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Fading Modeling in Indoor
THz Wireless Systems
Evangelos N. Papasotiriou, Alexandros-Apostolos A. Boulogeorgos and Angeliki Alexiou
Department of Digital Systems, University of Piraeus, Piraeus 18534, Greece.
E-mails: {vangpapasot, alexiou}@unipi.gr, al.boulogeorgos@ieee.org
Abstract—This contribution aims at experimentally val-
idating the suitability of well known fading distributions
in modeling the channel of indoor THz wireless systems.
In particular the suitability of α–µ, Rice and Nakagami-
m distributions is evaluated by fitting them to empir-
ical channel measurements. The fitting performance is
expressed in terms of the Kolmogorov-Smirnov (KS) and
Kullback-Leibler (KL) divergence tests. The results show
that the α–µand Rice distributions achieve a good fit to the
empirical data, wheras the Nakagami-m distribution fails
to provide an adequate fit in the majority of the examined
THz links.
I. INTRODUCTION
In the past decade the proliferation of wireless devices
has led to ever increasing bandwidth demands [1]. To-
wards this end the terahertz (THz) band, which spans
the range of 0.1–10 THz is envisioned as a promising
enabler [2], [3]. The future wireless technologies will
make use of the vast unallocated spectrum of the THz
band, in order to support novel usage scenarios and
provide unprecented performance excellence in the Tbps
regime.
The high frequencies of the THz band lead to severe
free space and molecular absorption attenuation of the
propagating signal [4]. This makes the channel modeling
of this band a demanding task. In this direction the
main focus regarding THz channel modeling was on de-
scribing the deterministic phenomena of the channel, i.e.
the free space and molecular absorption losses [5], [6].
In [5] and [6] a simplified line-of-sight (LoS) channel
model for the range of 100–450 GHz and 200–450 GHz,
respectively was presented. Meanwhile, the authors
in [7] and [8], [9] performed LoS and non-line-of-
sight (NLoS) wireless measurements at 90–200 GHz and
140–144 GHz, respectively. These works employed the
common deterministic wireless communications pathloss
model and by using their corresponding measurements,
the pathloss exponent and lognormal shadowing param-
eters were extracted.
The majority of the literature regarding the small-
scale fading channel modeling of the THz band has
been quite recent [10]–[15]. In [10], [11] the small-
scale fading of a THz wireless fiber extended link was
modeled by means of the α–µdistribution. In [12] a LoS
multipath THz channel model was introduced, where the
small-scale fading was modeled by means of the Rice
and Nakagami-m distributions. The proposed analytical
channel model was supported by employing experimen-
tal measurements. In [14], [16] a parametric Rice mul-
tipath THz fading channel model was developed, which
was based on a geometrical two dimensional indoor THz
propagation model. In [13] a channel model for THz
wireless systems operating at the range of 240–300 GHz
was developed, where the small-scale fading was mod-
eled as a mixture of Gamma distributions. The validity
of this model was backed up by channel measurements
performed in an anechoic chamber. In [15] the use of
Weibull and Nakagami-m fading distributions to model
the small-scale fading of 140 GHz systems was inves-
tigated and experimentally evaluated by employing LoS
and NLoS channel measurements.
Motivated by this, in this work the small-scale fading
statistics of THz wireless links is investigated, by em-
ploying experimental channel measurements of various
LoS and NLoS links. In more detail, the suitability of the
α–µ, Nakagami-m and Rice distributions to adequately
describe the small-scale fading statistics of THz wire-
less channels is examined. The goodness of fit of the
aforementioned distributions is evaluated in terms of the
Kolmogorov-Smirnov (KS) and Kullback-Leibler (KL)
divergence tests. By observing the results of the KS test,
which is set to the significance level of 5% all the distri-
butions provide an adequate fit to the empirical channel
distributions. However, from the more accurate KL test
it is observed that the α–µand Rice distributions achieve
an adequate fit to the empirical channel distributions of
all the measured links. Meanwhile, from the KL test it is
observed that the Nakagami-m distribution does not fit
quite well the tails of the empirical channel distributions
for the majority of the examined links.
The rest of this paper is organized as follows: Sec-
tion II revisits the fundamentals of α–µ, Rice and
Nakagami-m distributions. The experimental setup ac-
companied by the data preprocessing and fitting ap-
proach are described in Section III. Finally, concluding
remarks are given in Section IV.
II. BACKGROU ND KN OWL ED GE
TABLE I: Parameters of the α–µ, Nakagami-m and Rice distributions of the measured links.
TX d(m)α µ β m ΩNKΩRKLα–µKLRKLNKS–test LOS
0 4.05 4.89 0.67 48.9 2.93 2162.05 4.62 2090.13 0.01271 0.02354 0.03263 3 3
1 6.9 3.01 1.65 41.29 3.43 1651.36 5.53 1594.51 0.00237 0.00052 0.01313 3 3
2 6.22 2.22 0.9 18.58 1.05 337.96 0.45 329.99 0.00122 0.00114 0.00354 3 7
3 12.59 2.21 0.88 32.92 1.03 1062.4 0.34 1044.14 0.00047 0.00056 0.00152 3 3
4 12.82 2.02 0.98 12.74 1 162.03 0.11 161.61 0.00079 0.00079 0.00092 3 7
7 24.36 3.39 0.54 16.01 1.18 231.28 0.97 220.26 0.00224 0.01534 0.04199 3 7
8 24.13 2.32 0.84 13.36 1.05 173.25 0.48 168.7 0.00087 0.00086 0.00347 3 7
9 24.88 2.13 0.92 8.75 1.01 75.68 0.24 74.86 0.00111 0.00112 0.0019 3 7
10 31.33 2.59 0.77 5.57 1.14 29.53 0.79 28.28 0.00161 0.00274 0.02491 3 7
11 31.4 3.23 1.7 7.32 4.05 51.77 6.79 50.27 0.00888 0.01772 0.11281 3 7
12 33.5 2.05 0.98 6.38 1.02 40.51 0.24 40.12 0.00318 0.003 0.00372 3 7
13 41.07 2.74 0.7 8.43 1.13 66.91 0.79 64.06 0.00066 0.00337 0.02393 3 7
14 41.22 2.13 0.92 4.66 1.01 21.43 0.21 21.24 0.00233 0.00294 0.0053 3 7
18 49.09 2.66 0.7 5.26 1.08 26.03 0.6 25.19 0.01854 0.03233 0.06376 3 7
19 54.41 2.12 0.93 4.73 1.02 22.11 0.26 21.86 0.0016 0.00166 0.00321 3 7
20 56.36 3.24 0.65 4.85 1.38 21.69 1.4 20.49 0.007 0.02749 0.17457 3 7
21 58.68 2.1 0.95 4.98 1.02 24.57 0.27 24.28 0.00192 0.00158 0.00414 3 7
25 10.34 3.2 0.83 40.27 1.8 1522.4 2.27 1437.78 0.00207 0.00075 0.01965 3 3
A. The α–µfading distribution
The α–µfading distribution [17] has been exten-
sively employed in modelling the small-scale fading
characteristics of radio frequency (RF) wireless channels.
This distribution offers not only mathematical tractabil-
ity [18], but also encapsulates many important distribu-
tions of the statistical analysis as special cases. From
the α–µthe Rice, Nakagami-m, Gamma, Weibull, expo-
nential, Rayleigh and one-sided Gaussian distributions
can be obtained, by setting the parameters αand µto
the appropriate values [17]. The α–µprobability density
function (PDF) is expressed as [17]
f(x) =
αµµx
βαµ−1exp −µx
βα
βΓ (µ),(1)
where Γ (·)denotes the gamma function [19],
β=α
pE(Xα)and µ=E2(Xα)
V(Xα). The parameter X
represents a random variable following the α–µ
distribution. Furthermore, the signal non-linearity
caused by the propagation environment is expressed by
the parameter α, while the parameter µ > 0stands for
the number of multipath components of the received
signal [17]. The parameter µcan as well acquire
non-integer values. This can be attributed to non-zero
correlation among the in-phase and quadrature parts
of multipath components, or as non-zero correlation
among different clusters of multipath components, or
as non-Gaussianity of the in-phase and quadrature
components of the fading signal [17].
B. The Rice and Nakagami-m fading distributions
The Rice and Nakagami-m distributions are widely
employed in modeling the fading statistics of RF wireless
channels, while they have also been used in modeling
THz channels [12], [14], [15]. The PDF of the Rice
distribution is expressed as [20]
fR(x)=2xK+ 1
ΩR
exp −K−(K+ 1) x2
ΩR
×Io
2xsK(K+ 1)
ΩR
,
(2)
where Io(·)is the zero order modified Bessel of function
of first kind [21]. The parameters Kstands for the
power ratio of the LoS signal component to the other
NLoS signal components, while ΩRdenotes the average
received signal power. The PDF of the Nakagami-m
distribution is obtained as [20]
fm(x) =
2 exp −µx2
ΩN µ
ΩNµ
x−1+2µ
Γ (µ),(3)
where mand ΩNare the fading parameter and the
average received signal power, respectively.
III. EXP ER IM EN TAL SET UP & F ITTING
A. Experimental setup
The multipath THz measurements were conducted in
the premises of the Aalto University in Finland. A top-
view of the building is shown in fig. 1. Each measured
link consists of a single transmitter (TX) and a single
receiver (RX). In order to conduct measurements of
different links the single TX and the single RX antennas
are placed in different positions within the building.
During each measurement both the TX and RX are
static. The different TX indexes indicate the different
positions that the TX was placed. Also, the RX was
placed in three different positions within the building,
which are indicated by a corresponding index. Since the
measurements were conducted within an indoor envi-
ronment; some TX–RX links were in LoS conditions,
whereas others were either blocked by obstacles or the
Fig. 1: Top-view of the measurement environment.
transceivers were placed in different rooms and hence
they were measured in NLoS conditions. Also, in fig. 1,
the fan-shaped indication around the RX is an indicative
azimuth scanning range of the directional antenna.
The specifications of the THz channel sounding sys-
tem utilized in this work can be found in [8], [9]. The RX
and TX were fitted with a horn and a bicone antenna,
respectively. The gain of the RX and TX was 19 dBi
and 0dBi, respectively. In order for the RX antenna to
yield angularly resolved channel impulse responses, it
was rotated with an angular step of 10◦. The TX and RX
antenna heights was set equal to 1.85 m. The operational
frequency was set equal to 142 GHz having a bandwidth
of 4GHz. Finally, the transmit power at the TX was set
equal to 5dBm.
B. Preprocessing of the measurement data
In section III-A the described channel sounding yields
power angular delay profiles (PADP) for each TX–RX
link. The PADPs of a link can be represented by a set
of discrete propagation paths as
PADP (φ, t) =
I
X
i=1
GaPiδ(φ−φi)δ(t−ti),(4)
where φi, tiand Piare the azimuth angle at the RX, the
propagation delay time and gain of the i-th propagation
path, respectively, while 1≤i≤I. The parameter
Ga= 19 dBi, denotes the combined gains of the TX and
RX antennas.
In order to examine the fitting of the α–µ, Rice
and Nakagami-m small-scale fading distributions to THz
wireless channels, the measured gains of the detected
paths of each link will be employed. Hence, from the
propagation gain measurements of a link only their
stochastic behavior is of interest. In this direction, the
deterministic phenomenon of pathgain, i.e. the pathloss
must be omitted. To accomplish this, the link pathgain
measurements for each link are normalized to unity, i.e.
ζ2
i=Pi
PI
i=1 Pi
,(5)
where ζiis a pathloss normalized path amplitude.
C. Generation of different channel realizations
In the THz band the wavelength is significantly
smaller compared to the size of obstacles laid in the
propagation environment. As a consequence the prop-
agating THz electromagnetic (EM) waves are prone to
blockage [4]. Additionally, the THz wireless communi-
cations are severely attenuated by the free space loss and
the molecular absorption loss due to the water vapor [5].
As a result, the aforementioned phenomena lead to a sig-
nificant reduction of the multipath components arriving
to the RX from NLoS directions, which are still carrying
a significant amount of power.
In order to perform small-scale fading statistics there
is a need of multipath richness, which is not the case in
THz wireless communications. To surpass this limitation
different channel realizations of the transfer function can
be generated by changing the phases of the multipath
components [15], [22]. The assumed stochastic phases
follow the uniform distribution U(0,2π)[22]. Accord-
ingly, the channel coefficient of a single-input-single-
output (SISO) system can be obtained as [22]
h=X
i=1
ζiexp (−j2πf ti) exp (jψi),(6)
where ψi∈U(0,2π)stands for the uniformly dis-
tributed random phase of the i–th multipath component.
Furthermore, by assuming that the channel amplitude of
the coefficients does not change dramatically among the
progressing time ti, then the channel can be considered
as flat-fading, i.e. ti= 0 [22].
D. Fitting of the analytical distributions to the empirical
channel gains
The accurate description of the small-scale fading
statistics of THz wireless links is a field of ongoing
research [12]–[14]. Taking this into account, in this
work the suitability of well-known fading distributions
is examined in order to accurately model the fading
statistics of THz wireless links. In more detail, the α–µ,
Nakagami-m and Rice distributions are fitted to the pro-
vided channel gain measurements and the corresponding
parameters of each distribution are calculated. Their
corresponding parameters can be observed in Table I,
where the TX and dcolumns stand for the transmitter
index and the TX–RX separation distance of a link,
respectively. It should be noted that only the links with
the receiver at the position RX1are presented, because
for the positions of RX2and RX3similar parameter
values were obtained for the employed distributions.
The accuracy of the fitting of the analytical distribu-
tions of α–µ, Nakagami-m and Rice to the empirical
channel gain distributions is evaluated in terms of the
Kolmogorov-Smirnov (KS) goodness of fit test [23]
(with a significance level of 5%) and Kullback-Leibler
(KL) divergence test [24]. As shown by the KS–test
column of Table I all the examined distributions achieve
an adequate fit to the empirical ones. For the KL test
the less the value of the metric the better the fit of the
analytical distribution to the empirical one. From Table I
0 5 10 15 20 25 30 35 38
0
0.01
0.02
0.03
0.04
0.05
0.06
(a)
0 2 4 6 8 10 12
0
0.02
0.04
0.06
0.08
0.1
0.12
0.14
0.16
0.18
0.2
0.22
(b)
Fig. 2: Fitting of α–µ, Rice and Nakagami-m distribu-
tions to the empirical PDFs of links 7and 20.
the KL test values of the α–µ, Nakagami-m and Rice
distributions are obtained from the KLα–µ,KLNand
KLRcolumns, respectively.
Based on the KL test values it is observed that the α–µ
distribution provides a better fit than Rice and Nakagami-
m for the links 0,3,7,9,10,11,13,14,18,19, and 20.
Figures 2(a) and 2(b), serve as an illustrative example
of the achieved fitting of the analytical distributions to
the empirical ones for the links 7and 20, respectively.
In more detail, the blue circles stand for the empirical
PDF of the channel gain, whereas the red, green and
orange lines represent the α–µ, Nakagami-m and Rice
analytical distributions, respectively. From figures 2(a)
and 2(b), it is observed that α–µachieves the best fit.
In the meantime the right tail of Rice diverges from the
empirical distribution. Additionally, Nakagami-m shows
the greatest divergence from the empirical PDF.
According to the KL test the Rice distribution achieves
a better fit than α–µand Nakagami-m for the links 1,2,
8,12,21, and 25, respectively. The figures 3(a) and 3(b)
show the better fit to the empirical data achieved by the
Rice distribution in comparison with α–µand Rice for
the links 1and 25, respectively. The α–µdistribution
also achieves a good fit, but in both of these figures
the left tail slightly diverges from the empirical values,
whereas the Nakagami-m shows significant divergence
on both tails in comparison with the empirical distribu-
tion.
The KL test results of Table I show that the Nakagami-
0 10 20 30 40 50 60 70 80 90
0
0.005
0.01
0.015
0.02
0.025
0.03
0.035
0.04
(a)
0 10 20 30 40 50 60 70 80 90
0
0.005
0.01
0.015
0.02
0.025
0.03
(b)
Fig. 3: Fitting of α–µ, Rice and Nakagami-m distribu-
tions to the empirical PDFs of links 1and 25.
0 5 10 15 20 25 30
0
0.01
0.02
0.03
0.04
0.05
0.06
0.07
0.08
0.09
0.1
(a)
0 5 10 15 20
0
0.02
0.04
0.06
0.08
0.1
0.12
0.14
(b)
Fig. 4: Fitting of α–µ, Rice and Nakagami-m distribu-
tions to the empirical PDFs of links 9and 12.
m distribution can also achieve an adequate fit for the
links 2,3,4,8,9,12,14,19, and 21. In this direction,
the figures 4(a) and 4(b) serve as an illustrative exam-
ple of this observation, which shows the good fitting
achieved by the α–µ, Nakagami-m and Rice analytical
distributions to the empirical ones for the links 9and 12,
respectively. Finally, it should be noted that the method
described in section III-C could not be applied for the
links 15,16,17,22, and 23, because those links lack a
number of adequate measured paths. Hence, they are not
presented in Table I.
IV. CONCLUSIONS
In this work the small-scale fading statistics of indoor
THz wireless channels was investigated. In more detail,
the suitability of modeling the small-scale fading by
means of α–µ, Nakagami-m and Rice fading distribu-
tions was evaluated. The goodness of fit was assessed
by the means of the KS and KL tests. The results of the
KS test showed that all the distributions achieve a good
fit to the empirical channel gain distributions. However,
the more accurate KL test showed that the α–µand
Rice distributions accomplish a good fit to the empirical
channel gain distributions of all the examined links. On
the other hand, according to the KL test, for the majority
of the examined links the Nakagami-m failed to achieve
an adequate to the tails of the empirical distributions.
ACK NOW LE DG EM EN T
This work has received funding from the European
Commission Horizon 2020 research and innovation pro-
gramme ARIADNE under grant agreement No. 871464.
The multipath measured data used in this work were
provided by Nokia Bell-Labs and Aalto University.
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