# 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<EMAIL_ADDRESS>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.

###### Index Terms:

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 $\sim$ 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. 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. 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 $a^{*}$
transmits its data to aircraft $b^{*}$, while aircraft $a$, $a=1,2,\cdots,A$
are the interfering aircraft using the same frequency as aircraft $a^{*}$ and
$b^{*}$. 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 $N_{\rm total}$ antennas, which transmit and
receive signals on the same frequency. Explicitly, each aircraft utilizes
$N_{t}$ ($<N_{\rm total}$) antennas, denoted as data-transmitting antennas
(DTAs), for transmitting data and utilizes $N_{r}$ antennas, denoted as data-
receiving antennas (DRAs), for receiving data. In line with the maximum
attainable spatial degrees of freedom, generally, we have $N_{r}<N_{t}$.
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.

Figure 1: The proposed aeronautical communication system employing the RZF-TPC
aided and distance-based ACM scheme, where aircraft $a^{*}$ is transmitting
data to aircraft $b^{*}$ in the presence of co-channel interference.

###  II-A Channel State Information Acquisition

In order to transmit data from $a^{*}$ to $b^{*}$, aircraft $a^{*}$ needs the
CSI linking $a^{*}$ to aircraft $b^{*}$. Aircraft $a^{*}$ estimates the
reverse channel based on the pilots sent by $b^{*}$, 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 $a^{*}$
estimates the channel between the $N_{r}$ DRAs of $b^{*}$ and its $N_{t}$ DTAs
based on the pilots sent by $b^{*}$ in the presence of the interference
imposed by the aircraft $a$, $a=1,2,\cdots,A$. We consider the worst-case
scenario, where the interfering aircraft $a$ also transmits the same pilot
symbols as $b^{*}$, which results in the most serious co-channel interference.
Since the length of the cyclic prefix (CP) $N_{\rm cp}$ is longer than the
channel length $P$, 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 $a^{*}$,
$\widetilde{\bm{Y}}_{a^{*}}=\big{[}\widetilde{Y}^{a^{*}}_{1}~{}\widetilde{Y}^{a^{*}}_{2}\cdots\widetilde{Y}^{a^{*}}_{N_{t}}\big{]}^{\rm
T}\in\mathbb{C}^{N_{t}}$, received during the pilot training can be written as

$\displaystyle\widetilde{\bm{Y}}_{a^{*}}=\sqrt{P_{r,a^{*}}^{b^{*}}}\bm{H}_{a^{*}}^{b^{*}}\widetilde{\bm{X}}^{b^{*}}+\sum\limits_{a=1}^{A}\sqrt{P_{r,a^{*}}^{a}}\bm{H}_{a^{*}}^{a}\widetilde{\bm{X}}^{a}+\widetilde{\bm{W}}_{a^{*}},$
(1)

where
$\widetilde{\bm{X}}^{b^{*}}=\big{[}\widetilde{X}^{b^{*}}_{1}~{}\widetilde{X}^{b^{*}}_{2}\cdots\widetilde{X}^{b^{*}}_{N_{r}}\big{]}^{\rm
T}\in\mathbb{C}^{N_{r}}$ is the pilot symbol vector transmitted by $b^{*}$,
which obeys the complex Gaussian distribution with the mean vector of the
$N_{r}$-dimensional zero vector $\bm{0}_{N_{r}}$ and the covariance matrix of
the $N_{r}\times N_{r}$ identity matrix $\bm{I}_{N_{r}}$, denoted by
$\widetilde{\bm{X}}^{b^{*}}\sim\mathcal{CN}\left(\bm{0}_{N_{r}},\bm{I}_{N_{r}}\right)$,
and $\bm{H}_{a^{*}}^{a^{\prime}}\in\mathbb{C}^{N_{t}\times N_{r}}$ denotes the
FD channel transfer function coefficient matrix linking the $N_{r}$ DRAs of
$a^{\prime}$ to the $N_{t}$ DTAs of $a^{*}$, for $a^{\prime}=b^{*},a$, while
$\widetilde{\bm{W}}_{a^{*}}\sim\mathcal{CN}\left(\bm{0}_{N_{t}},\sigma_{w}^{2}\bm{I}_{N_{t}}\right)$
is the FD additive white Gaussian noise (AWGN) vector, and
$P_{r,a^{*}}^{b^{*}}$ and $P_{r,a^{*}}^{a}$ represent the received powers at a
single DTA of $a^{*}$ for the signals transmitted from $b^{*}$ and $a$,
respectively. Moreover, since the worst-case scenario is considered, aircraft
$a$ uses the same pilot symbol as $b^{*}$, and we have
$\widetilde{\bm{X}}^{a}=\widetilde{\bm{X}}^{b^{*}}$ for $1\leq a\leq A$.

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 $\bm{H}_{a^{*}}^{b^{*}}\in\mathbb{C}^{N_{t}\times N_{r}}$ is given by

$\displaystyle\bm{H}_{a^{*}}^{b^{*}}=$
$\displaystyle\nu\bm{H}_{\text{d},a^{*}}^{b^{*}}+\varsigma\bm{H}_{\text{r},a^{*}}^{b^{*}},$
(2)

where $\bm{H}_{{\rm d},a^{*}}^{b^{*}}\in\mathbb{C}^{N_{t}\times N_{r}}$ and
$\bm{H}_{{\rm r},a^{*}}^{b^{*}}\in\mathbb{C}^{N_{t}\times N_{r}}$ are the
deterministic and scattered channel components, respectively, while
$\nu=\sqrt{\frac{K_{\rm Rice}}{K_{\rm Rice}+1}}$ and $\varsigma=\sqrt{1-\nu}$,
in which $K_{\rm Rice}$ is the Rician $K$-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
$\bm{H}_{\text{r},a^{*}}^{b^{*}}$ can be expressed as [32]

$\displaystyle\bm{H}_{\text{r},a^{*}}^{b^{*}}=$
$\displaystyle\bm{R}_{a^{*}}^{\frac{1}{2}}\bm{G}_{a^{*}}^{b^{*}}\big{(}\bm{R}^{b^{*}}\big{)}^{\frac{1}{2}},$
(3)

where $\bm{R}^{b^{*}}\in\mathbb{C}^{N_{r}\times N_{r}}$ and
$\bm{R}_{a^{*}}\in\mathbb{C}^{N_{t}\times N_{t}}$ are the spatial correlation
matrices for the $N_{r}$ antennas of $b^{*}$ and the $N_{t}$ antennas of
$a^{*}$, respectively, while the elements of
$\bm{G}_{a^{*}}^{b^{*}}\in\mathbb{C}^{N_{t}\times N_{r}}$ follow the
independently identically distributed distribution $\mathcal{CN}(0,1)$. Thus,
$\mathcal{E}\left\\{\bm{vec}\left(\bm{H}_{{\rm
r},a^{*}}^{b^{*}}\right)\right\\}=\bm{0}_{N_{t}N_{r}}$, where
$\mathcal{E}\\{\cdot\\}$ is the expectation operator and $\bm{vec}(\bm{H})$
denotes the column stacking operation applied to $\bm{H}$, while the
covariance matrix $\bm{R}_{{\rm
r},a^{*}}^{b^{*}}=\mathcal{E}\left\\{\bm{vec}\left(\bm{H}_{{\rm
r},a^{*}}^{b^{*}}\right)\bm{vec}^{\rm H}\left(\bm{H}_{{\rm
r},a^{*}}^{b^{*}}\right)\right\\}\in\mathbb{C}^{N_{t}N_{r}\times N_{t}N_{r}}$
is given by $\bm{R}_{{\rm
r},a^{*}}^{b^{*}}=\bm{R}^{b^{*}}\otimes\bm{R}_{a^{*}}$, in which $\otimes$ is
the Kronecker product. Since all the aircraft are assumed to be equipped with
the same antenna array, we will assume that all the $\bm{R}_{a_{t}}$, $\forall
a_{t}\in\mathcal{A}=\\{1,2,\cdots,A,a^{*},b^{*}\\}$, 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
$\bm{R}_{a_{t}}=\bar{\bm{R}}_{t}$, $\forall a_{t}\in\mathcal{A}$ only holds
approximately., i.e., we have $\bm{R}_{a_{t}}=\bar{\bm{R}}_{t}$, $\forall
a_{t}\in\mathcal{A}$, and all the $\bm{R}^{a_{r}}$ are equal, namely,
$\bm{R}^{a_{r}}=\bar{\bm{R}}^{r}$, $\forall a_{r}\in\mathcal{A}$. Hence, all
the covariance matrices are equal, and they can be expressed as

$\displaystyle\bm{R}_{\text{r},a_{t}}^{a_{r}}=$
$\displaystyle\bar{\bm{R}}_{\text{r},t}^{r}=\bar{\bm{R}}^{r}\otimes\bar{\bm{R}}_{t},\,\forall
a_{t},a_{r}\in\mathcal{A}\text{ and }a_{t}\neq a_{r}.$ (4)

Note that in practice, $N_{r}\ll N_{t}$ and, therefore, the DRAs can always be
spaced sufficiently apart so that they become uncorrelated. Consequently, we
have $\bar{\bm{R}}^{r}=\bm{I}_{N_{r}}$.

According to [34], the received power $P_{r,a^{*}}^{b^{*}}$ at a single DTA
antenna of aircraft $a^{*}$ is related to the transmitted signal power
$P_{t}^{b^{*}}$ at a single DRA antenna of $b^{*}$ by

$P_{r,a^{*}}^{b^{*}}=P_{t}^{b^{*}}10^{-0.1L_{{\rm pathloss},a^{*}}^{b^{*}}}.$
(5)

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

$L_{\text{path
loss},a^{*}}^{b^{*}}\,[\text{dB}]=-154.06+20\log_{10}\left(f\right)+20\log_{10}\left(d\right),$
(6)

where $f$ [Hz] is the carrier frequency and $d$ [m] is the distance between
the communicating aircraft pair. For the received interference signal power
$P_{r,a^{*}}^{a}$, 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
$\widehat{\bm{H}}_{a^{*}}^{b^{*}}$ of $\bm{H}_{a^{*}}^{b^{*}}$ is given by
[36]

$\displaystyle\bm{vec}\left(\widehat{\bm{H}}_{a^{*}}^{b^{*}}\right)=$
$\displaystyle\bm{vec}\left(\nu\bm{H}_{\text{d},a^{*}}^{b^{*}}\right)+\varsigma^{2}\bar{\bm{R}}_{\text{r},t}^{r}\left(\frac{\sigma_{w}^{2}}{P_{r,a^{*}}^{b^{*}}}\bm{I}_{N_{r}N_{t}}+\varsigma^{2}\bar{\bm{R}}_{\text{r},t}^{r}+\sum\limits_{a=1}^{A}\frac{P_{r,a^{*}}^{a}}{P_{r,a^{*}}^{b^{*}}}\varsigma^{2}\bar{\bm{R}}_{\text{r},t}^{r}\right)^{-1}$
$\displaystyle\times\bigg{(}\bm{vec}\left(\varsigma\bm{H}_{\text{r},a^{*}}^{b^{*}}\right)+\sum\limits_{a=1}^{A}\sqrt{\frac{P_{r,a^{*}}^{a}}{P_{r,a^{*}}^{b^{*}}}}\bm{vec}\left(\varsigma\bm{H}_{\text{r},a^{*}}^{a}\right)+\frac{1}{\sqrt{P_{r,a^{*}}^{b^{*}}}}\bm{vec}\left(\widetilde{\bar{\bm{W}}}_{a^{*}}\Big{(}\widetilde{\bar{\bm{X}}}^{b^{*}}\Big{)}^{\rm
H}\right)\bigg{)}\\!,$ (7)

where $\widetilde{\bar{\bm{X}}}^{b^{*}}\in\mathbb{C}^{N_{r}\times N_{r}}$
consists of the $N_{r}$ consecutive pilot symbols with
$\widetilde{\bar{\bm{X}}}^{b^{*}}\big{(}\widetilde{\bar{\bm{X}}}^{b^{*}}\big{)}^{\rm
H}=\bm{I}_{N_{r}}$, and
$\widetilde{\bar{\bm{W}}}_{a^{*}}\in\mathbb{C}^{N_{r}\times N_{r}}$ is the
corresponding AWGN matrix over the $N_{r}$ consecutive OFDM symbols.
Explicitly, the distribution of the MMSE estimator (II-A) is [36]

$\displaystyle\bm{vec}\left(\widehat{\bm{H}}_{a^{*}}^{b^{*}}\right)\sim\mathcal{CN}\left(\bm{vec}\left(\nu\bm{H}_{\text{d},a^{*}}^{b^{*}}\right),\bm{\Phi}_{a^{*}}^{b^{*}}\right),$
(8)

whose covariance matrix
$\bm{\Phi}_{a^{*}}^{b^{*}}\in\mathbb{C}^{N_{t}N_{r}\times N_{t}N_{r}}$ is
given by

$\displaystyle\bm{\Phi}_{a^{*}}^{b^{*}}=$
$\displaystyle\varsigma^{2}\bar{\bm{R}}_{\text{r},t}^{r}\left(\frac{\sigma_{w}^{2}}{P_{r,a^{*}}^{b^{*}}}\bm{I}_{N_{r}N_{t}}\\!+\\!\varsigma^{2}\bar{\bm{R}}_{\text{r},t}^{r}\\!+\\!\sum\limits_{a=1}^{A}\frac{P_{r,a^{*}}^{a}}{P_{r,a^{*}}^{b^{*}}}\varsigma^{2}\bar{\bm{R}}_{\text{r},t}^{r}\right)^{-1}\varsigma^{2}\bar{\bm{R}}_{\text{r},t}^{r}.$
(9)

By defining $\bm{\Phi}_{a^{*}}=\mathcal{E}\left\\{\widehat{\bm{H}}_{{\rm
r},a^{*}}^{b^{*}}\left(\widehat{\bm{H}}_{{\rm r},a^{*}}^{b^{*}}\right)^{\rm
H}\right\\}\in\mathbb{C}^{N_{t}\times N_{t}}$ and
$\bm{\Phi}^{b^{*}}=\mathcal{E}\left\\{\left(\widehat{\bm{H}}_{{\rm
r},a^{*}}^{b^{*}}\right)^{\rm H}\widehat{\bm{H}}_{{\rm
r},a^{*}}^{b^{*}}\right\\}\in\mathbb{C}^{N_{r}\times N_{r}}$, where
$\widehat{\bm{H}}_{{\rm r},a^{*}}^{b^{*}}$ denotes the estimate of
$\bm{H}_{{\rm r},b^{*}}^{a^{*}}$, $\bm{\Phi}_{a^{*}}^{b^{*}}$ can be expressed
as

$\displaystyle\bm{\Phi}_{a^{*}}^{b^{*}}=$
$\displaystyle\bm{\Phi}_{a^{*}}\otimes\bm{\Phi}^{b^{*}}.$ (10)

According to Lemma 1 of [37], $\bm{\Phi}^{b^{*}}\to\bm{I}_{N_{r}}$ as
$N_{t}\to\infty$. Since $N_{t}$ is large, we have
$\bm{\Phi}^{b^{*}}\approx\bm{I}_{N_{r}}$. Hence, given
$\bm{\Phi}_{a^{*}}^{b^{*}}$, $\bm{\Phi}_{a^{*}}$ is uniquely determined. It is
well known that the computational complexity of this optimal MMSE channel
estimator is on the order of $\textsf{O}\big{(}N_{r}^{3}N_{t}^{3}\big{)}$.

###  II-B Data Transmission

During the data transmission, $a^{*}$ transmits the data vector
$\bm{X}^{a^{*}}=\big{[}X_{1}^{a^{*}}~{}X_{2}^{a^{*}}\cdots
X_{N_{r}}^{a^{*}}\big{]}^{\rm T}\in\mathbb{C}^{N_{r}}$ using its $N_{t}$ DTAs
to the $N_{r}$ DRAs of $b^{*}$, 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
$\bm{H}_{b^{*}}^{a^{*}}\in\mathbb{C}^{N_{r}\times N_{t}}$ encountered by
transmitting $\bm{X}^{a^{*}}$ is
$\bm{H}_{b^{*}}^{a^{*}}=\left(\bm{H}_{a^{*}}^{b^{*}}\right)^{\rm H}$ and its
estimate is given by
$\widehat{\bm{H}}_{b^{*}}^{a^{*}}=\left(\widehat{\bm{H}}_{a^{*}}^{b^{*}}\right)^{\rm
H}$, 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 $\bm{V}_{b^{*}}^{a^{*}}\in\mathbb{C}^{N_{t}\times N_{r}}$ is given by

$\displaystyle\bm{V}_{b^{*}}^{a^{*}}=$
$\displaystyle\bm{\Upsilon}_{b^{*}}^{a^{*}}(\widehat{\bm{H}}_{b^{*}}^{a^{*}})^{\rm
H},$ (11)

with

$\displaystyle\bm{\Upsilon}_{b^{*}}^{a^{*}}=$
$\displaystyle\left(\frac{1}{N_{t}}(\widehat{\bm{H}}_{b^{*}}^{a^{*}})^{\rm
H}\widehat{\bm{H}}_{b^{*}}^{a^{*}}+\xi_{b^{*}}^{a^{*}}\bm{I}_{N_{t}}\right)^{-1},$
(12)

where $\xi_{b^{*}}^{a^{*}}>0$ 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 $\textsf{O}\big{(}N_{t}^{3}\big{)}$. Given
$\bm{V}_{b^{*}}^{a^{*}}$, the received signal vector
$\bm{Y}_{b^{*}}\in\mathbb{C}^{N_{r}}$ of aircraft $b^{*}$ can be written as

$\displaystyle\bm{Y}_{b^{*}}=$
$\displaystyle\sqrt{P_{r,b^{*}}^{a^{*}}}\bm{H}_{b^{*}}^{a^{*}}\bm{V}_{b^{*}}^{a^{*}}\bm{X}^{a^{*}}\\!+\\!\sum\limits_{a=1}^{A}\sqrt{P_{r,b^{*}}^{a}}\bm{H}_{b^{*}}^{a}\bm{V}_{b^{a}}^{a}\bm{X}^{a}\\!+\\!\bm{W}_{b^{*}},$
(13)

where aircraft $a$ uses the RZF-TPC matrix
$\bm{V}_{b^{a}}^{a}\in\mathbb{C}^{N_{t}\times N_{r}}$ to transmit the data
vector $\bm{X}^{a}=\big{[}X_{1}^{a}~{}X_{2}^{a}\cdots
X_{N_{r}}^{a}\big{]}^{\rm T}$ to its desired receiving aircraft $b^{a}$ for
$1\leq a\leq A$, $b^{a}\neq b^{*}$ and $b^{a}\neq a$, and hence
$\sqrt{P_{r,b^{*}}^{a}}\bm{H}_{b^{*}}^{a}\bm{V}_{b^{a}}^{a}\bm{X}^{a}$ is the
interference imposed by $a$, while the AWGN vector
$\bm{W}_{b^{*}}=\big{[}W_{1}^{b^{*}}~{}W_{2}^{b^{*}}\cdots
W_{N_{r}}^{b^{*}}\big{]}^{\rm T}$ has the distribution
$\mathcal{CN}\big{(}\bm{0}_{N_{r}},\sigma_{w}^{2}\bm{I}_{N_{r}}\big{)}$. By
using $[\bm{A}]_{[n:~{}]}$ and $[\bm{A}]_{[~{}:m]}$ to denote the $n$-th row
and $m$-th column of $\bm{A}$, respectively, the signal received by the
$n_{r}^{*}$-th antenna of aircraft $b^{*}$ can be expressed as

$\displaystyle Y^{b^{*}}_{n_{r}^{*}}=$
$\displaystyle\sqrt{P_{r,b^{*}}^{a^{*}}}\left[\bm{H}_{b^{*}}^{a^{*}}\right]_{[n_{r}^{*}:~{}]}\left[\bm{V}_{b^{*}}^{a^{*}}\right]_{[~{}:n_{r}^{*}]}X_{n_{r}^{*}}^{a^{*}}+\sum\limits_{n_{r}\neq
n_{r}^{*}}\sqrt{P_{r,b^{*}}^{a^{*}}}\left[\bm{H}_{b^{*}}^{a^{*}}\right]_{[n_{r}^{*}:~{}]}\left[\bm{V}_{b^{*}}^{a^{*}}\right]_{[~{}:n_{r}]}X_{n_{r}}^{a^{*}}$
$\displaystyle+\sum\limits_{a=1}^{A}\sum\limits_{n_{r}=1}^{N_{r}}\sqrt{P_{r,b^{*}}^{a}}\left[\bm{H}_{b^{*}}^{a}\right]_{[n_{r}^{*}:~{}]}\left[\bm{V}_{b^{a}}^{a}\right]_{[~{}:n_{r}]}X_{n_{r}}^{a}+W^{b^{*}}_{n_{r}^{*}}.$
(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 $n_{r}$ antennas of aircraft
$b^{*}$ for $n_{r}\neq n_{r}^{*}$ on the desired signal, and the third term is
the interference imposed by aircraft $a$ for $1\leq a\leq A$ on the desired
signal.

##  III Analysis of Achievable Throughput of RZF-TPC

Since $b^{*}$ 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 $Y^{b^{*}}_{n_{r}^{*}}$ (II-B) received at the DRA $n_{r}^{*}$ of
$b^{*}$, the power of the desired signal $P_{{\rm
S}_{b^{*},n_{r}^{*}}^{a^{*}}}$ and the power of the interference pulse noise
$P_{{\rm I\&N}_{b^{*},n_{r}^{*}}^{a^{*}}}$ can be obtained respectively as

$\displaystyle P_{{\rm S}_{b^{*},n_{r}^{*}}^{a^{*}}}=$ $\displaystyle
P_{r,b^{*}}^{a^{*}}\left|\mathcal{E}\left\\{\left[\bm{H}_{b^{*}}^{a^{*}}\right]_{[n_{r}^{*}:~{}]}\left[\bm{V}_{b^{*}}^{a^{*}}\right]_{[~{}:n_{r}^{*}]}\right\\}\right|^{2},$
(15) $\displaystyle P_{{\rm I\&N}_{b^{*},n_{r}^{*}}^{a^{*}}}=$ $\displaystyle
P_{r,b^{*}}^{a^{*}}\text{Var}\left\\{\left[\bm{H}_{b^{*}}^{a^{*}}\right]_{[n_{r}^{*}:~{}]}\left[\bm{V}_{b^{*}}^{a^{*}}\right]_{[~{}:n_{r}^{*}]}\right\\}+P_{r,b^{*}}^{a^{*}}\sum\limits_{n_{r}\neq
n_{r}^{*}}\mathcal{E}\left\\{\left|\left[\bm{H}_{b^{*}}^{a^{*}}\right]_{[n_{r}^{*}:~{}]}\left[\bm{V}_{b^{*}}^{a^{*}}\right]_{[~{}:n_{r}]}\right|^{2}\right\\}$
$\displaystyle+\sum\limits_{a=1}^{A}P_{r,b^{*}}^{a}\sum\limits_{n_{r}=1}^{N_{r}}\mathcal{E}\left\\{\left|\left[\bm{H}_{b^{*}}^{a}\right]_{[n_{r}^{*}:~{}]}\left[\bm{V}_{b^{a}}^{a}\right]_{[~{}:n_{r}]}\right|^{2}\right\\}+\sigma_{w}^{2},$
(16)

where $\text{Var}\left\\{~{}\right\\}$ is the variance operator. Thus, the
SINR at $n_{r}^{*}$-th DRA of $b^{*}$ is given by

$\displaystyle\gamma_{b^{*},n_{r}^{*}}^{a^{*}}=$ $\displaystyle\frac{P_{{\rm
S}_{b^{*},n_{r}^{*}}^{a^{*}}}}{P_{{\rm I\&N}_{b^{*},n_{r}^{*}}^{a^{*}}}},$
(17)

and the achievable transmission rate per antenna between the transmitting
aircraft $a^{*}$ and the destination aircraft $b^{*}$ can be readily expressed
as

$\displaystyle C_{b^{*}}^{a^{*}}=$
$\displaystyle\frac{1}{N_{r}}\sum\limits_{n_{r}^{*}=1}^{N_{r}}\log_{2}\left(1+\gamma_{b^{*},n_{r}^{*}}^{a^{*}}\right).$
(18)

###  III-A Statistics of Channel Estimate

The MMSE channel estimate
$\left[\widehat{\bm{H}}_{b^{*}}^{a^{*}}\right]_{[n_{r}:~{}]}$ is related to
the true channel $\left[\bm{H}_{b^{*}}^{a^{*}}\right]_{[n_{r}:~{}]}$ by

$\displaystyle\left[\bm{H}_{b^{*}}^{a^{*}}\right]_{[n_{r}:~{}]}=$
$\displaystyle\left[\widehat{\bm{H}}_{b^{*}}^{a^{*}}\right]_{[n_{r}:~{}]}+\left[\widetilde{\bm{H}}_{b^{*}}^{a^{*}}\right]_{[n_{r}:~{}]},$
(19)

where the estimation error
$\left[\widetilde{\bm{H}}_{b^{*}}^{a^{*}}\right]_{[n_{r}:~{}]}$ is
statistically independent of both
$\left[\widehat{\bm{H}}_{b^{*}}^{a^{*}}\right]_{[n_{r}:~{}]}$ and
$\left[\bm{H}_{b^{*}}^{a^{*}}\right]_{[n_{r}:~{}]}$ [36]. Recalling the
distribution (8), we have

$\displaystyle\bm{vec}\left(\widehat{\bm{H}}_{b^{*}}^{a^{*}}\right)\sim\mathcal{CN}\left(\bm{vec}\left(\nu\bm{H}_{\text{d},b^{*}}^{a^{*}}\right),\bm{\Phi}_{b^{*}}^{a^{*}}\right),$
(20)

where $\bm{\Phi}_{b^{*}}^{a^{*}}$ is the covariance matrix of the MMSE
estimate $\bm{vec}\left(\widehat{\bm{H}}_{b^{*}}^{a^{*}}\right)$ given by

$\displaystyle\bm{\Phi}_{b^{*}}^{a^{*}}=$
$\displaystyle\varsigma^{2}\bar{\bm{R}}_{\text{r},r}^{t}\left(\frac{\sigma_{w}^{2}}{P_{r,b^{*}}^{a^{*}}}\bm{I}_{N_{r}N_{t}}\\!+\\!\varsigma^{2}\bar{\bm{R}}_{\text{r},r}^{t}\\!+\\!\sum\limits_{a=1}^{A}\frac{P_{r,a^{*}}^{a}}{P_{r,b^{*}}^{a^{*}}}\varsigma^{2}\bar{\bm{R}}_{\text{r},r}^{t}\right)^{-1}\varsigma^{2}\bar{\bm{R}}_{\text{r},r}^{t}.$
(21)

The spatial correlation matrix $\bar{\bm{R}}_{{\rm r},r}^{t}$ in (21) is given
by $\bar{\bm{R}}_{{\rm r},r}^{t}=\bar{\bm{R}}_{t}\otimes\bar{\bm{R}}^{r}$, and
we have $\bar{\bm{R}}^{r}=\bm{I}_{N_{r}}$.

The distribution of $\bm{vec}\left(\widetilde{\bm{H}}_{b^{*}}^{a^{*}}\right)$
is given by

$\displaystyle\bm{vec}\left(\widetilde{\bm{H}}_{b^{*}}^{a^{*}}\right)\sim\mathcal{CN}\left(\bm{0}_{N_{t}N_{r}},\bm{\Xi}_{b^{*}}^{a^{*}}\right),$
(22)

whose covariance matrix $\bm{\Xi}_{b^{*}}^{a^{*}}$ can be expressed as

$\displaystyle\bm{\Xi}_{b^{*}}^{a^{*}}=$
$\displaystyle\varsigma^{2}\bar{\bm{R}}_{{\rm
r},r}^{t}-\bm{\Phi}_{b^{*}}^{a^{*}}=\left[\begin{array}[]{cccc}\left[\bm{\Xi}_{b^{*}}^{a^{*}}\right]_{(1,1)}&\left[\bm{\Xi}_{b^{*}}^{a^{*}}\right]_{(1,2)}&\cdots&\left[\bm{\Xi}_{b^{*}}^{a^{*}}\right]_{(1,N_{r})}\\\
\vdots&\vdots&\cdots&\vdots\\\
\left[\bm{\Xi}_{b^{*}}^{a^{*}}\right]_{(N_{r},1)}&\left[\bm{\Xi}_{b^{*}}^{a^{*}}\right]_{(N_{r},2)}&\cdots&\left[\bm{\Xi}_{b^{*}}^{a^{*}}\right]_{(N_{r},N_{r})}\end{array}\right]\in\mathbb{C}^{N_{t}N_{r}\times
N_{t}N_{r}},$ (26)

where
$\left[\bm{\Xi}_{b^{*}}^{a^{*}}\right]_{(i,j)}=\mathcal{E}\left\\{\left[\widetilde{\bm{H}}_{b^{*}}^{a^{*}}\right]_{[i:~{}]}^{\rm
H}\left[\widetilde{\bm{H}}_{b^{*}}^{a^{*}}\right]_{[j:~{}]}\right\\}\in\mathbb{C}^{N_{t}\times
N_{t}}$, $\forall i,j\in\\{1,2,\cdots,N_{r}\\}$. This indicates that the
distribution of
$\left[\widetilde{\bm{H}}_{b^{*}}^{a^{*}}\right]_{[n_{r}:~{}]}$ is given by

$\displaystyle\left[\widetilde{\bm{H}}_{b^{*}}^{a^{*}}\right]_{[n_{r}:~{}]}^{\rm
T}\sim$
$\displaystyle\mathcal{CN}\left(\bm{0}_{N_{t}},\left[\bm{\Xi}_{b^{*}}^{a^{*}}\right]_{(n_{r},n_{r})}\right).$
(27)

Furthermore, the correlation matrix
$\mathcal{E}\left\\{\bm{vec}\left(\widehat{\bm{H}}_{a^{*}}^{b^{*}}\right)\bm{vec}\left(\widehat{\bm{H}}_{a^{*}}^{b^{*}}\right)^{\rm
H}\right\\}=\nu^{2}\bm{M}_{b^{*}}^{a^{*}}+\bm{\Phi}_{b^{*}}^{a^{*}}$, where

$\displaystyle\bm{M}_{b^{*}}^{a^{*}}=$
$\displaystyle\bm{vec}\left(\bm{H}_{\text{d},a^{*}}^{b^{*}}\right)\bm{vec}\left(\bm{H}_{{\rm
d},a^{*}}^{b^{*}}\right)^{\rm H}\in\mathbb{C}^{N_{t}N_{r}\times N_{t}N_{r}}.$
(28)

$\bm{M}_{b^{*}}^{a^{*}}$ can be expressed in a form similar to (26) having the
$(i,j)$-th sub-matrix of
$\left[\bm{M}_{b^{*}}^{a^{*}}\right]_{(i,j)}=\left[\bm{H}_{{\rm
d},a^{*}}^{b^{*}}\right]_{[i:~{}]}^{\rm H}\left[\bm{H}_{{\rm
d},a^{*}}^{b^{*}}\right]_{[j:~{}]}\in\mathbb{C}^{N_{t}\times N_{t}}$, $\forall
i,j\in\\{1,2,\cdots,N_{r}\\}$. Likewise, $\bm{\Phi}_{b^{*}}^{a^{*}}$ has a
form similar to that of (26) having the $(i,j)$th sub-matrix of
$\left[\bm{\Phi}_{b^{*}}^{a^{*}}\right]_{(i,j)}\in\mathbb{C}^{N_{t}\times
N_{t}}$, $\forall i,j\in\\{1,2,\cdots,N_{r}\\}$.

###  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

$\displaystyle\bm{\Upsilon}_{b^{*},\emptyset n_{r}^{*}}^{a^{*}}=$
$\displaystyle\left(\frac{1}{N_{t}}\big{(}\widehat{\bm{H}}_{b^{*}}^{a^{*}}\big{)}^{\rm
H}\widehat{\bm{H}}_{b^{*}}^{a^{*}}-\frac{1}{N_{t}}\left[\widehat{\bm{H}}_{b^{*}}^{a^{*}}\right]_{[n_{r}^{*}:~{}]}^{\rm
H}\left[\widehat{\bm{H}}_{b^{*}}^{a^{*}}\right]_{[n_{r}^{*}:~{}]}+\xi_{b^{*}}^{a^{*}}\bm{I}_{N_{t}}\right)^{-1}.$
(29)

Clearly, $\bm{\Upsilon}_{b^{*},\emptyset n_{r}^{*}}^{a^{*}}$ is independent of
$\left[\widehat{\bm{H}}_{b^{*}}^{a^{*}}\right]_{[n_{r}^{*}:~{}]}$. Recalling
$\bm{\Upsilon}_{b^{*}}^{a^{*}}$ of (12), we can express
$\bm{\Upsilon}_{b^{*}}^{a^{*}}\left[\widehat{\bm{H}}_{b^{*}}^{a^{*}}\right]_{[n_{r}^{*}:~{}]}^{\rm
H}$ as

$\displaystyle\bm{\Upsilon}_{b^{*}}^{a^{*}}\left[\widehat{\bm{H}}_{b^{*}}^{a^{*}}\right]_{[n_{r}^{*}:~{}]}^{\rm
H}=$ $\displaystyle\frac{\bm{\Upsilon}_{b^{*},\emptyset
n_{r}^{*}}^{a^{*}}\left[\widehat{\bm{H}}_{b^{*}}^{a^{*}}\right]_{[n_{r}^{*}:~{}]}^{\rm
H}}{1+\frac{1}{N_{t}}\left[\widehat{\bm{H}}_{b^{*}}^{a^{*}}\right]_{[n_{r}^{*}:~{}]}\bm{\Upsilon}_{b^{*},\emptyset
n_{r}^{*}}^{a^{*}}\left[\widehat{\bm{H}}_{b^{*}}^{a^{*}}\right]_{[n_{r}^{*}:~{}]}^{\rm
H}},$ (30)

according to Lemma 1. Furthermore,
$\left[\bm{H}_{b^{*}}^{a^{*}}\right]_{[n_{r}^{*}:~{}]}\left[\bm{V}_{b^{*}}^{a^{*}}\right]_{[~{}:n_{r}^{*}]}$
can be formulated as

$\displaystyle\left[\bm{H}_{b^{*}}^{a^{*}}\right]_{[n_{r}^{*}:~{}]}\left[\bm{V}_{b^{*}}^{a^{*}}\right]_{[~{}:n_{r}^{*}]}=$
$\displaystyle\frac{\left[\bm{H}_{b^{*}}^{a^{*}}\right]_{[n_{r}^{*}:~{}]}\bm{\Upsilon}_{b^{*},\emptyset
n_{r}^{*}}^{a^{*}}\left[\widehat{\bm{H}}_{b^{*}}^{a^{*}}\right]_{[n_{r}^{*}:~{}]}^{\rm
H}}{1+\frac{1}{N_{t}}\left[\widehat{\bm{H}}_{b^{*}}^{a^{*}}\right]_{[n_{r}^{*}:~{}]}\bm{\Upsilon}_{b^{*},\emptyset
n_{r}^{*}}^{a^{*}}\left[\widehat{\bm{H}}_{b^{*}}^{a^{*}}\right]_{[n_{r}^{*}:~{}]}^{\rm
H}}.$ (31)

Recalling Lemmas 2 to 4 and (19) as well as the fact that
$\left[\widetilde{\bm{H}}_{b^{*}}^{a^{*}}\right]_{[n_{r}^{*}:~{}]}$ is
independent of
$\left[\widehat{\bm{H}}_{b^{*}}^{a^{*}}\right]_{[n_{r}^{*}:~{}]}$, the
expectation of
$\left[\bm{H}_{b^{*}}^{a^{*}}\right]_{[n_{r}^{*}:~{}]}\bm{\Upsilon}_{b^{*},\emptyset
n_{r}^{*}}^{a^{*}}\left[\widehat{\bm{H}}_{b^{*}}^{a^{*}}\right]_{[n_{r}^{*}:~{}]}^{\rm
H}$ can be rewritten as

$\displaystyle\mathcal{E}\left\\{\left[\bm{H}_{b^{*}}^{a^{*}}\right]_{[n_{r}^{*}:~{}]}\bm{\Upsilon}_{b^{*},\emptyset
n_{r}^{*}}^{a^{*}}\left[\widehat{\bm{H}}_{b^{*}}^{a^{*}}\right]_{[n_{r}^{*}:~{}]}^{\rm
H}\right\\}=$
$\displaystyle\mathcal{E}\left\\{\left(\left[\widehat{\bm{H}}_{b^{*}}^{a^{*}}\right]_{[n_{r}^{*}:~{}]}+\left[\widetilde{\bm{H}}_{b^{*}}^{a^{*}}\right]_{[n_{r}^{*}:~{}]}\right)\bm{\Upsilon}_{b^{*},\emptyset
n_{r}^{*}}^{a^{*}}\left[\widehat{\bm{H}}_{b^{*}}^{a^{*}}\right]_{[n_{r}^{*}:~{}]}^{\rm
H}\right\\}$ $\displaystyle=$
$\displaystyle\vartheta_{b^{*},n_{r}^{*}}^{a^{*}}=\text{Tr}\left\\{\left[\bm{\Theta}_{b^{*}}^{a^{*}}\right]_{(n_{r}^{*},n_{r}^{*})}\bm{\Upsilon}_{b^{*},\emptyset
n_{r}^{*}}^{a^{*}}\right\\},$ (32)

in which $\text{tr}\\{\cdot\\}$ denotes the matrix-trace operation, and

$\displaystyle\vartheta_{b^{*},n_{r}^{*}}^{a^{*}}=$
$\displaystyle\left[\widehat{\bm{H}}_{b^{*}}^{a^{*}}\right]_{[n_{r}^{*}:~{}]}\bm{\Upsilon}_{b^{*},\emptyset
n_{r}^{*}}^{a^{*}}\left[\widehat{\bm{H}}_{b^{*}}^{a^{*}}\right]_{[n_{r}^{*}:~{}]}^{\rm
H},$ (33)
$\displaystyle\left[\bm{\Theta}_{b^{*}}^{a^{*}}\right]_{(n_{r}^{*},n_{r}^{*})}=$
$\displaystyle\nu^{2}\left[\bm{M}_{b^{*}}^{a^{*}}\right]_{(n_{r}^{*},n_{r}^{*})}+\left[\bm{\Phi}_{b^{*}}^{a^{*}}\right]_{(n_{r}^{*},n_{r}^{*})}.$
(34)

The following theorem is required for the asymptotic analysis of the
achievable data rate.

######  Theorem 1 (Deterministic equivalents [38, 39])

Let
$\bm{H}=\nu\bar{\bm{H}}+\varsigma(\bm{R})^{\frac{1}{2}}\frac{\bm{G}}{\sqrt{N}}(\widetilde{\bm{R}})^{\frac{1}{2}}\in\mathbb{C}^{N\times
K}$, where $\bar{\bm{H}}\in\mathbb{C}^{N\times K}$ is a deterministic matrix,
$\bm{R}\in\mathbb{C}^{N\times N}$ and
$\widetilde{\bm{R}}\in\mathbb{C}^{K\times K}$ are deterministic diagonal
matrices with non-negative diagonal elements, and
$\bm{G}\in\mathbb{C}^{N\times K}$ is a random matrix with each element obeying
the distribution $\mathcal{CN}(0,1)$, while $\nu\in[0,~{}1]$ and
$\varsigma\in[0,~{}1]$ with $\nu^{2}+\varsigma^{2}=1$ are the weighting
factors of $\bar{\bm{H}}$ and
$(\bm{R})^{\frac{1}{2}}\frac{\bm{G}}{\sqrt{N}}(\widetilde{\bm{R}})^{\frac{1}{2}}$,
respectively. Furthermore, $\bm{R}$ and $\widetilde{\bm{R}}$ have uniformly
bounded spectral norms with respect to $K$ and $N$. The matrices

$\displaystyle\bm{T}(\xi)=$
$\displaystyle\left(\xi\left(\bm{I}_{N}+\tilde{\delta}\varsigma^{2}\bm{R}\right)+\nu^{2}\bar{\bm{H}}\left(\bm{I}_{K}+\delta\widetilde{\bm{R}}\right)^{-1}\bar{\bm{H}}^{\rm
H}\right)^{-1},$ (35) $\displaystyle\widetilde{\bm{T}}(\xi)=$
$\displaystyle\left(\xi\left(\bm{I}_{K}+\delta\varsigma^{2}\widetilde{\bm{R}}\right)+\nu^{2}\bar{\bm{H}}^{\rm
H}\left(\bm{I}_{N}+\tilde{\delta}\bm{R}\right)^{-1}\bar{\bm{H}}\right)^{-1},$
(36)

are the respective approximations of the resolvent $\bm{Q}(\xi)$ and the co-
resolvent $\widetilde{\bm{Q}}(\xi)$

$\displaystyle\bm{Q}(\xi)=$ $\displaystyle\left(\bm{H}\bm{H}^{\rm
H}+\xi\bm{I}_{N}\right)^{-1},$ (37) $\displaystyle\widetilde{\bm{Q}}(\xi)=$
$\displaystyle\left(\bm{H}^{\rm H}\bm{H}+\xi\bm{I}_{K}\right)^{-1}.$ (38)

In (35) and (36), $(\delta,\tilde{\delta})$ admits a unique solution in the
class of Stieltjes transforms [40] of non-negative measures with the support
in $\mathbb{R}^{+}$, which are given by

$\displaystyle\delta=$
$\displaystyle\frac{1}{N}\text{Tr}\left\\{\varsigma^{2}\bm{R}\left(\xi\left(\bm{I}_{N}+\tilde{\delta}\varsigma^{2}\bm{R}\right)+\nu^{2}\bar{\bm{H}}\left(\bm{I}_{K}+\delta\varsigma^{2}\widetilde{\bm{R}}\right)^{-1}\bar{\bm{H}}^{\rm
H}\right)^{-1}\right\\},$ (39) $\displaystyle\tilde{\delta}=$
$\displaystyle\frac{1}{N}\text{Tr}\left\\{\varsigma^{2}\widetilde{\bm{R}}\left(\xi\left(\bm{I}_{K}+\delta\varsigma^{2}\widetilde{\bm{R}}\right)+\nu^{2}\bar{\bm{H}}^{\rm
H}\left(\bm{I}_{N}+\tilde{\delta}\varsigma^{2}\bm{R}\right)^{-1}\bar{\bm{H}}\right)^{-1}\right\\}.$
(40)

Then $\delta$ and $\tilde{\delta}$ can be numerically solved as

$\displaystyle\delta=$
$\displaystyle\lim\limits_{t\to\infty}\delta^{(t)}\text{ and
}\tilde{\delta}=\lim\limits_{t\to\infty}\tilde{\delta}^{(t)},$ (41)

by defining $\delta^{(t)}$ and $\tilde{\delta}^{(t)}$

$\displaystyle\delta^{(t)}=$
$\displaystyle\frac{1}{N}\text{Tr}\left\\{\varsigma^{2}\bm{R}\left(\xi\left(\bm{I}_{N}+\tilde{\delta}^{(t-1)}\varsigma^{2}\bm{R}\right)+\nu^{2}\bar{\bm{H}}\left(\bm{I}_{K}+\delta^{(t-1)}\varsigma^{2}\widetilde{\bm{R}}\right)^{-1}\bar{\bm{H}}^{\rm
H}\right)^{-1}\right\\},$ (42) $\displaystyle\tilde{\delta}^{(t)}=$
$\displaystyle\frac{1}{N}\text{Tr}\left\\{\varsigma^{2}\widetilde{\bm{R}}\left(\xi\left(\bm{I}_{K}+\delta^{(t)}\varsigma^{2}\widetilde{\bm{R}}\right)+\nu^{2}\bar{\bm{H}}^{\rm
H}\left(\bm{I}_{N}+\tilde{\delta}^{(t-1)}\varsigma^{2}\bm{R}\right)^{-1}\bar{\bm{H}}\right)^{-1}\right\\},$
(43)

with the initial values of $\delta^{(0)}=\tilde{\delta}^{(0)}=\frac{1}{\xi}$.

According to Theorem 1, $\bm{\Upsilon}_{b^{*}}^{a^{*}}$ can be approximated as

$\displaystyle\bm{\Upsilon}_{b^{*}}^{a^{*}}\approx$
$\displaystyle\bigg{(}\xi_{b^{*}}^{a^{*}}\left(\bm{I}_{N_{t}}+\tilde{\delta}_{b^{*}}^{a^{*}}\varsigma^{2}\bm{\Phi}_{a^{*}}\right)+\frac{1}{N_{t}}\nu^{2}\big{(}\bm{H}_{{\rm
d},b^{*}}^{a^{*}}\big{)}^{\rm
H}\left(\bm{I}_{N_{r}}+\delta_{b^{*}}^{a^{*}}\varsigma^{2}\bm{\Phi}^{b^{*}}\right)^{-1}\bm{H}_{{\rm
d},b^{*}}^{a^{*}}\bigg{)}^{-1}.$ (44)

By recalling (39) and (40), $\delta_{b^{*}}^{a^{*}}$ and
$\tilde{\delta}_{b^{*}}^{a^{*}}$ in (44) are given by

$\displaystyle\delta_{b^{*}}^{a^{*}}\\!=$
$\displaystyle\frac{1}{N_{t}}\text{Tr}\Bigg{\\{}\\!\varsigma^{2}\bm{\Phi}_{a^{*}}\bigg{(}\xi_{b^{*}}^{a^{*}}\left(\bm{I}_{N_{t}}\\!+\\!\tilde{\delta}_{b^{*}}^{a^{*}}\varsigma^{2}\bm{\Phi}_{a^{*}}\right)\\!+\\!\frac{1}{N_{t}}\nu^{2}\big{(}\bm{H}_{{\rm
d},b^{*}}^{a^{*}}\big{)}^{\rm
H}\left(\bm{I}_{N_{r}}\\!+\\!\delta_{b^{*}}^{a^{*}}\varsigma^{2}\bm{\Phi}^{b^{*}}\right)^{-1}\bm{H}_{{\rm
d},b^{*}}^{a^{*}}\bigg{)}^{-1}\\!\Bigg{\\}}\\!,\\!$ (45)
$\displaystyle\tilde{\delta}_{b^{*}}^{a^{*}}\\!=$
$\displaystyle\frac{1}{N_{t}}\text{Tr}\Bigg{\\{}\\!\varsigma^{2}\bm{\Phi}^{b^{*}}\bigg{(}\xi_{b^{*}}^{a^{*}}\left(\bm{I}_{N_{r}}\\!+\\!\delta_{b^{*}}^{a^{*}}\varsigma^{2}\bm{\Phi}^{b^{*}}\right)\\!+\\!\frac{1}{N_{t}}\nu^{2}\bm{H}_{{\rm
d},b^{*}}^{a^{*}}\left(\bm{I}_{N_{t}}\\!+\\!\tilde{\delta}_{b^{*}}^{a^{*}}\varsigma^{2}\bm{\Phi}_{a^{*}}\right)^{-1}\big{(}\bm{H}_{{\rm
d},b^{*}}^{a^{*}}\big{)}^{\rm H}\bigg{)}^{-1}\\!\Bigg{\\}}\\!.\\!$ (46)

Furthermore, given (44) and recalling (34), $\bm{\Upsilon}_{b^{*},\emptyset
n_{r}^{*}}^{a^{*}}$ can be approximated as

$\displaystyle\bm{\Upsilon}_{b^{*},\emptyset n_{r}^{*}}^{a^{*}}\\!\approx$
$\displaystyle\bigg{(}\\!\xi_{b^{*}}^{a^{*}}\\!\left(\\!\bm{I}_{N_{t}}\\!+\\!\tilde{\delta}_{b^{*}}^{a^{*}}\varsigma^{2}\bm{\Phi}_{a^{*}}\\!\right)\\!\\!+\\!\frac{1}{N_{t}}\nu^{2}\big{(}\bm{H}_{{\rm
d},b^{*}}^{a^{*}}\big{)}^{\rm
H}\\!\left(\\!\bm{I}_{N_{r}}\\!+\\!\delta_{b^{*}}^{a^{*}}\varsigma^{2}\bm{\Phi}^{b^{*}}\right)^{-1}\bm{H}_{{\rm
d},b^{*}}^{a^{*}}\\!-\\!\frac{1}{N_{t}}\left[\bm{\Theta}_{b^{*}}^{a^{*}}\right]_{(n_{r}^{*},n_{r}^{*})}\\!\bigg{)}^{-1}\\!\\!.$
(47)

Then $\vartheta_{b^{*},n_{r}^{*}}^{a^{*}}$ of (33) can be further expressed as

$\displaystyle\vartheta_{b^{*},n_{r}^{*}}^{a^{*}}=$
$\displaystyle\text{Tr}\bigg{\\{}\left[\bm{\Theta}_{b^{*}}^{a^{*}}\right]_{(n_{r}^{*},n_{r}^{*})}\bigg{(}\xi_{b^{*}}^{a^{*}}\left(\bm{I}_{N_{t}}+\tilde{\delta}_{b^{*}}^{a^{*}}\varsigma^{2}\bm{\Phi}_{a^{*}}\right)$
$\displaystyle+\frac{1}{N_{t}}\nu^{2}\big{(}\bm{H}_{{\rm
d},b^{*}}^{a^{*}}\big{)}^{\rm
H}\left(\bm{I}_{N_{r}}+\delta_{b^{*}}^{a^{*}}\varsigma^{2}\bm{\Phi}^{b^{*}}\right)^{-1}\bm{H}_{{\rm
d},b^{*}}^{a^{*}}-\frac{1}{N_{t}}\left[\bm{\Theta}_{b^{*}}^{a^{*}}\right]_{(n_{r}^{*},n_{r}^{*})}\bigg{)}^{-1}\bigg{\\}}.$
(48)

Hence, noting (31), (III-B) and (III-B), the desired signal power of (15) can
be expressed as

$\displaystyle P_{{\rm S}_{b^{*},n_{r}^{*}}^{a^{*}}}=$ $\displaystyle
P_{r,b^{*}}^{a^{*}}\bigg{(}\frac{\vartheta_{b^{*},n_{r}^{*}}^{a^{*}}}{1+\frac{1}{N_{t}}\vartheta_{b^{*},n_{r}^{*}}^{a^{*}}}\bigg{)}^{2}.$
(49)

###  III-C Interference Plus Noise Power

Recalling (33) and Lemma 1, we can express
$\big{[}\widehat{\bm{H}}_{b^{*}}^{a^{*}}\big{]}_{[n_{r}^{*}:~{}]}\big{[}\bm{V}_{b^{*}}^{a^{*}}\big{]}_{[~{}:n_{r}^{*}]}$
as

$\displaystyle\left[\widehat{\bm{H}}_{b^{*}}^{a^{*}}\right]_{[n_{r}^{*}:~{}]}\left[\bm{V}_{b^{*}}^{a^{*}}\right]_{[~{}:n_{r}^{*}]}=$
$\displaystyle\frac{\vartheta_{b^{*},n_{r}^{*}}^{a^{*}}}{1+\frac{1}{N_{t}}\vartheta_{b^{*},n_{r}^{*}}^{a^{*}}}.$
(50)

Thus, we have

$\displaystyle\text{Var}\left\\{\big{[}\bm{H}_{b^{*}}^{a^{*}}\big{]}_{[n_{r}^{*}:~{}]}\big{[}\bm{V}_{b^{*}}^{a^{*}}\big{]}_{[~{}:n_{r}^{*}]}\right\\}=$
$\displaystyle\mathcal{E}\Bigg{\\{}\Bigg{|}\frac{\big{[}\widetilde{\bm{H}}_{b^{*}}^{a^{*}}\big{]}_{[n_{r}^{*}:~{}]}\bm{\Upsilon}_{b^{*},\emptyset
n_{r}^{*}}^{a^{*}}\big{[}\widehat{\bm{H}}_{b^{*}}^{a^{*}}\big{]}_{[n_{r}^{*}:~{}]}^{\rm
H}}{1+\frac{1}{N_{t}}\big{[}\widehat{\bm{H}}_{b^{*}}^{a^{*}}\big{]}_{[n_{r}^{*}:~{}]}\bm{\Upsilon}_{b^{*},\emptyset
n_{r}^{*}}^{a^{*}}\big{[}\widehat{\bm{H}}_{b^{*}}^{a^{*}}\big{]}_{[n_{r}^{*}:~{}]}^{\rm
H}}\Bigg{|}^{2}\Bigg{\\}}$ $\displaystyle=$
$\displaystyle\text{Tr}\Bigg{\\{}\frac{\bm{\Upsilon}_{b^{*},\emptyset
n_{r}^{*}}^{a^{*}}\big{[}\bm{\Theta}_{b^{*}}^{a^{*}}\big{]}_{(n_{r}^{*},n_{r}^{*})}\bm{\Upsilon}_{b^{*},\emptyset
n_{r}^{*}}^{a^{*}}\big{[}\bm{\Xi}_{b^{*}}^{a^{*}}\big{]}_{(n_{r}^{*},n_{r}^{*})}}{\left(1+\frac{1}{N_{t}}\vartheta_{b^{*},n_{r}^{*}}^{a^{*}}\right)^{2}}\Bigg{\\}}.$
(51)

From Lemmas 1 and 3,
$\mathcal{E}\left\\{\big{|}\big{[}\bm{H}_{b^{*}}^{a^{*}}\big{]}_{[n_{r}^{*}:~{}]}\big{[}\bm{V}_{b^{*}}^{a^{*}}\big{]}_{[~{}:n_{r}]}\big{|}^{2}\right\\}$,
$n_{r}\neq n_{r}^{*}$, can be expressed as

$\displaystyle\mathcal{E}\left\\{\left|\left[\bm{H}_{b^{*}}^{a^{*}}\right]_{[n_{r}^{*}:~{}]}\left[\bm{V}_{b^{*}}^{a^{*}}\right]_{[~{}:n_{r}]}\right|^{2}\right\\}$
$\displaystyle=\mathcal{E}\Bigg{\\{}\Bigg{|}\frac{\big{[}\bm{H}_{b^{*}}^{a^{*}}\big{]}_{[n_{r}^{*}:~{}]}\bm{\Upsilon}_{b^{*},\emptyset
n_{r}}^{a^{*}}\big{[}\widehat{\bm{H}}_{b^{*}}^{a^{*}}\big{]}_{[n_{r}:~{}]}^{\rm
H}}{1+\frac{1}{N_{t}}\big{[}\widehat{\bm{H}}_{b^{*}}^{a^{*}}\big{]}_{[n_{r}:~{}]}\bm{\Upsilon}_{a^{*},\emptyset
n_{r}}^{b^{*}}\big{[}\widehat{\bm{H}}_{b^{*}}^{a^{*}}\big{]}_{[n_{r}:~{}]}^{\rm
H}}\Bigg{|}^{2}\Bigg{\\}},$
$\displaystyle=\mathcal{E}\Bigg{\\{}\frac{\left[\bm{H}_{b^{*}}^{a^{*}}\right]_{[n_{r}^{*}:~{}]}\bm{\Upsilon}_{b^{*},\emptyset
n_{r}}^{a^{*}}\left[\bm{\Theta}_{b^{*}}^{a^{*}}\right]_{(n_{r},n_{r})}\bm{\Upsilon}_{b^{*},\emptyset
n_{r}}^{a^{*}}\left[\bm{H}_{b^{*}}^{a^{*}}\right]_{[n_{r}^{*}:~{}]}^{\rm
H}}{(1+\frac{1}{N_{t}}\vartheta_{b^{*},n_{r}}^{a^{*}})^{2}}\Bigg{\\}},$ (52)

in which

$\displaystyle\vartheta_{b^{*},n_{r}}^{a^{*}}=$
$\displaystyle\text{Tr}\left\\{\left[\bm{\Theta}_{b^{*}}^{a^{*}}\right]_{(n_{r},n_{r})}\bm{\Upsilon}_{b^{*},\emptyset
n_{r}}^{a^{*}}\right\\},$ (53) $\displaystyle\bm{\Upsilon}_{b^{*},\emptyset
n_{r}}^{a^{*}}\\!\\!=$
$\displaystyle\bigg{(}\\!\xi_{b^{*}}^{a^{*}}\\!\\!\left(\\!\\!\bm{I}_{N_{t}}\\!\\!+\\!\frac{1}{N_{t}}\tilde{\delta}_{b^{*}}^{a^{*}}\varsigma^{2}\bm{\Phi}_{a^{*}}\\!\\!\right)\\!\\!+\\!\frac{1}{N_{t}}\nu^{2}\big{(}\bm{H}_{{\rm
d},b^{*}}^{a^{*}}\big{)}^{\rm
H}\\!\left(\bm{I}_{N_{r}}\\!\\!+\\!\delta_{b^{*}}^{a^{*}}\varsigma^{2}\bm{\Phi}^{b^{*}}\right)^{-1}\\!\bm{H}_{{\rm
d},b^{*}}^{a^{*}}\\!\\!-\\!\frac{1}{N_{t}}\\!\left[\bm{\Theta}_{b^{*}}^{a^{*}}\right]_{(n_{r},n_{r})}\\!\\!\bigg{)}^{-1}\\!\\!\\!.\\!$
(54)

According to Lemma 2, we have

$\displaystyle\bm{\Upsilon}_{b^{*},\emptyset n_{r}}^{a^{*}}=$
$\displaystyle\bm{\Upsilon}_{b^{*},\emptyset
n_{r},n_{r}^{*}}^{a^{*}}-\frac{\frac{1}{N_{t}}\bm{\Upsilon}_{b^{*},\emptyset
n_{r},n_{r}^{*}}^{a^{*}}\left[\widehat{\bm{H}}_{b^{*}}^{a^{*}}\right]_{[n_{r}^{*}:~{}]}^{\rm
H}\left[\widehat{\bm{H}}_{b^{*}}^{a^{*}}\right]_{[n_{r}^{*}:~{}]}\bm{\Upsilon}_{b^{*},\emptyset
n_{r},n_{r}^{*}}^{a^{*}}}{1+\frac{1}{N_{t}}\left[\widehat{\bm{H}}_{b^{*}}^{a^{*}}\right]_{[n_{r}^{*}:~{}]}\bm{\Upsilon}_{b^{*},\emptyset
n_{r},n_{r}^{*}}^{a^{*}}\left[\widehat{\bm{H}}_{b^{*}}^{a^{*}}\right]_{[n_{r}^{*}:~{}]}^{\rm
H}},$ (55)

where $\bm{\Upsilon}_{b^{*},\emptyset n_{r},n_{r}^{*}}^{a^{*}}$ is independent
of $\left[\widehat{\bm{H}}_{b^{*}}^{a^{*}}\right]_{[n_{r}^{*}:~{}]}$ and
$\left[\widehat{\bm{H}}_{b^{*}}^{a^{*}}\right]_{[n_{r}:~{}]}$, and it can be
approximated as

$\displaystyle\bm{\Upsilon}_{b^{*},\emptyset n_{r},n_{r}^{*}}^{a^{*}}=$
$\displaystyle\bigg{(}\xi_{b^{*}}^{a^{*}}\left(\bm{I}_{N_{t}}+\tilde{\delta}_{b^{*}}^{a^{*}}\varsigma^{2}\bm{\Phi}_{a^{*}}\right)+\frac{1}{N_{t}}\nu^{2}\big{(}\bm{H}_{{\rm
d},b^{*}}^{a^{*}}\big{)}^{\rm
H}\left(\bm{I}_{N_{r}}+\delta_{b^{*}}^{a^{*}}\varsigma^{2}\bm{\Phi}^{b^{*}}\right)^{-1}\bm{H}_{{\rm
d},b^{*}}^{a^{*}}$
$\displaystyle-\frac{1}{N_{t}}\left(\left[\bm{\Theta}_{b^{*}}^{a^{*}}\right]_{(n_{r},n_{r})}+\left[\bm{\Theta}_{b^{*}}^{a^{*}}\right]_{(n_{r}^{*},n_{r}^{*})}\right)\bigg{)}^{-1}.$
(56)

Then, we can rewrite
$\left[\bm{H}_{b^{*}}^{a^{*}}\right]_{[n_{r}^{*}:~{}]}\bm{\Upsilon}_{b^{*},\emptyset
n_{r}}^{a^{*}}\left[\bm{\Theta}_{b^{*}}^{a^{*}}\right]_{(n_{r},n_{r})}\bm{\Upsilon}_{b^{*},\emptyset
n_{r}}^{a^{*}}\left[\bm{H}_{b^{*}}^{a^{*}}\right]_{[n_{r}^{*}:~{}]}^{\rm H}$
as

$\displaystyle\left[\bm{H}_{b^{*}}^{a^{*}}\right]_{[n_{r}^{*}:~{}]}\bm{\Upsilon}_{b^{*},\emptyset
n_{r}}^{a^{*}}\left[\bm{\Theta}_{b^{*}}^{a^{*}}\right]_{(n_{r},n_{r})}\bm{\Upsilon}_{b^{*},\emptyset
n_{r}}^{a^{*}}\left[\bm{H}_{b^{*}}^{a^{*}}\right]_{[n_{r}^{*}:~{}]}^{\rm H}=$
$\displaystyle\hskip
28.45274pt\left[\bm{H}_{b^{*}}^{a^{*}}\right]_{[n_{r}^{*}:~{}]}\bm{\Upsilon}_{b^{*},\emptyset
n_{r},n_{r}^{*}}^{a^{*}}\left[\bm{\Theta}_{b^{*}}^{a^{*}}\right]_{(n_{r},n_{r})}\bm{\Upsilon}_{b^{*},\emptyset
n_{r},n_{r}^{*}}^{a^{*}}\left[\bm{H}_{b^{*}}^{a^{*}}\right]_{[n_{r}^{*}:~{}]}^{\rm
H}$ $\displaystyle\hskip
28.45274pt-2\Re\left\\{\frac{\frac{1}{N_{t}}\big{[}\bm{H}_{b^{*}}^{a^{*}}\big{]}_{[n_{r}^{*}:~{}]}\bm{\Upsilon}_{b^{*},\emptyset
n_{r},n_{r}^{*}}^{a^{*}}\big{[}\widehat{\bm{H}}_{b^{*}}^{a^{*}}\big{]}_{[n_{r}^{*}:~{}]}^{\rm
H}\big{[}\widehat{\bm{H}}_{b^{*}}^{a^{*}}\big{]}_{[n_{r}^{*}:~{}]}\widetilde{\bm{\Upsilon}}_{a^{*},\emptyset
n_{r},n_{r}^{*}}^{b^{*}}\big{[}\bm{H}_{b^{*}}^{a^{*}}\big{]}_{[n_{r}^{*}:~{}]}^{\rm
H}}{1+\frac{1}{N_{t}}\big{[}\widehat{\bm{H}}_{b^{*}}^{a^{*}}\big{]}_{[n_{r}^{*}:~{}]}\bm{\Upsilon}_{b^{*},\emptyset
n_{r},n_{r}^{*}}^{a^{*}}\big{[}\widehat{\bm{H}}_{b^{*}}^{a^{*}}\big{]}_{[n_{r}^{*}:~{}]}^{\rm
H}}\right\\}$ $\displaystyle\hskip
28.45274pt+\frac{\frac{1}{N_{t}^{2}}\left|\big{[}\bm{H}_{b^{*}}^{a^{*}}\big{]}_{[n_{r}^{*}:~{}]}\bm{\Upsilon}_{b^{*},\emptyset
n_{r},n_{r}^{*}}^{a^{*}}\big{[}\widehat{\bm{H}}_{b^{*}}^{a^{*}}\big{]}_{[n_{r}^{*}:~{}]}^{\rm
H}\right|^{2}\big{[}\widehat{\bm{H}}_{b^{*}}^{a^{*}}\big{]}_{[n_{r}^{*}:~{}]}\widetilde{\bm{\Upsilon}}_{b^{*},\emptyset
n_{r},n_{r}^{*}}^{a^{*}}\big{[}\widehat{\bm{H}}_{b^{*}}^{a^{*}}\big{]}_{[n_{r}^{*}:~{}]}^{\rm
H}}{\left(1+\frac{1}{N_{t}}\big{[}\widehat{\bm{H}}_{b^{*}}^{a^{*}}\big{]}_{[n_{r}^{*}:~{}]}\bm{\Upsilon}_{b^{*},\emptyset
n_{r},n_{r}^{*}}^{a^{*}}\big{[}\widehat{\bm{H}}_{b^{*}}^{a^{*}}\big{]}_{[n_{r}^{*}:~{}]}^{\rm
H}\right)^{2}},$ (57)

where $\Re\\{\cdot\\}$ denotes the real part of a complex number, and
$\widetilde{\bm{\Upsilon}}_{b^{*},\emptyset n_{r},n_{r}^{*}}^{a^{*}}$ is given
by

$\displaystyle\widetilde{\bm{\Upsilon}}_{b^{*},\emptyset
n_{r},n_{r}^{*}}^{a^{*}}=$ $\displaystyle\bm{\Upsilon}_{b^{*},\emptyset
n_{r},n_{r}^{*}}^{a^{*}}\left[\bm{\Theta}_{b^{*}}^{a^{*}}\right]_{(n_{r},n_{r})}\bm{\Upsilon}_{b^{*},\emptyset
n_{r},n_{r}^{*}}^{a^{*}}.$ (58)

Thus by recalling Lemmas 3 and 4, we have the following approximations

$\displaystyle\big{[}\widehat{\bm{H}}_{b^{*}}^{a^{*}}\big{]}_{[n_{r}^{*}:~{}]}\bm{\Upsilon}_{b^{*},\emptyset
n_{r},n_{r}^{*}}^{a^{*}}\big{[}\widehat{\bm{H}}_{b^{*}}^{a^{*}}\big{]}_{[n_{r}^{*}:~{}]}^{\rm
H}=\vartheta_{b^{*},n_{r}^{*}}^{a^{*}},$ (59)
$\displaystyle\big{[}\bm{H}_{b^{*}}^{a^{*}}\big{]}_{[n_{r}^{*}:~{}]}\bm{\Upsilon}_{b^{*},\emptyset
n_{r},n_{r}^{*}}^{a^{*}}\big{[}\widehat{\bm{H}}_{b^{*}}^{a^{*}}\big{]}_{[n_{r}^{*}:~{}]}^{\rm
H}=\vartheta_{b^{*},n_{r}^{*}}^{a^{*}},$ (60)
$\displaystyle\big{[}\bm{H}_{b^{*}}^{a^{*}}\big{]}_{[n_{r}^{*}:~{}]}\widetilde{\bm{\Upsilon}}_{b^{*},\emptyset
n_{r},n_{r}^{*}}^{a^{*}}\big{[}\bm{H}_{b^{*}}^{a^{*}}\big{]}_{[n_{r}^{*}:~{}]}^{\rm
H}=\text{Tr}\left\\{\big{[}\bm{\Omega}_{b^{*}}^{a^{*}}\big{]}_{(n_{r}^{*},n_{r}^{*})}\widetilde{\bm{\Upsilon}}_{b^{*},\emptyset
n_{r},n_{r}^{*}}^{a^{*}}\right\\},$ (61)
$\displaystyle\big{[}\widehat{\bm{H}}_{b^{*}}^{a^{*}}\big{]}_{[n_{r}^{*}:~{}]}\widetilde{\bm{\Upsilon}}_{b^{*},\emptyset
n_{r},n_{r}^{*}}^{a^{*}}\big{[}\widehat{\bm{H}}_{b^{*}}^{a^{*}}\big{]}_{[n_{r}^{*}:~{}]}^{\rm
H}=\text{Tr}\left\\{\big{[}\bm{\Theta}_{b^{*}}^{a^{*}}\big{]}_{(n_{r}^{*},n_{r}^{*})}\widetilde{\bm{\Upsilon}}_{b^{*},\emptyset
n_{r},n_{r}^{*}}^{a^{*}}\right\\},$ (62)
$\displaystyle\big{[}\widehat{\bm{H}}_{b^{*}}^{a^{*}}\big{]}_{[n_{r}^{*}:~{}]}\widetilde{\bm{\Upsilon}}_{b^{*},\emptyset
n_{r},n_{r}^{*}}^{a^{*}}\big{[}\bm{H}_{b^{*}}^{a^{*}}\big{]}_{[n_{r}^{*}:~{}]}^{\rm
H}=\text{Tr}\left\\{\big{[}\bm{\Theta}_{b^{*}}^{a^{*}}\big{]}_{(n_{r}^{*},n_{r}^{*})}\widetilde{\bm{\Upsilon}}_{b^{*},\emptyset
n_{r},n_{r}^{*}}^{a^{*}}\right\\},$ (63)

where
$\big{[}\bm{\Omega}_{b^{*}}^{a^{*}}\big{]}_{(n_{r}^{*},n_{r}^{*})}\in\mathbb{C}^{N_{t}\times
N_{t}}$ is given by

$\displaystyle\big{[}\bm{\Omega}_{b^{*}}^{a^{*}}\big{]}_{(n_{r}^{*},n_{r}^{*})}=$
$\displaystyle\nu^{2}\big{[}\bm{M}_{b^{*}}^{a^{*}}\big{]}_{(n_{r}^{*},n_{r}^{*})}+\big{[}\bm{R}_{b^{*}}^{a^{*}}\big{]}_{(n_{r}^{*},n_{r}^{*})}\,\,.$
(64)

First substituting (59) to (63) into (III-C) and then substituting the result
into (III-C), we obtain

$\displaystyle\mathcal{E}\left\\{\left|\big{[}\bm{H}_{b^{*}}^{a^{*}}\big{]}_{[n_{r}^{*}:~{}]}\big{[}\bm{V}_{b^{*}}^{a^{*}}\big{]}_{[~{}:n_{r}]}\right|^{2}\right\\}=\frac{\text{Tr}\left\\{\big{[}\bm{\Omega}_{b^{*}}^{a^{*}}\big{]}_{(n_{r}^{*},n_{r}^{*})}\widetilde{\bm{\Upsilon}}_{b^{*},\emptyset
n_{r},n_{r}^{*}}^{a^{*}}\right\\}}{(1+\frac{1}{N_{t}}\vartheta_{b^{*},n_{r}}^{a^{*}})^{2}}$
$\displaystyle\hskip
14.22636pt-\frac{2\\!\Re\\!\left\\{\\!\frac{1}{N_{t}}\vartheta_{b^{*},n_{r}^{*}}^{a^{*}}\text{Tr}\left\\{\big{[}\bm{\Theta}_{b^{*}}^{a^{*}}\big{]}_{(n_{r}^{*},n_{r}^{*})}\widetilde{\bm{\Upsilon}}_{b^{*},\emptyset
n_{r},n_{r}^{*}}^{a^{*}}\right\\}\\!\right\\}}{(1+\frac{1}{N_{t}}\vartheta_{b^{*},n_{r}^{*}}^{a^{*}})(1+\frac{1}{N_{t}}\vartheta_{b^{*},n_{r}}^{a^{*}})^{2}}\\!+\\!\frac{\left(\\!\frac{1}{N_{t}}\vartheta_{b^{*},n_{r}^{*}}^{a^{*}}\\!\right)^{2}\\!\text{Tr}\left\\{\\!\big{[}\bm{\Theta}_{b^{*}}^{a^{*}}\big{]}_{(n_{r}^{*},n_{r}^{*})}\\!\widetilde{\bm{\Upsilon}}_{b^{*},\emptyset
n_{r},n_{r}^{*}}^{a^{*}}\\!\right\\}}{\left(1+\frac{1}{N_{t}}\vartheta_{b^{*},n_{r}^{*}}^{a^{*}}\right)^{2}\big{(}1+\frac{1}{N_{t}}\vartheta_{b^{*},n_{r}}^{a^{*}}\big{)}^{2}}.$
(65)

Thirdly, according to Lemmas 1 and 3, we have

$\displaystyle\mathcal{E}\left\\{\left|\big{[}\bm{H}_{b^{*}}^{a}\big{]}_{[n_{r}^{*}:~{}]}\big{[}\bm{V}_{b^{a}}^{a}\right]_{[~{}:n_{r}]}\big{|}^{2}\right\\}=$
$\displaystyle\mathcal{E}\Bigg{\\{}\Bigg{|}\frac{\big{[}\bm{H}_{b^{*}}^{a}\big{]}_{[n_{r}^{*}:~{}]}\bm{\Upsilon}_{b^{a},\emptyset
n_{r}}^{a}\big{[}\widehat{\bm{H}}_{b^{a}}^{a}\big{]}_{[n_{r}:~{}]}^{\rm
H}}{1+\frac{1}{N_{t}}\big{[}\widehat{\bm{H}}_{b^{a}}^{a}\big{]}_{[n_{r}:~{}]}\bm{\Upsilon}_{b^{a},\emptyset
n_{r}}^{a}\big{[}\widehat{\bm{H}}_{b^{a}}^{a}\big{]}_{[n_{r}:~{}]}^{\rm
H}}\Bigg{|}^{2}\Bigg{\\}}$ $\displaystyle=$
$\displaystyle\frac{\text{Tr}\left\\{\bm{\Upsilon}_{b^{a},\emptyset
n_{r}}^{a}\left[\bm{\Theta}_{b^{a}}^{a}\right]_{(n_{r},n_{r})}\bm{\Upsilon}_{b^{a},\emptyset
n_{r}}^{a}\left[\bm{\Omega}_{b^{*}}^{a}\right]_{(n_{r}^{*},n_{r}^{*})}\right\\}}{(1+\frac{1}{N_{t}}\vartheta_{b^{a},n_{r}}^{a})^{2}},$
(66)

where $\vartheta_{b^{a},n_{r}}^{a}$,
$\left[\bm{\Theta}_{b^{a}}^{a}\right]_{(n_{r},n_{r})}\in\mathbb{C}^{N_{t}\times
N_{t}}$ and
$\left[\bm{\Omega}_{b^{*}}^{a}\right]_{(n_{r}^{*},n_{r}^{*})}\in\mathbb{C}^{N_{t}\times
N_{t}}$ are given respectively by

$\displaystyle\vartheta_{b^{a},n_{r}}^{a}=$
$\displaystyle\text{Tr}\left\\{\big{[}\bm{\Theta}_{b^{a}}^{a}\big{]}_{(n_{r},n_{r})}\bm{\Upsilon}_{b^{a},\emptyset
n_{r}}^{a}\right\\},$ (67)
$\displaystyle\big{[}\bm{\Theta}_{b^{a}}^{a}\big{]}_{(n_{r},n_{r})}=$
$\displaystyle\nu^{2}\left[\bm{M}_{b^{a}}^{a}\right]_{(n_{r},n_{r})}+\big{[}\bm{\Phi}_{b^{a}}^{a}\big{]}_{(n_{r},n_{r})},$
(68)
$\displaystyle\big{[}\bm{\Omega}_{b^{*}}^{a}\big{]}_{(n_{r}^{*},n_{r}^{*})}=$
$\displaystyle\nu^{2}\left[\bm{M}_{b^{*}}^{a}\right]_{(n_{r}^{*},n_{r}^{*})}+\big{[}\bm{R}_{b^{*}}^{a}\big{]}_{(n_{r}^{*},n_{r}^{*})},$
(69)

while $\bm{\Upsilon}_{b^{a},\emptyset n_{r}}^{a}$ has a similar form as
$\bm{\Upsilon}_{b^{*},\emptyset n_{r}}^{a^{*}}$, which is given by

$\displaystyle\bm{\Upsilon}_{b^{a},\emptyset n_{r}}^{a}\\!=$
$\displaystyle\bigg{(}\xi_{b^{a}}^{a}\\!\left(\\!\bm{I}_{N_{t}}\\!+\\!\tilde{\delta}_{b^{a}}^{a}\varsigma^{2}\bm{\Phi}_{a}\right)\\!+\\!\frac{1}{N_{t}}\nu^{2}\big{(}\bm{H}_{{\rm
d},b^{a}}^{a}\big{)}^{\rm
H}\\!\left(\bm{I}_{N_{r}}\\!+\\!\delta_{b^{a}}^{a}\varsigma^{2}\bm{\Phi}^{b^{a}}\right)^{-1}\\!\bm{H}_{{\rm
d},b^{a}}^{a}\\!-\\!\frac{1}{N_{t}}\big{[}\bm{\Theta}_{b^{a}}^{a}\big{]}_{(n_{r},n_{r})}\\!\bigg{)}^{-1}\\!\\!.\\!$
(70)

By substituting (III-C), (III-C) and (III-C) into (16), we obtain

$\displaystyle P_{{\rm
I\&N}_{b^{*},n_{r}^{*}}^{a^{*}}}=\sigma_{w}^{2}+P_{r,b^{*}}^{a^{*}}\text{Tr}\Bigg{\\{}\frac{\bm{\Upsilon}_{b^{*},\emptyset
n_{r}^{*}}^{a^{*}}\big{[}\bm{\Theta}_{b^{*}}^{a^{*}}\big{]}_{(n_{r}^{*},n_{r}^{*})}\bm{\Upsilon}_{b^{*},\emptyset
n_{r}^{*}}^{a^{*}}\big{[}\bm{\Xi}_{b^{*}}^{a^{*}}\big{]}_{(n_{r}^{*},n_{r}^{*})}}{\big{(}1+\vartheta_{b^{*},n_{r}^{*}}^{a^{*}}\big{)}^{2}}\Bigg{\\}}$
$\displaystyle\hskip 14.22636pt+P_{r,b^{*}}^{a^{*}}\sum\limits_{n_{r}\neq
n_{r}^{*}}\Bigg{(}\frac{\text{Tr}\Big{\\{}\big{[}\bm{\Omega}_{b^{*}}^{a^{*}}\big{]}_{(n_{r}^{*},n_{r}^{*})}\widetilde{\bm{\Upsilon}}_{b^{*},\emptyset
n_{r},n_{r}^{*}}^{a^{*}}\Big{\\}}}{\big{(}1+\frac{1}{N_{t}}\vartheta_{b^{*},n_{r}}^{a^{*}}\big{)}^{2}}-\frac{2\Re\Big{\\{}\frac{1}{N_{t}}\vartheta_{b^{*},n_{r}^{*}}^{a^{*}}\text{Tr}\big{\\{}\big{[}\bm{\Theta}_{b^{*}}^{a^{*}}\big{]}_{(n_{r}^{*},n_{r}^{*})}\widetilde{\bm{\Upsilon}}_{b^{*},\emptyset
n_{r},n_{r}^{*}}^{a^{*}}\big{\\}}\Big{\\}}}{\big{(}1+\frac{1}{N_{t}}\vartheta_{b^{*},n_{r}^{*}}^{a^{*}}\big{)}\big{(}1+\frac{1}{N_{t}}\vartheta_{b^{*},n_{r}}^{a^{*}}\big{)}^{2}}$
$\displaystyle\hskip
14.22636pt+\frac{\Big{(}\frac{1}{N_{t}}\vartheta_{b^{*},n_{r}^{*}}^{a^{*}}\Big{)}^{2}\text{Tr}\Big{\\{}\big{[}\bm{\Theta}_{b^{*}}^{a^{*}}\big{]}_{(n_{r}^{*},n_{r}^{*})}\widetilde{\bm{\Upsilon}}_{b^{*},\emptyset
n_{r},n_{r}^{*}}^{a^{*}}\Big{\\}}}{\Big{(}1+\frac{1}{N_{t}}\vartheta_{b^{*},n_{r}^{*}}^{a^{*}}\Big{)}^{2}\big{(}1+\frac{1}{N_{t}}\vartheta_{b^{*},n_{r}}^{a^{*}})^{2}}\Bigg{)}$
$\displaystyle\hskip
14.22636pt+\sum\limits_{a=1}^{A}P_{r,b^{*}}^{a}\sum\limits_{n_{r}=1}^{N_{r}}\frac{\text{Tr}\Big{\\{}\bm{\Upsilon}_{b^{a},\emptyset
n_{r}}^{a}\big{[}\bm{\Theta}_{b^{a}}^{a}\big{]}_{(n_{r},n_{r})}\bm{\Upsilon}_{b^{a},\emptyset
n_{r}}^{a}\big{[}\bm{\Omega}_{b^{*}}^{a}\big{]}_{(n_{r}^{*},n_{r}^{*})}\Big{\\}}}{\big{(}1+\frac{1}{N_{t}}\vartheta_{b^{a},n_{r}}^{a}\big{)}^{2}}.$
(71)

###  III-D Achievable Rate and Optimal Regularization Parameter

Finally, upon substituting (49) as well as (III-C) into (17) and then using
the result in (18), we arrive at the closed-form achievable transmission rate
per antenna $C_{b^{*}}^{a^{*}}$, which is our performance metric for designing
the distance thresholds for the distance-based ACM scheme.

Since it is intractable to obtain an analytic optimal regularization parameter
that maximizes the achievable transmission rate per antenna, we consider the
alternative mean-square-error for detecting the transmitted data vector
$\bm{X}^{a^{*}}$ by aircraft $b^{*}$, which is given by

$\displaystyle\mathcal{J}\big{(}\xi_{b^{*}}^{a^{*}}\big{)}=\mathcal{E}\Bigg{\\{}\Bigg{\|}\frac{1}{N_{t}}\bm{H}_{b^{*}}^{a^{*}}\bm{V}_{b^{*}}^{a^{*}}\bm{X}^{a^{*}}+\frac{1}{N_{t}}\sum\limits_{a=1}^{A}\sqrt{\frac{P_{r,b^{*}}^{a}}{P_{r,b^{*}}^{a^{*}}}}\bm{H}_{b^{*}}^{a}\bm{V}_{b^{a}}^{a}\bm{X}^{a}+\frac{1}{N_{t}\sqrt{P_{r,b^{*}}^{a^{*}}}}\bm{W}_{b^{*}}-\bm{X}^{a^{*}}\Bigg{\|}^{2}\Bigg{\\}}$
$\displaystyle\hskip
14.22636pt=\mathcal{E}\Bigg{\\{}\\!\bigg{\|}\frac{1}{N_{t}}\bm{H}_{b^{*}}^{a^{*}}\bm{V}_{b^{*}}^{a^{*}}\bm{X}^{a^{*}}\\!\\!-\\!\bm{X}^{a^{*}}\bigg{\|}^{2}\\!\\!+\\!\sum\limits_{a=1}^{A}\left\|\frac{1}{N_{t}}\sqrt{\frac{P_{r,b^{*}}^{a}}{P_{r,b^{*}}^{a^{*}}}}\bm{H}_{b^{*}}^{a}\bm{V}_{b^{a}}^{a}\bm{X}^{a}\right\|^{2}\\!\\!+\\!\Bigg{\|}\frac{1}{N_{t}\sqrt{P_{r,b^{*}}^{a^{*}}}}\bm{W}_{b^{*}}\Bigg{\|}^{2}\Bigg{\\}}\\!.\\!$
(72)

As detailed in Appendix -B, the closed-form optimal regularization parameter
that minimizes the mean-square data detection error (III-D) is given by

$\displaystyle\big{(}\xi_{b^{*}}^{a^{*}}\big{)}^{\star}=$
$\displaystyle\frac{1}{N_{t}}\sum\limits_{n_{t}=1}^{N_{t}}\widetilde{\varphi}_{n_{t}},$
(73)

with

$\displaystyle\widetilde{\varphi}_{n_{t}}=$
$\displaystyle\sum\limits_{k=1}^{N_{r}}\bm{\Xi}_{b^{*}}^{a^{*}}\lvert_{[n_{t}+(k-1)N_{t},n_{t}+(k-1)N_{t}]},$
(74)

in which $\bm{\Xi}_{b^{*}}^{a^{*}}\lvert_{[i,i]}$ denotes the $i$-th row and
$i$-th column element of $\bm{\Xi}_{b^{*}}^{a^{*}}$.

##  IV RZF-TPC Aided and Distance-Based ACM Scheme

The framework of the RZF-TPC aided and distance-based ACM scheme, which is
depicted in the middle of Fig. 1, is similar to that of the EB-TPC aided and
distance-based ACM scheme given in [19]. The main difference is that here we
adopt the much more powerful TPC solution at the transmitter. Given the system
parameters, including the total system bandwidth $B_{\rm total}$, the number
of subcarriers $N$, the number of CP samples $N_{\rm cp}$, the set of
modulation constellations and the set of channel codes, the number of ACM
modes $K$ together with the set of switching thresholds
$\\{d_{k}\\}_{k=0}^{K}$ can now be designed, where $d_{0}=D_{\max}$ and
$D_{\max}$ is the maximum communication distance, while $d_{K}=D_{\min}$ and
$D_{\min}$ is the minimum safe separation distance of aircraft. The online
operations of the RZF-TPC aided and distance-based ACM transmission can then
be summarized below.

  * 1)

In the pilot training phase, aircraft $a^{*}$ estimates the channel matrix
$\bm{H}^{b^{*}}_{a^{*}}$ between aircraft $b^{*}$ and aircraft $a^{*}$ based
on the pilots sent by $b^{*}$.

  * 2)

Aircraft $a^{*}$ acquires the channel $\bm{H}_{b^{*}}^{a^{*}}$ for data
transmission by exploiting the channel reciprocity, and generates the RZF-TPC
matrix $\bm{V}_{b^{*}}^{a^{*}}$ according to (11) and (12).

  * 3)

Based on the distance $d_{b^{*}}^{a^{*}}$ between aircraft $a^{*}$ and $b^{*}$
measured by its DME, aircraft $a^{*}$ selects an ACM mode for data
transmission according to

$\displaystyle\text{If }d_{k}\leq d_{b^{*}}^{a^{*}}<d_{k-1}:\text{ choose mode
}k,\,k\in\\{1,2,\cdots,K\\}.$ (75)

Note that the scenarios of $d_{b^{*}}^{a^{*}}\geq D_{\max}$ and
$d_{b^{*}}^{a^{*}}\leq D_{\min}$ are not considered, since there is no
available communication link, when two aircraft are beyond the maximum
communication range, while the minimum flight-safety separation must be
maintained.

TABLE I: A design example of RZF-TPC aided and distance-based ACM with $N_{t}=32$ and $N_{r}=4$. The other system parameters for this ACM are listed in Table III. Mode $k$ |  Modulation |  Code rate |  Spectral efficiency (bps/Hz) |  Switching threshold $d_{k}$ (km) |  Data rate per receive antenna (Mbps) |  Total data rate (Mbps)
---|---|---|---|---|---|---
1 |  BPSK |  0.488 |  0.459 |  550 |  2.756 |  11.023
2 |  QPSK |  0.533 |  1.003 |  450 |  6.020 |  24.079
3 |  QPSK |  0.706 |  1.329 |  300 |  7.934 |  31.895
4 |  8-QAM |  0.635 |  1.793 |  210 |  10.758 |  43.031
5 |  8-QAM |  0.780 |  2.202 |  130 |  13.214 |  52.857
6 |  16-QAM |  0.731 |  2.752 |  90 |  16.512 |  66.048
7 |  16-QAM |  0.853 |  3.211 |  35 |  19.258 |  77.071
8 |  32-QAM |  0.879 |  4.137 |  5.56 |  24.819 |  99.275
TABLE II: A design example of RZF-TPC aided and distance-based ACM with $N_{t}=64$ and $N_{r}=4$. The other system parameters for this ACM are listed in Table III. Mode $k$ |  Modulation |  Code rate |  Spectral efficiency (bps/Hz) |  Switching threshold $d_{k}$ (km) |  Data rate per receive antenna (Mbps) |  Total data rate (Mbps)
---|---|---|---|---|---|---
1 |  QPSK |  0.706 |  1.323 |  500 |  7.974 |  31.895
2 |  8-QAM |  0.642 |  1.813 |  400 |  10.876 |  43.505
3 |  8-QAM |  0.780 |  2.202 |  300 |  13.214 |  52.857
4 |  16-QAM |  0.708 |  2.665 |  190 |  15.993 |  63.970
5 |  16-QAM |  0.853 |  3.211 |  90 |  19.268 |  77.071
6 |  32-QAM |  0.831 |  3.911 |  35 |  23.464 |  93.854
7 |  64-QAM |  0.879 |  4.964 |  5.56 |  29.783 |  119.130

(a) $N_{t}=32,N_{r}=4$

(b) $N_{t}=64,N_{r}=4$

Figure 2: Two examples of designing the RZF-TPC aided and distance-based ACM
scheme. The switching thresholds and achievable system throughputs for (a) and
(b) are listed in Tables I and II, respectively.

Tables I and II provide two design examples of the RZF-TPC aided and distance-
based ACM in conjunction with $(N_{t},N_{r})=(32,4)$ and
$(N_{t},N_{r})=(64,4)$, respectively. The RZF-TPC aided and distance-based ACM
consists of $K$ ACM modes for providing $K$ data rates. The modulation schemes
and code rates are selected from the second generation VersaFEC [41], which is
well designed to provide high performance and low latency ACM. The SE of mode
$k$, ${\rm SE}_{k}$, is given by

$\displaystyle{\rm SE}_{k}=$ $\displaystyle r_{c}\log_{2}(M)\frac{N}{N+N_{\rm
cp}},$ (76)

where $M$ is the modulation order, $r_{c}$ is the coding rate and the data
rate $r_{{\rm pDRA},k}$ per DRA of mode $k$, is given by

$\displaystyle r_{{\rm pDRA},k}=$ $\displaystyle B_{\rm total}\cdot{\rm
SE}_{k},$ (77)

while the total data rate $r_{{\rm total},k}$ of mode $k$, is given by

$\displaystyle r_{{\rm total},k}=$ $\displaystyle N_{r}\cdot r_{{\rm
pDRA},k}.$ (78)

More explicitly, Fig. 2(a) illustrates how the $K=8$ ACM modes are designed
for the example of Table I. Explicitly, the 8 switching thresholds of Table I
are determined so that the SEs of the corresponding ACM modes is just below
the SE curve of the RZF-TPC. The $K=7$ ACM modes of Table II are similarly
determined, as illustrated in Fig. 2(b). Fig. 2 also confirms that the RZF-TPC
aided and distance-based ACM scheme significantly outperforms the EB-TPC aided
and distance-based ACM scheme of [19].

Figure 3: The accumulated transmitted data volume as the function of the
distance $d_{b^{*}}^{a^{*}}$ between communicating aircraft $a^{*}$ and
$b^{*}$. The distances between the interfering aircraft and the desired
receiving aircraft are uniformly distributed within the range of
$\big{[}d_{b^{*}}^{a^{*}},~{}D_{\max}\big{]}$, and other system parameters are
specified in Table III.

To further compare the RZF-TPC aided and distance-based ACM to the EB-TPC
aided and distance-based ACM of [19], we quantitatively calculate their
accumulated transmitted data volume $C_{\rm acc}$ that can be exchanged
between aircraft $a^{*}$ and aircraft $b^{*}$ both flying at a typical
cruising speed of $v=920$ km/h in the opposite direction from the minimum
separation distance $d_{\min}=5.56$ km to the maximum communication distance
$d_{\max}=740$ km. Let the current distance $d^{a^{*}}_{b^{*}}$ between
aircraft $a^{*}$ and aircraft $b^{*}$ be
$d^{a^{*}}_{b^{*}}\in\big{(}d_{K-(k^{*}-1)},~{}d_{K-k^{*}}\big{]}$. The
accumulated data volume transmitted from $a^{*}$ to $b^{*}$ for
$d_{K-(k^{*}-1)}<d^{a^{*}}_{b^{*}}\leq d_{K-k^{*}}$ can be calculated by

$\displaystyle C_{\rm acc}\big{(}d^{a^{*}}_{b^{*}}\big{)}=$
$\displaystyle\frac{d^{a^{*}}_{b^{*}}-d_{K-k^{*}}}{2v}r_{{\rm
total},(K-k^{*})}+\sum\limits_{k=1}^{k^{*}}\frac{d_{K-k}-d_{K-(k-1)}}{2v}r_{{\rm
total},(K-(k-1))}.$ (79)

The accumulated transmitted data volumes expressed in gigabyte (GB) of the
RZF-TPC aided and distance-based ACM are compared with the EB-TPC aided and
distance-based ACM in Fig. 3, for $(N_{t},N_{r})=(32,4)$ and
$(N_{t},N_{r})=(64,4)$. As expected, the achievable accumulated transmitted
data volume of the RZF-TPC aided and distance-based ACM is significantly
higher than that of the EB-TPC aided and distance-based ACM. In particular,
when aircraft $a^{*}$ and $b^{*}$ fly over the communication distance, from
$d_{b^{*}}^{a^{*}}=5.56$ km to $d_{b^{*}}^{a^{*}}=740$ km taking a period of
about 24 minutes, the RZF-TPC aided and distance-based ACM associated with
$(N_{t},N_{r})=(64,4)$ is capable of transmitting a total of about 77 GB of
data, while the EB-TPC aided and distance-based ACM with
$(N_{t},N_{r})=(64,4)$ is only capable of transmitting about 60 GB of data.
Note that (79) can be revised to include any other scenario, by introducing
the angle of bearing between two aircraft and their heading direction.

TABLE III: Default system parameters of the simulated AANET. Number of interference aircraft $A$ | 4
---|---
Number of DRAs $N_{r}$ | 4
Number of DTAs $N_{t}$ | 32
Transmit power per antennas $P_{t}$ | 1 watt
Number of total subcarriers $N$ | 512
Number of CPs $N_{\rm cp}$ | 32
Rician factor $K_{\rm Rice}$ | 5
System bandwidth $B_{\rm total}$ | 6 MHz
Carrier frequency | 5 GHz
Correlation factor between antennas $\rho$ | 0.1
Noise figure at receiver $F$ | 4 dB
Distance $d_{b^{*}}^{a^{*}}$ between communicating aircraft $a^{*}$ and $b^{*}$ | 10 km
Minimum communication distance $D_{\min}$ | 5.56 km
Maximum communication distance $D_{\max}$ | 740 km

##  V Simulation Study

To further evaluate the achievable performance of the proposed RZF-TPC aided
and distance-based ACM scheme as well as to investigate the impact of the key
system parameters, we consider an AANET consisting of $(A+2)$ aircraft, with
two desired communicating aircraft and $A$ interfering aircraft. Each aircraft
is equipped with $N_{t}$ DTAs and $N_{r}$ DRAs. The network is allocated
$B_{\rm total}=6$ MHz bandwidth at the carrier frequency of 5 GHz. This
bandwidth is reused by every aircraft and it is divided into $N=512$
subcarriers. The CP samples are $N_{\rm cp}=32$. The transmit power per
antenna is $P_{t}=1$ Watt. The default system parameters are summarized in
Table III. Unless otherwise specified, these default parameters are used. The
deterministic part of the Rician channel is generated according to the model
given in [42], which satisfies $\text{Tr}\left\\{\bm{H}_{{\rm
d},b}^{a}\big{(}\bm{H}_{{\rm d},b}^{a}\big{)}^{\rm H}\right\\}=N_{t}N_{r}$.
The scattering component of the Rician channel
$\bm{H}_{\text{r},b}^{a}\in\mathbb{C}^{N_{r}\times N_{t}}$ is generated
according to (3). As mentioned previously, the DRAs are uncorrelated and,
therefore, we have $\bm{R}_{b}=\bm{I}_{N_{r}}$. The spatial correlation matrix
of the DTAs is generated according to

$\displaystyle\bm{R}^{a}\lvert_{[m,n]}=$
$\displaystyle\big{(}\bm{R}^{a}\lvert_{[n,m]}\big{)}^{{\ddagger}}=(t\rho)^{|m-n|},\,1\leq
n,m\leq N_{t},$ (80)

where $(\cdot)^{{\ddagger}}$ denotes the conjugate operator, $t$ is a complex
number with $|t|=1$ and $\rho$ is the magnitude of the correlation coefficient
that is determined by the antenna element spacing [43].

In the following investigation of the achievable throughput by the RZF-TPC
aided and distance-based ACM scheme, ‘Theoretical results’ are the throughputs
calculated using (18) using the perfect knowledge of
$\left[\bm{M}_{b^{a}}^{a}\right]_{(n_{r},n_{r})}$ and
$\left[\bm{M}_{b^{*}}^{a}\right]_{(n_{r}^{*},n_{r}^{*})}$ in (III-C), and the
‘Approximate results’ are the throughputs calculated using (18) with both
$\left[\bm{M}_{b^{a}}^{a}\right]_{(n_{r},n_{r})}$ and
$\left[\bm{M}_{b^{*}}^{a}\right]_{(n_{r}^{*},n_{r}^{*})}$ substituted by
$\left[\bm{M}_{b^{*}}^{a^{*}}\right]_{(n_{r}^{*},n_{r}^{*})}$ in (III-C),
while the ‘Simulation results’ represent the Monte-Carlo simulation results.
For the EB-TPC aided and distance-based ACM scheme, the ‘Theoretical results’,
‘Approximate results’ and ‘Simulation results’ are defined similarly.

Figure 4: The achievable throughput per DRA as the function of the number of
interfering aircraft $A$. The distances between the interfering aircraft and
the desired receiving aircraft are uniformly distributed within the range of
$\big{[}d_{b^{*}}^{a^{*}},~{}D_{\max}\big{]}$, and the rest of the parameters
are specified in Table III.

In Fig. 4, we investigate the achievable throughput per DRA as the function of
the number of interfering aircraft $A$. Observe from Fig. 4 that for the RZF-
TPC aided and distance-based ACM, the ‘Theoretical results’ are closely
matched by the ‘Simulation results’, which indicates that our theoretical
analysis presented in Section III is accurate. Furthermore, there is about 0.4
bps/Hz gap between the ‘Theoretical results’ and the ‘Approximate results’. As
expected, the achievable throughput degrades as the number of interfering
aircraft increases. Moreover, the RZF-TPC aided and distance-based ACM scheme
is capable of achieving significantly higher SE than the EB-TPC aided and
distance-based ACM scheme.

Figure 5: The achievable throughput per DRA as a function of the distance
$d_{b^{*}}^{a^{*}}$ between the communicating aircraft $a^{*}$ and $b^{*}$.
The distances between the interfering aircraft and the desired receiving
aircraft are uniformly distributed within the range of
$\big{[}d_{b^{*}}^{a^{*}},~{}D_{\max}\big{]}$, and the rest of the parameters
are specified in Table III.

Fig. 5 portrays the achievable throughput per DRA as the function of the
distance $d_{b^{*}}^{a^{*}}$ between the communicating aircraft $a^{*}$ and
$b^{*}$. Compared to the EB-TPC aided and distance-based ACM, the RZF-TPC
aided and distance-based ACM is capable of achieving significantly higher SE,
particularly at shorter distances. At the minimum distance of
$d_{b^{*}}^{a^{*}}=5.56$ km, the SE improvement is about 3 bps/Hz, but the SE
improvement becomes lower as the distance becomes longer. When the distance
approaches the maximum communication range of 740 km, both the schemes have a
similar SE.

Figure 6: The achievable throughput per DRA as the function of the number of
DTAs $N_{t}$. The distances between the interfering aircraft and the desired
receiving aircraft are uniformly distributed within the range of
$\big{[}d_{b^{*}}^{a^{*}},~{}D_{\max}\big{]}$, and the rest of the parameters
are specified in Table III.

Fig. 6 shows the impact of the number of DTAs $N_{t}$ on the achievable
throughput. As expected, the achievable throughput increases upon increasing
$N_{t}$. Observe from Fig. 6 that for the RZF-TPC aided and distance-based
ACM, the ‘Theoretical results’ are closely matched by the ‘Simulation
results’, while the ‘Theoretical results’ are closely matched by the
‘Approximate results’, when $N_{t}\geq 80$, but there exists a small gap
between the ‘Theoretical results’ and the ‘Approximate result’ for $N_{t}<80$.
It can also be seen that the RZF-TPC aided and distance-based ACM achieves
approximately 3.0 bps/Hz SE improvement over the EB-TPC aided and distance-
based ACM.

Figure 7: The achievable throughput as the function of the number of DRAs
$N_{r}$. The distances between the interfering aircraft and the desired
receiving aircraft are uniformly distributed within the range of
$\big{[}d_{b^{*}}^{a^{*}},~{}D_{\text{max}}\big{]}$, and the rest of the
parameters are specified in Table III.

The impact of the number of DRAs $N_{r}$ on the achievable throughput is
studied in Fig. 7, where both the achievable throughput per antenna and the
achievable sum rate of all the $N_{r}$ DRAs are plotted. Observe that the
achievable throughput per antenna degrades upon increasing $N_{r}$, owing to
the increase of the interference amongst the receive antennas. On the other
hand, the achievable sum rate increases with $N_{r}$ due to the multiplexing
gain. But the sum rate becomes saturated for $N_{r}>12$, because the increase
in multiplexing gain is roughly cancelled by the increase of inter-antenna
interference. Not surprisingly, the RZF-TPC aided and distance-based ACM
significantly outperforms the EB-TPC aided and distance-based ACM.

Figure 8: The achievable throughput per DRA as the function of the correlation
factor of DTAs $\rho$. The distances between the interfering aircraft and the
desired receiving aircraft are uniformly distributed within the range of
$\big{[}d_{b^{*}}^{a^{*}},~{}D_{\max}\big{]}$, and the rest of the parameters
are specified in Table III. Figure 9: The achievable throughput per DRA as the
function of the Rician factor $K_{\rm Rice}$. The distances between the
interfering aircraft and the desired receiving aircraft are uniformly
distributed within the range of $\big{[}d_{b^{*}}^{a^{*}},~{}D_{\max}\big{]}$,
and the rest of the parameters are specified in Table III.

The effect of the correlation factor $\rho$ of the DTAs on the achievable
throughput per DRA is shown in Fig. 8. It can be observed that a higher
correlation between DTAs results in lower achievable throughput. For the RZF-
TPC aided and distance-based ACM, the simulated throughput and the theoretical
throughput are close for $\rho\leq 0.4$, but there is a clear performance gap
between the ‘Theoretical results’ and the ‘Simulation results’ for $\rho\geq
0.5$. For the EB-TPC aided and distance-based ACM, this performance gap
between the ‘Theoretical results’ and the ‘Simulation results’ is even bigger
and it exists clearly over the range of $\rho\geq 0.3$. This indicates that
for a higher correlation factor $\rho$, the simulated SINR, which is the
average over a number of realizations, may deviate considerably from the
theoretical SINR, which is the ensemble average. From Fig. 8, it is clear that
in addition to achieving a significantly better SE performance, the RZF-TPC
aided and distance-based ACM can better deal with the problem caused by strong
correlation among the DTAs than the EB-TPC aided and distance-based ACM.

Fig. 9 portrays the impact of the Rician factor $K_{\rm Rice}$ on the
achievable throughput per DRA. It can be seen from Fig. 9 that the achievable
throughput per DRA increases upon increasing the Rician factor $K_{\rm Rice}$.
Furthermore, the SE improvement of the RZF-TPC aided and distance-based ACM
over the EB-TPC aided and distance-based ACM also increases with $K_{\rm
Rice}$. Specifically, the SE enhancement is about 1.9 bps/Hz at $K_{\rm
Rice}=2.0$ and this is increased to about 2.6 bps/Hz for $K_{\rm Rice}=16.0$.

##  VI Conclusions

A RZF-TCP aided and distance-based ACM scheme has been proposed for large-
scale antenna array assisted aeronautical communications. For the design of
powerful RZF-TCP, our theoretical contribution has been twofold. For the first
time, we have derived the analytical closed-form achievable data rate in the
presence of both realistic channel estimation error and co-channel
interference. Moreover, we have explicitly derived the optimal regularization
parameter that minimizes the mean-square detection error. With the aid of this
closed-form data rate metric, we have designed a practical distance-based ACM
scheme that switches its coding and modulation mode according to the distance
between the communicating aircraft. Our extensive simulation study has
quantified the impact of the key system parameters on the achievable
throughput of the proposed RZF-TCP aided and distance-based ACM scheme. Our
simulation results have confirmed the accuracy of our analytical results.
Moreover, both our theoretical analysis and Monte-Carlo simulations have
confirmed that the RZF-TCP aided and distance-based ACM scheme substantially
outperforms our previous EB-TCP aided and distance-based ACM scheme. In the
scenario where two communicating aircraft fly at a typical cruising speed of
920 km/h in opposite direction all the way to the maximum horizon
communication distance 740 km, the RZF-TPC aided and distance-based ACM scheme
is capable of transmitting about 11.7 GB and 16.5 GB extra data volumes
compared to EB-TCP aided and distance-based ACM scheme for the configurations
of 32 DTAs/4 DRAs and 64 DTAs/4 DRAs, respectively. This study has therefore
offered a practical high-rate, high-SE solution for air-to-air communications.

###  -A Gallery of Lemmas

######  Lemma 1 (Matrix inversion lemma I [44])

Given the Hermitian matrix $\bm{A}\in\mathbb{C}^{N\times N}$, vector
$\bm{x}\in\mathbb{C}^{N}$ and scalar $\tau\in\mathbb{C}$, if $\bm{A}$ and
$\left(\bm{A}+\tau\bm{x}\bm{x}^{\rm H}\right)$ are invertible, the following
identity holds

$\displaystyle\left(\bm{A}+\tau\bm{x}\bm{x}^{\rm H}\right)^{-1}\bm{x}=$
$\displaystyle\frac{\bm{A}^{-1}\bm{x}}{1+\tau\bm{x}^{\rm
H}\bm{A}^{-1}\bm{x}}.$ (81)

######  Lemma 2 (Matrix inversion lemma II [44])

Given the Hermitian matrix $\bm{A}\in\mathbb{C}^{N\times N}$, vector
$\bm{x}\in\mathbb{C}^{N}$ and scalar $\tau\in\mathbb{C}$, if $\bm{A}$ and
$\left(\bm{A}+\tau\bm{x}\bm{x}^{\rm H}\right)$ are invertible, the following
identity holds

$\displaystyle\left(\bm{A}+\tau\bm{x}\bm{x}^{\rm H}\right)^{-1}=$
$\displaystyle\bm{A}^{-1}+\frac{\bm{A}^{-1}\tau\bm{x}\bm{x}^{\rm
H}\bm{A}^{-1}}{1+\tau\bm{x}^{\rm H}\bm{A}^{-1}\bm{x}}.$ (82)

######  Lemma 3 ([19])

Let $\bm{A}\in\mathbb{C}^{N\times N}$ and
$\bm{x}\sim\mathcal{CN}\left(\frac{1}{\sqrt{N}}\bm{m},\frac{1}{N}\bm{\Upsilon}\right)$,
where $\frac{1}{\sqrt{N}}\bm{m}\in\mathbb{C}^{N}$ and
$\frac{1}{N}\bm{\Upsilon}\in\mathbb{C}^{N\times N}$ are the mean vector and
the covariance matrix of the random vector $\bm{x}\in\mathbb{C}^{N}$,
respectively. Assuming that $\bm{A}$ has a uniformly bounded spectral norm
with respect to $N$ and $\bm{x}$ is independent of $\bm{A}$, we have

$\displaystyle\lim_{N\to\infty}\bm{x}^{\rm H}\bm{A}\bm{x}=$
$\displaystyle\text{Tr}\left\\{\left(\frac{1}{N}\bm{M}+\frac{1}{N}\bm{\Upsilon}\right)\bm{A}\right\\},$
(83)

where $\bm{M}=\bm{m}\bm{m}^{\rm H}$.

###### Lemma 4

Let $\bm{A}\in\mathbb{C}^{N\times N}$, and two independent random vectors
$\bm{x}\in\mathbb{C}^{N}$ and $\bm{y}\in\mathbb{C}^{N}$ have the distributions
$\bm{x}\sim\mathcal{CN}\left(\frac{1}{\sqrt{N}}\bm{m}_{x},\frac{1}{N}\bm{\Upsilon}_{x}\right)$
and
$\bm{y}\sim\mathcal{CN}\left(\frac{1}{\sqrt{N}}\bm{m}_{y},\frac{1}{N}\bm{\Upsilon}_{y}\right)$,
where $\frac{1}{\sqrt{N}}\bm{m}_{x}\in\mathbb{C}^{N}$ and
$\frac{1}{\sqrt{N}}\bm{m}_{y}\in\mathbb{C}^{N}$ are the mean vectors, while
$\frac{1}{N}\bm{\Upsilon}_{x}\in\mathbb{C}^{N\times N}$ and
$\frac{1}{N}\bm{\Upsilon}_{y}\in\mathbb{C}^{N\times N}$ are the covariance
matrices of $\bm{x}$ and $\bm{y}$, respectively. Assuming that $\bm{A}$ has a
uniformly bounded spectral norm with respect to $N$, and $\bm{x}$ and $\bm{y}$
are independent of $\bm{A}$, we have

$\displaystyle\lim_{N\to\infty}\bm{x}^{\rm H}\bm{A}\bm{y}=$
$\displaystyle\text{Tr}\left\\{\frac{1}{N}\bm{M}_{xy}\bm{A}\right\\},$ (84)

where $\bm{M}_{xy}=\bm{m}_{x}\bm{m}_{y}^{\rm H}$.

###### Proof 1

Let $\bm{z}_{x}=\sqrt{N}\bm{x}-\bm{m}$. Since
$\bm{x}\sim\mathcal{CN}\left(\frac{1}{\sqrt{N}}\bm{m}_{x},\frac{1}{N}\bm{\Upsilon}_{x}\right)$,
$\bm{z}_{x}\sim\mathcal{CN}\left(\bm{0}_{N},\bm{\Upsilon}_{x}\right)$. Let
$\bm{z}_{y}=\sqrt{N}\bm{y}-\bm{m}$. As
$\bm{y}\sim\mathcal{CN}\left(\frac{1}{\sqrt{N}}\bm{m}_{y},\frac{1}{N}\bm{\Upsilon}_{y}\right)$,
$\bm{z}_{x}\sim\mathcal{CN}\left(\bm{0}_{N},\bm{\Upsilon}_{y}\right)$.
Furthermore,

$\displaystyle\bm{x}^{\rm H}\bm{A}\bm{y}=$
$\displaystyle\left(\frac{1}{\sqrt{N}}\bm{m}_{x}+\frac{1}{\sqrt{N}}\bm{z}_{x}\right)^{\rm
H}\bm{A}\left(\frac{1}{\sqrt{N}}\bm{m}_{y}+\frac{1}{\sqrt{N}}\bm{z}_{y}\right)$
$\displaystyle=$ $\displaystyle\frac{1}{N}\bm{m}_{x}^{\rm
H}\bm{A}\bm{m}_{y}+\frac{1}{N}\bm{z}_{x}^{\rm
H}\bm{A}\bm{z}_{y}+\frac{1}{N}\bm{m}_{x}^{\rm
H}\bm{A}\bm{z}_{y}+\frac{1}{N}\bm{z}_{x}^{\rm H}\bm{A}\bm{m}_{y}.$ (85)

Since $\bm{z}_{x}\sim\mathcal{CN}\left(\bm{0}_{N},\bm{\Upsilon}_{x}\right)$
and $\bm{z}_{y}\sim\mathcal{CN}\left(\bm{0}_{N},\bm{\Upsilon}_{y}\right)$,
$\bm{z}_{x}$ and $\bm{z}_{y}$ do not depend on $\bm{m}_{x}$ and $\bm{m}_{y}$.
According to Lemma 1 of [37], we have

$\displaystyle\lim\limits_{N\to\infty}\frac{\bm{m}_{x}^{\rm
H}\bm{A}\bm{z}_{y}}{N}=$ $\displaystyle 0,$ (86)
$\displaystyle\lim\limits_{N\to\infty}\frac{\bm{z}_{x}^{\rm
H}\bm{A}\bm{m}_{y}}{N}=$ $\displaystyle 0.$ (87)

Since $\bm{z}_{x}$ and $\bm{z}_{y}$ are independent, according to the trace
lemma of [45], we have

$\displaystyle\lim\limits_{N\to\infty}\frac{1}{N}\bm{z}_{x}^{\rm
H}\bm{A}\bm{z}_{y}=0.$ (88)

Furthermore,

$\displaystyle\frac{1}{N}\bm{m}_{x}^{\rm
H}\bm{A}\bm{m}_{y}=\text{Tr}\left\\{\frac{1}{N}\bm{A}\bm{M}_{xy}\right\\}.$
(89)

Taking the limit $N\to\infty$ as well as substituting (86) to (89) into (1)
results in (84).

###  -B Derivation of the Optimal Regularization Parameter

Because the term
$\mathcal{E}\bigg{\\{}\sum\limits_{a=1}^{A}\Big{\|}\frac{1}{N_{t}}\sqrt{\frac{P_{r,b^{*}}^{a}}{P_{r,b^{*}}^{a^{*}}}}\bm{H}_{b^{*}}^{a}\bm{V}_{b^{a}}^{a}\bm{X}^{a}\Big{\|}^{2}+\Big{\|}\frac{1}{N_{t}\sqrt{P_{r,b^{*}}^{a^{*}}}}\bm{W}_{b^{*}}\Big{\|}^{2}\bigg{\\}}$
is independent of $\xi_{b^{*}}^{a^{*}}$,

$\displaystyle\frac{d\,\mathcal{J}\big{(}\xi_{b^{*}}^{a^{*}}\big{)}}{d\,\xi_{b^{*}}^{a^{*}}}=$
$\displaystyle\mathcal{E}\Bigg{\\{}\bigg{(}\frac{1}{N_{t}}\bm{H}_{b^{*}}^{a^{*}}\bm{V}_{b^{*}}^{a^{*}}\bm{X}^{a^{*}}-\bm{X}^{a^{*}}\bigg{)}^{\rm
H}\frac{d\,\big{(}\frac{1}{N_{t}}\bm{H}_{b^{*}}^{a^{*}}\bm{V}_{b^{*}}^{a^{*}}\bm{X}^{a^{*}}-\bm{X}^{a^{*}}\big{)}}{d\,\xi_{b^{*}}^{a^{*}}}$
$\displaystyle+\frac{d\,\big{(}\frac{1}{N_{t}}\bm{H}_{b^{*}}^{a^{*}}\bm{V}_{b^{*}}^{a^{*}}\bm{X}^{a^{*}}-\bm{X}^{a^{*}}\big{)}^{\rm
H}}{d\,\xi_{b^{*}}^{a^{*}}}\Big{(}\frac{1}{N_{t}}\bm{H}_{b^{*}}^{a^{*}}\bm{V}_{b^{*}}^{a^{*}}\bm{X}^{a^{*}}-\bm{X}^{a^{*}}\Big{)}\Bigg{\\}}$
$\displaystyle=\mathcal{E}\bigg{\\{}\bigg{(}\frac{1}{N_{t}}\bm{H}_{b^{*}}^{a^{*}}\bm{V}_{b^{*}}^{a^{*}}\bm{X}^{a^{*}}-\bm{X}^{a^{*}}\bigg{)}^{\rm
H}\bigg{(}-\frac{1}{N_{t}}\bm{H}_{b^{*}}^{a^{*}}\Big{(}\big{(}\bm{\Upsilon}_{b^{*}}^{a^{*}}\big{)}^{2}\big{(}\widehat{\bm{H}}_{b^{*}}^{a^{*}}\big{)}^{\rm
H}\Big{)}\bm{X}^{a^{*}}\bigg{)}$
$\displaystyle+\bigg{(}-\frac{1}{N_{t}}\bm{H}_{b^{*}}^{a^{*}}\Big{(}\big{(}\bm{\Upsilon}_{b^{*}}^{a^{*}}\big{)}^{2}\big{(}\widehat{\bm{H}}_{b^{*}}^{a^{*}}\big{)}^{\rm
H}\Big{)}\bm{X}^{a^{*}}\bigg{)}^{\rm
H}\bigg{(}\frac{1}{N_{t}}\bm{H}_{b^{*}}^{a^{*}}\bm{V}_{b^{*}}^{a^{*}}\bm{X}^{a^{*}}-\bm{X}^{a^{*}}\bigg{)}\bigg{\\}}$
$\displaystyle=\mathcal{E}\bigg{\\{}\\!\\!-\\!\frac{1}{N_{t}^{2}}\text{Tr}\Big{\\{}\\!E_{\rm
s}\widehat{\bm{H}}_{b^{*}}^{a^{*}}\bm{\Upsilon}_{b^{*}}^{a^{*}}\big{(}\bm{H}_{b^{*}}^{a^{*}}\big{)}^{\rm
H}\bm{H}_{b^{*}}^{a^{*}}\big{(}\bm{\Upsilon}_{b^{*}}^{a^{*}}\big{)}^{2}\big{(}\widehat{\bm{H}}_{b^{*}}^{a^{*}}\big{)}^{\rm
H}\\!\Big{\\}}\\!+\\!\frac{1}{N_{t}}\text{Tr}\Big{\\{}\\!E_{\rm
s}\bm{H}_{b^{*}}^{a^{*}}\big{(}\bm{\Upsilon}_{b^{*}}^{a^{*}}\big{)}^{2}\\!\big{(}\widehat{\bm{H}}_{b^{*}}^{a^{*}}\big{)}^{\rm
H}\\!\Big{\\}}$
$\displaystyle\\!\\!-\\!\frac{1}{N_{t}^{2}}\text{Tr}\Big{\\{}\\!E_{\rm
s}\widehat{\bm{H}}_{b^{*}}^{a^{*}}\big{(}\bm{\Upsilon}_{b^{*}}^{a^{*}}\big{)}^{2}\big{(}\bm{H}_{b^{*}}^{a^{*}}\big{)}^{\rm
H}\bm{H}_{b^{*}}^{a^{*}}\bm{\Upsilon}_{b^{*}}^{a^{*}}\big{(}\widehat{\bm{H}}_{b^{*}}^{a^{*}}\big{)}^{\rm
H}\\!\Big{\\}}\\!+\\!\frac{1}{N_{t}}\text{Tr}\Big{\\{}\\!E_{\rm
s}\widehat{\bm{H}}_{b^{*}}^{a^{*}}\big{(}\bm{\Upsilon}_{b^{*}}^{a^{*}}\big{)}^{2}\big{(}\bm{H}_{b^{*}}^{a^{*}}\big{)}^{\rm
H}\\!\Big{\\}}\\!\\!\bigg{\\}}\\!,\\!$ (90)

where $E_{\rm
s}=\mathcal{E}\big{\\{}\big{|}X_{n_{r}}^{a^{*}}\big{|}^{2}\big{\\}}$, $1\leq
n_{r}\leq N_{r}$. Setting
$\frac{d\,\mathcal{J}\big{(}\xi_{b^{*}}^{a^{*}}\big{)}}{d\,\xi_{b^{*}}^{a^{*}}}=0$
and followed by some further operations yields

$\displaystyle\text{Tr}\bigg{\\{}\bigg{(}\frac{1}{N_{t}}\widetilde{\bm{\Xi}}_{b^{*}}^{a^{*}}-\xi_{b^{*}}^{a^{*}}\bm{I}_{N_{t}}\bigg{)}\bm{\Upsilon}_{b^{*}}^{a^{*}}\big{(}\widehat{\bm{H}}_{b^{*}}^{a^{*}}\big{)}^{\rm
H}\widehat{\bm{H}}_{b^{*}}^{a^{*}}\big{(}\bm{\Upsilon}_{b^{*}}^{a^{*}}\big{)}^{2}\bigg{\\}}=0,$
(91)

where
$\widetilde{\bm{\Xi}}_{b^{*}}^{a^{*}}=\text{diag}\big{\\{}\widetilde{\varphi}_{1},\cdots,\widetilde{\varphi}_{N_{t}}\big{\\}}$.
This proves that (73) is an optimal regularization parameter.

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