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Comparative Analysis of Clustering Methods
for Power Delay Profile in MMW Bands and
In-Vehicle Scenarios
Radek Zavorka∗, Ales Prokes∗, Josef Vychodil ∗, Tomas Mikulasek
∗, Petr Horky ∗,Christoph Mecklenbr¨
auker †, Aniruddha Chandra
‡, Jan Marcin Kelner §, Cezary Henryk Zi´
ołkowski §
∗Department of Radio Electronics, Brno University of Technology, Brno, Czech Republic
†Institute of Telecommunications, TU Wien, Vienna, Austria
‡National Institute of Technology, Durgapur, India
§Institute of Communications Systems, Faculty of Electronics,
Military University of Technology, 00-908 Warsaw, Poland
e-mail: xzavor03@vutbr.cz
Abstract—The spatial statistics of radio wave propa-
gation in specific environments and scenarios, as well as
being able to recognize important signal components, are
prerequisites for dependable connectivity. There are several
reasons why in-vehicle communication is unique, including
safety considerations and vehicle-to-vehicle/infrastructure
communication.
The paper examines the characteristics of clustering
power delay profiles to investigate in-vehicle communica-
tion. It has been demonstrated that the Saleh-Valenzuela
channel model can also be adapted for in-vehicle com-
munication, and that the signal is received in clusters
with exponential decay. A measurement campaign was
conducted, capturing the power delay profile inside the
vehicle cabin, and the reweighted ℓ1minimization method
was compared with the traditional k-means clustering
techniques.
Index Terms—millimeter-waves; in-vehicle communica-
tion; clusters, k-means, reweighted ℓ1minimization
I. INT ROD UC TI ON
Channel modeling is a significant part of almost all
communication systems. We need to understand how the
signal propagates from point A to point B in a certain en-
vironment if reliable communication is required or new
standards or frequency bands should be implemented [1].
A prominent stochastic model for indoor multipath
propagation is the Saleh-Valenzuela (SV) model [2].
According to the SV model, the tap gain coefficients of
the channel impulse response (CIR) between transmitter
(TX) and receiver (RX) and also in individual clusters
have exponential decrease. A ragged environment, prop-
agation losses, and reflection potentially causes clusters
of multi path components (MPC) from the received sig-
nal. The model was proposed for the indoor environment
of mid-size rooms [3], but we show that the model is
applicable to in-vehicle communication.
Given that modern car wire harnesses often weigh
more than several tens of kilograms [4], millimeter-wave
(MMW) communication can also be used as a wireless
bus to support non-critical safety functions. It is essential
to systematically analyze the spatial signal distribution
inside the passenger cabin in various conditions to ensure
dependable connectivity.
In [5], researchers focused on examining and compar-
ing delay spread and path loss at UWB 3–11 GHz and
MMW 55–65 GHz. In the work [6], the in-vehicular
wireless 60 GHz channel for the aircraft cabin is de-
scribed. The measurement campaign outlined in the pa-
per [7] was used to analyze the effects of driving-related
vibrations and twisting on CIR and delay-Doppler spread
(DDS) inside the car cabin.
Data clustering proves invaluable in the identification
of outliers and determination of signal components,
particularly within complex datasets. A wide array of
methods, ranging from straightforward to highly in-
tricate, have been harnessed for this purpose. Among
these, the unsupervised k-means algorithm has gained
notable popularity [8], [9]. Nevertheless, a critical aspect
is specifying the optimal number of clusters that the
algorithm should delineate.
An intriguing alternative lies in the Sparsity-based
clustering technique, which adeptly segregates the Power
Delay Profile (PDP) into clusters without necessitating a
predetermined cluster count. Detailed discussions of this
method can be found in the works [10], [11], grounded
in the principles of reweighted ℓ1minimization.
A. Contribution of the Paper
This paper is focused on analyzing clustering methods
for measured millimeter waves in the delay domain in
frequency band 55-65 GHz and intra-vehicle scenario.
The benefits of paper can be divided into following areas:
•comparison of k-means method and Sparsity based
clustering based on reweighted ℓ1minimization
•analyzing of PDP for in-vehicular communication
from measurement campaign by the above men-
tioned algorithms
•validation of the SV model for and intra-vehicle
communication at MMW band
The paper is organized as follows:
The standard SV channel model is presented in sec-
tion II. The following sections go through clustering
techniques, ranging from basic k-means in section V to
more intricate Sparsity based clustering- Reweighted ℓ1
minimization in section VI. Section VII of the paper
regarding experimental measurement provides details
on the measurement setup and the application of the
aforementioned clustering algorithms to the observed
data. Conclusion section sums up the paper.
II. SA LE H-VALEN ZU EL A MO DE L
Channel models for indoor communication consist
of a large number of multipath components caused by
reflections, scattering, and diffraction. Given below is the
general, complex, low-pass channel impulse response:
h(τ) =
N
X
n=1
βnexp(jϕn)δ(τ−τn),(1)
where βnand ϕnare the amplitude gain and phase of
the nth path, respectively. Nis the total number of
propagation paths. τnis the propagation delay (arrival
time) of the nth path. δ(.) is the Dirac delta function.
[2], [10], [12]
CIR according to SV model [2] describing indoor
channel is expressed by
h(τ) =
∞
X
l=0
∞
X
k=0
βkl exp(jθkl)δ(t−Tl−τkl ),(2)
where βkl and θkl are the amplitude gain and phase of
the kth path within the lth cluster, respectively. Tlis
delay of lth cluster arrival. τkl is the excess arrival delay
of the kth path within the lth cluster. [2]
The most popular approach based on paper [2] as-
sumes a monotonically decreasing received power with
the function of Ttand τkl and average power of βkl is
described by equation
β2
kl =β2
0,0exp(−Tl
Γ) exp(−τkl
γ),(3)
where β2
0,0is the average power gain of the first path
of the first cluster, and Γand γare power-delay time
constants for the clusters and the paths, respectively [2].
The Figs. 1 and 2 represent the described model.
III. CHA NN EL D ES CR IP TI ON
In-vehicle radio channel measurements are typically
assumed to exhibit time-invariant characteristics. So, in
order to properly define the radio channel, the CTF is as
follows:
H(k) = s21(k),(4)
t
β2(t)
T0
T1
Tl
...
e-T/Γ
e-τ
/γ
Fig. 1. SV channel model - Exponentially decaying ray and cluster
average powers [2]
t
T0
T1
Tl
β00
β10...
β01
β11...
...
β0l
β1l...
βkl
τ
kl
Fig. 2. SV channel model - A realization of the impulse response [2]
where kis the measurement index that can be linked to
a specific frequency. We transform the CTF into a CIR
using an inverse Fourier transform in accordance with:
h(n) =
N−1
X
k=0
H(k)ejkn2π/N, (5)
where, in the delay domain, ndenotes a discrete time.
The PDP is then determined as the average of a several
realizations in accordance with:
P(n) = E{|h(n)|2}.(6)
IV. MET HO DS O F RA DI O SI GNAL CLUSTERING
In order to cluster radio signals, a variety of methods
are available. From fundamental machine learning tech-
niques like k-means [13] and enhanced k-PowerMeans
[14] up to sparsity-based techniques derived from
reweighted ℓ1minimization [10], [11].
Synthetic signals were generated in MATLAB for
validating the clustering approaches. Figure 3 shows an
exemplary realization of a synthetic signal. There is a
transmit signal and received signal that have been im-
pacted by noise, and a signal that has been reconstructed
via reweighted ℓ1minimization (explained below).
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5
time [-]
-0.2
0
0.2
0.4
0.6
0.8
1
1.2
Standardized power [dB]
Synthetic signal
orig
orig+noise
reconstructed
Fig. 3. Synthetic signal for verification oof clustering algorithms
V. K- ME AN S CL US TE RI NG A LG OR IT HM
Most likely one of the most well-known unsupervised
clustering algorithms, applied in many fields [8], [13].
The number of expected clusters must be specified for
proper working. The basic idea behind the algorithm
is to find a predetermined number of centroids in the
space among points. Measured data from the graph is
separated into distinct clusters. Centroids are placed in
order to reduce the distance between measured points
and clusters [15].
Simulations of the given algorithm are shown in
Fig. 4. It is obvious that k-means clustering produces
inaccurate results because includes the tail of the previ-
ous cluster in the new one.
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5
time [-]
-0.2
0
0.2
0.4
0.6
0.8
1
1.2
Standardized power [dB]
K-means clustering
Fig. 4. Clustering based on k-means algorithm on synthetic signal
VI. SPARSI TY B AS ED C LU STERING-REWEIGHTED L1
MI NI MI ZATI ON
The SV model is based on the assumption that multi-
path propagation occurs in clusters with an exponential
decrease in received power. Therefore, the proposed
approach partitions the measured PDP into several seg-
ments, or distinct clusters. The concept of the algorithm
proceeds in two steps [10]:
•reconstruction of transmit signal ˆ
Pfrom measured
PDP.
•identification of the clusters from the shape of ˆ
P.
The following non-convex optimization problem is for-
mulated from the aforementioned concept [16]:
min
ˆ
P
P−ˆ
P
2
2+λ
Ω2·Ω1·ˆ
P
0,(7)
where P= [P(0), . . . , P (N−1)]Tdenotes the received
signal and similarly for the reconstructed signal ˆ
P, and
λis a positive regularization parameter. The feasible set
Xfor the minimization problem (7) is
X={ˆ
P∈RN
0|
Ω2·Ω1·ˆ
P
1≤Lmax}.(8)
The finite-difference operator Ω1is used for estimat-
ing the slope of ˆ
P. The turning point, where the slope
changes significantly, is determined using the finite-
difference operator Ω2, represented by the filter [1, -1],
defined in [10] as:
Ω2=
1−1 0 . . . . . . 0
0 1 −1. . . . . . 0
.
.
..............
.
.
0 0 ...1−1 0
0 0 . . . . . . 1−1
(N−2)×(N−1)
.
(9)
The reweighted ℓ1norm minimization strategy [17]
is used by [10] to numerically approximate a sparse
solution and a more accurate clustering result.
The reweighted ℓ1minimization [17] is applied to
the optimization problem (7) for iteratively increasing
the sparsity of the solution by the weighted norm. The
algorithm consists of the following phases [10].
1) Set the iteration count mto zero and the initial
weights w(0)
n= 1,n= 1, ..., N . In the reweighted
ℓ1minimization, the weight parameter is later
employed to ensure a sparse solution.
2) The weighted ℓ1minimization problem is as fol-
lows:
ˆ
P(m)= arg min
ˆ
P∈RN
0
P−ˆ
P
2
s.t.
Wm·Ω2·Ω1·ˆ
P
1≤Lmax
(10)
where Lmax is the upper bound on the number of
clusters in a PDP. W(m)is a matrix with weights
at diagonal.
3) The weights are updated according to
w(m+1)
n=1
ˆ
P(m)(n) + ϵ,(n= 1, . . . , N ).
(11)
In order to maintain stability and make sure that a
zero-valued component in ˆ
P(m)does not strictly
preclude a nonzero estimate at the following phase,
the constant ϵis utilized. Epsilon can be selected
as the smallest value permitted in the computing
environment or a value significantly smaller than
the anticipated nonzero magnitudes of P. In our
case for testing, the ϵwas equal to 10−9.
4) When the method has run for the specified number
of iterations Mor when the weights have con-
verged, the algorithm terminates. If not, increment
mand move on to step 2.
The recovered ˆ
Psignal (red color) for the simulation
case is depicted in Fig. 3.
It is feasible to locate the beginnings of clusters by
using the recovered signal ˆ
P, which is piecewise linear.
The following vector
Φ = hΩ2·Ω1·ˆ
Pi(N−2)×1,(12)
denotes the pivot point of ˆ
Pwhere the slope dramati-
cally changes and can be used to pinpoint clusters with
accuracy. In Fig. 6 there is a synthetic signal from Fig. 3
divided into clusters automatically by presented sparsity
based clustering algorithm which uses reweighted ℓ1
minimization according gained decision boundary shown
in Fig. 5
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5
time [-]
-0.8
-0.6
-0.4
-0.2
0
0.2
0.4
0.6
[-]
decision boundary
1 . P-rec
Ω1
Fig. 5. Decision boundary for synthetic signal according to Φby
reweighted ℓ1minimization
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5
time [-]
-0.2
0
0.2
0.4
0.6
0.8
1
1.2
Standardized power [dB]
Sparsity based Clustering - Reweighted l1 Minimization
Fig. 6. Synthetic signal divided into clusters by based clustering
algorithm with the use of reweighted ℓ1minimization
VII. EXP ER IM EN TAL ME AS UR EM EN TS
The measurements were taken as part of a measuring
campaign [18] to analyze in-vehicle communication and
signal propagation inside car cabin.
The transmit and receive open-ended waveguide an-
tennas are used for measurements inside a mid-sized
vehicle Skoda Octavia III. The RX antenna is positioned
on the front windshield adjacent to the rear-view mirror.
The TX antenna is positioned in the middle of the
back seat. A complete description of the measurement
campaign may be found in [18].
A. 55-65 GHz measurement apparatus
In the 55-65 GHz frequency range, the transmission
coefficient between two antennas is measured using the
R&S ZVA67 four-port VNA. A broadband power ampli-
fier (QuinStar QPW-50662330, measured gain of 35 dB
in the band of interest) was used on the transmitting side
of the measurement setup to increase the system dynamic
range. Antennas for transmitting and receiving signals
were WR15 open waveguide, which has a radiation
pattern identical to [18] (similarly as in [19]). This
measurement setup has a system dynamic range of about
50 dB. The IF bandwidth of 100 Hz and the VNA output
power of 5 dBm were selected.
A 10 MHz frequency step is used during the measure-
ment, which is carried out in the frequency domain. The
inverse Fourier transform with Blackman windowing is
used to convert the channel frequency response into
the time domain. Phase-stable coaxial cables were uti-
lized to prevent the TX antenna motions from reducing
the observed phase accuracy. The power amplifier was
calibrated together with the all four ports. In order to
equalize the VNA transmission in the forward direction,
the TX and RX antennas flanges were coupled together.
The open-ended waveguide utilized for the experiment
at 55-65 GHz has a radiation pattern that is 120° wide
in the E-plane and 60° wide in the H-plane (for 3 dB
decay).
B. Application clustering methods on measured PDP
based on in-vehicle communication
A significant number of PDPs in a variety of sce-
narios were gathered during the measurement campaign
outlined above. It is straightforward to show that the
channel inside the car cabin and the measurement inside
the room are quite comparable, allowing an SV model
to be used. Therefore, in-vehicle PDP can be clustered
using the suggested approaches.
The illustrative PDPs from measurement configuration
are shown in Fig. 7, Scenario 1- the transceiver in the
location behind the driver, the trunk door was open and
car cabin without passengers and Fig. 8 Scenario 2-
with open windows instead trunk door. The reconstructed
signal ˆ
Pis shown with red color, and it is used to
detect crucial turning points by using the sparsity-based
clustering-reweighted ℓ1minimization approach.
Decision boundary Φaccording to (12) is shown in
Fig. 7 and 8 below, respectively. There is no denying
that some pikes are noticeably higher than others. Once
the decision level is set to -0.35, value optimized for
this measurement scenario, clusters can be found and
the PDP divided into clusters of MPC.
0 20 40 60 80 100 120 140 160 180 200
time [ns]
-150
-100
-50
Relative power [dB]
PDP of in-vehicle measurement
0 20 40 60 80 100 120 140 160 180 200
time [ns]
-0.5
0
0.5
Decision boundary for signal clustering
Fig. 7. PDP of measured in-vehicle communication (Scenario 1) (blue)
and reconstructed ˆ
Psignal (red) using reweighted ℓ1minimization and
Decision boundary for clustering of measured signal
PDPs clustered using k-means and sparsity-based
clustering, respectively, are displayed in Figs. 9 and
0 20 40 60 80 100 120 140 160 180 200
time [ns]
-150
-100
-50
0
Relative power [dB]
PDP of in-vehicle measurement
0 20 40 60 80 100 120 140 160 180 200
time [ns]
-0.5
0
0.5 Decision boundary for signal clustering
Fig. 8. PDP of measured in-vehicle communication (Scenario 2) (blue)
and reconstructed ˆ
Psignal (red) using reweighted ℓ1minimization and
Decision boundary for clustering of measured signal
10. The common k-means approach is obviously in-
appropriate for time domain data since it divides PDP
into a horizontal plane rather than separating clusters
into a vertical plane. Compared to that, Sparsity based
clustering segregated the signal with excellent precision
in accordance with the decision boundary in Figs. 7 and
8 using Reweighted ℓ1minimization.
In scenario 1, points that indicate the beginning of
new clusters have a markedly higher intensity than other
points. This is due to the fact that the decrease in PDP is
practically exponential (a straight line on a logarithmic
scale) and exhibits minimal deviation. Additionally, the
reconstructed signal for scenario 2 appears to be very
smooth, but as can be seen in the graph of the decision
boundary for scenario 2, the signal Φcontains more
intense spikes, making it crucial to set the horizontal
level for point selection very carefully.
0 20 40 60 80 100 120 140 160 180 200
time [ns]
-150
-100
-50
Relative power [dB]
PDP of in-vehicle measurement clustered by Sparsity Based Clustering
0 20 40 60 80 100 120 140 160 180 200
time [ns]
-150
-100
-50
Relative power [dB]
CIR of in-vehicle measurement clustered by k-means
PDP
PDP
Fig. 9. PDP of in-vehicle measurement (Scenario 1) clustered by
k-means algorithm and Sparsity Based Clustering- Reweighted ℓ1
minimitazion
0 20 40 60 80 100 120 140 160 180 200
time [ns]
-150
-100
-50
0
Relative power [dB]
PDP of in-vehicle measurement clustered by Sparsity Based Clustering
0 20 40 60 80 100 120 140 160 180 200
time [ns]
-150
-100
-50
0
Relative power [dB]
PDP of in-vehicle measurement clustered by k-means
Fig. 10. PDP of in-vehicle measurement (Scenario 2) clustered by
k-means algorithm and Sparsity Based Clustering- Reweighted ℓ1
minimitazion
VIII. CO NC LU SI ON S
We investigated clustering for partitioning MPCs into
distinct segments. The clustering of the PDP is highly
advantageous for estimating easily interpretable channel
parameters. PDP data was collected during an extensive
measurement campaign focused on in-vehicle communi-
cation at 60 GHz. The delay domain representation of
the PDP was subjected to both k-means and sparsity-
based clustering techniques. The k-means technique is
less suited for analyzing time domain data, primarily due
to its horizontal rather than vertical delay data clustering.
However, the sparsity-based clustering approach proved
highly effective for accurately segmenting real-world
PDP data obtained from in-vehicle measurements. It was
confirmed that the Saleh-Valenzuela model is adequate
for in-vehicle communication at 60 GHz.
ACK NOW LE DG ME NT
This work was supported by the Czech Science Foun-
dation, Lead Agency Project No. 22-04304L, Multi-
band prediction of millimeter-wave propagation effects
for dynamic and fixed scenarios in rugged time varying
environments. The infrastructure of the SIX Center was
used.
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