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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

N−1

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(jθm

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=φk−1+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=

k−1

X

i=k−M

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

e−x2

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

e−σ2

p

2

2 erf π−jσ2

p

√2σp!+erf π+jσ2

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 e−σ2

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

i∈Φ

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

i∈Φ

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

A−a

Z

0−a

B−b

Z

0−b

C−c

Z

0−c

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 e−σ2

Tx/Rx .(20)

Then, the expected interference level can be evaluated as

E[Iaggr] = c2

8πpλNTxe−σ2

Tx

×

A−a

Z

0−a

B−b

Z

0−b

C−c

Z

0−c

r−1Z

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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