these spontaneous electronic transfers in interfacial processes even if the XPS spectra of manganese oxides in terms of Mn oxidation states can be complex [START_REF] Nesbitt | XPS study of reductive dissolution of 7Åbirnessite by H3AsO3, with constraints on reaction mechanism[END_REF][START_REF] Banerjee | XPS study of reductive dissolution of birnessite by oxalate: Rates and mechanistic aspects of dissolution and redox processes[END_REF][START_REF] Banerjee | XPS study of dissolution of birnessite by humate with constraints on reaction mechanism[END_REF][START_REF] Nesbitt | Interpretation of XPS Mn (2p) spectra of Mn Oxyhydroxides and constraints on the mechanism of MnO2 precipitation[END_REF][START_REF] Ardizzone | Mn3O4 and γ-MnOOH powders, preparation, phase composition and XPS characterization[END_REF][30]. The great advantage of the electrodeposited thin films compared to powder samples is a direct characterization without any preparation due to their good conductivity (no charge effect). In addition, Open Circuit Potential (OCP) was also followed during these different interactions with Mn(II) solutions in presence and in absence of oxygen.

To complete this work, interactions with pollutants were realized in conditions previously studied to investigate the behavior of our samples in real settings [START_REF] Ndjeri | Degradation of glyphosate and AMPA (AminoMethylPhosphonic Acid) solutions by thin films of birnessite electrodeposited: A new design of material for remediation processes? Colloids Surf[END_REF][START_REF] Pensel | Efficient electrochemical treatment based on electrodeposited thin films of birnessite for mineralisation of AMPA (Aminomethylphosphonic Acid) in very soft conditions[END_REF]. Glyphosate and AMPA (aminomethylphosphonic acid) were chosen because these compounds represent respectively the pesticide the most used in the world and its main metabolite, which becomes a real problem due to its persistence and its toxicity [START_REF] Stuart | Review of risk from potential emerging contaminants in UK groundwater[END_REF][START_REF] Li | Degradation and Isotope Source Tracking of Glyphosate and Aminomethylphosphonic Acid[END_REF]. Samples were systematically characterized at the extreme near surface by XPS after the interaction in order to validate the real robustness of the birnessite thin film, a key point for future applications.

EXPERIMENTAL SECTION

Chemical products

Reagents, MnSO4.H2O (ACS Reagent Aldrich), Na2SO4.10H2O (ACS Reagent Sigma Aldrich), glyphosate (PESTANAL® analytical standard Fluka), AMPA (aminomethylphosphonic acid) (99% Sigma Aldrich) were used without further purification. Solutions were prepared with Milli-Q water and the pH was free.

Electrodeposition of birnessite thin films

The electrochemical synthesis of pure birnessite thin films was performed according to our previous studies from oxidation of Mn(II) [START_REF] Choumane | Electrodeposited birnessite thin film: An efficient eco-friendly sorbent for removing heavy metals from water[END_REF][START_REF] Ndjeri | Degradation of glyphosate and AMPA (AminoMethylPhosphonic Acid) solutions by thin films of birnessite electrodeposited: A new design of material for remediation processes? Colloids Surf[END_REF][START_REF] Pensel | Efficient electrochemical treatment based on electrodeposited thin films of birnessite for mineralisation of AMPA (Aminomethylphosphonic Acid) in very soft conditions[END_REF][START_REF] Larabi-Gruet | Studies of electrodeposition from Mn(II) species of thin layers of birnessite onto transparent semiconductor[END_REF]. Electrodeposition was carried out in a solution prepared with Milli-Q water (Na2SO4 0.4 M; [Mn(II)] = 1.6 10 -3 M; pH free (5 -6), with no buffer) in presence of dissolved oxygen (no deaeration). Electrodeposition was performed at room temperature in a classical electrochemical cell set-up using three electrodes: a Mercurous Sulfate Electrode (MSE: +0.656 V vs Normal Hydrogen Electrode (NHE)) as reference electrode, a twisted wire of platinum as counter electrode, and a fluorine doped tin oxide coated glass substrate (SnO2:F or FTO from SOLEMS; 120 Ω/cm²; S = 3 cm²) as working electrode.

Electrodeposition were performed by chronoamperometry (+0.6 V vs MSE) with a fixed electric charge thanks to a Microautolab potentiostat from Metrohm and GPES software. In these experimental conditions, pure H-Birnessite thin film was obtained whatever the value of the fixed electrical charge [START_REF] Choumane | Electrodeposited birnessite thin film: An efficient eco-friendly sorbent for removing heavy metals from water[END_REF]30,[START_REF] Ndjeri | In situ grazing-Incidence X-Ray diffraction during electrodeposition of birnessite thin films: Identification of solid precursors[END_REF][START_REF] Pensel | In situ XANES measurements during electrodeposition of thin film: Example of birnessite, a promising material for environmental applications[END_REF]. For this study, the electrical charge was fixed to Q = 1.8 C, which is the optimal value for samples dedicated to spontaneous degradation of organic pollutants [START_REF] Ndjeri | Degradation of glyphosate and AMPA (AminoMethylPhosphonic Acid) solutions by thin films of birnessite electrodeposited: A new design of material for remediation processes? Colloids Surf[END_REF][START_REF] Pensel | Efficient electrochemical treatment based on electrodeposited thin films of birnessite for mineralisation of AMPA (Aminomethylphosphonic Acid) in very soft conditions[END_REF]. After electrodeposition, thin films were rinsed with Milli-Q water and dried in ambient air. They may be used several days after their preparation.

Interaction studies

One sample of birnessite thin film was introduced into a cell containing 20 mL of interaction solution. Various concentrations of Mn(II) (0, 10 -3 M, 10 -2 M and 10 -1 M), with Na2SO4 at 10 -5 M as electrolyte, were studied. The initial pH of the solutions was always free and no buffer was added. The initial value of pH varied from 3.8 to 5.8 according to the concentration of Mn(II), Table 1.

For each concentration, Open Circuit Potential (OCP) was measured in continuous during the first day, and punctually (several hours) during long durations (until 95 days). For that, a twoelectrode setup was used: the reference electrode was MSE and the working electrode was the birnessite sample in interaction with the studied solution. Between each measurement, the cells were put in the dark and covered with a paraffin film to avoid evaporation of the solution. The pH were measured at the end of the interaction after 95 days, Table 1. The influence of oxygen was also studied. For that, some interactions were performed into degassed solutions of Mn(II) at 10 -1 M with Na2SO4 at 10 -5 M as electrolyte. In this case, the solution was degassed by a continuous flow of argon and OCP was measured in continuous during 8 days. Some interactions were also carried out in presence of pollutants, such as glyphosate and AMPA, in presence of oxygen. For that, solutions with 110 µM of each pollutant, corresponding to 18.5 mg/L for glyphosate and 12.2 mg/L for AMPA respectively, were prepared from powders of glyphosate and AMPA. The pH of the solution was free (≈ 4 -5) and no buffer was added. Two durations were studied based on our previous studies, e.g. 90 h and 168 h for glyphosate and AMPA respectively, corresponding to the optimal conditions of degradation of each pollutant [START_REF] Ndjeri | Degradation of glyphosate and AMPA (AminoMethylPhosphonic Acid) solutions by thin films of birnessite electrodeposited: A new design of material for remediation processes? Colloids Surf[END_REF][START_REF] Pensel | Efficient electrochemical treatment based on electrodeposited thin films of birnessite for mineralisation of AMPA (Aminomethylphosphonic Acid) in very soft conditions[END_REF]. All interactions were performed at room temperature.

Characterization of thin films

The characterizations were made directly without preparation. The surface morphology of the films was observed by Scanning Electron Microscopy (SEM) using a Zeiss Merlin compact microscope (applied voltage fixed to 10 kV). The crystalline structure of the films was identified by X-Ray diffraction (XRD) with a Bruker diffractometer using Cu Kα1 radiation (λ = 1.5405 Å), in the classical Bragg-Brentano set-up. The extreme surface of the thin films (few nanometers) was analyzed using a Thermofisher Scientific K-Alpha X-Ray Photoelectron Spectroscopy (XPS) system, with Al Kα micro-focused monochromator. The monochromatic Al Kα line was used as X-ray excitation with 1486.68 eV. X-ray spots (400 µm) were usually used to probe the surface.

The interpretation of the XPS spectra of manganese oxides in terms of Mn oxidation states was based on studies of Nesbitt et al. [START_REF] Nesbitt | XPS study of reductive dissolution of 7Åbirnessite by H3AsO3, with constraints on reaction mechanism[END_REF][START_REF] Banerjee | XPS study of reductive dissolution of birnessite by oxalate: Rates and mechanistic aspects of dissolution and redox processes[END_REF][START_REF] Banerjee | Oxidation of aqueous Cr (III) at birnessite surfaces: Constraints on reaction mechanism[END_REF][START_REF] Banerjee | XPS study of dissolution of birnessite by humate with constraints on reaction mechanism[END_REF][START_REF] Nesbitt | Interpretation of XPS Mn (2p) spectra of Mn Oxyhydroxides and constraints on the mechanism of MnO2 precipitation[END_REF]. Semi-quantitative analyzes were extracted from standardized XPS spectra and on the base on peak deconvolution allowing the determination of different chemical states of the metal [START_REF] Ardizzone | Mn3O4 and γ-MnOOH powders, preparation, phase composition and XPS characterization[END_REF]30].

RESULTS AND DISCUSSION

OCP measurements: influence of Mn(II) concentration and oxygen.

Fig. 1 presents OCP measurements performed punctually during 8 days when electrodeposited birnessite thin films were in contact with solutions of Na2SO4 10 -5 M (as electrolyte) containing various Mn(II) concentrations, from 0 to 0.1 M, in presence of oxygen. In absence of oxygen, OCP measurements were performed in continuous during 8 days. Some measurements were also realized until 95 days, and the global results in term of pH and potential values, are regrouped in Table 1.

In absence of Mn(II), OCP increases to reach a constant value around 0.2 V vs MSE during the first 24 hours (no peak). After 95 days, OCP drops to zero Volts with a significant increase of pH (from 5.8 to 7.5). The evolutions of potentials are different when measurements were performed in presence of Mn(II) species. Indeed, the values of OCP increase more rapidly, and in direct relation with the augmentation of Mn(II) concentration. Moreover, the OCP amplification is reflected systematically by "a peak" in presence of Mn(II), with OCP maximum values equal to 0.140, 0.250 and 0.325 V vs MSE after 12 hours of interaction in solution containing 10 -3 M, 10 -2 M and 0.1 M Mn(II) respectively. In addition, OCP and pH stay stable in time and particularly for the higher Mn(II) concentrations (10 -2 and 0.1 M).

The evolution of OCP in absence of oxygen by argon degassing during 8 days, is similar to the experiment performed in presence of oxygen with a similar concentration (e.g. 0.1 M Mn(II)). This result signifies that the absence of dioxygen in solution did not affect the behavior of the solid. Furthermore, the thin films stay systematically adherent and visually unchanged during all experiments whatever the conditions signifying that they are stable in these solutions during long durations, and even in unfavorable conditions. However, it is necessary to confirm if the material was modified by characterization at macroscopic and microscopic scales, particularly in the more unfavorable conditions (e.g. 0.1 M Mn(II) in presence or in absence of oxygen).

Characterization of materials after interaction with concentrated solutions of Mn(II) in presence or in absence of dissolved oxygen

Observations by Scanning Electron Microscopy (SEM)

SEM observations (x 3000; inset: x 20000), reported on Fig. 2, were performed directly on the conductive thin films after synthesis (Fig. 2A), and after interactions with 0.1 M Mn(II) solutions in presence (Fig. 2B) or in absence of dissolved oxygen (Fig. 2C). The surface of the electrodeposited birnessite thin film after synthesis and before interaction presents a very covering texture (Fig. 2A). Large nano-sheets are visible with homogeneous grain size around 2 µm as already observed previously [START_REF] Zaied | Spontaneous oxidative degradation of indigo carmine by thin films of birnessite electrodeposited onto SnO2[END_REF][START_REF] Zaied | Studies of N-Demethylation oxidative and degradation of methylene blue by thin layers of birnessite electrodeposited onto SnO2[END_REF][START_REF] Ndjeri | Degradation of glyphosate and AMPA (AminoMethylPhosphonic Acid) solutions by thin films of birnessite electrodeposited: A new design of material for remediation processes? Colloids Surf[END_REF][START_REF] Pensel | Efficient electrochemical treatment based on electrodeposited thin films of birnessite for mineralisation of AMPA (Aminomethylphosphonic Acid) in very soft conditions[END_REF]30]. Despite the interaction with concentrated solutions of Mn(II) (0.1 M) during a long time (112 days) (Fig. 2B), and even in unfavorable conditions in absence of dissolved oxygen during 8 days (Fig. 2C), the surface morphology was not modified.

Moreover, the shapes and sizes of the birnessite sheets are preserved. These observations show that these electrodeposited thin films seem robust at this macroscopic scale.

Identification of the crystalline structure by X-Ray Diffraction after interaction

Fig. 3 presents XRD measurements recorded on the birnessite thin films after interaction with 0.1 M Mn(II) solutions in absence (b) or in presence (c) of dissolved oxygen in comparison to one thin film after synthesis (a). Only the characteristic peaks of birnessite can be identified on the different diffractograms, according to JCPDS 23-1239 card, expected the peaks attributed to SnO2 substrate (JCPDS 41-1445 card). However, some modification was observed on diffractograms after interaction. The peak at 12.13° disappears but the other ones are always present, and no characteristic peaks of other manganese compounds are present. This point is crucial because this loss of peak has been already observed in previous studies performed by Elzinga [26], after the interaction of birnessite powder with Mn(II) solutions at 2 10 -3 M and 4 10 -3 M during 10 days at fixed pH 7, and other materials such as feitknechtite (β-Mn III OOH) and manganite (γ-Mn III OOH) had been respectively identified by XRD measurements. The presence of others manganese oxides confirm the real redox transformation of birnessite powder by Mn(II) at this pH. In the present study, the experiments were reproduced in extreme conditions with very high Mn(II) concentration (0.1 M) and during long durations in presence (112 days) or in absence of oxygen (9 days), and neither the film morphology nor the structure of birnessite change after interaction in such conditions.

Characterization of the extreme surface by X-ray Photoelectron Spectroscopy (XPS)

To complete the characterization of the bulk of material by SEM and XRD, the extreme surface of birnessite thin films was probed directly without preparation by X-ray Photoelectron Spectroscopy (XPS). XPS spectra of the Mn(2p3/2) and O(1s) regions of electrodeposited birnessite thin films were collected for various samples depending on the interaction conditions, Fig. 4A,4B and 4C. In these figures, the red solid curve corresponds to the fit of the XPS data and the colored areas represent the fitted Mn(2p3/2) and O(1s) multiplet peaks. The corresponding compositions are gathered in Table 2.

First, the unreacted birnessite was characterized, Fig. 4A. Previous studies had reported in the literature analyzes of Mn(2p3/2) and O(1s) spectra of birnessite and the attribution of the different peaks [START_REF] Nesbitt | XPS study of reductive dissolution of 7Åbirnessite by H3AsO3, with constraints on reaction mechanism[END_REF][START_REF] Banerjee | XPS study of reductive dissolution of birnessite by oxalate: Rates and mechanistic aspects of dissolution and redox processes[END_REF][START_REF] Banerjee | Oxidation of aqueous Cr (III) at birnessite surfaces: Constraints on reaction mechanism[END_REF][START_REF] Banerjee | XPS study of dissolution of birnessite by humate with constraints on reaction mechanism[END_REF][START_REF] Nesbitt | Interpretation of XPS Mn (2p) spectra of Mn Oxyhydroxides and constraints on the mechanism of MnO2 precipitation[END_REF]30]. The energy peak position and the feature of the Mn(2p3/2) spectrum obtained on the electrodeposited thin film was typical of birnessite reported in the literature [START_REF] Nesbitt | Interpretation of XPS Mn (2p) spectra of Mn Oxyhydroxides and constraints on the mechanism of MnO2 precipitation[END_REF]. Indeed, a maximum is observed near to 642 eV, attributed to Mn(IV) (orange peak) but with a contribution at higher binding energy (centered at 643.5 eV), attributed to the presence of Mn(III) (pink peak), which is characteristic of birnessite. At lower binding energy, another contribution centered at 640 eV, reveals also the presence of Mn(II) (grey peak). Other peaks are present without specific attribution because they may result from the contribution of mixed oxidation degrees of manganese, as it has been already reported in the literature [START_REF] Nesbitt | Interpretation of XPS Mn (2p) spectra of Mn Oxyhydroxides and constraints on the mechanism of MnO2 precipitation[END_REF].

The O(1s) spectrum of unreacted birnessite, Fig. 4A, shows a well-defined peak, centered on 530 eV, corresponding to oxide O 2-(blue peak) in largest majority, OH -(purple peak) centered at 531.5 eV, and minor attached water H2O (red and green peaks) as expected. This O(1s) spectrum is typical of birnessite according to the literature [START_REF] Nesbitt | Interpretation of XPS Mn (2p) spectra of Mn Oxyhydroxides and constraints on the mechanism of MnO2 precipitation[END_REF]. These results are in good agreement with our previous in situ XRD and XAS measurements during electrodeposition of thin films, which confirmed that H-birnessite in particular was synthesized in these experimental conditions [START_REF] Ndjeri | In situ grazing-Incidence X-Ray diffraction during electrodeposition of birnessite thin films: Identification of solid precursors[END_REF][START_REF] Pensel | In situ XANES measurements during electrodeposition of thin film: Example of birnessite, a promising material for environmental applications[END_REF].

The corresponding compositions are gathered in Table 2. Fig. 4B and4C show respectively the Mn(2p3/2) and O(1s) spectra of birnessite thin films after interaction with a Mn(II) solution at 0.1 M, in presence and in absence of dioxygen in solution. Mn(2p3/2) and O(1s) spectra show the same peak positions and features that those analyzed on the birnessite thin film before interaction (Fig. 4A). The additional contribution of birnessite, near the peak centered at 642 eV on Mn(2p3/2) spectra, is always observed, as a welldefined peak corresponding to the oxide species O 2-on the O(1s) spectra. Moreover, the surface composition does not change significantly and stays similar after the interaction with Mn(II) solution in presence or in absence of oxygen, Table 2. Nevertheless, the surface composition in atomic percentage of Mn(II) tends to increase lightly after the interaction of the thin film with the solutions. This could be due to the possible adsorption of Mn(II) present in high concentration in solution, which is more pronounced in absence of oxygen (no oxidation reaction of Mn(II)).

Despite these extreme conditions imposed to birnessite thin films in terms of Mn(II) concentration and duration, the XPS spectra stay comparable to those of the unreacted birnessite thin film, indicating that the extreme surface is not modified. This observation is different from the results reported in the literature with birnessite powder, which has been reduced into manganite (γ-Mn III OOH) after interaction with Mn(II) [START_REF] Elzinga | Reductive transformation of birnessite by aqueous Mn(II)[END_REF]. This latter compound, characterized by Nesbitt and Banerjee by XPS [START_REF] Nesbitt | Interpretation of XPS Mn (2p) spectra of Mn Oxyhydroxides and constraints on the mechanism of MnO2 precipitation[END_REF], has a O(1s) spectrum with an equal proportion of O 2-and OH -and also a Mn(2p3/2) spectrum with multiple peaks with equal intensity. These spectra are strongly different from those reported in Fig. 4A and4B and confirm the absence of the transformation of birnessite thin films.

Characterization of the extreme surface of thin films after interaction with high concentrated solutions of pollutants: Examples of glyphosate and AMPA

Similar XPS characterizations were performed on birnessite thin films after their interaction with solutions containing pollutants such as glyphosate and AMPA during 90 hours and 168 hours respectively. These conditions were chosen according to our previous studies, which report the degradation of these pollutants by electrodeposited birnessite thin films by a simple contact [START_REF] Ndjeri | Degradation of glyphosate and AMPA (AminoMethylPhosphonic Acid) solutions by thin films of birnessite electrodeposited: A new design of material for remediation processes? Colloids Surf[END_REF] or as electrode materials [START_REF] Pensel | Efficient electrochemical treatment based on electrodeposited thin films of birnessite for mineralisation of AMPA (Aminomethylphosphonic Acid) in very soft conditions[END_REF].

In the case of a simple contact, the interaction leads to the degradation of the pollutants and the reduction of birnessite with the production of Mn(II) in solution due to spontaneous redox reactions, as shown by Ndjeri et al. [START_REF] Ndjeri | Degradation of glyphosate and AMPA (AminoMethylPhosphonic Acid) solutions by thin films of birnessite electrodeposited: A new design of material for remediation processes? Colloids Surf[END_REF]. However, the concentration of Mn(II) found in solution were lower than expected for this rate of degradation and mineralization of the pollutant. This suppose that Mn(II) ions were adsorbed onto the solid, and (i) can be oxidized by dissolved oxygen present in solution to regenerate birnessite, or (ii) they can induce the reduction of birnessite by redox reactions. The aim of these experiments was to identify formally the effect of these spontaneous degradation reactions on birnessite thin film, to determine the chemical composition of the extreme surface after reaction, and the real effect of Mn(II) produced. These points are very important for future applications.

Fig. 5A and 5B

show respectively the XPS spectra of electrodeposited birnessite thin films after interaction with glyphosate and AMPA, and the corresponding compositions of the extreme surface in each case are gathered in Table 3. No difference is observed on the Mn(2p3/2) and O(1s) spectra compared to the initial XPS spectra of birnessite thin film after synthesis and before interaction with solution (Fig. 4A). This result is significant since it proves the lack of the surface transformation. Only the proportion of Mn(II) increases slightly (Table 3), due to the adsorption of Mn(II) produced during the degradation of pollutants as it has been proposed [START_REF] Ndjeri | Degradation of glyphosate and AMPA (AminoMethylPhosphonic Acid) solutions by thin films of birnessite electrodeposited: A new design of material for remediation processes? Colloids Surf[END_REF]. The proof of adsorbed Mn(II) on the film surface is then in good agreement with the degradation process.

Moreover, the fact that the extreme surface of birnessite is unchanged even after several days of interaction, while pollutants are degraded, are very encouraging for the development of this innovative treatment.

CONCLUSION

Electrodeposited birnessite thin film is a very interesting material for developing innovative treatment. Remarkable properties of degradation have been already established by previous studies and to complete them it has been shown here that birnessite thin film is also a robust material. The film is able to resist to extreme and unfavorable conditions without being altered. Nature and structure of the bulk and the near surface of the material stay unchanged after long times of interaction with a Mn(II) concentrated solution, and even in absence of oxygen. Same results were observed in presence of pollutants as glyphosate and AMPA at high concentrations, due to a very efficient in continuous regeneration of birnessite, which is very encouraging for future applications.

FUNDING SOURCES:

The present study has been supported by the funding of LabEx CHARMMMAT in its policy of innovative project.

Figure captions: Fig. 1 :

 1 Figure captions:

Fig. 2 :

 2 Fig.2: SEM observations (x 3000; inset: x 20000) of birnessite thin films synthesized by

Fig. 3 :

 3 Fig.3: XRD measurements of electrodeposited birnessite thin films: (a) After synthesis and before interaction; (b) After interaction in anoxic Mn(II) solution at 0.1 M with Na2SO4 10 -5 M as electrolyte during 8 days; (c) After interaction in oxic solution of 0.1 M Mn(II) with Na2SO4 10 -5

Fig. 4 :

 4 Fig.4: XPS Mn(2p3/2) and O(1s) spectra of electrodeposited birnessite thin film (A) After

Fig. 5 :

 5 Fig.5: XPS Mn(2p3/2) and O(1s) spectra of electrodeposited birnessite thin films (A) After

Fig. 1 22Fig. 2

 12 Fig.1

Fig. 3

 3 Fig.3

Fig. 4

 4 Fig.4

Fig. 5 Tables: Table 1 :

 51 Fig.5

Table 2 :

 2 Manganese and Oxygen proportions in electrodeposited birnessite thin film near-surface before and after interaction with 0.1 M Mn(II) solutions, with Na2SO4 10 -5 M as electrolyte, in presence or in absence of oxygen.

		After synthesis and	After interaction in oxic	After interaction in anoxic
	Species	before interaction	conditions during 112 days	conditions during 8 days
		(%)	(%)	(%)
	Mn(II)	6.8	7.9	8.3
	Mn(III)	31.7	34.2	34.7
	Mn(IV)	61.5	57.9	57.0
	O 2-	89.6	78.0	81.0
	OH -	7.5	11.0	12.5
	H2O	2.9	11.0	8.0

Table 3 .

 3 Manganese and Oxygen proportions in electrodeposited birnessite near surface before and after interaction with aqueous solutions containing glyphosate (110 µM) or AMPA (110 µM).

		Before	After interaction for 90 hours	After interaction for 168 hours
	Species	interaction	with a solution of glyphosate	with a solution of AMPA
		(%)	(%)	(%)
	Mn(II)	6.8	9.0	8.0
	Mn(III)	31.7	33.0	33.0
	Mn(IV)	61.5	58.0	59.0
	O 2-	89.6	91.8	86.0
	OH -	7.5	6.2	7.0
	H2O	2.9	2.0	7.0

ACKNOWLEDGMENT:

Thanks to the LabEx CHARMMMAT for financing these researches.