# Airplane-Aided Integrated Next-Generation Networking

Muralikrishnan Srinivasan1, Sarath Gopi2, Sheetal Kalyani2, Xiaojing Huang3,
Lajos Hanzo4 1\. Muralikrishnan Srinivasan is with ETIS UMR8051, CY
University, ENSEA, CNRS, Cergy, France.
(Email:muralikrishnan.srinivasan@ensea.fr)2. Sarath Gopi and Sheetal Kalyani
are with the Dept. of Electrical Engineering, Indian Institute of Technology,
Madras, India. (Emails:ee14d007@ee, skalyani@ee.iitm.ac.in).3. Xiaojing Huang
is with the University of Technology Sydney. (Email:
Xiaojing.Huang@uts.edu.au)4. Lajos Hanzo is with the School of Electronics and
Computer Science, University of Southampton.(Email<EMAIL_ADDRESS>Muralikrishnan Srinivasan and Sarath Gopi are co-first authors.L. Hanzo would
like to acknowledge the financial support of the Engineering and Physical
Sciences Research Council projects EP/P034284/1 and EP/P003990/1 (COALESCE) as
well as of the European Research Council’s Advanced Fellow Grant QuantCom
(Grant No. 789028).This work was also supported in part by the Australian
Research Council Discovery Project (DP200101532).This article has been
accepted for publication in IEEE Transactions on Vehicular Technology, but has
not been fully edited. Content may change prior to final publication. Citation
information: DOI 10.1109/TVT.2021.3098098, IEEE Transactions on Vehicular
Technology

###### Abstract

A high-rate yet low-cost air-to-ground (A2G) communication backbone is
conceived for integrating the space and terrestrial network by harnessing the
opportunistic assistance of the passenger planes or high altitude platforms
(HAPs) as mobile base stations (BSs) and millimetre wave communication. The
airliners act as the network-provider for the terrestrial users while relying
on satellite backhaul. Three different beamforming techniques relying on a
large-scale planar array are used for transmission by the airliner/HAP for
achieving a high directional gain, hence minimizing the interference among the
users. Furthermore, approximate spectral efficiency (SE) and area spectral
efficiency (ASE) expressions are derived and quantified for diverse system
parameters.

##  I Introduction

Next-generation wireless standards are expected to cope with increased traffic
demands and support emerging applications even in remote locations such as
rural hinterlands, mountains, deserts and even for vessels such as cruise-
ships in the oceans [1]. However, the operational fifth-generation (5G)
standards have predominantly been designed for terrestrial communications. One
of the promising techniques of augmenting cellular communication is through
air-based platforms such as unmanned aerial vehicles (UAV) or high-altitude
platforms (HAP). Hence, extensive research has been dedicated to the design,
channel modelling, and security of UAV-based cellular communications [2, 3, 4,
5], as well as to their performance analysis [6, 7, 8, 9, 10, 11, 12, 13, 14,
15]. Sakhaee and Jamalipour [16] along with Kato [17] showed as early as 2006
that the probability of finding at least two but potentially up to dozens of
aircraft capable of establishing an AANET above-the-cloud is close to $100\%$.
It was inferred by investigating a snapshot of flight data over the United
States (US). They also proposed a quality of service (QoS) based so-called
multipath Doppler routing protocol by jointly considering both the QoS and the
relative velocity of nodes in order to find stable routing paths.

Integrating the aerial networks with the terrestrial networks has the
potential of increasing both the data rate and the coverage quality of
terrestrial networks [18, 19, 20, 21, 22, 23]. There have also been some
attempts to integrate the space networks with terrestrial networks or to
provide Internet coverage for airliner [24, 25, 26, 27, 28, 29, 30, 31]. The
applicability of these prior contributions are tabulated in I. However, most
of these contributions rely on reusing the existing long-term evolution (LTE)
bands, which are already congested in the sub-6GHz bands [32, 33]. Therefore,
the creation of a high-capacity integrated space terrestrial network (ISTN) or
a space-air-ground integrated network (SAGIN) is still elusive at the time of
writing both due to the bandwidth limitation of aerial backbones and owing to
the limited area spectral efficiency (ASE) of the air-to-ground (A2G) systems,
given their large footprint on the ground. Therefore, it is imperative to
explore new architectures integrating the existing terrestrial networks with
space networks.

Works | Applicability
---|---
[18] | Emergency networks
[19] | Vehicular networks
[20] | Broadbad connectivity
[21] | Cellular networks
[22] | 5G Cellular networks
[23] | Secure networks
[24] | Multi-media systems
[25] | Integration of space and HAP systems
[26] | Emergency networks
[27] | Cognitive communications
TABLE I: Integrated aerial-terrestrial networks in Sub-6GHz band

###  I-A Related Contributions:

A critical cornerstone of the next-generation systems is the potential
exploitation of millimetre wave (mmWave) carrier frequencies to benefit from
their broad unused spectrum. Hence the authors of [34, 35, 36, 37, 38, 39]
have investigated the challenges of mmWave based A2G and air-to-air (A2A)
communications. For example, the two-ray propagation model’s applicability in
different scenarios employing UAVs was explored in [34]. The pros and cons of
UAV-BS in complementing the mmWave backhaul were demonstrated in [35]. The
concept of mmWave A2A networks was first explored by Cuvelier and Heath [37],
while mmWave based HAPs in terrestrial transmissions were studied in [38, 39].

As a further development, Huang et al. [1] have proposed the ISTN concept
relying on civil airliner networks and mmWave communication to form a high-
capacity yet low-cost A2A and A2G communications backbone employing high-gain
antenna arrays. In this concept, the airliners act as an efficient network-
provider for terrestrial users, since the distance from the planes to the
ground is much shorter than that from the satellite. Furthermore, to provide
A2G cellular coverage for small cells that can support high ASE, adaptive
beamforming is proposed. In areas where the civil-airliners cannot be used,
dedicated HAPs would be used as the backbone. A similar topology is presented
in [40, Fig 1].

| Our Scheme | [41] | [42] | [43] | [44] | [45] | [46] | [47] | [48] | [49]
---|---|---|---|---|---|---|---|---|---|---
mmWave | $\checkmark$ |  |  | $\checkmark$ | $\checkmark$ | $\checkmark$ |  | $\checkmark$ | $\checkmark$ | $\checkmark$
Airliner/UAV backbone | Airliner | UAV |  | UAV | UAV | UAV |  | UAV | UAV | HAP
Adaptive null steering | $\checkmark$ |  | $\checkmark$ |  |  |  | $\checkmark$ |  |  |
Channel unaware |  |  |  |  |  |  |  |  |  |
precoding | $\checkmark$ |  | $\checkmark$ |  |  | $\checkmark$ | $\checkmark$ |  |  |
Rician channel | $\checkmark$ |  |  |  |  | $\checkmark$ |  |  |  | $\checkmark$
Simulated Metric | ASE/SE | SE | Array | SE | SE | SE | SE | Beam | Beam | SE
|  |  | response |  |  |  |  | coverage | coverage |
Theoretical expressions | ASE/SE |  |  |  |  | SE |  |  |  | SE
TABLE II: Comparison of the proposed scheme with existing works

###  I-B Design challenges:

To actually design such a high-capacity airline-aided integrated network,
several challenges have to be addressed. For example, a cruising airliner
maintains an altitude of at least $10~{}km$ from the ground, while solar-
charged unmanned aircraft are envisioned to circle above $20~{}km$ for
avoiding civilian planes. The mmWave channel suffers from substantial pathloss
owing to raindrops, high-absorption and other atmospheric effects, especially
at a carrier frequency of $73.5$ GHz. Another challenge to overcome is the
huge channel estimation overhead, which results from the rapidly fluctuating
high-Doppler channel between the cruising airliner and ground users. Hence, a
careful selection of the channel model, antenna dimensions, Rician factor and
other system parameters is required for investigating a realistic stand-alone
model.

###  I-C Contributions

To overcome the above-mentioned challenges, we design a high-performance
system having a high data-rate and ASE and provide theoretical performance
guarantees with the aid of approximate expressions. We consider a planar-array
aided stand-alone airliner/HAP in a macro-cell communicating with the
terrestrial BS/users. Although a strong line-of-sight (LoS) component exists
between the airplane and the ground users, the non-line-of-sight (NLoS)
component fluctuates drastically over time. Hence, to achieve a high
directional gain while minimizing the users’ interference, we propose three
different channel-agnostic transmit precoding schemes for avoiding the massive
pilot overhead required for estimating the channel-state information (CSI) or
the NLoS component. Explicitly, our Transmit Precoders (TPC) rely only on the
users’ position relative to the airliner and they are quite robust to
incorrect Doppler compensation and position vector mismatches.

We also derive approximate expressions for the SE/ASE of the users.
Furthermore, the proposed schemes are evaluated through extensive simulations,
and its performance is compared to the analytically obtained values.
Additionally, depending on the dimensions of the planar array and of the LoS
factor, the ASE achieved by our system becomes several times higher than that
of conventional terrestrial networks [50, 51, 52, 53] capable of providing
data rates on the order of several Gbps. The authors of [41, 42, 43, 44, 45,
46, 47, 48, 49] have considered 3D beamforming in the context of UAV
communications, where most of them tended to rely either on UAVs flying at a
modest altitude or on channel-aware TPCs. Specifically, the authors of [49]
designed a TPC relying on the effective channel matrix between the access
point (AP) and the users. Therefore, a dedicated downlink channel estimation
phase is required by the TPC. By contrast, in our case, the channel between
the airliner and the user is dominated by the LoS component due to the
airliner’s altitude. Typically the aeronautical model for such a scenario has
a strong LoS path and a much weaker NLoS ground-reflected path [54]. Hence
using a TPC vector relying on the estimated channel matrix is
counterproductive. Furthermore, in such a highly mobile scenario, accurate
channel estimation requires a potentially excessive pilot overhead.
Explicitly, compared to the pedestrian walking across a mmWave cell at
$5~{}km/hr$, a plane travelling at $1000~{}km/hr$ would require a $200$ times
higher pilot overhead. Even if we disregard the huge training overhead
requirement, the CSI is prone to estimation error.

Against the above backdrop, we boldly contrast our novel contributions to the
prior art in Table II. The theoretical analysis of the proposed system and our
extensive simulations indicate that our design leads to high capacity
airplane-aided integrated networks that are eminently suitable for filling the
coverage-holes of next-generation wireless systems.

The rest of the paper is structured as follows. In Section II, the proposed
system design is discussed in detail, along with different beamforming. In
Section III, theoretical expressions are derived for the ASE/SE using the
popular use and forget bound. In Section IV, our simulation results and
interesting design guidelines are discussed, while in Section V, some future
research directions are provided.

##  II Proposed System Design

UserReuse Distance $D$Macro-cell centerymacro-cell radius
$R$xOrigin$\theta^{z}$ zenith angle$\theta^{a}$ azimuth anglez

Figure 1: The Airliner/HAP is assumed to be at the origin. The macro-cell
centre is the point on the ground directly below the airliner. A circular
macro-cell of radius $R$ is considered around the macro-cell centre. The
zenith-azimuth angle pair $(\theta^{z},\theta^{a})$ of the ’User’ in a micro-
cell of interest (MCI), is marked with respect to the airliner. Each micro-
cell has a radius $r$. The first tier of $N_{I}$ interfering micro-cells is
shown at a reuse distance $D$. In this figure, $N_{I}=6$.

In this section, we design the ISTN backbone and propose a system model using
various adaptive beamforming for supporting high data-rates and seamless
connectivity. Consider a circular macro-cell of radius $R$ with an airliner at
its centre at an altitude $H_{t}$. The radius $R$ is chosen to be at least
$5~{}km$, so a minimum inter-airliner distance of $10~{}km$ is maintained. In
remote locations outside the regular flight path, HAPs can be installed for
providing seamless connectivity to satellites. Free-space optical (FSO) links
connect them to Low-Earth orbit (LEO)/ medium-Earth orbit (MEO) satellites as
their high-speed backhaul.

Each airliner/HAP is equipped with a planar antenna having $M\times M$ equally
spaced elements. Without loss of generality, it can be assumed that the
antenna elements are parallel to the ground and the centre of the antenna is
the origin $(0,0,0)$. The macro-cell is further divided into several tightly
packed micro-cells of radius $r<<R$. Each micro-cell supports a single time-
frequency block. The users can either be a cellular user equipment (CUE) or
even an LTE base station, which in turn supports several UEs. The intended
user is at position $(x_{0},y_{0},-H_{t})$, and the micro-cell containing it
is referred to as the micro-cell of interest (MCI).

Assume that there are $N_{1}$, $N_{2}$ …$N_{J}$ interfering micro-cells in the
first $J$ tiers using the same time-frequency block. The centres of these
micro-cells are located at distances $D$, $2D$, …, $JD$ from the MCI, where
$D$ denotes the reuse distance. Let $N_{I}=N_{1}+N_{2}+...+N_{J}$ be the total
number of interfering cells and let the coordinates of these interfering users
be $(x_{i},y_{i},-H_{t})$ $i=1,...,N_{I}$. All the users are assumed to have a
single antenna. Let $(\theta_{i}^{z},\theta_{i}^{a})$ represent the zenith and
azimuth angle pair for the $i^{th}$ user, which are:

$\displaystyle\theta_{i}^{z}=\tan^{-1}\left(\frac{\sqrt{x_{i}^{2}+y_{i}^{2}}}{-H_{t}}\right),\quad
i=0,...,N_{I}$ (1)

and

$\displaystyle\theta_{i}^{a}=\tan^{-1}\left(\frac{y_{i}}{x_{i}}\right),\quad
i=0,...,N_{I}.$ (2)

Here $i=0$ represents the user under consideration, while $i=1,...,N_{I}$
represent the interferers. The entire system is shown in Fig. 1.

Note that the typical cruising airliner altitudes are in the $9-12~{}km$
range. At such distances, the mmWave channels’ attenuation, say at
$73.5~{}GHz$, is significant. Existing contributions, such as [43, 44, 45, 47,
48], which deal with aerial mmWave networks, consider only low-flying UAVs or
low altitude platforms (LAPs) and hence suffer from relatively low
attenuation. In the absence of beamforming at the transmitter, the users in
the macro-cell suffer from mutual interference, resulting in a reduced data-
rate. Therefore, to reduce the mutual interference amongst the users and
improve the spectral efficiency, some form of adaptive beamforming must be
used by the airliner’s planar array. Furthermore, the adaptive beamforming
schemes must be robust to incorrect Doppler compensation and position vector
mismatches. We therefore propose three different beamforming or TPC schemes at
the Airliner/HAP. The design of the beamforming vectors is described in the
subsequent paragraphs.

###  II-A Null-steered beamforming (NSB) design

NSB relies on signal processing techniques for creating transmit nulls and
maxima in the undesired and desired receivers’ directions, respectively, for
mitigating the interference [42, 55, 56]. For $i=0,1,..,N_{I}$, let
${\mathbf{e}}_{i}$ be the $M^{2}\times 1$ vector representation of the $i$th
user’s steering vector with respect to each of the components of the planar
array. The three-dimensional Cartesian co-ordinates of the $(m,n)$th component
of the planar array are represented by $(x_{m},y_{n},0)$, where
$x_{m}=\left[-\frac{M-1}{2}+(m-1)\right]\frac{\lambda}{2}$ and
$y_{n}=\left[-\frac{M-1}{2}+(n-1)\right]\frac{\lambda}{2}$, for $m=1,..M$ and
$n=1,...,M$. Note that the inter-elemental spacing is $\lambda/2$, where
$\lambda$ is the wavelength of the carrier. Thus the entry of
${\mathbf{e}}_{i}$, which corresponds to the position of the user with respect
to the $(m,n)$th element of the planar array, is given by
$\exp\left[j\frac{2\pi}{\lambda}\left(x_{m}\psi_{i}^{x}+y_{n}\psi_{i}^{y}\right)\right]$,
where we have

$\displaystyle\psi_{i}^{x}=\sin\theta_{i}^{z}\cos\theta_{i}^{a},$ (3)

and

$\displaystyle\psi_{i}^{y}=\sin\theta_{i}^{z}\sin\theta_{i}^{a},$ (4)

while $\theta_{i}^{z}$ and $\theta_{i}^{a}$ are defined in (1) and (2),
respectively, which are functions of the user location. Now the null-steered
beamforming vector $\tilde{{\mathbf{e}}}_{i}$ used by the airliner is

$\displaystyle\tilde{{\mathbf{e}}}_{i}={\mathbf{e}}_{i}-{\mathbf{E}}_{i}\left({\mathbf{E}}_{i}^{H}{\mathbf{E}}_{i}\right)^{-1}{\mathbf{E}}_{i}^{H}{\mathbf{e}}_{i},\forall
i=0,1,...,N_{I},$ (5)

where ${\mathbf{E}}_{i}$ is the matrix whose columns are the steering vectors,
except for ${\mathbf{e}}_{i}$, which is given by:

$\displaystyle{\mathbf{E}}_{i}=\left[{\mathbf{e}}_{0}~{}{\mathbf{e}}_{2}~{}...~{}{\mathbf{e}}_{i-1}~{}{\mathbf{e}}_{i+1}~{}...~{}{\mathbf{e}}_{N_{I}}\right].$
(6)

This can be obtained by solving the following optimization problem [57]:

$\displaystyle\underset{\tilde{{\mathbf{e}}}_{i}}{\min}\|\tilde{{\mathbf{e}}}_{i}-{\mathbf{e}}_{i}\|^{2}$
(7) such that
$\displaystyle\tilde{{\mathbf{e}}}_{i}^{H}{\mathbf{E}}_{i}=\bm{0}.$

NSB may also be interpreted as a zero-forcing precoder, where the precoding
vectors only require the knowledge of the user location.

###  II-B Null-Steered Beamformer with Derivative Constraints (NSB-D)

The NSB detailed in Section II-A can be extended by adding additional
derivative constraints [57, 3.7.2]. The higher order derivatives of the
directional pattern in the directions of nulls are set to zero to broaden the
beam-widths, to make the TPC robust to mismatches in the steering vector. In
this paper, we explicitly add the following first-order derivative constraints
to (7):

$\displaystyle\bar{{\mathbf{e}}}_{i}^{a}=\frac{\partial}{\partial\theta_{i}^{a}}{\mathbf{e}}_{i}$
$\displaystyle=0~{}\forall~{}i=0,1,...,N_{I}$
$\displaystyle\bar{{\mathbf{e}}}_{i}^{z}=\frac{\partial}{\partial\theta_{i}^{z}}{\mathbf{e}}_{i}$
$\displaystyle=0~{}\forall~{}i=0,1,...,N_{I}.$ (8)

Solving the optimization problem yields the beamforming vector
$\tilde{{\mathbf{e}}}_{i}$ as [57]:

$\displaystyle\tilde{{\mathbf{e}}}_{i}={\mathbf{e}}_{i}-\tilde{\mathbf{E}}_{i}\left(\tilde{\mathbf{E}}_{i}^{H}\tilde{\mathbf{E}}_{i}\right)^{-1}\tilde{\mathbf{E}}_{i}^{H}{\mathbf{e}}_{i}~{}\forall~{}i=0,1,...,N_{I},$
(9)

where

$\displaystyle\tilde{\mathbf{E}}_{i}=\left[{\mathbf{e}}_{0}~{}...~{}{\mathbf{e}}_{i-1}~{}{\mathbf{e}}_{i+1}~{}...~{}{\mathbf{e}}_{N_{I}}~{}\bar{{\mathbf{e}}}_{0}^{a}~{}\bar{{\mathbf{e}}}_{0}^{z}~{}...~{}\bar{{\mathbf{e}}}_{N_{I}}^{a}~{}\bar{{\mathbf{e}}}_{N_{I}}^{z}\right].$
(10)

Note that (9) is similar to (6) except that ${\mathbf{E}}$ is replaced with
$\tilde{\mathbf{E}}$, which includes the derivative constraints of all the
users.

###  II-C Minimum-Power Distortionless Response Beamformer (MPDRB)

Another popular beamformer is the MPDRB, where the total output power is
minimized subject to a distortionless constraint [57]. In other words, one can
determine the steering vectors $\tilde{{\mathbf{e}}}_{i}$ so that the total
power $\|\tilde{{\mathbf{e}}}_{i}^{H}\hat{\mathbf{E}}_{i}\|^{2}$ subject to a
distortionless constraint is minimized, which is formulated as :

$\displaystyle\underset{\tilde{{\mathbf{e}}}_{i}}{\min}~{}\|\tilde{{\mathbf{e}}}_{i}^{H}\hat{\mathbf{E}}_{i}\|^{2}~{}\text{such
that}~{}\tilde{{\mathbf{e}}}_{i}^{H}{\mathbf{e}}_{i}=\bm{1},$ (11)

where
$\hat{\mathbf{E}}_{i}=\left[{\mathbf{e}}_{0}~{}{\mathbf{e}}_{1}~{}...~{}{\mathbf{e}}_{i-1}~{}{\mathbf{e}}_{i}~{}{\mathbf{e}}_{i+1}~{}...~{}{\mathbf{e}}_{N_{I}}\right]$.
Solving the optimization problem results in [57]:

$\tilde{{\mathbf{e}}}_{i}=\frac{\left[\hat{\mathbf{E}}_{i}\hat{\mathbf{E}}_{i}^{H}\right]^{-1}{\mathbf{e}}_{i}}{{\mathbf{e}}_{i}^{H}\left[\hat{\mathbf{E}}_{i}\hat{\mathbf{E}}_{i}^{H}\right]^{-1}{\mathbf{e}}_{i}}.$
(12)

However, note that specific to our application, even for a $200\times 200$
planar array, since the position vectors are of dimensions $40000\times 1$,
the dimension of the matrix $\hat{\mathbf{E}}_{i}\hat{\mathbf{E}}_{i}^{H}$ is
$40000\times 40000$, and hence ill-conditioned. To circumvent ill-
conditioning, one can apply the singular value decomposition (SVD) of
$\hat{\mathbf{E}}_{i}$ to obtain the inverse in the numerator of (12). Let
${\mathbf{U}}$ and ${\mathbf{D}}$ be the left singular value matrix and
diagonal matrix of singular values, respectively. Since there are $N_{I}+1$
users, there will be $N_{I}+1$ non-zero singular values. If
${\mathbf{D}}_{N_{I}+1}$ and ${\mathbf{U}}_{N_{I}+1}$ are the diagonal matrix
of non-zero singular values and the corresponding left singular value matrix,
then we can compute $\tilde{{\mathbf{e}}}_{i}$ in (12) as

$\displaystyle\tilde{{\mathbf{e}}}_{i}=\frac{{\mathbf{U}}_{N_{I}+1}{\mathbf{D}}_{N_{I}+1}^{-2}\hat{{\mathbf{e}}_{i}}}{\hat{{\mathbf{e}}_{i}}^{H}{\mathbf{D}}_{N_{I}+1}^{-2}\hat{{\mathbf{e}}_{i}}^{H}},$
(13)

where, $\hat{{\mathbf{e}}_{i}}={\mathbf{U}}_{N_{T}+1}^{H}{\mathbf{e}}_{i}$.
Note that, the MPDRB relies on all the user locations in the minimization
criterion, including the actual desired receiver location. Similar to the case
of NSB, additional derivative constraints can be added to the MDPRB
formulation to make it robust. However, evaluating the solution in this case
will be time-consuming using popular software such as MATLAB, owing to the
$M\times M$ dimensional $\hat{\mathbf{E}}_{i}\hat{\mathbf{E}}_{i}^{H}$.

The system’s efficacy under the different beamforming schemes is determined by
a pair of popular metrics, namely the ASE and the SE, which are functions of
various system parameters, like the Rician factor $K$, the array dimensions
$M\times M$, or the micro-cell radius, etc. In the next section, we derive the
approximate expressions of the performance metrics, followed by extensive
simulation results for characterizing the system.

##  III Theoretical approximations for ASE/SE

It is essential for us to characterize the SINR and then derive theoretical
expressions for our metrics, such as the ASE and SE. To begin with, let
$\alpha_{i}$ $i=0,1,...,N_{I}$ represent the symbol intended for the $i$th
user. Without loss of generality, let $i=0$ denote the user under
consideration and $i=1,..,N_{I}$ denote the interferers in the other micro-
cells. The symbol received by the user is

$y=\sqrt{P_{r}}{\mathbf{h}}_{0,Ric}^{H}\tilde{{\mathbf{e}}}_{0}\alpha_{0}+\sqrt{P_{r}}\sum_{i=1}^{N_{I}}{\mathbf{h}}_{0,Ric}^{H}\tilde{{\mathbf{e}}}_{i}\alpha_{i}+n,$
(14)

where $n$ represents the complex Gaussian noise having the power of
$\sigma^{2}=kTBN_{F}$, with $k=1.374\times 10^{-23}$ being Boltzmann’s
constant, $T$ the temperature in Kelvins, $B$ the bandwidth, and $N_{F}$ the
noise figure of the receiver. The received power is given by:

$\displaystyle P_{r}=\frac{P_{t}G_{t}G_{r}}{\tilde{\nu}\nu},$ (15)

where $P_{t}$ is the power transmitted from the Airliner/HAP, $G_{t}$ and
$G_{r}$ are the transmitter and receiver antenna gains, while $\tilde{\nu}$
includes the frequency-dependent atmospheric loss also including the back-off
loss of the modulation scheme as well as other transmitter and receiver
losses. Finally, the term $\nu$ represents the path-loss given by [58],

$\displaystyle\nu$ $\displaystyle=20\log\left(\frac{4\pi
d~{}f_{c}}{c}\right)~{}dB=\left(\frac{40\pi d~{}f_{c,GHz}}{3}\right)^{2},$
(16)

where $c=3\times 10^{8}~{}m/s$ is the speed of the light, $d$ is the distance
from the airliner/HAP to the user in meters and $f_{c,GHz}$ is the carrier-
frequency in $GHz$. The Rician fading channel between the Airliner/HAP and the
user is represented by:

$\displaystyle{\mathbf{h}}_{i,Ric}=\sqrt{\frac{K}{1+K}}{{\mathbf{e}}}_{i}+\sqrt{\frac{1}{1+K}}h_{i}{\mathbf{1}},$
(17)

where $K$ is the Rician factor and $h_{i}$ is the NLoS component, while
${\mathbf{1}}$ is the vector of ones. Furthermore, the NLoS component
${\mathbf{h}}_{i}$ is a complex Gaussian random variable (RV) with zero mean
and unit variance. The instantaneous SINR is now given by the following
theorem.

######  Theorem 1.

The instantaneous signal to interference plus noise (SINR) $\gamma_{SINR}$, of
the desired user under NSB is given by 111 Very similar expressions can be
derived for the other two schemes, but omitted here given the page limit. :

$\displaystyle\gamma_{SINR}$
$\displaystyle=\frac{P_{r}|X_{s}|^{2}}{\sum_{i=1}^{N_{I}}P_{r}|X_{i}|^{2}+\sigma^{2}},$
(18)

where $X_{s}\sim{\mathcal{CN}}(\mu,\sigma_{s}^{2})$, with

$\displaystyle\mu=\sqrt{\frac{K}{1+K}}\left(M^{2}-{\mathbf{e}}_{0}^{H}{\mathbf{E}}_{0}\left({\mathbf{E}}_{0}^{H}{\mathbf{E}}_{0}\right)^{-1}{\mathbf{E}}_{0}^{H}{\mathbf{e}}_{0}\right),$
(19)

and

$\displaystyle\sigma_{s}^{2}=\frac{1}{1+K}\left\|{\mathbf{1}}^{H}\left(1-{\mathbf{E}}_{0}\left({\mathbf{E}}_{0}^{H}{\mathbf{E}}_{0}\right)^{-1}{\mathbf{E}}_{0}^{H}\right){\mathbf{e}}_{0}\right\|^{2}.$
(20)

Furthermore, still referring to (18), we have
$X_{i}\sim{\mathcal{CN}}(0,\sigma^{2}_{i})$, where

$\displaystyle\sigma^{2}_{i}=\frac{1}{1+K}\left\|{\mathbf{1}}^{H}\left(1-{\mathbf{E}}_{i}\left({\mathbf{E}}_{i}^{H}{\mathbf{E}}_{i}\right)^{-1}{\mathbf{E}}_{i}^{H}\right){\mathbf{e}}_{i}\right\|^{2}.$
(21)

###### Proof.

For the proof, please see Appendix A. ∎

Assuming that the users in a cell are allocated identical bandwidths, the Area
Spectral Efficiency (ASE) is defined as the sum of the maximum bit
rate/Hz/unit area supported by the cell’s Airliner/HAP [50]:

$ASE=\frac{C}{\pi(D/2)^{2}},$ (22)

where $C$ is the capacity of the intended user in Bps/Hz. Given
$\gamma_{SINR}$, the average channel capacity is formulated as:

$\displaystyle\bar{C}_{SINR}$
$\displaystyle=\mathbb{E}[\text{log}_{2}(1+\gamma_{SINR})]$ (23)
$\displaystyle=\int_{0}^{\infty}\text{log}_{2}(1+\gamma_{SINR})f(\gamma_{SINR})\text{d}\gamma_{SINR},$

where $E[.]$ represents the expectation and $f(\gamma_{SINR})$ denotes the pdf
of $\gamma_{SINR}$. By applying the popular use-and-forget bound of [59], the
approximate average channel capacity is given by

$\displaystyle\bar{C}_{SINR}^{app}$
$\displaystyle=\mathbb{E}[\text{log}_{2}(1+\gamma_{SINR})]$ (24)
$\displaystyle\approx\text{log}_{2}\left(\frac{P_{r}\mathbb{E}[|X_{s}|^{2}]}{\sum_{i=1}^{N_{I}}P_{r}\mathbb{E}[|X_{i}|^{2}]+\sigma^{2}}\right),$

where we have $E[|X_{s}|^{2}]=\sigma_{s}^{2}+\mu^{2}$ and
$E[|X_{i}|^{2}]=\sigma_{i}^{2}$. Thus, the average ASE in $bps/Hz/m^{2}$ is
formulated as:

${ASE}_{SINR}^{app}=\frac{4\bar{C}_{SINR}^{app}}{\pi D^{2}}.$ (25)

Similarly, the average SE of the user is given by

${SE}_{SINR}^{app}=\bar{C}_{SINR}^{app}.$ (26)

Note that the average capacity is a function of $\mu$, $\sigma_{s}^{2}$ and
$\sigma^{2}_{i}$, $i=1,...,N_{I}$, which are parameterized by $\theta_{i}^{z}$
and $\theta_{i}^{a}$ for $i=0,...,N_{I}$, the zenith and azimuth angles of all
the users. Furthermore, the angles themselves are functions of the relative
locations of the desired user and the interferers through (1) and (2),
respectively. Therefore, the total average capacity is obtained by averaging
the expressions over the user locations. Recall that the beamforming vectors
are only dependent on the user positions, but not on the channel-gains. Note
that, the approximations are universal, straightforward and are applicable for
any value of $K$, received power $P_{r}$ and other system parameters.

##  IV Simulation Results

Parameter | Value
---|---
Macro-cell radius $R$ | $5-20~{}km$
Micro-cell radius $r$ |  $50~{}m$, $75~{}m$, $100~{}m$
Vertical airliner/HAP distance $H_{t}$ |  $10~{}km$, $21~{}km$
Carrier frequency $f_{c}$ | $73.5~{}GHz$
Total Bandwidth B | $5~{}GHz$
Reuse factor | 7
Bandwidth per User | $714~{}MHz$
Dimensions of the planar array $M$ |  $200$, $300$, $400$, $500$
Rician factor $K$ | $10,15,30~{}dB$
Back-Off | $10~{}dBm$
Transmitter loss | $1.8~{}dB$
Transmitter antenna gain | $10\log(M^{2})$
Atmospheric and cloud loss | $7.9~{}dB$
Receiver antenna gain | $60.2~{}dB$
Receiver noise figure | $6~{}dB$
Other receiver loss | $1.8~{}dB$
TABLE III: Simulation parameters Figure 2: Average ASE of NSB vs. distance of
MCI from the macro-cell centre for $M=500$ and $H_{t}=10~{}km$. The
theoretical result is based on (25). Figure 3: Average ASE of NSB vs. distance
of MCI from the macro-cell centre for $r=75~{}m$. The theoretical result is
based on (25).

Extensive Monte-Carlo simulations have been carried out for evaluating the ASE
and SE of the system, assuming that the Airliner/HAP is located at $(0,0,0)$.
The desired UE is positioned uniformly in the MCI of radius $r$ with its
centre located at $(x,y,-H_{t})$. For a frequency reuse factor of $7$, the
reuse distance is fixed at approximately $D=4r$. We consider the five tiers of
interfering micro-cells with their centres set to $4kr,\forall k=1,..,5$ from
the MCI222The number of interfering tiers to be chosen is based on an
engineering trade-off that varies with the distance from the macro-cell
centre. For example, for MCI at the centre of macro-cell (directly below the
airliner), considering three tiers of interferers is sufficient. In the case
of the macro-cell edge, 5 tiers of interferers are needed for $r=50~{}m$.. The
interfering UEs are also uniformly placed in the interfering micro-cell of
radius $r$. The interferences imposed by the more distant tiers of micro-cells
and macro-cells are assumed to be negligible. To represent a strong LoS
component, we consider the Rician factors $K$ to be $10$, $15$ and $30$ dB.
All the other transmission parameters were proposed initially in [1] and are
summarized in Table III for completeness.

In Fig. 2, the simulated and approximate ASE of NSB are plotted vs. the
horizontal distance of the MCI from the airliners, for several Rician factors
$K$ and micro-cell radii $r$. Naturally, upon increasing $K$, the ASE/SE
increases. Since the ASE is inversely proportional to the cell-radius, it
decreases upon increasing the micro-cell radius $r$. However, the ASE fails to
reach its maximum, when the MCI is directly below the airliner, namely when
the MCI is at the macro-cell centre. The ASE is a function of both the MCI
distance from the airliners as well as of $K$ and of the micro-cell radius
$r$. For example, for $r=50~{}m$ and $K=30$ dB, an ASE as high as
$1200~{}bps/Hz/km^{2}$ is achieved when the user is at a distance of $5~{}km$
from the macro-cell centre.333The signal and interference powers are
approximately of the order of $10^{6}$ and $10^{-6}$ respectively. At a reuse
distance $D=200~{}m$, the SE and ASE are $38~{}bps/Hz$ and
$1200~{}bps/Hz/km^{2}$ approximately. At the macro-cell centre, the ASE is
reduced to $600~{}bps/Hz/km^{2}$.

When the desired user is at the macro-cell centre, the interferers are
distributed in all the quadrants; hence their azimuth angles are uniformly
distributed in $[0,360]$ degrees. Now, as the desired user moves away from the
macro-cell centre, there are two effects. Firstly, the range of azimuth angles
of the interferers decreases because the first five tiers of interferers we
consider are concentrated in a single quadrant. Secondly, the absolute value
of the zenith angles ($\theta^{z}$) of the interference increases from $0^{0}$
as we move away from the macro-cell centre. Both these effects increase the
correlation between the desired and interfering users’ steering vectors, hence
reducing the power of the null-steering vectors. Therefore, both the signal
and interference powers are reduced as the micro-cell centre moves away from
the macro-cell centre. Near the macro-cell centre the reductions of both the
signal and interference powers become similar and hence the ASE fluctuates.
However, as the micro-cell centre moves further away, the interference power
reduction is more substantial than the signal power reduction and hence the
ASE increases. Further away, the desired power reduction becomes substantial
and hence the ASE decreases. The SE variations vs. the distance can also be
explained using similar reasoning. Furthermore, for higher altitudes of the
airliner, the SINR of the users decreases owing to their higher path-loss.
Therefore, the ASE decreases, as observed in Fig. 3. Note that the ASE
achieved by the massive MIMO scheme of [51] is on the order of
$10~{}bps/Hz/km^{2}$, while our scheme achieves in the order of
$1000~{}bps/Hz/km^{2}$, which is comparable to the ASE achieved by the HetNet
and DenseNet of [52] and [53], respectively. The transmit (Tx) power at each
of the antenna elements needs to be carefully chosen depending on the noise
power, the number of antenna elements, and the power consumption allowed at
the transmitter. For a Tx power of $30~{}dBm$, power consumption of $40~{}kW$
is incurred for $M=200$. Whereas for a Tx power of $5~{}dBm$, power
consumption of only $0.126~{}kW$ is incurred, without any drop in the ASE
444Though a smaller Tx power can be chosen with no drop in the ASE, we have
chosen $5~{}dBm$ to account for unforeseen practical losses.. Even if we
choose $M=500$, the power consumption does not exceed $0.8~{}kW$ for a Tx
power of $5~{}dBm$.

Figure 4: Average ASE of NSB vs. $M$ for $H_{t}=10~{}km$ and $r=50~{}m$. The
theoretical result is based on (25). Figure 5: Average SE of NSB vs. $M$ for
MCI at $2~{}km$ from the macro-cell center. The theoretical result is based on
(26).

The ASE and SE variations of NSB vs. $M$ are portrayed in Fig. 4 and 5. An
array dimension of $500\times 500$ provides the highest ASE. For an antenna
element spacing of $\lambda/2$, where $\lambda=4~{}mm$ is the career
wavelength, the array dimensions will not exceed $1~{}m^{2}$. The increase in
ASE vs. $M$ remains marginal compared to that vs. the Rician factor $K$. For
example, for an increase in $M$ from $200$ to $500$, the ASE improves from
approximately $300$ to $400$ $bps/Hz/km^{2}$ respectively. On the other hand,
for an increase in $K$ from $10$dB to $30$dB, the ASE improves from $300$ to
$550$ $bps/Hz/km^{2}$, respectively. With an increase in micro-cell radius, we
observe a SE reduction in Fig. 5. However, the reduction is only marginal
compared to the ASE reduction seen in Fig.2.

Figure 6: Average ASE vs. distance of MCI from the macro-cell centre for
$M=300$, $r=75~{}m$, $K=30dB$ and $H_{t}=10~{}km$. Figure 7: Average SE vs.
distance of MCI from the macro-cell centre for $M=300$, $r=75~{}m$ and
$H_{t}=10~{}km$.

In Fig. 6, the simulated ASE is plotted vs the horizontal distance of the MCI
from the airliners for various beamforming techniques. We can observe that
NSB-D performs nearly as well as NSB, provided that the MCI is near the
macrocell centre. However, recall that as the distance of the MCI increases,
the correlation between the desired and interfering users’ steering vectors
increases, hence reducing the beamforming vectors’ power. Coupled with the
widening of beam by the derivative constraints, a drastic output SINR, and ASE
reduction, is observed. For example, for an MCI at a distance of $10$ km from
the macrocell centre, the ASE of NSB-D is nearly half of that of NSB. A
similar trend can be observed in SE for different $K$ values, as seen in Fig.
7. On the other hand, MPDRB provides the lowest ASE of the three beamformers.
For example, for an MCI at a distance of $2.5$ km from the macrocell centre,
the ASE of MPDRB is nearly third of that of NSB. However, note that MPDRB is
more robust to variations in the distance of the MCI from the macrocell
centre. The correlation between the desired and interfering users’ steering
vectors does not contribute to a significant ASE reduction because it does not
explicitly create nulls in the interfering users’ directions, unlike NSB and
NSB-D.

###  IV-A Doppler compensation

We note that all the above simulations considered an idealized setting for
which the Doppler spread was neglected. However, in a practical system, it is
necessary to compensate for the Doppler frequency offset (DFO) caused by the
airliner’s radial velocity component with respect to the user location. Since
the velocity reading of the high-speed airliner are perfectly known at the
transmitter, the Doppler shift due to this motion can be pre-compensated [60].
Assuming that the airliner is heading in the $0^{0}$ azimuth-direction, the
radial component of the velocity vector towards the angle pair
$(\theta_{i}^{z},\theta_{i}^{a})$ is

$\displaystyle v_{r}=v_{a}\cos\theta_{i}^{a}\sin\theta_{i}^{z},$ (27)

where $v_{a}$ is the airplane’s speed. Hence, the Doppler-induced frequency
deviation in the $(\theta_{i}^{z},\theta_{i}^{a})$-direction is given by

$\displaystyle
f_{D}=\frac{\left(1-\frac{v_{r}}{c}\right)}{\left(1+\frac{v_{r}}{c}\right)}f_{T},$
(28)

where $c$ is the speed of light and $f_{T}$ is the carrier frequency. To pre-
compensate for the spread in frequency, the airplane can transmit at a
frequency of

$\displaystyle{f}_{T}=\frac{\left(1+\frac{v_{r}}{c}\right)}{\left(1-\frac{v_{r}}{c}\right)}f_{c},$
(29)

so that at the user end, the received signal frequency will be $f_{c}$ after
Doppler shift. Note that for an average airplane speed of $200~{}$m/s and a
high carrier frequency of $73.5GHz$, the Doppler offset is negligible, since
$\left(1\pm\frac{v_{r}}{c}\right)\approx 1$. In some cases, the radial
component of the velocity vector towards the angle pair can be incorrectly
estimated as

$\displaystyle\tilde{v_{r}}=v_{a}\cos\theta_{i}^{a}\sin\theta_{i}^{z}+\Delta
v_{r}v_{a}\cos\theta_{i}^{a}\sin\theta_{i}^{z},$ (30)

where $\Delta v_{r}$ is the offset/error in the estimation with $\Delta
v_{r}=0$ indicating perfect estimation. In such a case, we can observe that
the pre-compensation results in the airplane transmitting at a frequency of

$\displaystyle\hat{f}_{T}=\frac{\left(1+\frac{\tilde{v}_{r}}{c}\right)}{\left(1-\frac{\tilde{v}_{r}}{c}\right)}f_{c},$
(31)

which after the Doppler spread results in

$\displaystyle\hat{f}_{c}=\frac{\left(1-\frac{v_{r}}{c}\right)}{\left(1+\frac{v_{r}}{c}\right)}\frac{\left(1+\frac{\tilde{v}_{r}}{c}\right)}{\left(1-\frac{\tilde{v}_{r}}{c}\right)}f_{c}$
(32)

at the user end. In Table IV, the average ASE is tabulated for the different
beamformers operating with and without incorrect frequency offset estimation.
It can be noted that even for a $50\%$ error in estimating the offset,
corresponding to $\Delta v_{r}=\pm 1$, the degradation in the ASE is moderate.
The degradation is the lowest for NSB-D, because the derivative constraints
increase the beam-width towards the user directions and increase the
robustness to steering vector mismatches.

Beamformer | $\Delta v_{r}=-1$ | $-0.5$ | $0$ | $0.5$ | $1$
---|---|---|---|---|---
NSB | 963 | 969 | 969 | 969 | 965
NSB-D | 907 | 908 | 909 | 908 | 907
MPDRB | 215 | 215 | 217 | 216 | 215
TABLE IV: Average ASE expressed in $bps/Hz/km^{2}$ vs $\Delta v_{r}$ for
$M=200$, $K=30~{}dB$, $r=50$ m, $H_{t}=10$ km and MCI at a distance of $2.5$
km from the macrocell centre.

###  IV-B Position vector mismatches

One of the essential requirements for the proposed technique is the signalling
of the user locations at the airplane. The users can estimate their location
using a global positioning system (GPS) and communicate it to the airplane
with a local base station’s aid. A high precision technique such as real-time
kinematic (RTK) processing or carrier phase differential tracking can be
employed [61]. Alternatively, it is possible to continuously track the user
from the airplane by periodically sending signals to the users. In other
words, the established methods of precise positioning of high-speed vehicles
can be used to get an accurate estimate of the user location [62, 63]. Given
precise estimates of the users’ locations and the airplane’s velocity, sub-
meter position accuracy can be obtained for accurate beamforming [64].

We study the performance of the beamformers for up to $5$ m offset in the
users’ position. Let us assume that the coordinates of the users and the
interferers are incorrectly measured as $(x_{i}+\delta
cos\beta_{i},y_{i}+\delta sin\beta_{i},-H_{t})$ $i=0,1,...,N_{I}$, where
$\delta$ is the magnitude of the offset and $\beta_{i}$ is the random phase.
The average ASE for various values of $\delta$ (in metres) is shown in Table
V. We observe that NSB-D is more robust to position vector mismatches than
NSB. NSB-D also outperforms NSB in the presence of position vector mismatches
due to its broader beamwidth. For example, for an MCI at $3.5$ km, with no
position vector mismatch, i.e., $\delta=0$, the ASE of NSB-D is
$763~{}bps/Hz/km^{2}$, and that of NSB is $1025~{}bps/Hz/km^{2}$. For
$\delta=1$ m, the ASE is nearly halved in the case of NSB, whereas the
degradation is negligible for NSB-D. Although MPDRB is also resistant to
position vector mismatches compared to NSB, the overall ASE obtained still
remains lower.

Beamformer | MCI (km) | $\delta=0$ | $\delta=0.5$ | $\delta=1$ | $\delta=5$
---|---|---|---|---|---
NSB | $1$ | 637 | 528 | 474 | 329
| $2.5$ | 969 | 544 | 480 | 335
| $3.5$ | 1025 | 538 | 478 | 332
NSB-D | $1$ | 580 | 580 | 580 | 550
| $2.5$ | 908 | 908 | 867 | 599
| $3.5$ | 763 | 763 | 746 | 496
MPDRB | $1$ | 221 | 221 | 221 | 220
| $2.5$ | 220 | 220 | 218 | 217
| $3.5$ | 215 | 215 | 214 | 214
TABLE V: Average ASE expressed in $bps/Hz/km^{2}$ vs $\delta$ for $M=200$,
$K=30~{}dB$, $H_{t}=10$ km and $r=50$ m.

Both the theory and simulations indicate substantial ASE gains for our
proposed airplane-aided integrated network. However, our work only represents
the first step towards realizing a high-capacity ISTN; hence it relies on some
idealized simplifying assumptions to be eliminated by future research.

##  V Conclusions and Directions for Future Research

In this treatise, we considered a planar-array aided stand-alone airliner/HAP
in a macro-cell communicating with the terrestrial BS/users. To achieve a high
directional gain and to minimize the interference among the users, we invoked
three beamforming techniques, namely NSB, NSB with derivative constraints, and
MPDRB, for transmission from the airliner/HAP and provided approximate SE and
ASE expressions. We also studied the performance of the system for incorrect
Doppler compensation and position vector mismatches. NSB was the least complex
TPC scheme that provided maximum ASE, in the absence of position vector
mismatches. However, NSB-D was more robust to position vector mismatches and
MPDRB more robust to variations in the distance of the MCI from the macrocell
centre.

We considered a simple system, where the airliner is the network provider for
the terrestrial users. The routing protocols, traffic-offloading and
optimizing the tele-traffic resources in conjunction with the existing
terrestrial and space networks require careful further study. An interesting
future study would quantify the effect of the beam-squint encountered by the
proposed schemes [65, 66, 67]. A possible solution to overcome this problem
could be to partition the entire band into multiple narrower bands and apply a
phase shift based beamformer separately. Furthermore, the hybrid beamforming
techniques discussed in [67, 68, 69, 70, 71, 72, 73] could be adopted for
overcoming the beam-squint effects and for reducing the power consumption.
Another critical topic that can be explored is the impact of mobility and real
flight schedule on the handovers between the different macro-cells and the
existing networks. Finally, the optimal radius of micro-cells used at
different distances from the macro-cell centre is another exciting future
research direction.

##  Appendix A Proof for Theorem 1

The array gain of the desired user is
$X_{s}={\mathbf{h}}_{0,Ric}^{H}\tilde{{\mathbf{e}}}_{0}$ and that of the $i$th
interferer is $X_{i}={\mathbf{h}}_{0,Ric}^{H}\tilde{{\mathbf{e}}}_{i}$. In
order to determine the distribution of the SINR, we have to determine the
distribution of the components in the numerator and the denominator. Upon
expanding $X_{s}$, we arrive at,

$\displaystyle X_{s}$
$\displaystyle={\mathbf{h}}_{0,Ric}^{H}\tilde{{\mathbf{e}}}_{0}$
$\displaystyle=\sqrt{\frac{K}{1+K}}\left({\mathbf{e}}_{0}^{H}{\mathbf{e}}_{0}-{\mathbf{e}}_{0}^{H}{\mathbf{E}}_{0}\left({\mathbf{E}}_{0}^{H}{\mathbf{E}}_{0}\right)^{-1}{\mathbf{E}}_{0}^{H}{\mathbf{e}}_{0}\right)$
$\displaystyle\>+\sqrt{\frac{1}{1+K}}h_{0}{\mathbf{1}}^{H}\left(1-{\mathbf{E}}_{0}\left({\mathbf{E}}_{0}^{H}{\mathbf{E}}_{0}\right)^{-1}{\mathbf{E}}_{0}^{H}\right){\mathbf{e}}_{0}.$
(33)

Observe that by construction, we have:

${\mathbf{e}}_{j}^{H}{\mathbf{e}}_{i}=\begin{cases}M^{2};\qquad\qquad\quad
i=j\\\
\frac{\sin\left(\frac{M}{2}\left(\psi_{j}^{x}-\psi_{i}^{x}\right)\right)}{\left(\frac{1}{2}\sin\left(\psi_{j}^{x}-\psi_{i}^{x}\right)\right)}\frac{\sin\left(\frac{M}{2}\left(\psi_{j}^{y}-\psi_{i}^{y}\right)\right)}{\left(\frac{1}{2}\sin\left(\psi_{j}^{y}-\psi_{i}^{y}\right)\right)};\>i\neq
j,\end{cases}$ (34)

where $\psi_{x}$ and $\psi_{y}$ are defined in (3) and (4), respectively.
Since, $h$ is a zero-mean Gaussian RV with unit variance, it can be readily
seen that $X_{s}$ is a Gaussian RV with a mean given by (19) and variance by
(20). Now the $i$th interference component is formulated as:

$\displaystyle X_{i}$
$\displaystyle={\mathbf{h}}_{0,Ric}^{H}\tilde{{\mathbf{e}}}_{i},$
$\displaystyle=\left(\sqrt{\frac{K}{1+K}}{\mathbf{e}}_{0}^{H}+\sqrt{\frac{1}{1+K}}h_{0}{\mathbf{1}}^{H}\right)\tilde{{\mathbf{e}}}_{i}$
$\displaystyle=\left(\sqrt{\frac{K}{1+K}}{\mathbf{e}}_{0}^{H}+\sqrt{\frac{1}{1+K}}h_{0}{\mathbf{1}}^{H}\right)$
$\displaystyle\quad\times\left(1-{\mathbf{E}}_{i}\left({\mathbf{E}}_{i}^{H}{\mathbf{E}}_{i}\right)^{-1}{\mathbf{E}}_{i}^{H}\right){\mathbf{e}}_{i},$
(35)

where the null-steering beamforming vectors $\tilde{e}_{j}$ are designed in
such a way that ${\mathbf{e}}_{i}^{H}\tilde{{\mathbf{e}}}_{j}\approx 0$,
$\forall~{}i~{}\neq~{}j$. Hence, the RV $X_{i}$ has a negligible mean
component. On the other hand, $X_{i}$ has a variance given by (21).

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