Regularized Zero-Forcing Precoding Aided Adaptive Coding and Modulation for Large-Scale Antenna Array Based Air-to-Air Communications

# Regularized Zero-Forcing Precoding Aided Adaptive Coding and Modulation for Large-Scale Antenna Array Based Air-to-Air Communications

Jiankang Zhang, , Sheng Chen, , Robert G. Maunder, , Rong Zhang, , Lajos Hanzo,  The authors are with School of Electronics and Computer Science, University of Southampton, U.K. (E-mails: {jz09v, sqc, rm, rz, lh}@ecs.soton.ac.uk. S. Chen is also with King Abdulaziz University, Jeddah, Saudi Arabia.The financial support of the European Research Council’s Advanced Fellow Grant and of the Royal Society Wolfson Research Merit Award as well as of the EPSRC project EP/N004558/1 are gratefully acknowledged. The research data for this paper is available at https://doi.org/10.5258/SOTON/D0592.
###### Abstract

We propose a regularized zero-forcing transmit precoding (RZF-TPC) aided and distance-based adaptive coding and modulation (ACM) scheme to support aeronautical communication applications, by exploiting the high spectral efficiency of large-scale antenna arrays and link adaption. Our RZF-TPC aided and distance-based ACM scheme switches its mode according to the distance between the communicating aircraft. We derive the closed-form asymptotic signal-to-interference-plus-noise ratio (SINR) expression of the RZF-TPC for the aeronautical channel, which is Rician, relying on a non-centered channel matrix that is dominated by the deterministic line-of-sight component. The effects of both realistic channel estimation errors and of the co-channel interference are considered in the derivation of this approximate closed-form SINR formula. Furthermore, we derive the analytical expression of the optimal regularization parameter that minimizes the mean square detection error. The achievable throughput expression based on our asymptotic approximate SINR formula is then utilized as the design metric for the proposed RZF-TPC aided and distance-based ACM scheme. Monte-Carlo simulation results are presented for validating our theoretical analysis as well as for investigating the impact of the key system parameters. The simulation results closely match the theoretical results. In the specific example that two communicating aircraft fly at a typical cruising speed of 920 km/h, heading in opposite direction over the distance up to 740 km taking a period of about 24 minutes, the RZF-TPC aided and distance-based ACM is capable of transmitting a total of 77 Gigabyte of data with the aid of 64 transmit antennas and 4 receive antennas, which is significantly higher than that of our previous eigen-beamforming transmit precoding aided and distance-based ACM benchmark.

Aeronautical communication, Rician channel, large-scale antenna array, adaptive coding and modulation, transmit precoding, regularized zero-forcing precoding

## I Introduction

The vision of the ‘smart sky’ [1] in support of air traffic control and the ‘Internet above the clouds’ [2] for in-flight entertainment has motivated researchers to develop new solutions for aeronautical communications. The aeronautical ad hoc network (AANET) [3] exchanges information using multi-hop air-to-air radio communication links, which is capable of substantially extending the coverage range over the oceanic and remote airspace, without any additional infrastructure and without relying on satellites. However, the existing air-to-air communication solutions can only provide limited data rates. Explicitly, the planed L-band digital aeronautical communications system (L-DACS) [4, 5] only provides upto 1.37 Mbps air-to-ground communication rate, and the aeronautical mobile airport communication system [6] only offers 9.2 Mbps air-to-ground communication rate in the vicinity of the airport. Finally, the L-DACS air-to-air mode [7] is only capable of providing 273 kbps net user rate for direct air-to-air communication, which cannot meet the high-rate demands of the emerging aeronautical applications.

The existing aeronautical communication systems mainly operate in the very high frequency band spanning from 118 MHz to 137 MHz [8], and there are no substantial idle frequency slots for developing broadband commercial aeronautical communications. Moreover, the ultra high frequency band has almost been fully occupied by television broadcasting, cell phones and satellite communications [1, 9]. However, there are many unlicensed-frequencies in the super high frequency (SHF) band spanning from 3 GHz to 30 GHz, which may be explored for the sake of developing broadband commercial aeronautical communications. Explicitly, the wavelength spans from 1 cm to 10 cm for the SHF band, which results in 0.5 cm  5 cm antenna spacing by utilizing the half-wavelength criterion for designing the antenna array. This antenna spacing is capable of accommodating a large-scale antenna array on commercial aircraft, which offers dramatic throughput and energy efficiency benefits [10]. To provide a high throughput and a high spectral efficiency (SE) for commercial air-to-air applications, we propose a large-scale antenna array aided adaptive coding and modulation (ACM) based solution in the SHF band.

As an efficient link adaptation technique, ACM [11, 12] adaptively matches the modulation and coding modes to the conditions of the propagation link, which is capable of enhancing the link reliability and maximizing the throughput. The traditional ACM relies on the instantaneous signal-to-noise ratio (SNR) or signal-to-interference-plus-noise ratio (SINR) to switch the ACM modes, which requires the acquisition of the instantaneous channel state information (CSI). Naturally, channel estimation errors are unavoidable in practice, especially at aircraft velocities [13]. Furthermore, the CSI-feedback based ACM solution may potentially introduce feedback errors and delays [14]. Intensive investigations have been invested in robust ACM, relying on partial CSI [13] and imperfect CSI [15], or exploiting non-coherent detection for dispensing with channel estimation all together [16]. However, all these ACM solutions are designed for terrestrial wireless communications and they have to frequently calculate the SINR and to promptly change the ACM modes, which imposes heavy mode-signaling overhead. Therefore, for air-to-air communications, these ACM designs may become impractical.

Unlike terrestrial channels, which typically exhibit Rayleigh characteristics, aeronautical communication channels exhibit strong line-of-sight (LOS) propagation characteristics [17, 18], and at cruising altitudes, the LOS component dominates the reflected components. Furthermore, the passenger planes typically fly across large-scale geographical distances, and the received signal strength is primarily determined by the pathloss, which is a function of communication distance. In [19], we proposed an eigen-beamforming transmit precoding (EB-TPC) aided and distance-based ACM solution for air-to-air aeronautical communication by exploiting the aeronautical channel characteristics. EB-TPC has the advantage of low-complexity operation by simply conjugating the channel matrix, and it also enables us to derive the closed-form expression of the attainable throughput, which facilitates the design of the distance-based ACM [19]. However, its achievable throughput is far from optimal, since EB-TPC does not actively suppress the inter-antenna interference. Zero-forcing transmit precoding (ZF-TPC) [20] by contrast is capable of mitigating the inter-antenna interference, but it is challenging to provide a closed-form expression for the achievable throughput, particularly for large-scale antenna array based systems. Tataria et al. [21] investigated the distribution of the instantaneous per-terminal SNR for the ZF-TPC aided multi-user system and approximated it as a gamma distribution. Additionally, ZF-TPC also surfers from rate degradation in ill-conditioned channels. By introducing regularization, the regularized ZF-TPC (RZF-TPC) [22] is capable of mitigating the ill-conditioning problem by beneficially balancing the interference cancellation and the noise enhancement [23]. Furthermore, owing to the regularization, it becomes possible to analyze the achievable throughput for the Rayleigh fading channel. Hoydis et al. [24] used the RZF-TPC as the benchmark to study how many extra antennas are needed for the EB-TPC in the context of Rayleigh fading channels.

However, the Rician fading channel experienced in aeronautical communications, which has a non-centered channel matrix due to the presence of the deterministic LOS component, is different from the centered Rayleigh fading channel. This imposes a challenge on deriving a closed-form formula of the achievable throughput, which is a fundamental metric of designing ACM solutions. Few researches have tackled this challenge. Nonetheless, recently three conference papers [25, 26, 27] have investigated the asymptotic sum-rate of the RZF-TPC in Rician channels. Explicitly, Tataria et al. [25] investigated the ergodic sum-rate of the RZF-TPC aided single-cell system under the idealistic condition of uncorrelated Rician channel and the idealistic assumption of perfect channel knowledge. Falconet et al. [26] provided an asymptotic sum-rate expression for RZF-TPC in a single-cell scenario by assuming identical fading-correlation for all the users. Sanguinetti et al. [27] extended this work from the single-cell to the coordinated multi-cell scenario under the same assumption. But crucially, the authors of [27] did not consider the pilot contamination imposed by adjacent cells during the uplink channel estimation [28, 29]. Moreover, the study [27] assumed Rician fading only within the serving cell, while the interfering signals arriving from adjacent cells were still assumed to suffer from Rayleigh fading. This assumption has limited validity in aeronautical communications. Most critically, the asymptotic sum-rates provided in [26] and [27] were based on the assumption that both the number of antennas and the number of served users tend to infinity. The essence of the ‘massive’ antenna array systems is that of serving a small number of users on the same resource block using linear signal processing by employing a large number of antenna elements. Assuming that the number of users on a resource block tends to infinity has no physical foundation at all.

Against this background, this paper designs an RZF-TPC scheme for large-scale antenna array assisted and distance-based ACM aided aeronautical communications, which offers an appealing solution for supporting the emerging Internet above the clouds. Our main contributions are:

1. We derive the closed-form expression of the achievable throughput for the RZF-TPC in the challenging new context of aeronautical communications. Our previous contribution work relying on EB-TPC [19] invoked relatively simple analysis, since it did not involve the non-centered channel matrix inverse. By contrast, the derivation of the closed-form throughput of our new RZF-TPC has to tackle the associated non-centered matrix inverse problem. Moreover, in contrast to the EB-TPC, the regularization parameter of the RZF-TPC has to be optimized for maximizing the throughput. In this paper, we derive the closed-form asymptotic approximation of the SINR for the RZF-TPC in the presence of both realistic channel estimation errors and co-channel interference imposed by the aircraft operating in the same frequency band. We also provide the associated detailed proof. Moreover, we explicitly derive the optimal analytical regularization parameter that minimizes the mean square detection error. Given this asymptotic approximation of the SINR, the fundamental metric of the achievable throughput as the function of the communication distance is provided for designing the distance-based ACM.

2. We develop the new RZF-TPC aided and distance-based ACM design for the application to the large antenna array assisted aeronautical communication in the presence of imperfect CSI and co-channel interference, first considered in [19]. Like our previous EB-TPC aided and distance-based ACM scheme [19], the RZF-TPC aided and distance-based ACM scheme switches its ACM mode based on the distance between the communicating aircraft pair. However, the RZF-TPC is much more powerful, and the proposed design offers significantly higher SE over the previous EB-TPC aided and distance-based ACM design. Specifically, the new design achieves up to 3.0 bps/Hz and 3.5 bps/Hz SE gains with the aid of 32 transmit antennas/4 receive antennas and 64 transmit antennas/4 receive antennas, respectively, over our previous design.

## Ii System Model

We consider an air-to-air communication scenario at cruising altitude. Our proposed time division duplex (TDD) based aeronautical communication system is illustrated Fig. 1. In the communication zone considered, aircraft transmits its data to aircraft , while aircraft , are the interfering aircraft using the same frequency as aircraft and . The aeronautical communication system operates in the SHF band and we assume that the carrier frequency is 5 GHz, which results in a wave-length of 6 cm. Thus, it is practical to accommodate a large-scale high-gain antenna array on the aircraft for achieving high SE. We assume furthermore that all the aircraft are equipped with the same large-scale antenna array. Specifically, each aircraft has antennas, which transmit and receive signals on the same frequency. Explicitly, each aircraft utilizes () antennas, denoted as data-transmitting antennas (DTAs), for transmitting data and utilizes antennas, denoted as data-receiving antennas (DRAs), for receiving data. In line with the maximum attainable spatial degrees of freedom, generally, we have . Furthermore, the system adopts orthogonal frequency-division multiplexing (OFDM) for improving the SE and the TDD protocol for reducing the latency imposed by channel information feedback. Each aircraft has a distance measuring equipment (DME), e.g., radar, which is capable of measuring the distance to nearby aircraft. Alternatively, the GPS system may be utilized to provide the distance information required.

### Ii-a Channel State Information Acquisition

In order to transmit data from to , aircraft needs the CSI linking to aircraft . Aircraft estimates the reverse channel based on the pilots sent by , and then exploits the channel’s reciprocity of TDD protocol to acquire the required CSI. Explicitly, this pilot training phase is shown at the top of Fig. 1, where estimates the channel between the DRAs of and its DTAs based on the pilots sent by in the presence of the interference imposed by the aircraft , . We consider the worst-case scenario, where the interfering aircraft also transmits the same pilot symbols as , which results in the most serious co-channel interference. Since the length of the cyclic prefix (CP) is longer than the channel length , inter-symbol interference is completely eliminated, and the receiver can process the signals on a subcarrier-by-subcarrier basis. Thus, the frequency-domain (FD) signal vector of , , received during the pilot training can be written as

 ˜Ya∗=√Pb∗r,a∗Hb∗a∗˜Xb∗+A∑a=1√Par,a∗Haa∗˜Xa+˜Wa∗, (1)

where is the pilot symbol vector transmitted by , which obeys the complex Gaussian distribution with the mean vector of the -dimensional zero vector and the covariance matrix of the identity matrix , denoted by , and denotes the FD channel transfer function coefficient matrix linking the DRAs of to the DTAs of , for , while is the FD additive white Gaussian noise (AWGN) vector, and and represent the received powers at a single DTA of for the signals transmitted from and , respectively. Moreover, since the worst-case scenario is considered, aircraft uses the same pilot symbol as , and we have for .

Typically, the aeronautical channel consists of a strong LOS path and a cluster of reflected/delayed paths [17, 30, 31]. Hence, the channel is Rician, and is given by

 Hb∗a∗= νHb∗d,a∗+ςHb∗r,a∗, (2)

where and are the deterministic and scattered channel components, respectively, while and , in which is the Rician -factor of the channel. When aircraft are at cruising altitude, the deterministic LOS component dominates, and the scattered component is very weak which may come from the reflections from other distant aircraft or tall mountains. Note that when an aircraft is at cruising altitude, there is no local scatters at all, because a minimum safe distance is enforced among aircraft, and there exists no shadowing effect either. For an aircraft near airport space for landing/takeoff, the scattering component is much stronger than at cruising, but the LOS component still dominates. The scattering component in this case includes reflections from ground, and shadowing effect has to be considered. The scattered component can be expressed as [32]

 Hb∗r,a∗= R12a∗Gb∗a∗(Rb∗)12, (3)

where and are the spatial correlation matrices for the antennas of and the antennas of , respectively, while the elements of follow the independently identically distributed distribution . Thus, , where is the expectation operator and denotes the column stacking operation applied to , while the covariance matrix is given by , in which is the Kronecker product. Since all the aircraft are assumed to be equipped with the same antenna array, we will assume that all the , , are equal111 The local scattering in the aeronautical channel is not as rich as in the terrestrial channel [33], and the difference in the local scatterings amongst different aircraft may be omitted. Furthermore, at the cruising altitude, there exists no local scattering at all. However, even though it is reasonable to assume that all jumbo jets are equipped with identical antenna arrays, the geometric shapes of different types of jumbo jets are slightly different, and thus , only holds approximately., i.e., we have , , and all the are equal, namely, , . Hence, all the covariance matrices are equal, and they can be expressed as

 Rarr,at= ¯Rrr,t=¯Rr⊗¯Rt,∀at,ar∈A and at≠ar. (4)

Note that in practice, and, therefore, the DRAs can always be spaced sufficiently apart so that they become uncorrelated. Consequently, we have .

According to [34], the received power at a single DTA antenna of aircraft is related to the transmitted signal power at a single DRA antenna of by

 Pb∗r,a∗=Pb∗t10−0.1Lb∗pathloss,a∗. (5)

Since we mainly consider air-to-air transmissions, there exists no shadowing, and the pathloss model can be expressed as [34]

 Lb∗path loss,a∗[dB]=−154.06+20log10(f)+20log10(d), (6)

where [Hz] is the carrier frequency and [m] is the distance between the communicating aircraft pair. For the received interference signal power , we have a similar pathloss model. For air-to-ground communication near airport space, it may need to consider shadowing effect, and the shadow fading standard deviation in dB should be added to the pathloss model [35].

The minimum mean square error (MMSE) estimate of is given by [36]

 vec(ˆHb∗a∗)= vec(νHb∗d,a∗)+ς2¯Rrr,t(σ2wPb∗r,a∗INrNt+ς2¯Rrr,t+A∑a=1Par,a∗Pb∗r,a∗ς2¯Rrr,t)−1 ×(vec(ςHb∗r,a∗)+A∑a=1 ⎷Par,a∗Pb∗r,a∗vec(ςHar,a∗)+1√Pb∗r,a∗vec(˜¯Wa∗(˜¯Xb∗)H)), (7)

where consists of the consecutive pilot symbols with , and is the corresponding AWGN matrix over the consecutive OFDM symbols. Explicitly, the distribution of the MMSE estimator (II-A) is [36]

 (8)

whose covariance matrix is given by

 Φb∗a∗= ς2¯Rrr,t(σ2wPb∗r,a∗INrNt+ς2¯Rrr,t+A∑a=1Par,a∗Pb∗r,a∗ς2¯Rrr,t)−1ς2¯Rrr,t. (9)

By defining and , where denotes the estimate of , can be expressed as

 Φb∗a∗= Φa∗⊗Φb∗. (10)

According to Lemma 1 of [37], as . Since is large, we have . Hence, given , is uniquely determined. It is well known that the computational complexity of this optimal MMSE channel estimator is on the order of .

### Ii-B Data Transmission

During the data transmission, transmits the data vector using its DTAs to the DRAs of , in the presence of the co-channel interference imposed by other aircraft, as shown at the bottom of Fig. 1. Owing to the TDD channel reciprocity, the channel encountered by transmitting is and its estimate is given by , which is used for designing the transmit precoding (TPC) for mitigating the inter-antenna interference (IAI). We adopt the powerful RZF-TPC whose TPC matrix is given by

 Va∗b∗= Υa∗b∗(ˆHa∗b∗)H, (11)

with

 Υa∗b∗= (1Nt(ˆHa∗b∗)HˆHa∗b∗+ξa∗b∗INt)−1, (12)

where is the regularization parameter. It can be seen that the complexity of calculating the TPC matrix for the RZF-TPC scheme is on the order of . Given , the received signal vector of aircraft can be written as

 Yb∗= √Pa∗r,b∗Ha∗b∗Va∗b∗Xa∗+A∑a=1√Par,b∗Hab∗VabaXa+Wb∗, (13)

where aircraft uses the RZF-TPC matrix to transmit the data vector to its desired receiving aircraft for , and , and hence is the interference imposed by , while the AWGN vector has the distribution . By using and to denote the -th row and -th column of , respectively, the signal received by the -th antenna of aircraft can be expressed as

 Yb∗n∗r= √Pa∗r,b∗[Ha∗b∗][n∗r: ][Va∗b∗][ :n∗r]Xa∗n∗r+∑nr≠n∗r√Pa∗r,b∗[Ha∗b∗][n∗r: ][Va∗b∗][ :nr]Xa∗nr +A∑a=1Nr∑nr=1√Par,b∗[Hab∗][n∗r: ][Vaba][ :nr]Xanr+Wb∗n∗r. (14)

where the first term in the right-hand side of (II-B) is the desired signal, the second term represents the IAI imposed by the antennas of aircraft for on the desired signal, and the third term is the interference imposed by aircraft for on the desired signal.

## Iii Analysis of Achievable Throughput of RZF-TPC

Since does not know the estimated CSI, the achievable ergodic rate is adopted. We will also take into account the channel estimation error. From the signal (II-B) received at the DRA of , the power of the desired signal and the power of the interference pulse noise can be obtained respectively as

 PSa∗b∗,n∗r= Pa∗r,b∗∣∣∣E{[Ha∗b∗][n∗r: ][Va∗b∗][ :n∗r]}∣∣∣2, (15) PI&Na∗b∗,n∗r= +A∑a=1Par,b∗Nr∑nr=1E{∣∣[Hab∗][n∗r: ][Vaba][ :nr]∣∣2}+σ2w, (16)

where is the variance operator. Thus, the SINR at -th DRA of is given by

 γa∗b∗,n∗r= PSa∗b∗,n∗rPI&Na∗b∗,n∗r, (17)

and the achievable transmission rate per antenna between the transmitting aircraft and the destination aircraft can be readily expressed as

 Ca∗b∗= 1NrNr∑n∗r=1log2(1+γa∗b∗,n∗r). (18)

### Iii-a Statistics of Channel Estimate

The MMSE channel estimate is related to the true channel by

 [Ha∗b∗][nr: ]= (19)

where the estimation error is statistically independent of both and [36]. Recalling the distribution (8), we have

 (20)

where is the covariance matrix of the MMSE estimate given by

 Φa∗b∗= ς2¯Rtr,r(σ2wPa∗r,b∗INrNt+ς2¯Rtr,r+A∑a=1Par,a∗Pa∗r,b∗ς2¯Rtr,r)−1ς2¯Rtr,r. (21)

The spatial correlation matrix in (21) is given by , and we have .

The distribution of is given by

 vec(˜Ha∗b∗)∼CN(0NtNr,Ξa∗b∗), (22)

whose covariance matrix can be expressed as

 Ξa∗b∗= ς2¯Rtr,r−Φa∗b∗=⎡⎢ ⎢ ⎢ ⎢ ⎢ ⎢⎣[Ξa∗b∗](1,1)[Ξa∗b∗](1,2)⋯[Ξa∗b∗](1,Nr)⋮⋮⋯⋮[Ξa∗b∗](Nr,1)[Ξa∗b∗](Nr,2)⋯[Ξa∗b∗](Nr,Nr)⎤⎥ ⎥ ⎥ ⎥ ⎥ ⎥⎦∈CNtNr×NtNr, (23)

where , . This indicates that the distribution of is given by

 [˜Ha∗b∗]T[nr: ]∼ CN(0Nt,[Ξa∗b∗](nr,nr)). (24)

Furthermore, the correlation matrix , where

 Ma∗b∗= (25)

can be expressed in a form similar to (23) having the -th sub-matrix of , . Likewise, has a form similar to that of (23) having the th sub-matrix of , .

### Iii-B Desired Signal Power

Four useful lemmas are collected in Appendix -A. In order to exploit Lemma 1 for calculating the desired signal power, we define

 Υa∗b∗,∅n∗r= (26)

Clearly, is independent of . Recalling of (12), we can express as

 Υa∗b∗[ˆHa∗b∗]H[n∗r: ]= Υa∗b∗,∅n∗r[ˆHa∗b∗]H[n∗r: ]1+1Nt[ˆHa∗b∗][n∗r: ]Υa∗b∗,∅n∗r[ˆHa∗b∗]H[n∗r: ], (27)

according to Lemma 1. Furthermore, can be formulated as

 [Ha∗b∗][n∗r: ][Va∗b∗][ :n∗r]= [Ha∗b∗][n∗r: ]Υa∗b∗,∅n∗r[ˆHa∗b∗]H[n∗r: ]1+1Nt[ˆHa∗b∗][n∗r: ]Υa∗b∗,∅n∗r[ˆHa∗b∗]H[n∗r: ]. (28)

Recalling Lemmas 2 to 4 and (19) as well as the fact that is independent of , the expectation of can be rewritten as

 E{[Ha∗b∗][n∗r: ]Υa∗b∗,∅n∗r[ˆHa∗b∗]H[n∗r: ]}= E{([ˆHa∗b∗][n∗r: ]+[˜Ha∗b∗][n∗r: ])Υa∗b∗,∅n∗r[ˆHa∗b∗]H[n∗r: ]} = ϑa∗b∗,n∗r=Tr{[Θa∗b∗](n∗r,n∗r)Υa∗b∗,∅n∗r}, (29)

in which denotes the matrix-trace operation, and

 ϑa∗b∗,n∗r= [ˆHa∗b∗][n∗r: ]Υa∗b∗,∅n∗r[ˆHa∗b∗]H[n∗r: ], (30) [Θa∗b∗](n∗r,n∗r)= ν2[Ma∗b∗](n∗r,n∗r)+[Φa∗b