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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 movement 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.
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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 dierent 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 eort 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 eective antenna gains and the
eective 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
eects on beam misalignment in mmW backhauling se-
tups were investigated. They proposed a computation-
ally ecient 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 dierent antenna motion models.
To the best of the authors’ knowledge, a compara-
tive study between the impact of beam misalignment,
caused by dierent 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 dierent 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 aect
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 coecient 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 )2eκa(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 coecient, 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
coecient can be expressed as
E[H( f,r)] =c2
(4πr f )2eκa(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 eects such as wind, small earth-
quakes, trac 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 dierent 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
suer 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 eective area,
A, of radius α. Similarly, the footprint of the TX circular
beam at distance requals ρ. Note that ρbelongs in the
interval 0 ρwr, where wrrepresents the maximum
radius of the beam at distance r. Furthermore, both the
RX antenna eective areas and the TX beam footprint
are considered on the positive xyplane 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 coecient, 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 eective 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 oers 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
N1
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 dierent 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 xθ. 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 suers 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 dierent 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 suer 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.
Figure 4: The expected antenna gain with and without antenna move-
ment for 32-antenna array.
Figure 5: The expected antenna gain with and without antenna move-
ment for 256-antenna array.
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 suer 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 dierence 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 dierent 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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11
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... Similar to the asymptotic analysis on the outage probability in (40), average BER in the high SNR for the FG-multihop can be derived which has a similar dependence on the average SNR resulting the diversity of the system as ...
Preprint
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Conference Paper
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