I Introduction
In the last few years, unmanned aerial vehicles (UAVs) have attracted a lot of attention, due to the availability of compact, smallsize, energyefficient models able to perform many critical tasks efficiently and in an automated manner. The integration of UAVs in wireless communication networks has become a hot research area, mainly with two different approaches [1, 2]. The first research approach focuses on the services that UAVs can bring to wireless networks, since UAVs can be regarded as moving access points (APs). With this perspective, UAVs can be used to increase the network capacity ondemand, fill network coverage holes, fastly deploy a mobile network architecture in the presence of a catastrophic event, etc. The second research approach focuses on the services that the network can bring to UAVs, and in particular on the use of a wireless network to support communications with UAVs [3, 4, 5, 6]. Considering the latter approach, [7, 8, 9]
have recently investigated the use of massive MIMO (mMIMO) to support UAVs cellular communications, showing that equipping base stations (BSs) with large antenna arrays dramatically increases—with respect to a traditional cellular deployment—the probability of meeting the stringent reliability requirements of the UAVs command and control (C
C) link.In this paper, we investigate the use of cellfree (CF) and usercentric (UC) network deployments for the support of UAV communications. In a CF massive MIMO architecture [10]
, large base stations with colocated massive MIMO arrays are replaced by a much larger number of APs, with a small number of antennas and reduced processing capabilities. The APs are connected via a backhaul network to a central processing unit (CPU), which sends to the APs the data symbols to be transmitted to the users and receives soft estimates of the received data symbols from all APs. Neither channel estimates nor beamforming vectors are propagated through the backhaul network, and the timedivisionduplex protocol is used to exploit uplink/downlink channel reciprocity. The results in
[10] show that the CF approach provides better performance than a smallcell system in terms of 95likely peruser throughput. More recently, [11, 12] have introduced a usercentric (UC) virtualcell massive MIMO approach to CF massive MIMO, assuming that each AP does not serve all the users in the system, but only a subset of them. The UC approach has been shown to provide better performance than the pure CF approach to the vast majority of the users in the network, since it allows APs to focus their available resources on the users that will benefit the most.Following on such a track, in this paper we evaluate the capability of a CF UC massive MIMO deployment to support UAV communications in the presence of legacy ground users (GUEs) using the same frequency band. Assuming a Ricean channel model, and linear minimum mean square error (LMMSE) channel estimation, the paper derives a lower bound to the achievable spectral efficiency for both the uplink and downlink. The numerical results reveal the superiority of the considered architecture with respect to a traditional multicell massive MIMO network, for both UAVs and GUEs.
Ii System model
Iia CellFree Network Topology
We consider a network that consists of outdoor APs, GUEs, and UAVs—as depicted in Fig. 1—, whose sets are denoted by , , and , and have cardinalities , , and , respectively. In the following, we let the term users denote both GUEs and UAVs. We assume that all users are equipped with a single antenna and that each AP is equipped with a uniform linear array (ULA) with antennas. We define , use to denote the number of users in the system, and let with cardinality be the set of users served by the th AP on a given physical resource block (PRB). Moreover, we denote by the set of APs serving the th user, with representing its cardinality.
The APs are connected by means of a backhaul network to a CPU wherein datadecoding is performed. Building on the approach of [10], all communications take place on the same frequency band; uplink and downlink are separated through timedivisionduplex (TDD). The coherence interval is thus divided into three phases: (a) uplink channel estimation, (b) downlink data transmission, and (c) uplink data transmission. In phase (a), users send pilot data in order to enable channel estimation at the APs. In phase (b), APs use channel estimates to perform channelmatched beamforming and send data symbols on the downlink. Finally, in phase (c), users send uplink data symbols to the APs. Note that no pilots are transmitted on the downlink and no channel estimation is performed at the users.
IiB Propagation Channel
We denote by the channel between the th user and the th AP. We assume Ricean fading channel, which consist of a dominant lineofsight (LOS) component on top of a Rayleighdistributed component modelling the scattered multipath. The channel from the th user to the th AP is modelled as
(1) 
where is a scalar coefficient modelling the channel pathloss and shadowing effects, is the Ricean factor, and the dimensional vector contains the i.i.d. smallscale fading coefficients between the th AP and the th user. Moreover, is a random variable representing a phase rotation. The vector is the AP antenna array steering vector corresponding to the directionofarrival . Letting denote the vector containing the coordinates of the th antenna at the th AP, and denoting by the vector containing the coordinates of the th user, the th entry of the vector can be expressed as
(2) 
In the following we further describe the above parameters depending on the specific link type:
IiB1 GUEAP parameters
With regard to the GUEtoAP channel, i.e., when , we assume that all the GUEs channels are Rayleighdistributed, i.e., . For the large scale coefficients we adopt the model of [10], i.e.
(3) 
where represents the path loss (expressed in dB) from the th GUE to the th AP, evaluated using the threeslope path loss model of [10, 11]. Moreover,
represents the shadow fading with standard deviation
, where takes into account the correlation of the shadow fading between APs and GUEs that are in close proximity [10, 11].IiB2 UAVAP parameters
With regard to the UAVtoAP channel, i.e., when , the Ricean factor is assumed to be a function of the UAVAP distance [13], i.e.
(4) 
where is the distance between the th user, and the th AP and is the LOS probability evaluated according to [14, Table B1] for the UMi scenario. For the large scale fading we assume
(5) 
with the pathloss evaluated according to [14, Table B2] for the UMi scenario.
IiC User Association and Scheduling
The set of users associated to the th AP can be determined according to several criteria. In this paper, we consider the two following approaches.
IiC1 CF approach
In the CF approach, each AP communicates with all the users in the system, i.e. we have that and the set .
IiC2 UC approach
In the UC approach, the th user is served by the APs that it receives with best average channel conditions. The set , contains the APs with the largest slow fading coefficients to the th user.
Iii The Communication Process
(16)  
Iiia Uplink Training
We denote by the length (in timefrequency samples) of the channel coherence time, and by the length (in timefrequency samples) of the uplink training phase, where we must ensure that . Denote by the dimensional column pilot sequence sent by the th user, and assume that , . The signal received at the th AP during the training phase can be expressed through the following dimensional matrix
(6) 
with denoting the power employed by the th user during the training phase, and a dimensional matrix with i.i.d. entries containing the thermal noise contribution at the th AP. Based on the observable , and exploiting the knowledge of the users’ pilot sequences, the th AP performs estimation of the channel vectors . We assume here knowledge of the user transmit powers . Assuming knowledge of the largescale fading coefficients as in [10] and of the vectors , we form a LMMSE estimate of based on the dimensional statistics
(7) 
The LMMSE channel estimate of the channel is thus written as where the dimensional matrix can be written as
(8) 
with
(9) 
and
(10) 
IiiB Downlink Data Transmission
(23)  
The APs treat the channel estimates as the true channels and perform conjugate beamforming on the downlink. The signal transmitted by the th AP in a generic symbol interval is the following dimensional vector
(11) 
with the downlink datasymbol for the th user, and a scalar coefficient controlling the power transmitted by the th AP to the th user. Letting denote the overall transmitted power by the th AP, the normalized transmitted power must satisfy the constraint
(12) 
where , and denotes the trace operator.
Subsequently, each user receives phasealigned contributions from all APs and does not need to perform channel estimation. The generic th user receives the soft estimate for the data symbol
(13) 
with being the additive white Gaussian noise (AWGN).
Given the expression in (13) an upper bound (UB) for the achievable spectral efficiency can be obtained as [15]
(14) 
where and are the lengths (in timefrequency samples) of the downlink and uplink data transmission phases in each coherence interval, respectively. The expectation in (14) is made over the fast fading channel realizations.
In this paper we also derive a LB of the downlink spectral efficiency, .
Lemma 1: A LB of the downlink spectral efficiency is given by
(15) 
where is shown in (16) at the bottom of this page, and
(17) 
Proof: The proof of Lemma 1 is based on the application of the useandthenforget (UatF) bound [16]. The details of the derivation are omitted due to lack of space.
IiiC Uplink Data Transmission
Since users do not perform channel estimation, they just send their data symbols without any channeldependent phase offset. The dimensional vector received at the th AP in a generic symbol interval is expressed as
(18) 
with and representing the uplink transmit power and the data symbol of the th user, respectively, and is the dimensional AWGN vector.
Each AP decodes the data transmitted by users in . The th AP thus forms, for each , the statistics and sends them to the CPU. Accordingly, the CPU is able to perform the soft estimates for the data sent by the users as follows
(19) 
Using straightforward manipulations, (19) can be rewritten as
(20) 
Similarly to procedure followed for deriving (20), an UB for the achievable spectral efficiency can be obtained as [15]
(21) 
Lemma 2: A LB for the uplink spectral efficiency can be expressed as
(22) 
where in (23) is shown at the bottom of this page, and
(24) 
Proof: The details of the proof, which is also based on the UatF bound, are again omitted due to space constraints.
Iv Power Allocation Strategies
Iva Downlink Power Control
IvA1 Proportional power allocation
For the downlink data transmission, the first power allocation strategy that we consider is the proportional power allocation (PPA):
(25) 
where is the power transmitted by the th AP to the th user. This power allocation rule is such that the generic th AP divides its power in a way that is proportional to the estimated channel strengths. This way, users with good channel coefficients will receive a larger share of the transmit power than users with bad channels.
IvA2 Waterfilling power control
The second power allocation that we consider is a modified waterfilling power control (WFPC), where the “noise” level for the communication between the th AP and the th user is written as
(26) 
The WFPC gives the following power allocation
(27) 
where is the water level and denotes the positive part operator, with the constraint
(28) 
This heuristic power allocation rule can be seen as a sort of APcentric approach to the CF massive MIMO system and is based on the wellknown waterfilling algorithm
[17], which allocates a larger amount of power to the users with better channels conditions, i.e., to those with the lower “noise” levels.IvB Uplink Power Control
For the uplink data transmission, we consider standard fractional power control (FPC) [18, 14], where the th user transmit power is given by
(29) 
and the parameter is obtained considering the channels from the th user to all the APs in the set as
(30) 
V Numerical Results and Key Insights
Deployment  

AP distribution  Horizontal: uniform, vertical: 15 m 
GUE distribution  Horizontal: uniform, vertical: 1.65 m 
UAV distribution  Horizontal: uniform, vertical uniform between 22.5 m and 300 m [14] 
PHY and MAC  
Carrier freq., bandwidth  GHz, MHz 
AP antenna array  Fourelement ULA with spacing 
User antennas  Omnidirectional with 0 dBi gain 
Power control  DL: proportional power allocation (PPA) or waterfilling power control (WFPC) 
UL: FPC with , dBm  
Thermal noise  174 dBm/Hz spectral density 
Noise figure  9 dB at APs/GUEs/UAVs 
User association  Cell free (CF) or user centric (UC) 
Traffic model  Full buffer 
We consider a square area of 1 km with APs, GUEs and UAVs. To avoid boundary effects, and to emulate a network with an infinite area, the square area is wrapped around at the edges. We assume that and that orthogonal pilots are randomly assigned to the users in the system, i.e., our results account for the impact of pilot contamination. The uplink transmit power during training is , with mW . Regarding power allocation, we assume that the maximum downlink power transmitted by the th AP is mW, , and the maximum uplink power transmitted by the th user is mW, . We consider samples, corresponding to a coherence bandwidth of kHz and a coherence time of ms [10], and . The remaining system parameters are detailed in Table I. In the following, we report the rate per user, obtained as the product of the spectral efficiency—as per Section III—and the system bandwidth . We also show the benchmark performance in the case of (i) perfect channel state information (PCSI), and (ii) a multicell massive MIMO (mMIMO) system with four 100antenna BSs transmitting 8 W each.
V1 Uplink performance
report the cumulative distribution functions (CDFs) of the uplink (UL) rate for GUEs and UAVs, respectively, under: (i) a cellfree architecture (CF), (ii) a usercentric architecture (UC)—both under FPC—, and (iii) a benchmark multicell mMIMO deployment. Both figures show the advantages granted by the use of CF and UC schemes with respect to a classical multicell mMIMO deployment. In particular:

[leftmargin=*]

Due to the UL interference caused by UAVs—each in LOS with multiple BSs—the rate of many GUEs under a multicell mMIMO setup is limited. The percentage of GUEs in outage is reduced under perfect CSI, but the overall performance still remains negatively affected by the residual UAVtoBS interference.

A distributed network architecture significantly improves the rates of the most vulnerable GUEs by bringing the APs in close proximity with them. Similar gains are achieved under CF and UC approaches, and they amount to over one order of magnitude for the 95likely rate. Only the best GUEs, which happen to be located close to their serving BS, are better off under a multicell mMIMO setup.

While for UAVs the baseline performance of multicell mMIMO is not as bad as it is for GUEs, similar observations can be made. The most vulnerable UAVs strongly benefit from a distributed architecture that turns UAVtoBS interference into useful signal. Additionally, a CF approach is preferable to UC, since UAVs experience good LOS propagation conditions with a large number of APs, and thus benefit from having many—rather than a subset of—APs serving them.
V2 Downlink performance
Figs. 4 and 5 show the CDFs of the downlink (DL) rate for GUEs and UAVs, respectively, for the following architectures: (i) cellfree with proportional power allocation (CFPPA), (ii) usercentric with and proportional power allocation (UCPPA), (iii) cellfree with waterfilling power control (CFWFPC), and (iv) multicell mMIMO with uniform power allocation (mMIMOUni). Based on these figures, the following observations can be made:

[leftmargin=*]

The DL GUE performance under multicell mMIMO is affected by pilot contamination caused by the UAVs in the UL channel estimation phase. This is illustrated by the gap between the lower bound (LB) and the rates obtained under perfect CSI (PCSI). A usercentric approach with a fair power allocation policy (CFPPA and UCPPA) provides substantial gains.

A cellfree network brings significant benefits to the UAV DL—particularly under WFPC—owed to a large number of APs that serve each UAV and thus generate useful signal from what would otherwise be intercell interference.

A greedy waterfilling power control (WFPC) favors UAVs over GUEs, since UAVs end up being allocated more power due to their better channel conditions.
Vi Conclusions
In this paper, we have investigated the use of cellfree and usercentric architectures for supporting wireless communications with UAVs. From the derived spectral efficiency bounds, we have demonstrated that usercentric and cellfree network deployments can outperform multicell mMIMO networks, and that the improvements are particularly noticeable for the users with worst performance. An extension of this study will consider the use of more sophisticated power control rules, introducing strict reliability requirements for UAV communications, as well as the derivation of spectral efficiency formulas suited for the finite blocklength regime.
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