I Introduction
Due to a rapidly growing market, unmanned aerial vehicles (UAVs) have recently gained attentions in many applications. UAVs can be used in cellular and satellite communication systems to improve data connections between a basetransceiver station (BTS) and users that are far from the BTS or obstructed by surrounding objects such as tall buildings and mountains [5, 3, 1, 4, 2]. UAVs can improve data transmission in various ways. Firstly, due to its height in the air, a UAV can have lineofsight (LoS) to the BTS which increases the received signaltonoise ratio (SNR). Secondly, a UAV can use a higher transmit power with equipped largecapacity battery or with solarcharging systems [6]. Thirdly, a UAV can easily adjust its gesture in the sky to beamform the relayed data into a better direction to the BTS. Lastly, UAVs can appear anywhere when there is a need which yields flexible and lowcost network deployments.
A typical scenario of UAVassisted cellular communication system is depicted in Fig. 1. In a simple form, UAVs can be used as amplifyandforward (AF) relays [7] to assist users when they are at cell edges or in deep shadow fading. The UAVs can also be more advanced such as with capabilities of beamforming with gesture adjustments and digital precoding. To simplify the analysis, we model the UAVassisted cellular communication system as a Rayleigh product model when UAVs are beyond LoS. The Rayleigh product channel arises from a general doublescattering model [8], and has been considered before in other contexts e.g., [9, 10, 11, 12, 13, 14]
. However, these works are more focused on analyzing the statistics of eigenvalues and in general have complex expressions for the ergodic capacity.
In this paper, we take a special interest in comparing the ergodic capacity between the UAVassisted communication and the one without it. We derive a lowerbound for the ergodic capacity of the UAVassisted model in closedform, which is shown to be tight. The lowerbound provides insights about the tradeoff between the transmit power and the number of antennas needed of the UAVs. It is also helpful to aid in the designs of UAVassisted cellular networks, for tasks such as specifying the number of antennas and transmit power of the UAVs to achieve certain ergodic capacity, or maximizing the utility of each spentantenna with a given transmit power.
Ii Capacity with UAVAssisted Communication
Let’s consider two different approaches for users connecting to a BTS. The first one is that users, with a total number of transmit antennas, are directly communicating with the BTS, which is equipped with receive antennas. The received signal at BTS reads
(1) 
where comprises the transmitted signal from one or multiple users. The Rayleigh multiinput multioutput (MIMO) channel is of size
and comprises independent and identically distributed (i.i.d.) complexvalued Gaussian elements with zeromean and unitvariance. For simplicity, we assume that
is additive white Gaussian noise (AWGN) with an identity covariance matrix. We let denote the transmitpower^{1}^{1}1In this paper, we abuse the term “transmit power” by including distancedependent power attenuation and largescale fading. on each user antenna.The capacity in this case equals
(2) 
When users are in deep fading or at cell edges, they suffer from poor data connections due to a low received SNR at the BTS, which equals . Increasing transmit power has several disadvantages as it raises interference to neighboring users and also consumes more batterypower. In extreme situations such as natural hazards, the batterylife is important for users to maintain longterm connections to the rescuers. On the other hand, UAVs can be used as AF relays to improve datatransmission of the users through at least two possible means: amplify the received signal from the users with a higher transmit power; and redirect the signal into the direction of the BTS.
Assuming such a UAVassisted cellular communication scenario, where the UAVs are equipped with a total transmit and receive antennas^{2}^{2}2The antennas can belong to a a single UAV or multiple UAVs., the received signal in this case yields a Rayleigh product channel model
(3) 
where , and and are the MIMO channels from users to the UAV with size , and from the UAV to the BTS with size , respectively. Similarly to (1), we denote as the transmit power and model and as Rayleigh channels that comprise i.i.d. complexvalued Gaussian elements with zeromean and unitvariance, and the noise is the same as in (1).
The capacity corresponding to (3) equals
(4) 
where denotes the additional timedelay in UAVassisted transmissions [9]. In a pipelined scheme can be negligible [15], and we let in the discussions.
The received SNR at the BTS in this case equals . To have a higher SNR than the case with direct transmission, it requires
(5) 
Instead of the received SNR, we are also interested in comparing the ergodic capacities in these two cases. Especially when , that is, the number of antennas equipped with the UAVs is less than that of the users. In this case, the spatial multiplexing gain is reduced. This can be due to a large number of users in difficult situations at the same time, or the UAVs have lowcost designs with limit numbers of antennas. According to (5), when is small the transmit power has to increase. But as UAVs use builtin battery, the powercapacity can also be limited. Therefore, it is of interest to evaluate the ergodic capacity in relation to parameters and for the UAVassisted systems, and understand when it is beneficial to use UAVs for assistances.
Iii A LowerBound on Ergodic Capacity
In this section we derive a lowerbound for the Rayleigh product channel model (3). A similar analysis can be carried out for multitier connections through UAVs, that is, the product channel comprises more than two components. At the begining we assume , but as it will become clear later, such an assumption is not needed for the validity of the derived lowerbound.
Iiia The Case
For the purpose of comparison, we first find an upperbound for the direct communication between the users and the BTS. By Jensen’s inequality, the ergodic capacity corresponding to the direct approach (2) is upper bounded as
(6)  
which is tight when the number of receiver antennas is large such as with massive MIMO systems [16].
To derive a lowerbound for the UAVassisted case, we fist note that
where and are matrices.
Using Minkowski’s inequality
the ergodic capacity , corresponding to the UAVassisted approach (4), satisfies
Again by Jensen’s inequality, it holds that
(7) 
Since
(8) 
where , and are complex Wishart distributed, it is readily seen from [20, 21] that
(9)  
(10) 
where is the digamma function [27] and is the EulerMascheroni constant.
IiiB The Case
Although is more interesting, we next also consider the case
. Denote the singularvalue decomposition (SVD)
(12) 
where matrices is unitary, and is diagonal with the last diagonal elements being 0s.
Then, it holds that
where denotes the submatrix of obtained by removing the last columns, and is the submatrix by removing both the last rows and columns of . The last equality in (IIIB) holds since and have identical nonzero eigenvalues. As has the same distribution as , the elements in
are also i.i.d. complex Gaussian distributed, and the same is true for
. That is to say, the ergodic capacity of the new product channel obtained by switching and , i.e., the numbers of antennas of the UAV and the users in Fig. 2, is identical to the original case. Similarity, when , one can also switch the antennas numbers of the UAV and the BTS, while the ergodic capacity remains the same. These arguments lead to a below lemma.Lemma 1.
The ergodic capacity of the Rayleigh product model (3) is invariant under permutations of the antenna parameters .
Following Lemma 1 and the analysis in Sec. IIIA, we have Proposition 1 that states the lowerbound of the ergodic capacity for arbitrary setting of .
Proposition 1.
The ergodic capacity of the Rayleigh product model (3) is lowerbounded as
(14) 
where
and , , and is the remaining element in .
IiiC Transmission with Optimal linear Precoding
Next we consider the case with optimal linear precoding. That is, we assume that the UAV knows both the channel and
. This requires the UAVs to be more than just AF relays, since channel estimation is needed and the slotdelay
will increase. However, we can also assume that the UAV can adjust its gesture to gradually find an optimal beamforming direction based on, e.g., measured received signal strength, and the channel estimation is not required. Nevertheless, in this section we assume that the UAV can apply an optimal linear precoder to improve the performance.With an optimal precoding matrix , the received signal in (3) changes to
(15) 
To optimize the capacity in (15), the precoder is set to
where the unitary matrices is defined in (12), and is obtained from the SVD
(16) 
The diagonal matrix (with being its th diagonal element) denotes the power allocation with a totalpower constraint .
With such a precoder, the capacity in (15) equals
(17)  
where and are the th diagonal elements of and , respectively. The optimal th diagonal element of can be optimized through waterfilling [7]
. However, evaluating the optimal ergodic capacity needs to consider joint probability distribution functions (pdfs) of
and [16, 17]. As we are interested in deriving a lowerbound of the ergodic capacity, to simplify the analysis^{3}^{3}3Although MarchenkoPastur law [18] can be used to simplify the eigenvalue distribution, it requires (as well as and ) to be sufficiently large, which does not hold for practical cases with a finite number of UAVs. an equal power allocation for all transmit antennas of the UAV is assumed. That is, setting and the capacity equals(18) 
Clearly, the number of nonzero eigenvalues in (18) is , and the ergodic capacity is then lowerbounded as
(19) 
where the eigenvalues and has the pdf [16] shown in (20) and (21), respectively, where the coefficient is the associated Laguerre polynomial of order [19]. Inserting them back to (19), the ergodic capacity with optimal precoding is lower bounded by the following double integral,
(20)  
(21) 
(22) 
IiiD Discussions on the Parameter Designs for the UAV
With the derived lowerbound, in order for , it is sufficient to have (assuming )
(23) 
That is,
(24) 
Assuming , in which case,
the condition (24) becomes
(25) 
Hence, for a given set of and , the required transmit power for the UAV exponentially decreases in the number of antennas . This makes intuitive sense according to the MIMO capacity formula [16]. Secondly, when users are in deep fading, we can assume is rather small, and it holds that
Then, the condition (25) is identical to (5). This is because is sufficiently large, which yields according to the channel hardening and favorable propagation properties [17, 26] in massive MIMO systems. Therefore, the differences between the capacities and is the same as the differences in the received SNRs for these two cases.
To design such a UAV assisted communication system, it is of interest to optimize the number of antenna for a given total transmit power constraint
(26) 
Although it may not be true in practical scenarios, theoretically it is always beneficial to have more transmit antennas than to have higher transmit power per antenna under Rayleigh fading. Therefore, we consider the optimization problem to find a maximal for a given such that the capacityincrement ratio is above a certain threshold . That is, with (26) we solve
(27) 
where uses the lowerbound in (11) and specifies the number of antennas of the UAVs. Such an optimization is meaningful in a case that each UAV is equipped with a singleantenna, and the objective is to maximize the utility of each UAV for assisting the users.
Iv Numerical Results
In this section, we provide simulation results to show the performance of UAVassisted cellular communications, as well as the effectiveness of the derived lowerbound for ergodic capacity. We also elaborate on the tradeoffs between the transmit power and the number of antennas used for the UAVs.
Iva Tightness of the LowerBound
In Fig. 3, we compare the ergodic capacities and for received signal models (1) and (3), respectively, and with settings and . The upperbound for and lowerbounds for with different values of are also plotted. As can be seen, the derived lowerbound for in (11) is quite tight when is smaller than . When is larger, it is also asymptotically tight as SNR increases. Furthermore, as expected, when is small, i.e., users are in deepfading propagation, the UAVassisted communication even with can provide higher capacities than a direct approach.
IvB Power Increment for a Small
In Fig. 4, we test the same cases in Fig. 3 with and , and aim at finding the minimal such that the ergodic capacity . We use with two different approaches. The first one is based on the numerical results of the ergodic capacities and the is exact. The second approach is using the derived closedfrom lowerbound in (11) and the value of is computed directly according to (24). As can be seen, these two approaches are quite close, which validates the effectiveness of derived lowerbound.
In Fig. 5, we plot the ratio of for and in relation to . An interesting observation is that when is small, the required power is even less than
in order to have the same ergodic capacity. This is because the Rayleigh product channel has more degrees of freedom in the channel elements, which justifies the use of UAVs for improving the throughput of the cellular network. As predicted by (
25), when increases, the required power is exponentially increased in , and the UAVassisted communication becomes less powerefficient.IvC Ergodic Capacity with Optimal Precoding
In Fig. 6, we compare the ergodic capacities for received signal model (3) with settings and , and a fixed total transmit power . As can be seen, the ergodic capacity with provides substantial gains compared to the case , due to higher spatial multiplexing gains. Further, with optimal precoding (based on both waterfilling and equal powerallocation), the capacities are boosted in the low SNR regime. For a large or at high SNR, the gains with precoding become marginal, due to a large value of . Therefore, the derived lowerbound in (11) is still a good approximation for cases with linear precoding, as it is close to the lowerbound (with equal powerallocation) in (22) and also the optimal precoding (with waterfilling).
IvD Tradeoff Between Power and Number of Antennas
Lastly in Fig. 7, we show the capacityincrement ratio with and using the derived lowerbound in (11) (the numerical results are quite close and therefore not shown). If we set the utility threshold to , the maximal values of are 3, 4, and 4 for at 10, 0, and 10 dB, respectively. Further increasing the number of antennas (with unchanged) will have an utility less than . Another observation is that when increases, the capacity incrementratio also increases, but the gaps also gets smaller. That also means that the solution of (27) will converge.
V Summary
We have considered an unmanned aerial vehicle (UAV) assisted cellular communication system, where the UAV is used as an amplifyandforward relay to improve the data transmissions between a basetransceiver station (BTS) and users at cell edges or in deep shadow fading. We have modeled the channel as a Rayleigh product channel in this case, and derived a tight lowerbound of the ergodic capacity in closedfrom for it. With the obtained lowerbound, analytical results has been simplified, and the behaviors of the ergodic capacity can be clearly seen in terms of the transmit power and the number of antennas of the UAV.
References
 [1] Y. Zeng, R. Zhang, and T. J. Lim, “Wireless communications with unmanned aerial vehicles: opportunities and challenges,” IEEE Commun. Mag., vol. 54, no. 5, pp. 3642, May, 2016.
 [2] S. Hayat, E. Yanmaz, and C. Bettstetter, “Experimental analysis of multipointtopoint UAV communications with IEEE 802.11n and 802.11ac,” IEEE Int. Symp. Personal, Indoor, and Mobile Radio Commun. (PIMRC), Hong Kong, Sep. 2015, pp. 19911996.
 [3] P. Chandhar, D. Danev, and E. G. Larsson, “Massive MIMO for communications with drone swarms,” IEEE Trans. Wireless Commun., vol. 17, no. 3, pp. 16041629, Mar. 2018.
 [4] Q. Wu, Y. Zeng, and R. Zhang, “Joint trajectory and communication design for multiUAV enabled wireless networks,” IEEE Trans. Wireless Commun., vol. 17, no. 3, pp. 21092121, Mar. 2018
 [5] J. Chen and D. Gesbert, “Optimal positioning of flying relays for wireless networks: A LOS map approach,” IEEE Int. Conf. Commun. (ICC), Paris, France, May 2017, pp. 16.
 [6] S. Morton, R. D’Sa, and N. Papanikolopoulos, “Solar powered UAV: Design and experiments,“ IEEE/RSJ Int. Conf. Intell. Robots Syst. (IROS), Hamburg, Germany, Sep. 2015, pp. 24602466.
 [7] A. J. Goldsmith and P. P. Varaiya, “Capacity of fading channels with channel side information,” IEEE Trans. Inf. Theory, vol. 43, no, 6, pp. 19861992, Nov. 1997.
 [8] D. Gesbert, H. Bölcskei, D. A. Gore, and A. J. Paulraj, “Outdoor MIMO wireless channels: Models and performance prediction,” IEEE Trans. Commun., vol. 50, no. 12 pp. 19261934, Dec. 2002.
 [9] A. Firag, P. J. Smith, and M. R. McKay, “Capacity analysis of MIMO three product channels,” IEEE Commun. Theory Workshop (CTW), Australian Feb. 2010, pp. 1318.
 [10] S. Yang and J. C. Belfiore, “Optimal spacetime codes for the MIMO amplifyandforward cooperative channel,” IEEE Trans. Inf. Theory, vol. 53, no. 2, pp. 647663, Jan. 2007.
 [11] S. Jin, M. McKay, K. Wong, and X. Gao, “Transmit beamforming in Rayleigh product MIMO channels: Capacity and performance analysis,” IEEE Trans. Signal Process., vol. 56, no. 10, pp. 52045221, Oct. 2008.
 [12] F. Xue and J. Shi, “On the productdeterminantsum of central Wishart matrices and its application to wireless networks,” IEEE Trans. Inf. Theory, vol. 56, no. 7, pp. 34133421, Jul. 2010.
 [13] Z. D. Baia, B. Miaoc, and B. Jin, “On limit theorem for the eigenvalues of product of two random matrices,” J. Multivariate Anal., vol. 98, no. 1, pp. 76101, Aug. 2006.
 [14] C. K. Lo, S. Vishwanath, and R. W. Heath, “Rate bounds for MIMO relay channels using precoding,” IEEE Global Commun. Conf. (GLOBECOM), St. Louis, MO, USA, Dec. 2005, pp. 11721176.
 [15] R. U. Nabar, H. Bölcskei, and F. Kneubuhler, “Fading relay channels: Performance limits and spacetime signal design,” IEEE J. Select. Areas Commun., vol. 22, no. 6, pp. 1099–1109, Aug. 2004
 [16] I. E. Telatar, “Capacity of multiantenna Gaussian channels,” Trans. Emerg. Telecommun. Technol., vol. 10, no. 6, pp. 585595, Nov. 1999.
 [17] B. M. Hochwald, T. L. Marzetta, and V. Tarokh, “Multipleantenna channel hardening and its implications for rate feedback and scheduling,” IEEE Trans. Inf. Theory, vol. 50, no. 9, pp. 18931909, Sep. 2004.
 [18] V. A. Marčenko and L. A. Pastur, “Distribution of eigenvalues for some sets of random matrices,” Math. of the USSRSbornik, vol. 1, no. 4, pp. 457483, 1967.
 [19] I. S. Gradshteyn and I. M. Ryzhik, Table of integrals, series, and products, New York: Academic Press, 1980.
 [20] O. Oyman, R. U. Nabar, H. Bölcskei, and A. J. Paulraj, “Tight lower bounds on the ergodic capacity of Rayleigh fading MIMO channels,” IEEE Global Commun. Conf. (GLOBECOM), Nov. 2002, pp. 11721176.
 [21] N. R. Goodman, “The distribution of the determinant of a complex Wishart distributed matrix,” Ann. Math. Stat., vol. 34, no. 1, pp. 178180, Mar. 1963.
 [22] R. A. Horn and C. R. Johnson, Matrix Analysis, Cambridge University Press, New York, 1985.
 [23] D. Chizhik, G. Foschini, M. Gans, and R. Valenzuela, “Keyholes, correlations, and capacities of multielement transmit and receive antennas,” IEEE Trans. Wireless Commun., vol. 1, no. 2, pp. 361368, Apr. 2002.

[24]
R. R. Müller, “On the asymptotic eigenvalue distribution of concatenated vectorvalued fading channels,”
IEEE Trans. Inf. Theory., vol. 48, no. 7, pp. 20862091, Jul. 2002.  [25] A. Müller and J. Speidel, “Capacity of multipleinput multipleoutput keyhole channels with antenna selection”, Proc. European Wireless Conf., Paris, France, Apr. 2007.
 [26] T. L. Marzetta, E. G. Larsson, H. Yang, and H. Q. Ngo, Fundamentals of massive MIMO, Cambridge University Press, 2016.
 [27] A. Erdélyi, W. Magnus, F. Oberhettinger, and F. G. Tricomi, Higher transcendental functions, vol. III, McGrawHill, New York, 1955.
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