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The paper studies the joint impact of phase noise (PN) and co-channel interference (CCI) in indoor terahertz (THz) uplink. We formulate the theoretical framework that quantifies the impact of PN on the transceiver antenna directivity by extracting exact closed-form and low-complexity tight approximations for the expected gains. Additionally, by employing stochastic geometry, we model the propagation environment of indoor THz wireless systems and provide the analytical characterization of the CCI in the presence of PN, in terms of its expected value. The analysis is verified through computer simulations that reveal the accuracy of the presented theory with moderate numbers of users. The paper provides readily available tools for analyzing and designing indoor THz networks.
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Stochastic Analysis of Indoor THz Uplink with
Co-channel Interference and Phase Noise
Joonas Kokkoniemi, Alexandros-Apostolos A. Boulogeorgos, Mubarak Umar Aminu,
Janne Lehtom¨
aki, Angeliki Alexiou, and Markku Juntti
Centre for Wireless Communications (CWC), University of Oulu, P.O. Box 4500, 90014 Oulu, Finland
Email: joonas.kokkoniemi@oulu.fi
Department of Digital Systems, University of Piraeus, Piraeus 18534, Greece
Email: al.boulogeorgos@ieee.org
Abstract—The paper studies the joint impact of phase noise
(PN) and co-channel interference (CCI) in indoor terahertz (THz)
uplink. We formulate the theoretical framework that quantifies
the impact of PN on the transceiver antenna directivity by ex-
tracting exact closed-form and low-complexity tight approxima-
tions for the expected gains. Additionally, by employing stochastic
geometry, we model the propagation environment of indoor THz
wireless systems and provide the analytical characterization of
the CCI in the presence of PN, in terms of its expected value.
The analysis is verified through computer simulations that reveal
the accuracy of the presented theory with moderate numbers of
users. The paper provides readily available tools for analyzing
and designing indoor THz networks.
I. INTRODUCTION
The spectrum scarcity of the low radio frequency (RF)
band has aspired the investigation of higher spectrum bands
in millimeter wave (mmWave) and terahertz (THz) regimes
for the design of innovative beyond the fifth generation (B5G)
wireless systems and applications [1]–[3]. The potential use
cases range from short distance massive connectivity and
ultra high-speed data kiosks to long distance backhaul links.
Communications in these bands can offer an unprecedented
increase in the bandwidth and support ultra high data rates,
but they suffer from severe path attenuation [4], which can
significantly limit the transmission range of the system. More-
over, the hardware imperfections of transceivers [5] can also
constrain their reliability.
To counterbalance the large path loss even at short distances
and to support larger transmission distances required in several
B5G applications, such as backhauling, directional antennas
have to be employed [6]–[8]. A single-input and single-output
wireless communication system operating at 237.5 GHz for
transmitting data over 20 m at a data rate of 100 Gbit/s
with antenna gains equal to 50 dBi was reported in [6].
Furthermore, Kallfass et al. [7] implemented an 850 m link at
240 GHz, in which 55 dBi antennas were deployed. Finally,
the feasibility of mmWave and THz high-directive wireless
systems for railways was theoretically proven in [8].
From the RF design point of view, the direct conversion
architecture (DCA) has received considerable attention in
high frequency/data-rate systems due to its low-complexity
and cost-efficient configuration [6], [9], [10]. However, DCA
deployments are typically sensitive to front-end impairments,
which are often inevitable due to component mismatches and
manufacturing defects [11]–[13]. An indicative example is the
phase noise (PN), which is caused by local oscillator (LO)
imperfections and results in considerable performance degra-
dation (see, e.g., [14] and references therein). Consequently,
several published works have verified and studied the impact
of PN in mmWave and THz wireless systems [15], [16]. For
instance, Sum et al. [15] investigated the error and throughput
performance of multi-Gb/s mmWave wireless personal area
networks that experience multipath fading, whereas, in [16],
its impact on channel estimation and carrier recovery was
highlighted.
From the network point of view, there are several studies
that employ stochastic geometry to model the random node
location and to extract the interference profile [17]–[27]. Most
of these works assumed that the network was deployed in an
infinite space. There are only two works [26], [27] that con-
sidered finite network size. In more detail, in [26], the authors
employed stochastic geometry to model the interference levels
in indoor visible light communications and in [27] it was used
for interference characterization in outdoor mmWave systems.
Despite the paramount importance of interference and PN,
all the contributions in this area, including the mentioned ones,
neglected its impact on the main lobe antenna gain as well
as its effect on the co-channel interference (CCI), which can
be caused by neighbor transceivers. Motivated by the above,
this paper focuses on evaluating the joint impact of PN and
CCI in indoor THz wireless systems. In this direction, we
first present a model for the stochastic main antenna lobe
gain reduction due to PN, and extract an exact closed-from
expression and tight low-complexity approximation for the
antenna gain expected value. Building upon this model and
by employing a stochastic geometry approach, we characterize
the CCI in terms of its expected value. It is worth noting
that stochastic geometry is a powerful mathematical tool that
can be used to replace high-complexity simulations in infinite
spherically or circularly symmetric networks [23]–[25]. In
this work, we consider indoor scenarios, i.e., spatially limited
spaces, with dimensions considerably larger than the trans-
mission wavelength, and we verify through respective Monte
Carlo simulations the accuracy of the proposed approach
and the derived expressions. Finally, our results highlight the
importance of taking into account both the impact of PN and
CCI, when analyzing and designing indoor THz systems.
The rest of this paper is organized as follows. Section II
presents the system model. Section III delivers the stochastic
phase noise model as well as the closed-form expressions for
the evaluation of the expected value of the antenna gains.
Section IV covers the stochastic geometry model for inter-
ference characterization in indoor locations. Finally, Section V
provides numerical results that verify the presented theoretical
framework and insightful discussions concerning the perfor-
mance of indoor THz systems and closing remarks are given
in Section VI.
II. SY ST EM MO DE L
Illustrated in Fig. 1, we consider an indoor uplink of a
THz network that consists of a single receiver (Rx), i.e., the
THz access point (TAP), and multiple transmitters (Txs). The
network is assumed to be deployed within a room, which
is modeled as a three dimensional rectangular space of size
A×B×Cm3. The Rx resides at certain coordinates (a, b, c)
in the Cartesian space limited by the size of the room. The
interfering Txs are randomly distributed around the room.
Besides the random interfering Txs, we have a desired Tx,
which determines the signal-to-interference-plus-noise ratio
(SINR) levels. It is assumed that all the transmit beams of
Txs are perfectly pointed towards the TAP; as a consequence,
the Rx experiences interference from all the nodes. However,
the receive beam of the TAP is pointed towards the desired
Tx, which decreases the interference level from the random
interfering Txs due to the Rx sees the interference mostly
through the side lobes because of highly directional antennas.
This makes the system uplink as in the downlink direction the
Rx would experience interference from the side lobes of the
other Tx. Thus, also the interference level is expected to be
smaller due to both sides see the interference through the side
lobes in contrast to the uplink where in average the Rx sees
the Txs’ main lobes through the side lobes. This case will
be studied in the future work. Moreover, we assume that the
ALOHA transmission scheme is employed for simplicity of
analysis. The ALOHA assumption mainly contributes to the
fact that the Txs are sending randomly on the same channel
without any specific resource allocation.
We assume that all the transceivers are equipped with uni-
form linear array (ULA) antennas that consist of Nidentical
antenna elements equally spaced by distance d. Hence, in
the absence of PN, the complex array factor (AF ) can be
obtained as
AF (α) = β(Γ)a(α) = 1
N
N1
X
n=0
ej2π
λdn sin(Γ)ej2π
λdn sin(α),
(1)
where β(Γ) is the beamformer, Γis the beamforming direc-
tion, a(α)is the antenna array response, αis the angle of
observation, n∈ {0,1, . . . , N 1}is the antenna index, λ
is the wavelength, and dis the antenna element spacing. The
Fig. 1. The indoor system model illustration, where the Rx is assumed to
be in the upper corner of the room in order to have a maximum visibility to
the room.
array power gain is then given by
G(α) = |AF (α)|2.(2)
The maximum gain of a ULA antenna is equal to the number
of the antenna elements in the antenna array, i.e., G(Γ) = NTx.
It is assumed that THz transceivers experience PN, which
influences the AF as
AFp(α) = AF (α)γm
p,(3)
where γm
pis the complex PN of the mth RF chain and is
modeled as
γn
p= exp(m
k),(4)
with θm
kbeing the PN angle of RF chain m. The phase
noise is random and unique for each Rx chain, and, thus,
γm
pcorrupts each RF chain mindependently in (1). In the
numerical results, we assume that the number of RF chains
is equal to the number of antenna elements. This represents
a full digital beamformer. In the THz frequencies cheaper
choice would be a hybrid structure where a single RF chain
controls number of analogue phase shifters. For the sake of the
analysis, full digital beamformer is considered in this paper
and a comparison of the digital and hybrid beamforming is
provided in the future work. The LOs in the system can
either be phased-locked or frequency-locked. When the LO
is phased-locked, i.e., phased-locked loop (PLL) is employed
in the system, the PN causes a small mismatch and is normally
well modeled by a Gaussian distribution. In case the system is
frequency-locked, the LO in the system is tuned to the carrier
frequency but it is free-running. The PN in this case is modeled
as a Wiener process [28], i.e.,
φi=φk1+wk,(5)
where wkis Gaussian random variable. Moreover, by assum-
ing that the memory length of the Wiener process is M, the
experienced PN can be expressed as
θk=
k1
X
i=kM
φi+wk.(6)
Angle [degr]
Antenna Gain [dBi]
No phase noise
p
2 = 0.01 rad2
p
2 = 0.25 rad2
Fig. 2. Illustration of an antenna gain of the ULA model with 128 antenna
elements with and and without the phase noise.
When the phase noise is assumed to be zero mean Gaussian,
this can be written as
θk∼ N(0, M σ2
p)(7)
due to effective sum of multiple Gaussian distributions. Note
that in (7) σpstands for the PN standard deviation.
From (3), it is evident that the PN decreases the main
antenna lobe gain by effectively distributing the transmitted
energy to random directions and mostly to the side lobes
of the antenna. This occurs through the PN modulating the
beamformer β(Γ) (see eq. (1)). Should the PN be absolutely
random, the antenna gain would start to resemble to an
omnidirectional antenna. This indicates that the distribution of
PN is crucial to the depth of damage it does to the beamformer.
The higher the PN standard deviation becomes, the more it
impacts on the antenna phases. However, in realistic cases,
the PN is expected to be small, but can be increased by the
Wiener-type probability process with memory. An indicative
example of the impact of PN in the antenna gain is provided
in Fig. 2, where an antenna pattern of 128 antenna elements
is considered. From this figure, we can observe that the most
evident impact of the PN is on the side lobes. The impact of
the phase noise on the main lobe is derived and discussed in
the next section.
Finally, we utilize a line of sight (LOS) path loss model
that takes into account the free space path loss (FSPL) and
molecular absorption loss [29]. The LOS channel without
multipath components is a valid choice in the THz band with
high gain antennas because of the highly directional antennas
with perfect beam alignment are not very efficiently radiating
in NLOS directions. Similarly, the Rx does not receive much
power from the NLOS directions where the Tx sends very little
power in the first place. The LOS path gain of the channel can
be obtained as
l(r, f) = c2exp(κa(f)r)
(4πrf )2,(8)
where κa(f)is the absorption coefficient at frequency f,ris
the distance from Tx to Rx, and cis the speed of light. The
absorption coefficient is calculated with the help of databases
[30], and as presented, e.g., in [29].
III. EXP EC TE D MAIN LOB E GAIN
The stochastic impact of the PN can be described by a
mapping from the angular distribution into a unit circle. This
is because the real part of the complex PN γpdescribes the
depth of the power degradation (or amplification). This is a
consequence of the real part giving all the information of the
power fluctuations due to the fact that it is directly linked to the
imaginary part by the Kramers–Kronig relation. An example
would be that if θkis zero, γpis one. If θkis fully random
(0 to 2π), γpis zero mean. Therefore, we can obtain the PN
impact on the main lobe gain by calculating the expected value
of the real axis of unit circle by using the PDF of the PN
E[pGpn] = pNTx
π
Z
π
cos(x)
q2πσ2
p
ex2
2σ2
pdx, (9)
where E[pGpn]is the expected amplitude antenna gain, Ntx
is the number of antennas and the maximum power gain of
an ideal ULA, and cos(x)maps the angles xon the real axis
of the unit circle. Solving this yields
E[pGpn] =
pNTx
eσ2
p
2
2 erf π2
p
2σp!+erf π+2
p
2σp!! (10)
where erf(x)is the error function. From this we can calculate
that the expected antenna gain is very well approximated by
E[Gpn] = NTx eσ2
p(11)
for small values of σp. The phase variations need to be small
enough to prevent the random phase from rotating around
the unit circle for this approximation to be accurate. This
approximation will be demonstrated in the numerical results to
give the expected value of the antenna gain. The exact proof,
alternative approximation with error, and more discussion will
be addressed in future work. Other antenna patterns could
be utilized as well with this model. This would be done
by replacing the antenna gain term (NTx herein) with the
appropriate gain term describing the desired antenna pattern’s
main lobe gain.
IV. INTERFERENCE CHARACTERIZATION
The main focus of this section is to model the indoor prop-
agation environment and to characterize the interference of
the indoor THz system. The indoor propagation environment
is confined by walls, which limits user distribution around
the so called typical node of the network. The typical node
is often assumed to be at the origin of an infinite network
experiencing a similar interference as any node surrounded
by random source nodes. In the derivation of the stochastic
model below, we change the integration bounds to take into
account the location of the typical node, or the desired Rx,
in the finite network. Due to the shape of a typical room,
we use Cartesian coordinate system. Note that, in the typical
stochastic geometry models, the space is usually spherically
symmetric about the typical node. This gives a very straight-
forward way to integrate the space just over the radial distance.
We will show in the numerical results by simulations that the
Cartesian coordinate based finite systems can be analysed by
stochastic geometry equally accurately.
The aggregate interference at Rx can be analytically evalu-
ated as [23]–[25]
Iaggr =X
iΦ
l(ri),(12)
where
l(ri) = Z
W
PTx
WEΘ[GTx(Θ)]EΘ[GRx (Θ)]l(ri, f )df. (13)
with Φbeing the set of interfering nodes, PTx representing the
transmit power of the Txs, and Wdenoting the communication
bandwidth. Moreover, EΘ[GTx(Θ)] and EΘ[GRx(Θ)] are the
expected antenna gains of the Txs and the Rx, and Θis
the direction of the antenna in three dimensional space. The
expected antenna gains in the context of this work are the max-
imum transmit powers of all Txs (desired and interference),
i.e., the main lobe gains with possible phase noise impacts
included. The expected antenna gain for Rx is the maximum
gain towards the desired Tx, and random with respect to the
interfering Txs due to the fact that the Rx is pointed at the
desired Tx.
The moments of the interference can be calculated from the
Laplace transform of the aggregate interference [17], [18]
LIaggr (s) = EΦ"exp sX
iΦ
l(ri)!#,(14)
which can be calculated as
LIaggr (s) = exp
2πpλ Z
R
r(1 exp(sl(r)))dr
,(15)
where λis the density of the nodes and pis the probability
of a node to transmit. Notice that Rin (15) refers to the three
dimensional Cartesian coordinate space in this paper. Thus,
r=px2+y2+z2.(16)
This allows us to confine the integration into a room sized
A×B×Cfor x,y, and zaxes, respectively. The nth raw
moment of the aggregate interference can be obtained from
the nth derivative of the Laplace transform as [20]
E[In] = (1)ndn
dsnLIaggr (s)s=0,(17)
where
LIaggr (s) = exp(L(s)).(18)
The term L(s)in (18) is
L(s) =
2πpkλ
Aa
Z
0a
Bb
Z
0b
Cc
Z
0c
r(1 exp(sl(r)))dxdydz. (19)
Before performing the integration, the typical node is moved
to the origin in order to simplify the calculation of the
distances. As a consequence, the coordinate system itself is
also moved with respect to the typical node and hence the
above integration bounds.
The expected interference level can be obtained as in (17)
and (11) under the assumption of all the Tx antennas pointing
towards the access point, i.e.
EΘ[GTx(Θ)] = NT x eσ2
Tx/Rx .(20)
Then, the expected interference level can be evaluated as
E[Iaggr] = c2
8πpλNTxeσ2
Tx
×
Aa
Z
0a
Bb
Z
0b
Cc
Z
0c
r1Z
W
PTx
W f 2exp(κa(f)r)df dxdydz,
(21)
where Ntx is the number of Tx elements of the interfering
Txs, and σ2
Tx is the corresponding Tx PN variance. Notice
that because of the Rx being pointed at the desired Tx, the
random interfering Txs experience random Rx antenna gain.
Therefore, the expected antenna gain is EΘ[GRx(Θ)] = 1 due
to the preservation of the transmit energy and is not visible in
the above equation. This behavior is validated by a simulation
model where the stochastic model is utilized with unit receiver
gain and the simulation model is run with actual antenna gain
and random interfering Tx locations.
V. NUMERICAL EX AM PL ES
This section is focused on demonstrating the joint impact of
PN and CCI in indoor THz systems and validating the theoret-
ical framework with Monte Carlo computer simulations. The
simulations were performed by dropping a Poisson distributed
number of users with mean Nuin random locations in a three-
dimensional rectangular volume, which is limited in x,y, and
zaxes by A,B, and C, respectively. The stochastic geometry
itself has been proven to be exact in the previous works [23]–
[25]. The main purpose of the simulations is to check the
validity of the antenna gain model herein. We assume that
the random interfering Txs point at the Rx at random angles
determined by their locations per simulation. The user transmit
beams are all perfectly pointed at the Rx, but the AP receive
beam is pointed toward the desired Tx.
For the sake of simplicity and without loss of generality,
the PN variance is assumed to be the same for all the
transceivers as is the memory length of the Wiener processes.
The following insightful scenario is examined. The center
frequency is set to 300 GHz and the Tx powers at equal 0 dBm
for all nodes. Morever, the number of antenna elements for the
Rx and all the Txs is 128, the PN variance per unit memory
length is set to 0.017 rad2, and the memory length of the
Wiener process is assumed to vary from 1 to 80. The Rx noise
figure is 10 dB, the communication bandwidth is 5 GHz, and
the probability of transmission is 50%. Furthermore, the room
is assumed to be a typical small room sized 400×600×240
0 0.2 0.4 0.6 0.8 1 1.2
Std of the Wiener process [rad]
0
20
40
60
80
100
120
140
Linear antenna gain
Simulated 128 ant. element gain
Theoretical 128 ant. element gain
Simulated 32 ant. element gain
Theoretical 32 ant. element gain
Fig. 3. Simulated and theoretical antenna gains as a function of the phase
noise standard deviation.
0 0.2 0.4 0.6 0.8 1 1.2
Total PN standard deviation [rad]
-75
-70
-65
-60
-55
-50
-45
-40
-35
Signal power [dBm]
Desired signal, theo
Desired signal, sim
Interference, N=20, theo
Interference, N=20, sim
Interference, N=16, theo
Interference, N=16, sim
Interference, N=12, theo
Interference, N=12, sim
Interference, N=8, theo
Interference, N=8, sim
Interference, N=4, theo
Interference, N=4, sim
Noise floor
Fig. 4. Simulated and theoretical received powers for the interfering links
and the desired link, as well as the noise floor as a function of the phase noise
standard deviation.
cm3(A×B×C). The desired user’s Tx is at 90 cm away from
the Rx, and the number of interfering users is varied from 4
to 20. The desired Rx is located at coordinates (20 cm, 20 cm,
150 cm), i.e., 20 cm away from the walls and at 150 cm height
from the floor. The Monte Carlo simulations were performed
over 10,000 network realizations for all the parameters.
Figure 3 shows the simulated and theoretical antenna gains
as a function of the standard deviation of the PN for different
numbers of antenna elements. We observe that as the PN
standard deviation increases, the expected main lobe antenna
gain decreases for a fixed number of antenna elements. For
instance, for 128 antenna elements, as the PN standard de-
viation shifts from 0.2 to 0.4 rad, an approximately 12.5%
antenna gain degradation occurs. Moreover, in the extreme
case, in which the PN standard deviation changes from 0 to 1.2
rad, the antenna gain degradation equals 72.7%. On the other
hand, for 32 antenna elements, as the PN standard deviation
increases from 0.2 to 0.4 rad, an about 6.67% antenna gain
reduction is observed, while, as the PN standard deviation
shifts from 0 to 1.2 rad, the antenna gain degradation is
approximately equal to 68.5%. This example reveals that the
same PN standard deviation shift causes a more significant
antenna gain degradation as the number of antenna elements
increases. We can also see that the phase noise fluctuates the
No. of interfering Txs
Total PN standard deviation [rad]
8
12
0
16
SINR [dB]
20
0.2
24
0.4 4
0.6 8
0.8 12
1.0 16
1.2 20
Fig. 5. Theoretical SINR as a function of the phase noise standard deviation
and number of users.
antenna gain. This is best shown in the simulated antenna
gains where even the averaged antenna gains fluctuate more
and more as the PN variance increases. This is caused by
the uncertainty the PN introduces to the designed beamformer
phases.
Figure 4 illustrates the impact of the PN to the received
signal powers for different values of interfering nodes. More
precisely, the signal power of the desired signal accompanied
by the aggregated received power of the interferers as a
function of the PN standard deviation are plotted. As a bench-
mark, the noise floor is also depicted. Note that the markers
denote the simulation results, while the continuous lines are
the theoretical ones. We observe that the simulation and
analytical results are identical; hence, the theoretical analysis
is validated. Moreover, it is evident that for a fixed number
of interfering nodes the expected antenna gains decrease as
the PN standard deviation increases. Thus, the received power
also decreases. The Rx has a random antenna gain with respect
to the interfering Txs. This causes slightly less impact of the
phase noise on the interference compared to the desired link
with the main lobe gains at both ends. This has a small impact
on the expected SINR as a function of the PN, which is also
shown in Fig. 5 showing the stochastic SINR as a function
of the PN variance and the number of the interfering users.
The additional loss by PN may drive the SNR/SINR below
the operational region depending on the bit error rate require-
ments. However, in general and for small phase noise values,
the impact of the phase noise on the system performance can
be considered small. The correct interpretation of the phase
noise on the antenna gains is important in order to be able to
model the system and its performance accurately.
VI. CONCLUSION
This paper examined the joint impact of PN and CCI in
indoor THz wireless systems. We provided closed-form ex-
pressions and low-complexity approximations for the expected
transmit and receive antenna gain degradation due to PN.
We also proved that stochastic geometry can be used for
indoor confined finite THz networks and we capitalized this
remark by extracting the analytical framework that evaluates
the expected value of interference. Our results highlighted
the importance of taking into account both the transceiver
imperfections and the interference levels when analyzing and
designing indoor mmWave and THz wireless networks.
The model herein was derived for an uplink system and we
are planning to extend this work to a downlink scenario. We
have also planned to update the antenna models from ULA
into more representative ones used in the real mmWave and
THz communications. In the downlink model, the desired Rx
experiences the interference from the side lobes of the other
TXs pointed at the TAP (acting as a Tx for the desired Rx).
Then the impact of the phase noise is most likely larger,
because the phase noise’s relative gain on the side lobes is
larger than that on the narrow and large gain main lobe. The
main lobe, on the other hand, does experience larger power
loss. That causes gain loss on the desired links, but also causes
the increased side lobe interference. Understanding the sum
impact of those phenomena requires dedicated analysis.
ACK NOW LE DG EM EN T
This work was supported by Horizon 2020, European
Union’s Framework Programme for Research and Innovation,
under grant agreement no. 761794 (TERRANOVA). It was
also supported in part by the Academy of Finland 6Genesis
Flagship under grant no. 318927.
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... In [28], 2D finite mmW systems with sectored antenna patterns are investigated; once again, a uniform distribution is assumed for the steering angles of the interfering beams. The uplink of an indoor wireless system operating at the terahertz band is analyzed in [29], considering a 3D rectangular region. The probe receiver is arbitrarily placed whereas the interfering transmitters are randomly placed within the 3D region. ...
... where ℎ for = {1, . . . , 8} are given in Lemma 2 and the terms , with ∈ {1, 2} are given in (29). ...
... If we consider for instance the downlink with a BS placed at and random nodes, the joint distribution of Theorem 2 would model the transmit angles Θ , Ψ and the distance , whereas the receive angles, Θ , Ψ , would be obtained from the transmit angles after simple trigonometric transformations. Importantly, our results do not make any assumption on the radiation patterns, as opposed to previous works restricted to a sector model, e.g., [28], or assuming perfect beam alignment, e.g., [27,29]. Our results hold for any radiation patterns t (•) and r (•), which can be related either to single-element antennas e.g., horn antennas, [40], or antenna arrays, e.g., uniform planar arrays (UPAs). ...
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Directional beamforming will play a paramount role in 5G and beyond networks in order to combat the higher path losses incurred at millimeter wave bands. Appropriate modeling and analysis of the angles and distances between transmitters and receivers in these networks are thus essential to understand performance and limiting factors. Most existing literature considers either infinite and uniform networks, where nodes are drawn according to a Poisson point process, or finite networks with the reference receiver placed at the origin of a disk. Under either of these assumptions, the distance and azimuth angle between transmitter and receiver are independent, and the angle follows a uniform distribution between $0$ and $2\pi$. Here, we consider a more realistic case of finite networks where the reference node is placed at any arbitrary location. We obtain the joint distribution between the distance and azimuth angle and demonstrate that these random variables do exhibit certain correlation, which depends on the shape of the region and the location of the reference node. To conduct the analysis, we present a general mathematical framework which is specialized to exemplify the case of a rectangular region. We then also derive the statistics for the 3D case where, considering antenna heights, the joint distribution of distance, azimuth and zenith angles is obtained. Finally, we describe some immediate applications of the present work, including the analysis of directional beamforming, the design of analog codebooks and wireless routing algorithms.
... Stochastic geometry has been shown to provide an efficient set of tools for network performance evaluations [5]- [11]. Traditionally, network performance has been studied with simulation models. ...
... The aggregate interference at Rx can be analytically evaluated as [8]- [11] I ...
... if the interfering users are pointed towards the RIS. Given these channel gains and the expression for the aggregate interference in (3), we can derive the aggregated interference levels similarly as we did in past works [8]- [11]. Otherwise the assumptions in modelling are as follows. ...
Preprint
In this paper, we utilize tools from stochastic geometry to estimate the interference propagation via reconfigurable intelligent surface (RIS) in the millimeter wave (mmWave, 30-300 GHz) band and specifically on the D band (110-170 GHz). The RISs have been of great interest lately to maximize the channel gains in non-line-of-sight (NLOS) communication situations. We derive expressions for stochastic interference level in RIS powered systems and validate those with simulations. It will be shown that the interference levels via RIS link are rather small compared to the designed RIS link or the LOS interference as the random interference loses significant part of the RIS gain. We also analyse the validity of far field channel and antenna gains in the near field of a large array. It is shown that, while the high frequency systems require large arrays that push the far field far away from the antenna, the far field equations are very accurate up to about half way of the near field.
... To address the reliability degradation caused by selfblockage and dynamic human blockage in THz communication systems, [11] evaluated the performance achieved by multi-connectivity strategies. In addi-tion, the impacts of APs' heights on the uplink and downlink coverage performance of 3D THz networks were highlighted in [12] and [13], respectively. The authors in [14] derived the coverage probability and ASE of unmanned aerial vehicular (UAV)-aided THz networks under the assumptions of HPPP-distributed UAVs and users. ...
... The cdf and probability density function (pdf) of the distance from the typical receiver to its serving AP are (12) Remark 1. Specifically, for the central receiver (V = 0), since Xd (riO) = -ffi-, (5) and (6) ...
... Specifically, for the central receiver, (11) and (12) reduce to ...
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