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Impact of beam misalignment on THz wireless systems

Joonas Kokkoniemia,∗, Alexandros-Apostolos A. Boulogeorgosb, Mubarak Aminua, Janne Lehtom¨

akia, Angeliki

Alexioub, Markku Junttia

aCentre for Wireless Communications (CWC), University of Oulu, P.O. Box 4500, 90014 Oulu, Finland

bDepartment of Digital Systems, University of Piraeus, Piraeus 18534, Greece

Abstract

This paper focuses on deriving expected values for the transmit (TX) and receive (RX) antenna gains in terahertz

(THz) wireless fronthaul and backhaul links under stochastic beam misalignment, which is created by antenna move-

ment coming from the building or antenna mast swaying. In particular, four diﬀerent antenna movement models are

considered: (i) Gaussian motion of a single antenna; (ii) Gaussian motion of both the TX and RX antennas; (iii)

2-dimensional (2D) Gaussian motion of a single antenna; and (iv) 2D Gaussian motion of the one antenna and one-

dimensional Gaussian motion of the other. Models (i) and (iii) depict fronthaul scenarios, in which the access point is

usually installed in high buildings or in road-sides. Models (ii) and (iv) may model backhaul applications. To verify

our analysis and quantify the impact of beam misalignment, we provide analytic and simulations results, which reveal

that the antenna motion can cause a signiﬁcant degradation on the expected value of the TX and RX antenna gains.

Moreover, we use the derived models to modify the link budget assessment and provide insightful results for a number

of realistic scenarios. These result clearly highlight the impact of beam misalignment in the received signal quality as

well as the importance of taking into account the beam misalignment and using the correct antenna movement model,

when evaluating its impact.

Keywords:

Antenna misalignment, THz backhauling, THz communications, THz fronhauling

1. Introduction

In order to counterbalance the spectrum scarcity

problem, the beyond ﬁfth generation wireless networks

are expected to use pencil-beam high-frequency mil-

limeter wave (mmW) and terahertz (THz) band links

for fronthaul and backhaul applications [1, 2]. The THz

band provides an extremely large amount of license-free

spectrum, but the communication links in this band suf-

fer from high channel attenuation. This forces utiliza-

∗Corresponding author

Email addresses: joonas.kokkoniemi@oulu.fi

(Joonas Kokkoniemi), al.boulogeorgos@ieee.org

(Alexandros-Apostolos A. Boulogeorgos),

mubarak.aminu@oulu.fi (Mubarak Aminu),

janne.lehtomaki@ee.oulu.fi (Janne Lehtom¨

aki),

alexiou@unipi.gr (Angeliki Alexiou),

markku.juntti@oulu.fi (Markku Juntti)

1This project (TERRANOVA) was supported by Horizon 2020,

European Union’s Framework Programme for Research and Innova-

tion, under grant agreement no. 761794. This work was also sup-

ported in part by the Academy of Finland 6Genesis Flagship under

grant no. 318927.

tion of extremely high-gain antennas and thus the em-

ployment of pencil-beamforming approaches2. How-

ever, the pencil-beamforming comes with challenges on

beam acquisition, which, in realistic scenarios, is not

perfect, causing signiﬁcant link quality degradation [4].

As a consequence, beam misalignment is considered

one of the main bottlenecks of THz wireless networks.

To bring a solution to the aforementioned problem,

a great deal of research eﬀort has been put on theoreti-

cally and experimentally analysing the impact of beam

misalignment on mmW and THz wireless systems, as

well as proposing beam tracking approaches [5–9]. The

impact of deterministic beam misalignment on the er-

godic capacity of a mmW ad-hoc network was studied

in [5]. On the other hand, in [6], Priebe et al. evaluated

the impact of beam misalignment through Monte Carlo

simulations in terms of eﬀective antenna gains and the

eﬀective resulting path losses. They assumed that the

2To establish a 850 m link in the 240 GHz band, 50 dBi transmis-

sion (TX) and reception (RX) antenna gains are required [3].

Preprint submitted to Nano Communication Networks April 17, 2020

misalignment angles in the azimuth and elevation with

respect to the main beam direction can be modeled as

uncorrelated Gaussian random processes. In [7], Lee

et al. experimentally assessed the power losses due to

beam steering misalignment. Their results veriﬁed that

even a small beam misalignment causes a notable re-

ceived power degradation. Additionally, in [8], wind

eﬀects on beam misalignment in mmW backhauling se-

tups were investigated. They proposed a computation-

ally eﬃcient beam-alignment technique that samples the

channel subspace adaptively by employing hierarchical

codebooks. Finally, in [9], Boulogeorgos et al. investi-

gated the impact of beam misalignment in THz wireless

backhaul systems assuming that the RX antenna expe-

riences 2-dimensional (2D) Gaussian motion while the

TX is static. To sum up, most of the aforementioned

contributions studied the impact of the beam misalign-

ment in mmW and THz wireless backhauling systems,

assuming diﬀerent antenna motion models.

To the best of the authors’ knowledge, a compara-

tive study between the impact of beam misalignment,

caused by diﬀerent antenna motions has not been ad-

dressed in the technical literature. Aspired by this, in

the present contribution we focus on extracting expected

values for the TX and RX antenna gains in THz wire-

less fronthaul and backhaul links under stochastic beam

misalignment assuming four diﬀerent antenna motion

models. Namely: (i) Gaussian motion of a single an-

tenna; (ii) Gaussian motion of both the TX and RX

antennas; (iii) 2-dimensional (2D) Gaussian motion of

a single antenna; and (iv) 2D Gaussian motion of the

one antenna and one-dimensional Gaussian motion of

the other. Note that models (i) and (iii) accommodates

fronthaul scenarios, in which the access point is usually

installed in high buildings or in road-sides, while (ii)

and (iv) may model backhaul applications. Our anal-

ysis is veriﬁed through respective Monte Carlo simu-

lations that quantify the impact of beam misalignment

and reveals that the antenna motion can cause signiﬁcant

degradation on the expected value of the TX and RX an-

tenna gains. Moreover, we translate the derived antenna

gains models into link-budget (LB) in order to evaluate

the received signal quality. In this direction, we present

insightful results that assess the LB in a number of real-

istic scenarios. Our results clearly highlight the impact

of beam misalignment in the received signal quality as

well as the importance of taking into account the beam

misalignment and using the appropriate antenna motion

model when assessing its impact.

In general, the problem of random motion to antennas

becomes very important problem in the high frequency,

high gain systems that depend on constantly available

high antenna gains. This paper analyses the impact of

the motion statistics on the performance. These aﬀect

communications of all scales. We mainly discuss back-

haul and fronthaul communications. However, it will be

shown that the movement statistics herein have greater

impact on short distances than on longer distances.

The rest of this paper is organized as follows. Sec-

tion 2 gives the path loss model of this paper, 3 goes

into the antenna movement and misalignment models,

and Section 4 derives the expected antenna gains from

the stochastic antenna movement models. In Section 5,

numerical results are provided accompanied by the re-

spective discussion and insightful observations and ﬁ-

nally, Section 6 summarizes and concludes the paper.

2. Path Loss Model

Due to focus on backhaul and fronthaul communica-

tions, we assume the common line-of-sight (LOS) path

loss model to calculate the channel path gain, i.e., the

channel coeﬃcient H(f,r, θ). In the THz band, this is

formed of the free space path loss (FSPL) according to

the Friis transmission equation and the molecular ab-

sorption loss [10]

H( f,r, θ)=c2

(4πr f )2e−κa(f)rGT x (θ)GRx(θ),(1)

where cis the speed of light, ris the distance between

Tx and Rx, fis the frequency, exp(−κa(f)r) is molecu-

lar absorption loss where κa(f) is the frequency depen-

dent molecular absorption loss coeﬃcient, which can

be estimated, e.g., as we have done in [11] or by uti-

lizing the full line-by-line models [12], and GT x(θ) and

GRx (θ) are the Tx and Rx gains, where θis the observa-

tion angle. In the systems where the antennas are per-

fectly pointed at each other, the antenna gains are usu-

ally taken as the maximum gains. We focus on the case

where the antennas are not stationary but moving. Then

the expected antenna gain needs to be evaluated over the

movement statistics. With uncertainty in the antenna

gains and assuming that antenna gains at transmitter

and receiver are uncorrelated (i.e., E[XY]=E[X]E[Y]

where E[·] is the expected value), the expected channel

coeﬃcient can be expressed as

E[H( f,r)] =c2

(4πr f )2e−κa(f)rEΦ[GT x(Θ)]EΦ[GRx (Θ)],

(2)

where EΦ[GT x(Θ)] and EΦ[GRx (Θ)] are the expected Tx

and Rx antenna gains, respectively, over certain antenna

misalignment PDF Φand with the random antenna di-

rections Θdrawn from Φ. The antenna misalignment

2

PDFs are derived below along with how to obtain these

expectations.

We utilize the common Johnson-Nyquist thermal

noise term

Pn=ZW

kBT d f ≈kBT W,(3)

where Wis the bandwidth of the signal, kBis the Boltz-

mann constant, and Tis the temperature. Putting the

channel model together, the signal-to-noise ratio (SNR)

at the receiver becomes

SNR( f,r)=PTxE[H( f,r)]

Pn

,(4)

Where PTx is the transmitted power.

3. Beam Misalignment Models

The antennas are usually assumed to be stationary

and the problem is mostly in acquiring the correct di-

rection to maximize the antenna gain and thereby also

the link quality and performance. However, in prac-

tice, the antennas experience stochastic motion mainly

due to environmental eﬀects such as wind, small earth-

quakes, traﬃc etc., which are translated into beam mis-

alignment. When the gain of the antenna gets higher,

such as in the millimeter wave and THz systems, the

narrow antenna beams become more vulnerable to ran-

dom antenna movement. Motivated by this, we present

stochastic antenna motion models in this section. We

consider four diﬀerent types of antenna movements:

Gaussian, double-Gaussian, Rayleigh, and Gaussian-

Rayleigh type movement. The Gaussian case corre-

sponds to a scenario in which one antenna is stationary

and the other experiences one dimensional (1D) sway-

ing that can be modeled as a Gaussian process. In other

words, we assume that in each time slot, the 1D point-

ing error can be modeled as a random variable (RV)

that follows a Gaussian distribution3. To further elab-

orate this assumption (which is also true for the other

motion statistics herein), the antenna movement is very

low compared to the sampling frequency of the chan-

nel. Therefore, the antenna motion is highly correlated

in time. However, in this paper, we calculate the ex-

pected antenna gains in the presence of antenna motion.

The expected gain is taken as a long term average. This

leads to expected gain appear as if there was no time

3This is a commonly accepted assumption that have been used in

several previously published works (see, for example, [9, 13–19] and

references therein.

correlation. In reality, the antenna gain in the presence

of the antenna motion would vary from very good to

very low. The average expected gain, however, is ob-

tained from the long time average.

The Gaussian motion corresponds to antenna swing-

ing at one end of the link. The double-Gaussian case

refers to a scenario where both the TX and the RX

suﬀer from independent 1D Gaussian swaying. The

Rayleigh case models 2D, i.e., up-down and left-right,

independent swaying or shaking. This would corre-

spond roughly to an earthquake or some other physical

motion shaking the antenna. Finally, the last case corre-

sponds to the scenario in which the other end is swaying

according to Gaussian motion and the other one is shak-

ing according to Rayleigh motion. The considered an-

tenna movement cases are illustrated in Fig. 1. The cor-

responding models for the four motion types and com-

binations analysed in this paper are detailed below.

3.1. Gaussian Movement

The PDF of the Gaussian movement can be ob-

tained as

fg(x)=1

p2πσ2

s

e

x2

2σ2

s,(5)

where xstands for the antenna displacement, and σ2

sis

the spatial jitter variance. Notice that we assume that

the Gaussian movement is perfectly horizontal and the

jitter variance describes the antenna swaying variance.

3.2. Double-Gaussian Movement

Both ends of the link swing in the double-Gaussian

model. Due to the antenna motion independence, this

case can be modelled similarly as in the case of single

side movement, but by applying the Gaussian move-

ment PDFs to the both antennas. Otherwise the PDFs

are the same as in above Gaussian movement.

3.3. Two Dimensional Shaking

In this section, we examine an interesting scenario in

which only the TX antenna experiences 2D shaking. We

assume that the RX antenna has a circular eﬀective area,

A, of radius α. Similarly, the footprint of the TX circular

beam at distance requals ρ. Note that ρbelongs in the

interval 0 ≤ρ≤wr, where wrrepresents the maximum

radius of the beam at distance r. Furthermore, both the

RX antenna eﬀective areas and the TX beam footprint

are considered on the positive x−yplane of a Cartesian

coordinate system and zis the pointing error, which can

be expressed as the radial distance of the transmission

and reception beams. Due to the symmetry of the beam

3

Figure 1: Illustration of the four considered cases antenna movement cases.

shapes, the beam misalignment coeﬃcient, hp, which

returns the fraction of power that can be collected by

the RX antenna, depends only on the radial distance,

i.e., z=|z|.

By considering independent and identical Gaussian

distributions for the elevation and horizontal displace-

ment [20–22], the radial displacement at the RX is

proven to follow a Rayleigh distribution with PDF that

can be obtained as

fz(x)=x

σ2

s

exp x2

2σ2

s!,(6)

where σ2

sis the variance of the pointing errors. Note

that this model was extensively used in several studies

in free space optical systems (see e.g., [23, 24] and ref-

erences therein) as well as in THz wireless systems [22].

Based on [20], hpcan be approximated as

hp(x;r)≈Aoexp

−2x2

w2

eq

,(7)

where weq stands for the equivalent beamwidth, ris the

transmission distance, and Aois the fraction of the col-

lected power at x=0, which can be calculated as

Ao=(erf (v))2,(8)

with

v=√πa

√2wr

.(9)

In (9), ais the radius of the RX eﬀective area and wris

the radius of the TX beam footprint at distance r. More-

over, w2

eq is related to the squared beamwidth at distance

r,w2

r, through

w2

eq =w2

r

√πerf (v)

2vexp −v2,(10)

where erf (·)stand for the error function.

3.4. Gaussian-Rayleigh Movement

In the Gaussian-Rayleigh case, the one end of the link

is swinging according to the Gaussian motion and the

other end is shaking according to the Rayleigh motion.

As in the double Gaussian, due to independence of the

antennas, the motion statistics are applied to the both

ends independently. We could also in the same way

study a double Rayleigh scenario, where both transmis-

sion ends would experience 2D motion. This can be

obtained by utilizing the Rayleigh distribtion indepen-

dently to the Rx and Tx antenna gains. However, this

scenario is seen as the most rare among the cases here

and is not analyzed in this paper. It could be possible

in the case of two handheld devices, but with limited

motion variance. This cases is left for the speciﬁc case

study for some research group that might consider a sce-

nario where this motion would be relevant.

3.5. The Parameters for the Simulations

The two fundamental motion mechanisms considered

in this paper, namely, the PDFs of the Gaussian and

Rayleigh motions are illustrated in Fig. 2. The x-axis in

Fig. 2 shows the physical displacement of the antenna

from the zero-position due to movement with few mo-

tion variances. In this paper, we utilize jitter variances

from 0.01 to 0.2 m2. Those correspond to extreme dis-

placement from about 25 cm to about 1.5 m. The cho-

sen range for the jitter variance is mostly based on illus-

trating antenna movement from low to extreme. These

values, however, ﬁt quite well to movement of the real

structures, such as high buildings, high antenna masts,

an antenna on top of a cruise ship on stormy ocean, or

even a user holding a device. There are large numbers of

possible application where the models presented herein

can be utilized. The focus herein is on the stochastic

modelling of the antenna movement rather than any spe-

ciﬁc scenario. However, the chosen range of jitter vari-

4

-2 -1.5 -1 -0.5 0 0.5 1 1.5 2

Displacement [m]

0

1

2

3

4

5

6

7

PDF

10-4

Gaussian

Rayleigh

Movement

2 = 0.20 m2

2 = 0.01 m2

2 = 0.05 m2

2 = 0.05 m2

2 = 0.01 m2

2 = 0.20 m2

Figure 2: PDFs of the Gaussian and Rayleigh misalignment for jitter

variances of 0.05, 0.1 and 0.2 m2.

ances cover the most of the applications where the an-

tenna movement may be an issue. The maximum vari-

ance of 0.2 m2with a maximum swing range of roughly

±1.5 meters is quite extreme and the most of the real

world applications will experience lower movement.

4. Stochastic Antenna Gain

We assume uniform linear array (ULA) antennas at

both the TX and RX whereby the antenna elements are

placed linearly and separated by equal distance d. The

ULA assumption oﬀers an easy way to compare the im-

pact of antenna movement to the expected antenna gain.

Mathematically, the antenna gain at a certain azimuth

angle of observation αis given as

G(α)=|AF(α)|2,(11)

where AF(α) is the array factor. The array factor pro-

vides the far-ﬁeld radiation pattern of ULA given as

AF(α)=β(Γ)Ha(α)

=1

√N

N−1

X

n=0

ej2π

λdn(sin(α)−sin(Γ)),(12)

where a(α) is the steering vector, and λand d(=λ/2),

where λis the wavelength, are the carrier wavelength

and spacing between the antenna elements, respectively.

β(Γ) is a vector consisting of the beamforming weights

and Γis the desired beam steering direction. Example

responses of such array (Γ = 0) for 32, 256, and 1024

antenna elements are given in Fig. 3. These patterns

are also considered in the numerical results as they give

good range of diﬀerent antenna gains and beamwidths.

-8 -6 -4 -2 0 2 4 6 8

Angle [°]

-20

-10

0

10

20

30

40

Antenna gain [dBi]

1024 antenna elements

128 antenna elements

32 antenna elements

Figure 3: Antenna array factors of the ULA model utilized in this

paper for 32, 256, and 1024 element antenna arrays.

In real life, with λ/2 antenna spacing, these antenna ar-

rays would have sizes ranging from 1.6 cm to 51.2 cm

for 32 to 1024 element arrays, respectively, at 300 GHz

carrier frequency. The linear phased array antenna’s

maximum achievable gain is equivalent to the number

of antenna elements due to constructive summation of

the individual antenna element gains at the beamform-

ing direction (given the antenna element phases are per-

fectly adjusted). In the ideal case, the antenna patterns

of each antenna element can be considered on the av-

erage equal, and thus, the maximum gain is a sum over

the antenna elements. This assumption is very accurate

if the number of antenna elements is large [25]. Fur-

thermore, the ULA antenna elements are assumed to be

placed horizontally with each antenna element being a

vertical dipole element giving omnidirectional pattern

in azimuth angle. The elevation radiation pattern of

such ULA antenna is very wide. Therefore, the mo-

tion statistics described above are in valid for the mo-

tion seen on the azimuth angle from the broadside of

the antenna.

4.1. Stochastic Antenna Misalignment

As mentioned above, the antenna movement causes

the antenna pattern to deviate from the maximum gain

given the antennas at the Tx and Rx are perfectly

pointed toward each other. The expected antenna gain

is calculated as an expectation over the antenna pattern

and the movement statistics. The movement statistics

were discussed in the previous section as well as the as-

sumptions on those. From the expected antenna gain

point of view, the statistics could be provided by any

5

PDF describing the antenna gain pattern and the move-

ment of the antenna(s). The expected antenna gain is

obtained as

EΦ[G(Θ)] =Zθ

Φ(θ)G(θ)dθ, (13)

i.e., as an integral over the antenna pattern and the PDF

Φ(θ) of the antenna movement.

The PDFs for the movement scenarios herein are

Φg(θ)=fg(θ) and Φr(θ)=fr(θ) for the Gaussian and

Rayleigh distributions, respectively. Due to the inde-

pendence of the Rx and Tx antennas, the combinations

of the antenna statistics are calculated separately for the

both ends, Tx and Rx. The distributions f(x) in this pa-

per are for the radial displacement of the antenna gain

due to the motion. We need to map those to the antenna

directions by calculating the transformation x→θ. This

map can be done simply by trigonometry

θ=arctan x

r,(14)

where ris the distance between Tx and Rx, and xis the

displacement. Figure 2 above shows the displacement

for the considered Gaussian and Rayleigh distributions

with 0.01, 0.05 and 0.2 m2jitter variances. The above

equation also gives away the fact that the antenna’s an-

gular displacement will be less severe for the same ra-

dial displacement as the distance is increased. This

is intuitively expected as the same, e.g., lateral move-

ment looks the larger the closer-by the observer is to the

moving object. If, however, one would consider piv-

otal movement, the situation would change. The further

away subject would experience a larger deviation in the

antenna gain. Otherwise, the basic modelling would be

the same: the expected antenna would be an integral

over the pivotal angle and the movement statistic. How-

ever, in this paper and in the numerical results we focus

on the radial displacement type of a movement.

One could visualize the impact of the movement PDF

versus the expected antenna gain with the given antenna

patterns in Fig. 3. Assume that the 1024 element ULA

gives the antenna pattern and the 32 element ULA de-

picts the PDF of the antenna motion. It is obvious that

the maximum gain of the antenna would drop sharply

due to averaging over such large angular range. Turning

the situation other way around, i.e., the 1024 element

ULA depicting the PDF of the movement and the 32

element ULA the antenna gain, the impact of the mo-

tion would be very small. The minor deviation from the

maximum gain still keeps the average gain at feasible

level. This is also shown in the numerical results where

the 32 antenna array with low gain suﬀers considerably

Table 1: Simulation parameters used in this paper

Antenna parameters

Parameter Value(s)

Antenna gains 32, 256, 1024

Antenna gains [dBi] 15, 24, 30

Antenna HPBWs [◦] 3.2, 0.4, 0.1

Antenna movement variance [m2] 0.01 – 0.2

Antenna element spacing [m] λ/2

Total antenna gains [dBi] 30, 48, 60

Other parameters

Parameter Value(s)

Center frequency [GHz] 300

Noise temperature [K] 296

Noise power [dBm] -73.9

Transmission bandwidth [GHz] 10

Transmit power [dBm] 0

less of the motion than the 1024 element array. This is

an expected result, but shows some interesting SNR and

capacity behavior between the diﬀerent antenna gains.

5. Numerical Results

The impact of the antenna misalignment to the ex-

pected antenna gain and performance of the point-to-

point link is analysed in this section. The simulation

parameters used to calculate the numerical results are

listed in Table 1. The center frequency of the trans-

mission was 300 GHz with 10 GHz bandwidth. The

transmit power was 0 dB. The noise power was calcu-

lated for 296 K noise temperature. The antenna conﬁg-

urations were 32, 256, and 1024 antenna element ULAs

that correspond to single side antenna gains of about 15,

24, and 30 dBi, respectively. The half power beamwidth

(HPBW) of a phased array can be estimated in radians

by [26]

α3dB =0.886 λ

NT xd=1.772

NT x

,(15)

where α3dB is the HPBW and the last term is for the uti-

lized antenna element spacing of d=λ/2. This approx-

imation was veriﬁed to be accurate for the phased array

described above and for the antenna conﬁgurations de-

scribed on Table 1. Notice that the approximation of the

HPBW is dependent on the antenna type and the above

one is valid for linear phased arrays. In the simulations,

each of the motion statistics were calculated assuming

the same antenna conﬁguration in both ends (Rx and

Tx).

Figs. 4 to 6 show the expected antenna gains for 32,

256, and 1024 element antenna arrays, respectively, for

6

the four movement cases presented above and as a func-

tion of the movement variance. The illustrated losses

are for the combined Rx and Tx gains. The gains of the

antenna arrays were kept equal at both sides. The sever-

ity of the movement to the total antenna gain strongly

depends on the antenna gain, i.e., how narrow is the

main lobe beam and on the distance between Tx and

Rx. The narrower is the beam, the more the movement

has impact on the gain. As the distance increases, the

relative motion becomes smaller and the impact of the

movement to the antenna gain is also smaller. This is

because of relative motion of an object to an observer

seems smaller as the distance increases. On the other

hand, from the antenna gain point of view, over longer

distances the energy spreads more and the Tx illumi-

nates larger area.

The antenna gain plays an important role in the gain

degradation. We use antenna arrays in the analysis

herein. The antenna arrays are strongly focused towards

the steered direction. We can see in Figs. 5 and 6 that

the relative gain loss is much higher in the case 1024

element array compared to the 256 element array. Over

a 100 meter link, the 1024-element array gives on bar

or lower gain than 256 element array. This is because

the 3-dB beamwidth of the larger array is four times

smaller (0.1 vs. 0.4 degrees). We can see in Fig. 7 how

even very small movement causes a severe impact on

the total antenna gain of the 1024 element array and es-

pecially compared to 256 element array. Especially the

most severe movement, the Gaussian-Rayleigh is very

sensitive to the movement with the gain dropping very

fast for high number of antenna elements.

For the 32 element array, the impact of the move-

ment over long distances is negligible. This follows

the much higher 3-dB beamwidth of 3.2 degrees. In

this case, similarly as for the higher antenna element

cases, the movement causes larger loss for the shorter

distances. Comparing the worst case scenarios, namely

the Gaussian-Rayleigh cases for 20 meters at 0.2 m2

motion variance, the total antenna gain losses for the

link budget are 4, 25, and 40 dBs for the 32, 256, and

1024 element arrays.

In general, it can be concluded that the 3-dB

beamwidth plays an important role in how severe loss

the the antenna movement causes. Very high gain an-

tennas therefore suﬀer more from the movement as it

could be expected. It should be remembered that the

3-dB beamwidth does not only depend on the antenna

gain, but also on the antenna structure and the shape of

the beam. Thus, the possible impact of the movement

to the link budget need to be considered for each appli-

cation and the type of antennas that are utilized therein.

0.01 0.05 0.1 0.15 0.2

Variance of antenna motion [m 2]

25

26

27

28

29

30

31

Expected antenna gain [dB]

No movement

Gaussian, single

side, 20 m

Rayleigh, 2D

movement, 20 m

Gaussian, both

sides, 20 m

Gaussian-Rayleigh,

20 m

Gaussian, single

side, 100 m

Rayleigh, 2D

movement, 100 m

Gaussian, both

sides, 100 m

Gaussian-Rayleigh,

100 m

Figure 4: The expected antenna gain with and without antenna move-

ment for 32-antenna array.

0.01 0.05 0.1 0.15 0.2

Variance of antenna motion [m 2]

20

25

30

35

40

45

50

Expected antenna gain [dB]

No movement

Gaussian, single

side, 20 m

Rayleigh, 2D

movement, 20 m

Gaussian, both

sides, 20 m

Gaussian-Rayleigh,

20 m

Gaussian, single

side, 100 m

Rayleigh, 2D

movement, 100 m

Gaussian, both

sides, 100 m

Gaussian-Rayleigh,

100 m

Figure 5: The expected antenna gain with and without antenna move-

ment for 256-antenna array.

0.01 0.05 0.1 0.15 0.2

Variance of antenna motion [m 2]

10

20

30

40

50

60

70

Expected antenna gain [dB]

No movement

Gaussian, single

side, 20 m

Rayleigh, 2D

movement, 20 m

Gaussian, both

sides, 20 m

Gaussian-Rayleigh,

20 m

Gaussian, single

side, 100 m

Rayleigh, 2D

movement, 100 m

Gaussian, both

sides, 100 m

Gaussian-Rayleigh,

100 m

Figure 6: The expected antenna gain with and without antenna move-

ment for 1024-antenna array.

Figs. 8 to 10 show the above total antenna gains’ im-

pact on the SNR as a function of distance for the move-

ment variance of 0.05 m2and 0.2 m2. As expected from

above, the antenna gain has a major impact on the SNR

and how it behaves. As it was shown above, the severity

7

0 0.5 1 1.5 2 2.5 3 3.5 4

Variance of antenna motion [m 2]10-3

40

45

50

55

60

65

Expected antenna gain [dB]

Gaussian, single side, 20 m

Rayleigh, 2D movement, 20 m

Gaussian, both sides, 20 m

Gaussian-Rayleigh, 20 m

1024 antenna elements

Stationary antennas

256 antenna elements

Figure 7: Comparison of the expected antenna gains for very small

antenna movement between 256-antenna array and the 1024-antenna

array.

of the movement and the 3-dB beamwidth have major

impact on the expected gain of the system. The expected

antenna gain tends to be higher at larger distances and

this is also visible in Figs. 8 to 10. The longer distance

in general suﬀer relatively lower gain loss in the pres-

ence of movement. This in the extreme cases leads to

SNR increasing as a function distance as the antenna

gain increases faster than than the path loss. However,

this is only true for the extreme movement cases with

very large antenna arrays, such as Gaussian-Rayleigh

movement with 256 and 1024 element antenna arrays.

The antenna gains considered here are rather low for

long distance links at 300 GHz. This shows as nega-

tive and low SNRs at long distance links. Especially the

32 element array provided gains that would be adequate

mostly for very short distance link, such as in-room

communications. However, these ﬁgures also show the

vulnerability of the THz frequency link budgets to any

unwanted additional losses. The designed system may

operate well in ideal conditions, but adding, for in-

stance, a storm that shakes the antenna may render the

link unusable. the results herein do not take into ac-

count the coding that would be able to correct burst er-

rors. On the other hand, high bandwidth systems sample

the channel very fast compared to a physical movement.

Therefore, the error bursts can be expected to be very

long compared to channel codes’ ability to ﬁx the errors.

The correlation of the movement and the channel sam-

pling is not considered here. The results herein depict

long term average expected values. Thus, the negative

SNR does not mean that the channel is constantly un-

usable, it is just unusable for the most of the time. The

20 30 40 50 60 70 80 90 100

Distance [m]

-20

-18

-16

-14

-12

-10

-8

-6

-4

SNR [dB]

Stationary antennas

Gaussian, single side,

2=0.05m2

Rayleigh, 2D movement,

2=0.05m2

Gaussian, both sides,

2=0.05m2

Gaussian-Rayleigh,

2=0.05m2

Gaussian, single side,

2=0.2m2

Rayleigh, 2D movement,

2=0.2m2

Gaussian, both sides,

2=0.2m2

Gaussian-Rayleigh,

2=0.2m2

Figure 8: The expected SNR with and without antenna movement for

32 element antenna array.

antenna movement therefore sets application speciﬁc re-

quirements on movement tracking, tracking speed, over-

all antenna gain, and the channel coding. These then ul-

timately set the boundaries for the highest modulation

orders that can be utilized and therefore the achievable

throughput.

We further included Table 2 to show the link bud-

get calculations for the 1024 element antenna array and

with the movement variance of 0.05 m2. This gives a

better insight on the impact of the antenna movement

to the link budget. It can be seen that even with rel-

atively low movement, the SNR decreases quite a bit

when movement is introduced. From the link budget

point of view, the antenna gain of about 30 dBi per

side would be very tight for such link distances, but it

also shows that if there is not much room to move in

the link budget, an external forces can take the system

below operational link quality. A minor note here is

that by choosing the center frequencies correctly, for in-

stance here at 300 GHz, we can neglect the impact of

the molecular absorption loss, which is at most 0.3 dB

at 100 meter link distance.

Finally, Fig. 11 shows the calculated capacities for

the 1024 element antenna array as a function of the dis-

tance and the antenna movement variance. We show the

results for this particular antenna conﬁguration since it

best depicts the peculiar SNR behavior in the presence

of antenna motion as shown above. The capacity is cal-

culated as the familiar Shannon capacity

C=Wlog2(1 +SNR),(16)

where Cis the capacity and Wis the bandwidth. We

can see that the overall capacity is mostly following the

expected trend with decreasing capacity with distance

and movement variance. And exception is made by

8

20 30 40 50 60 70 80 90 100

Distance [m]

-15

-10

-5

0

5

10

15

SNR [dB]

Stationary antennas

Gaussian, single side,

2=0.05m2

Rayleigh, 2D movement,

2=0.05m2

Gaussian, both sides,

2=0.05m2

Gaussian-Rayleigh,

2=0.05m2

Gaussian, single side,

2=0.2m2

Rayleigh, 2D movement,

2=0.2m2

Gaussian, both sides,

2=0.2m2

Gaussian-Rayleigh,

2=0.2m2

Figure 9: The expected SNR with and without antenna movement for

256 element antenna array.

20 30 40 50 60 70 80 90 100

Distance [m]

-20

-15

-10

-5

0

5

10

15

20

25

30

SNR [dB]

Stationary antennas

Gaussian, single side,

2=0.05m2

Rayleigh, 2D movement,

2=0.05m2

Gaussian, both sides,

2=0.05m2

Gaussian-Rayleigh,

2=0.05m2

Gaussian, single side,

2=0.2m2

Rayleigh, 2D movement,

2=0.2m2

Gaussian, both sides,

2=0.2m2

Gaussian-Rayleigh,

2=0.2m2

Figure 10: The expected SNR with and without antenna movement

for 1024 element antenna array.

Table 2: Link budget values for the 1024 element antenna array for

antenna movement variance of 0.05 m2

Parameter 20 meter link 100 meter link

Transmit power 0 dBm 0 dBm

FSPL 108.1 dB 122.3 dB

Absorption loss 0.06 dB 0.3 dB

Noise power -73.9 dBm -73.9 dBm

Expected total antenna gain

Stationary antennas 60.2 dBi 60.2 dBi

SNR 26.0 dB 11.8 dB

Gaussian 48.5 dB 55.1 dB

SNR 14.5 dB 6.7 dB

Rayleigh 38.4 dBi 50.2 dBi

SNR 4.2 dB 1.8 dB

Gaussian-Rayleigh 26.7 dBi 45.1 dBi

SNR -7.5 dB -3.3 dB

the Gaussian-Rayleigh movement that causes so large

gain loss at short distances that the capacity actually

increases as a function of distance. This is somewhat

expected based on the above SNR values since the

Shannon capacity mirrors the SNR performance. This

is of course the extreme case that would require very

large movement and equally interesting results are given

by Rayleigh and Gaussian-Gaussian cases. These two

movement cases ﬂatten the SNR such that the capac-

ity diﬀerence at lowest movement variance here at 0.01

m2was dropping from 35 Gbps to 26 Gbps when mov-

ing from 20 meter to 100 meter link distance for the

Rayleigh movement and from 33 to 26 Gbps for the

Gaussian-Gaussian movement. This is very low drop

compared to for instant single side Gaussian movement

where for the same movement variance leads to capacity

drop from about 59 Gbps to 33 Gbps when moving from

20 meters to 100 meters. This is very interesting behav-

ior, which is attributed partially to the extremely highly

directional antennas used in the paper. However, this is

where the world is going for in the future mmWave and

THz systems and the antenna movement has to be taken

into account in the link budget analysis as well as in the

tracking algorithms.

These examples show that the antenna movement

may degrade the expected antenna gain signiﬁcantly.

This makes the back- or fronthaul link more vulnerable

to increased bit errors, causes need for higher coding

and lower modulation orders, and causes lower overall

capacity. This type of a random motion may require

new tracking algorithms in order to minimize the link

performance hit in those scenarios and cases when ex-

ternal forces causes additional movement. This type of

random motion can be caused by many source such as

in the case where a car is driving on rough surface (gen-

eral movement plus shaking), during windy weather

(antenna swaying), or during an earthquake (shaking).

Herein we considered displacement type movement, but

there is still more work to be done in the future to prop-

erly model the pivotal movement. Also, the antenna mo-

tion tracking during the random motion is an important

problem as it is also the case when, for instance, person

holds a phone. The device’s orientation is constantly

and randomly changing.

6. Conclusion

We studied the stochastic behavior of the antenna

misalignment. Three diﬀerent movement models were

utilized in the analysis. It was shown that the distance

between the antennas is one important factor herein,

since the relative moment of the antenna depends on

9

0

0

20

Variance of antenna motion [m 2]

0.1

(a) Capacity [Gbps] Gaussian

20

40

Distance [m]

60

40

80 0.2

100

60

0

0

Variance of antenna motion [m 2]

20 0.1

(b) Capacity [Gbps] Rayleigh

40

Distance [m]

60

20

80 0.2

100

40

0

0

20

Variance of antenna motion [m 2]

0.1

40

Distance [m]

60

(c) Capacity [Gbps] Gaussian-Gaussian

80

20

0.2

100

40

0

0

Variance of antenna motion [m 2]

0.1

20

(d) Capacity [Gbps] Gaussian-Rayleigh

40

Distance [m]

60

10

80 0.2

100

20

0

10

20

30

40

50

60

Figure 11: The expected capacity of the link as a function of distance and antenna movement variance for 1024 element antenna array.

the distance. The same movement is relatively more se-

vere at short distances than in the longer ones due to the

angular dispersion of the radial displacement of the an-

tenna is less signiﬁcant. Also, the antenna gain plays

an important role. Very wide beamwidths do not ex-

perience the antenna displacement as strongly as very

highly directional antennas.

The results in this paper have many use cases, such as

in the backhaul link performance and link budget esti-

mation, stochastic geometry models, and in other mod-

els where movement of the antennas is present. In the

future work, we will extend the models to cover more

severe movement in the cases where the antenna is held,

e.g., by a user. Also, based on the models herein, it

is easy to extend those to include any type of arbitrary

movement and various realistic antenna patterns for dif-

ferent applications.

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