UAV Communications with WPT-aided Cell-Free Massive MIMO Systems

04/23/2021
by   Jiakang Zheng, et al.
BEIJING JIAOTONG UNIVERSITY
0

Cell-free (CF) massive multiple-input multiple-output (MIMO) is a promising solution to provide uniform good performance for unmanned aerial vehicle (UAV) communications. In this paper, we propose the UAV communication with wireless power transfer (WPT) aided CF massive MIMO systems, where the harvested energy (HE) from the downlink WPT is used to support both uplink data and pilot transmission. We derive novel closed-form downlink HE and uplink spectral efficiency (SE) expressions that take hardware impairments of UAV into account. UAV communications with current small cell (SC) and cellular massive MIMO enabled WPT systems are also considered for comparison. It is significant to show that CF massive MIMO achieves two and five times higher 95%-likely uplink SE than the ones of SC and cellular massive MIMO, respectively. Besides, the large-scale fading decoding receiver cooperation can reduce the interference of the terrestrial user. Moreover, the maximum SE can be achieved by changing the time-splitting fraction. We prove that the optimal time-splitting fraction for maximum SE is determined by the number of antennas, altitude and hardware quality factor of UAVs. Furthermore, we propose three UAV trajectory design schemes to improve the SE. It is interesting that the angle search scheme performs best than both AP search and line path schemes. Finally, simulation results are presented to validate the accuracy of our expressions.

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I Introduction

The demand for data throughput has been rapidly growing for decades due to a large amount of user equipments (UEs) and various wireless applications [2, 3]. In order to fullfill this demand, cell-free (CF) massive multiple-input multiple-output (MIMO) has been recently proposed as a promising technology for providing a higher and more uniform spectral efficiency (SE) to the UEs in wireless networks [4, 5]. CF massive MIMO systems consist of many geographically distributed access points (APs) connected to a central processing unit (CPU) for coherently serving the UEs by spatial multiplexing on the same time-frequency resource [5, 6].

Compared with small-cell (SC) systems and cellular massive MIMO systems, there are no cell boundaries and many more APs than UEs in CF massive MIMO systems. The APs only serve UEs within their own cell in conventional SC and cellular massive MIMO systems, which may cause large inter-cell interference. The results in [5, 6] showed that the CF massive MIMO system provides better performance than SC and cellular massive MIMO systems in terms of 95%-likely per-user SE. Based on these seminal works, a plethora of papers of CF massive MIMO have been published in recent years. For instance, the performance of CF massive MIMO system with Rician fading channels was investigated in [7]. A thorough investigation of the channel hardening and favorable propagation phenomena in CF massive MIMO systems from a stochastic geometry perspective was provided in [8]

. A practical framework based on deep learning was proposed in

[9]

to perform channel estimation in CF millimeter-wave massive MIMO systems. The authors in

[10] investigated the effect of hardware impairments on the performance of CF massive MIMO systems and point out that hardware impairments at the transmitter has larger effect on the uplink average SE compared with the one at the receiver.

Recently, there are several important and practical applications of unmanned aerial vehicles (UAV), such as mobile base stations, mobile relays, and mobile data collections [11]. In addition, UAVs have been found many promising applications, such as aerial inspection, photography, precision agriculture, traffic control, search and rescue, package delivery, and telecommunications [12]. In addition, there are many significant advantages for UAV-aided wireless communications [13, 14]. For examples, UAVs can be deployed rapidly as aerial base stations or aerial mobile relays to provide improved performance for existing wireless communication networks and to support emergent service in disaster areas [15]. Furthermore, UAVs are useful for data collection and dissemination in wireless sensor networks [16]. Many important and fundamental aspects of UAV-aided wireless communications have been studied in last decade. The authors in [17] introduced a novel concept of three-dimensional (3D) cellular networks and proposed latency-optimal cell association to improve the SE. The authors in [18] proposed new inter-cell interference coordination designs to mitigate the strong uplink interference in cellular-connected UAV communication. By leveraging the use of UAVs for data offloading, the authors in [19] provided a new hybrid network architecture for cellular systems. However, UAV communication with CF massive MIMO systems is rarely investigated. The numerical results in [20, 21] revealed that, in UAV-aided wireless communications, CF massive MIMO may provide better performance than a traditional cellular massive MIMO network. In addition, an UAV is used as the mobile base station (BS) to further improve the performance of the CF massive MIMO system [22].

Although with promising benefits, wireless communications with UAVs are also faced with several challenges [13, 23, 24]. The performance and operational duration of a UAV system is fundamentally constrained by the limited onboard energy [13]

. More specifically, energy consumption of the UAVs can be broadly classified into mobility energy consumption and communication energy consumption, which aim to satisfy the movement of the UAVs and the communication requirement, respectively

[25]. Using radio frequency energy harvesting to achieve wireless power transfer (WPT) and wireless information transfer has been proposed as an effective method to replenish the energy of UAV communication requirement [26]. The authors in [27] investigated the throughput of an UAV-enabled wireless powered communication network with downlink WPT and uplink wireless information transfer. An UAV enabled mobile edge computing wireless powered system was studied in [28] to maximize the achievable computation rate. Moreover, the authors in [29] studied the robust joint design for an energy-constrained UAV secure communication system with WPT to maximize the minimum secrecy rate. Another main challenge stems from the size, weight, and power constraints of UAVs, which could limit their communication, computation, and endurance capabilities [13]. Therefore, the UAV hardware impairment acts as a non-linear filter in practice, because of limited resolutions of analog-to-digital and digital-to-analog converters, power amplifier and oscillator phase noise [30]. However, most of works consider the residual hardware impairments model, which is uncorrelated with input signal [30, 31]. Considering the hardware impairments correlated with input signal, the authors in [32] proposed a more realistic hardware impairments model.

Different from conventional wireless systems in a state of stillness, UAV-enabled wireless communications require proper trajectory design schemes due to energy constraints [33]. Therefore, trajectory design is one important aspect for the success of UAV-enabled wireless communications [34]. The authors in [35] studied the joint design of the 3D aerial trajectory and the wireless resource allocation for maximization of the system sum throughput. The authors in [36] and [37] exploited the mobility of UAV via its trajectory design to tackle the information security in the physical layer. A flight time minimization problem was solved in [38], considering a scenario where an UAV collects data from a set of sensors on a straight line. The authors in [39] investigated the optimal energy trade-off between the UAV and its served ground terminal via UAV trajectory designs. A general UAV-enabled radio access network was studied in [40], while each periodic flight duration of the UAV and the mission completion time for saving UAV time were minimized via optimizing the UAV trajectory in periodic and one-time operation scenarios, respectively. Moreover, the authors in [41] exploited the mobility of the UAV to maximize the energy transferred to all energy receivers by optimizing the UAV trajectory in an UAV-enabled multiuser WPT system.

Motivated by the aforementioned observations, we study the UAV communication with CF massive MIMO enabled WPT systems, where the harvested energy (HE) from the downlink WPT is used to support the uplink data and pilot transmission. In addition, the hardware impairments effect at UAV is also considered. We investigate the uplink energy harvesting and downlink SE performance of the considered systems. The performance of the corresponding SC system and cellular massive MIMO are analyzed for comparison. Moreover, a useful angle search trajectory design scheme is proposed to improve the SE. The heuristic AP search and line path trajectory design schemes are also considered for comparison. The specific contributions of our work are listed as follows:

  • We first derive closed-form expressions for the downlink HE and uplink SE of the UAV communication with CF massive MIMO enabled WPT systems taking into account a realistic hardware impairment model at UAV. Our results show that CF massive MIMO performs better than cellular massive MIMO both in terms of downlink HE and uplink SE. For a fair comparison, we multiply the downlink transmit power by the number of APs, but the uplink SE of SC is still smaller than CF massive MIMO.

  • We find that the maximum SE can be observed by changing the ratio of downlink WPT and uplink data transmission. Increasing the number of antennas and decreasing the altitude of UAV both can improve the uplink SE and lead to longer uplink data transmission are preferred for the optimal operating point of SE. It is also found that hardware impairment of UAV has a bad effect on the SE and reduces the optimal downlink WPT time which we obtain maximum SE. Moreover, the large-scale fading decoding (LSFD) receiver cooperation can reduce the interference of the terrestrial UE (TUE).

  • We investigate the SE-improved UAV trajectory design taking into account a destination. Angle search trajectory design scheme is proposed to improve the SE. Compared with AP search and line path trajectory design schemes, we find that angle search trajectory design scheme performs better in CF massive MIMO. In addition, the angle search scheme can bypass the APs with large interference in both CF massive MIMO and SC systems, but can not in cellular massive MIMO systems.

The rest of this paper is organized as follows. In Section II, we model the channel mode, uplink channel estimation, downlink energy harvesting and uplink data transmission of CF massive MIMO. In Section III, we analyze the downlink HE and uplink SE of the UAV-aided wireless communication enabled WPT systems with CF massive MIMO, SC and cellular massive MIMO architectures, respectively. In Section IV, we provide angle search and AP search trajectory design schemes to achieve performance gain, and line path trajectory design scheme is considered for comparison. The specific simulation results are given in Section V. Finally, we summarize the full text in Section VI.

Notations:

Column vectors and matrices are represented by boldface lowercase letters

and boldface uppercase letters , respectively. We use superscripts and to represent conjugate and conjugate transpose, respectively. is the identity matrix, and denotes the logarithm with base 2. The Euclidean norm, the expectation operators and the definitions are denoted by , , and , respectively. Finally,

represents a circularly symmetric complex Gaussian random variable

with variance

.

Ii System Model

Fig. 1: UAV communications with CF massive MIMO enabled WPT systems.
Fig. 2: Coherence block structure

As illustrated in Fig. 1, we consider a wireless powered CF massive MIMO system consisting of APs and one energy-harvesting UAV. Each AP is equipped with antennas and UAV is equipped with a single antenna111Large signal processing complexity as well as the power consumption make it quite costly to employ multiple antennas in UAVs.. The APs are connected to a CPU via fronthaul links. We assume that all APs simultaneously serve the UAV on the same time-frequency resource, and the hardware impairment at UAV acts as a non-linear filter [32]. Besides, the effect of TUE interference will be investigated in Section III-D.

We consider frame-based transmissions over flat-fading channels on a single frequency band. The communication is divided into coherence blocks consisting of channel uses as shown in Fig. 2. We assume that the uplink training phase occupies channel uses for channel estimation, the downlink WPT phase occupies channel uses and uplink data transmission phase occupies channel uses. We define as time-splitting fraction, which reflects the proportion of downlink energy harvesting and uplink data transmission.

Ii-a Propagation Model

A 3D Euclidean space is used to determine the locations of APs and the UAV. We assume horizontal plane coordinates of APs are and the horizontal plane coordinate of UAV is . In addition, we assume the high of APs are zero and the UAV flies at a fixed altitude . Considering the UAV trajectory, we define the coherence block as time slot, which is the minimum flight time interval. At each time slot, the UAV does channel estimation, downlink wireless energy harvesting and uplink data transmission. Then, the position of UAV at the th time slot can be expressed as

(1)

Due to the high probability of line-of-sight (LoS) link in UAV communication, the large-scale fading between the UAV and the AP

can be modeled by the free-space path loss as

(2)

where denotes the received power at the reference distance (e.g., m) between the transmitter and the receiver.

Take the impact of buildings and obstructions into account, we need to use more refined channel models to reflect the change of propagation environment. For the small-scale fading of UAV communications, we use spatially correlated and altitude dependent Rician fading channels, which include a probabilistic LoS component on top of a Rayleigh distributed component modeling the scattered multipath. Therefore, multiplying the large-scale fading by the small-scale fading effect, the channel gain between UAV and AP at time slot is expressed as

(3)

where is the LoS component, and is the positive semi-definite covariance matrix describing the spatial correlation of the non-line-of-sight (NLoS) components. It is worth noting that channel gain also can be expressed as , where is a Rayleigh distributed component.

Ii-B Uplink Channel Estimation

Due to one UAV is considered, we assume that the UAV only send one pilot () to each AP for uplink channel estimation. Then, the received signal in AP at time slot is

(4)

where is the pilot transmit power of UAV and is the receiver noise in AP at time slot . In addition, is the hardware quality factor of the UAV, and is the hardware impairment of the UAV. Using linear minimum mean square error (LMMSE) estimation, each AP can compute the LMMSE estimate of the channel coefficient as

(5)

where

(6)
(7)

The estimate and the estimation error are distributed as and , where

(8)

Ii-C Downlink Energy Harvesting

Using the beamforming scalar , the transmitted signal from AP at time slot is

(9)

where is the downlink transmission power, and is different energy symbols sent to the UAV at time slot . Then, the received signal at the UAV is expressed as

(10)

where is the hardware impairment of UAV, which is distributed as . is the received power in UAV at time slot expressed as222It is worth noting that the received power is only used for the communication power of UAV, which includes the information signal power and the constant circuit power caused by hardware impairment. In addition, the flight power of UAVs could come from the battery or solar energy [35].

(11)

We assume the energy due to the hardware impairment and the noise in (10) cannot be harvested. Therefore, the HE by UAV is . Note that the energy used for pilot transmission is drawn from , which is the HE in the last time slot333The UAV has an initial energy dBm to send pilot for connecting to the wireless powered networks.. At steady state, we assume a fraction of the HE is used by UAV to send pilot at the next time slot, and the left HE is used for uplink data transmission.

Ii-D Uplink Data Transmission

During the uplink data transmission, the received complex baseband signal in AP at time slot is

(12)

where is the uplink data transmission power, and is the hardware impairment of the UAV. is the receiver noise.

To detect the symbol transmitted from the UAV, the th AP multiplies the received signal with the conjugate of its (locally obtained) channel estimate as

(13)
Remark 1.

During the UAV movement, the channel estimation, the beamforming scalar and MR combing vector change in real time. TABLE I summarizes the computational complexity in terms of complex multiplications per coherence block for different communication phases.

Communication Phase Operation Complex Multiplications
Channel Estimation Computing the precomputed statistical matrix
Multiplying the precomputed statistical matrix
with the received signal vector
Downlink Transmission Computing the beamforming scalar vectors
Multiplying beamforming scalar vectors with
the transmit energy symbols
Uplink Reception Computing MR combing vectors ——
Multiplying MR combing vectors
with the received signal
TABLE I: Computational complexity per coherence block.

Iii Performance Analysis

In this section, we investigate the downlink HE of UAV and uplink performance of CF massive MIMO systems within each time slot. Meanwhile, we consider SC and cellular massive MIMO systems for comparison. Novel expressions for the downlink HE and uplink SE are derived for both systems.

Iii-a CF massive MIMO

Iii-A1 Downlink HE

Using the beamforming scalar , and assume the energy due to hardware impairment and the noise in (10) cannot be harvested. The downlink HE is derived in the following theorem.

Theorem 1.

Based on (11), we can derive the downlink HE by UAV as

(14)

where

(15)
Proof:

Please refer to Appendix A. ∎

Note that the downlink HE at the time slot is divided into two parts by the fraction . Therefore, the uplink data transmission power is , and the next slot pilot transmit power is . Then, we can find that .

Iii-A2 Uplink SE

In CF massive MIMO systems, the obtained quantity (13) is sent to the CPU via the fronthaul as

(16)

where represents the desired signal, represents the beamforming gain uncertainty, represents the hardware impairment effect, and represents the noise term, respectively.

Theorem 2.

A lower bound on the capacity of UAV is

(17)

where is given by

(18)
Proof:

Please refer to Appendix B. ∎

Iii-B Small cell

Iii-B1 Downlink HE

In SC systems, the UAV harvests energy from the energy-maximizing AP. Then, the received signal from AP is

(19)

where is the hardware impairment of UAV. is the received power from AP expressed as

(20)
Corollary 1.

Considering the energy due to hardware impairment and the noise in (19) cannot be harvested and using the beamformer , the HE by UAV at time slot can be derived as

(21)
Proof:

It follows the similar steps in Theorem 1. ∎

In SC systems, we can obtain the uplink data transmission power and the next slot pilot transmit power as and , respectively.

Iii-B2 Uplink SE

In the uplink of a SC system, each AP first estimates the channels based on signals sent from the UAV, as described earlier. The so-obtained channel estimate of UAV is used to multiply the received signal for detecting the desired signal. The combined uplink signal at the AP is

(22)

where the first term denotes the desired received signal from UAV. The remaining terms , and are uncorrelated and represent interference caused by channel estimation errors, the hardware impairment effect, and receiver noise at AP.

Corollary 2.

Using the maximum-ratio combining, the capacity of UAV is lower bounded as

(23)

where

(24)
Proof:

It follows similar steps in [32, Theorem 4.1] for cellular massive MIMO. ∎

Iii-C Cellular massive MIMO

We consider a cellular network with one cell and antennas for cellular BS. It means that the UAV is always served by the cellular BS when flying from the initial position to the destination. The block-fading channel from BS to UAV is modeled as

(25)

where is the LoS component, and is the spatial correlation matrix of the NLoS components. When using standard LMMSE estimation, the LMMSE estimate of and the independent estimation error are respectively given by

(26)
(27)

where

(28)
(29)

Iii-C1 Downlink HE

In cellular massive MIMO system with one cell, the UAV harvests energy from the cellular BS. Then, the received signal from BS is

(30)

where is the hardware impairment of UAV. is the received energy from cellular BS expressed as

(31)
Corollary 3.

Considering the energy due to hardware impairment and the noise in (30) cannot be harvested and using the beamformer , the HE by UAV at time slot can be derived as

(32)

where

(33)
Proof:

It follows the similar steps in Theorem 1. ∎

In cellular systems, we can obtain the uplink data transmission power and the next slot pilot transmit power as and , respectively.

Iii-C2 Uplink SE

The received signal at BS is

(34)

where the first term denotes the desired received signal from UAV. The remaining terms , and are uncorrelated and represent interference caused by channel estimation errors, the hardware impairment effect, and receiver noise at AP.

Corollary 4.

Using the maximum-ratio combining, the capacity of UAV is lower bounded as

(35)

where is given by

(36)
Proof:

It follows similar steps in [32, Theorem 4.1] for cellular massive MIMO. ∎

Iii-D Interference From One Terrestrial UE

The terrestrial interference has a significant impact on UAV communications [12]. In the following, we assume that there is a single-antenna TUE in considered systems. The TUE and UAV respectively use one channel use for channel estimation. In addition, there is no hardware impairment and HE at TUE, and TUE has the same length of uplink and downlink phases with UAV for data transmission. Let denote the channel coefficient between the TUE and the th AP. The fading channel is modelled as

(37)

where is the spatial correlation matrix and is the large-scale fading coefficient. Using minimum mean square error (MMSE) estimation, the channel estimation can be obtained as

(38)

where is the signal transmit power of TUE, , and is the received signal between AP and TUE. The channel estimation is distributed as , where

(39)

Iii-D1 Downlink HE

For CF massive MIMO systems with maximum ratio precoding to TUE, the transmitted signal from AP at time slot can be expressed as

(40)

Then, the received power in UAV at time slot can be expressed as

(41)

Following similar steps in Theorem 1, we can obtain the downlink HE as

(42)

For SC systems, we assume that the TUE is served by AP . Then, we can write as

(43)

For cellular massive MIMO systems, we can write as

(44)

Iii-D2 Uplink SE

For CF massive MIMO systems with the TUE, the received complex baseband signal in AP at time slot can be expressed as

(45)

where is the uplink data transmission power of the TUE. Based on (III-A2), we can obtain the received data at CPU as

(46)

where represents the interference caused by transmitted data from TUE. Following similar steps in Theorem 2, the capacity of UAV is lower bounded by

(47)

where

(48)
Corollary 5.

With the help of LSFD receiver cooperation [6], when there is a TUE as interference, we can derive the maximum SE as

(49)

with is given by

(50)

where