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Signal processing for channel sounding: parameter estimation and calibration

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Signal processing for channel sounding: parameter estimation and calibration

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Content SIGNAL PROCESSING FOR CHANNEL SOUNDING:
PARAMETER ESTIMATION AND CALIBRATION
by
Rui Wang
A Dissertation Presented to the
FACULTY OF THE USC GRADUATE SCHOOL
UNIVERSITY OF SOUTHERN CALIFORNIA
In Partial Fulfillment of the
Requirements for the Degree
DOCTOR OF PHILOSOPHY
(ELECTRICAL ENGINEERING)
December 2018
Copyright 2018 Rui Wang
Dedication
To my dad (Xiaoming Wang), mom (Lizhen Huang), grandparents and wife
(Wenya Chen),
ii
Acknowledgments
First and foremost I would like to express my sincere gratitudes to my Doc-
toral advisor, Prof. Andreas F. Molisch. This thesis and related works cannot be
completed without his constant advice and encouragement. I am still thankful for
his patience and trust to me when I was new to the field of channel measurements
and modeling. Participating in various projects and discussions provided the time
and experiences to me for growth. Under his guidance I have mastered the tech-
nical knowledge and skills to carry out the works included in this thesis, but more
importantly he has also helped me form the right attitudes and habits towards
scientific research, which I will benefit from for in my future career.
I also would like to thank Prof. Moghaddam and Prof. Razaviyayn for their
invaluable input into improving this thesis. Many thanks to all my colleagues
and staff members at Communication Science Institute (CSI). It has been a great
pleasure to work and study with so many talented scholars. Particularly I would
like to thank Dr. Umit Bas, with whom I have the fortune to collaborate on many
research projects that lead to several key results in this thesis. His passion about
research and positive attitudes toward life has inspired me daily in the lab. I also
would like to thank Dr. Olivier Renaudin. We worked closely together on the
METRANs project for his two-year Postdoc at University of Southern California
(USC). His sharpe insights into challenging research problems and excellence in
iii
the project management have set us in the right position to finish a challenging
project. I also would like to mention other colleagues at WiDeS, Dr. Junyang
Shen, Dr. Joongheon Kim, Dr. Hao Feng, Dr. Vinod Kristem, Dr. Umit Bas,
Dr. Zheda Li, Dr. Seun Sangodoyin, Dr. Vishnu Ratnam, Dr. Sundar Aditya,
Dr. Daoud Burghal, Prof. Ruisi He, Prof. Tingting Zhang, Prof. Shengqian Han,
Dr. Zhiyuan Jiang, Dr. Fengyu Luan, Dr. Somasundaram Niranjayan, Dr. Jussi
Salmi, Ming-chun Lee, Jorge Gomez, Zihang Cheng, Thomas Choi for your daily
inspirations that stimulate me to continue this journey.
I would like to thank several sponsors that financially support my PhD work.
I was grateful to receive the Viterbi graduate fellowship upon my admission to
USC. Throughout these years my research work has been funded by InterDigital,
NationalScience Foundation(NSF),Caltrans, NationalInstituteofStandardsand
Technology (NIST), Samsung.
Ialsowanttosincerelythankmyfamily,particularlymyparentsandgrandpar-
ents. Without their encouragement and support I wouldn’t have had the freedom
to pursue my dream and become who I am today.
Finally thank you Wenya for all the love, trust and sharing with all the joys
and hardships along this journey.
Rui Wang, December 2018.
iv
Contents
Dedication ii
Acknowledgments iii
Abstract ix
1 Introduction 1
1.1 Motivation of the Thesis . . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 Scope of the Thesis . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
1.3 Contributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
1.4 Structure of the Thesis . . . . . . . . . . . . . . . . . . . . . . . . . 14
1.5 Summary of Publications . . . . . . . . . . . . . . . . . . . . . . . . 16
2 V2V Channel Sounder at 5.9 GHz 21
2.1 Introduction and Motivation . . . . . . . . . . . . . . . . . . . . . . 21
2.2 Sounder Setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
2.2.1 Baseband Channel Sounder . . . . . . . . . . . . . . . . . . 24
2.2.2 Antenna Array . . . . . . . . . . . . . . . . . . . . . . . . . 27
2.2.3 Data logging. . . . . . . . . . . . . . . . . . . . . . . . . . . 30
3 HRPE for V2V Channel Evaluation 33
3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
3.2 Signal Data Model . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
3.2.1 Double Directional Channel Model . . . . . . . . . . . . . . 37
3.2.2 MIMO Channel Sounding Data Model . . . . . . . . . . . . 38
3.3 Estimation Algorithm. . . . . . . . . . . . . . . . . . . . . . . . . . 42
3.3.1 Reduced Data Model . . . . . . . . . . . . . . . . . . . . . . 44
3.3.2 Initialization Method . . . . . . . . . . . . . . . . . . . . . . 46
3.3.3 Score function and FIM . . . . . . . . . . . . . . . . . . . . 49
3.4 Validation with Data . . . . . . . . . . . . . . . . . . . . . . . . . . 53
3.4.1 Simulation with Synthetic Channel Responses . . . . . . . . 53
3.4.2 Evaluation with V2V MIMO Measurement Data . . . . . . . 54
v
3.5 Summary and Discussion . . . . . . . . . . . . . . . . . . . . . . . . 60
4 Switching Sequences for Fast Time-varying Channels 63
4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64
4.2 Signal Model and Ambiguity Function . . . . . . . . . . . . . . . . 67
4.2.1 Signal Data Model . . . . . . . . . . . . . . . . . . . . . . . 67
4.2.2 Spatio-temporal Ambiguity Function . . . . . . . . . . . . . 69
4.2.3 Simplified Signal Data Model . . . . . . . . . . . . . . . . . 71
4.3 Switching Pattern Optimization . . . . . . . . . . . . . . . . . . . . 73
4.3.1 Problem Formulation . . . . . . . . . . . . . . . . . . . . . . 76
4.3.2 Solution and Results . . . . . . . . . . . . . . . . . . . . . . 79
4.4 Parameter Extraction Algorithm. . . . . . . . . . . . . . . . . . . . 80
4.4.1 Path Parameter Initialization Method . . . . . . . . . . . . . 81
4.4.2 Parameter Joint Optimization Method . . . . . . . . . . . . 82
4.5 Validation with Data . . . . . . . . . . . . . . . . . . . . . . . . . . 83
4.5.1 Simulation with Synthetic Channel Responses . . . . . . . . 83
4.5.2 Evaluation with Measurement Data . . . . . . . . . . . . . . 85
4.6 Summary and Discussion . . . . . . . . . . . . . . . . . . . . . . . . 90
5 HRPE for MmWave Channel Evaluation 93
5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94
5.2 Signal Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100
5.2.1 Phased Arrays with Steerable Beams . . . . . . . . . . . . . 100
5.2.2 Rimax Data Model . . . . . . . . . . . . . . . . . . . . . . . 102
5.3 Limits in Angular Estimation . . . . . . . . . . . . . . . . . . . . . 105
5.4 Calibration Procedures . . . . . . . . . . . . . . . . . . . . . . . . . 108
5.5 Calibration Practical Limitations . . . . . . . . . . . . . . . . . . . 118
5.5.1 Center Misalignment . . . . . . . . . . . . . . . . . . . . . . 119
5.5.2 Phase Noise . . . . . . . . . . . . . . . . . . . . . . . . . . . 123
5.6 Measurement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127
5.6.1 Preprocessing with Calibration Results . . . . . . . . . . . . 128
5.6.2 Two-path Experiment . . . . . . . . . . . . . . . . . . . . . 129
5.6.3 Two-pole Experiment . . . . . . . . . . . . . . . . . . . . . . 132
5.7 Summary and Discussion . . . . . . . . . . . . . . . . . . . . . . . . 133
6 Conclusions 136
6.1 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136
6.2 Future Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 138
Reference List 140
vi
A Supplemental Materials to Chapter 3 151
A.1 3D tensors T
1
and T
2
in 4D-RiMAX . . . . . . . . . . . . . . . . . . 151
A.2 Jacobian Matrix and FIM in 4D-RiMAX . . . . . . . . . . . . . . . 154
B Supplemental Materials to Chapter 4 157
B.1 Derivation of Separability of the Ambiguity Function . . . . . . . . 157
B.2 A Simpler Expression of X
T
. . . . . . . . . . . . . . . . . . . . . . 158
B.3 Ambiguity function and Correlation function . . . . . . . . . . . . . 159
B.4 3D Tensors T
1
andT
2
in RiMAX-RS . . . . . . . . . . . . . . . . . 160
B.5 Jacobian Matrix in RiMAX-RS . . . . . . . . . . . . . . . . . . . . 162
vii
viii
Abstract
Connected Vehicles and millimeter wave (mmWave) communications are both
key parts of the upcoming 5G wireless communication systems. In order to study
the performance of these new systems, improved channel models that capture the
unique characteristics of vehicle-to-vehicle (V2V) and mmWave channels are of
great importance. This thesis introduces several contributions in the field of signal
processing tools for evaluating channel measurement data of V2V and mmWave.
Firstly, we focus on the differences between the V2V channel and the typi-
cal cellular channel, and between the mmWave channel and the centimeter wave
(cmWave) channel. To perform large scale V2V multiple-input multiple-output
(MIMO) channel measurements, a real-time MIMO channel sounder based on NI-
USRPs is designed, constructed and calibrated. Our sounder design emphasizes
improvements ofthestabilityofthesetupandincreasingtheMIMOsnapshotrate.
A novel high resolution parameter estimation (HRPE) algorithm to evaluate fast
time-varying channels is proposed, which is an extension of the state-of-the-art
RiMAX algorithm. In order to improve the estimation accuracy, the signal model
of specular paths (SPs) includes the phase rotation due to the Doppler effect.
Secondly, we study the fundamental limit on the number of antennas that
a time-division multiplex (TDM) channel sounder can deploy in fast time-varying
channels,whichisposedbythecommonly-usedsequentialswitching(SS)sequence.
ix
A novel spatio-temporal ambiguity function is introduced to characterize the per-
formance of different non-sequential switching (non-SS) sequences that takes into
account the non-idealities of real-world arrays. An optimization problem is formu-
lated tosearch fora better switching pattern and an algorithmbased onsimulated
annealing (SA) is proposed as the solution. As a result, we can extend the estima-
tion range of Doppler shifts by eliminating ambiguities in parameter estimation.
Both simulation and measurement results show that the improved performance of
the new switching sequence over the conventional SS sequence.
Finally, we investigate the capability of mmWave channel sounder with phased
arrays to perform the super-resolution parameter estimation. Our novel multi-
beam mmWave channel sounder is capable to conduct MIMO measurements in
dynamic environments. We also study the limits of direction finding of multipath
components (MPCs) with phased arrays and establish its connection with virtual
arrays. A novel two-step calibration procedure is proposed for our mmWave chan-
nel sounder, in order to extract the system frequency responses and the frequency-
independent beam patterns. We verify and validate the Rimax algorithm and
calibration results in an anechoic chamber with added reflectors.
x
Chapter 1
Introduction
1.1 Motivation of the Thesis
It is envisioned that the development of communication technology from the
fourth-generation (4G) to the fifth-generation (5G) requires a major paradigm
change [1]. The mobile data traffic in 5G will be between 100x and 1000x of that
in the 4G technology.
Thanks to the large available bandwidth, the mmWave technology is a promis-
ing candidate of 5G to fulfill the requirement on the data rate [2]. Different
from 4G whose main focus is on the cellular technology, 5G will also embrace
new and diverse applications such as machine-to-machine (M2M) and vehicle-to-
vehicle (V2V) communications, which require datalinks with very lowlatency and
high reliability. Autonomous driving and other intelligent transportation systems
(ITSs) can thus greatly benefit from the development and maturation of 5G. It is
envisionedthatthevehiclesontheroadcangatherdataabouttrafficandroadcon-
ditions, and exchange these data among themselves through V2V communication,
or with the road infrastructure, i.e. vehicle-to-infastructure (V2I) communication.
Through exchanging the information, the vehicles may improve their braking aid
and lane assist systems, which helps reduce the accident rate. The vehicles can
also keep a shorter distance between each other, which helps reduce the fuel con-
sumption and traffic jams.
1
Channelmeasurementandmodelingareprerequisitesfordesigningandimprov-
ing any new communication system for mmWave and V2V [3, 4]. Realistic and
accurate channel models are important to study the theoretical limits of the new
transceiver designsforanycommunicationsystem. Theyprovide valuabletoolsfor
other researchers and engineers to quickly assess the performance of their system
design without prototyping it fully in hardware. To ensure that the models are
realistic and relevant, they must be based on measurements. Measurements in
turn require the construction and calibration of channel sounders and the devel-
opment of evaluation techniques. An accurate and efficient parameter estimation
algorithm is the bridge between the channel measurement results and the channel
model.
Sounders and evaluation techniques have to be tuned to the systems that are
to be operated in the channel, because the models have to reflect all the channel
properties relevant for the system. The MIMO technology is a key contributor
to the success of the 4G Long-Term Evolution (LTE) and WiFi. It enables the
transmitter (TX) to send parallel data streams at the same time to increase the
capacity significantly. Redundant antennas also provide diversity and robustness
to the system to combat the channel fading [3, 5]. Both V2V and mmWave can
potentially benefit greatly from MIMO, so there has always been a strong interest
instudyingtheV2VandmmWaveMIMOchannels. EvaluationsofMIMOsystems
require thus channel measurements and parameter evaluations suitable for multi-
antenna systems, as outlined next.
The double-directional channel model (DDCM) presents a fundamental frame-
work of channel models that could be independent of system details of the channel
sounder, such as the transmitted waveform, the system bandwidth and antenna
configurations[6]. Thisfeaturemakesthempreferabletowirelesssystemengineers,
2
Figure 1.1: An illustration of multipath channel with the presence of LOS and
several reflected paths
because they can easily modify the channel model to reflect the system settings
that they are interested in. In these models, the channel is assumed to consist of
severalMPCs, andeachMPCismodeledasaplanewave, andanexampleisshown
in Fig.1.1. The double-directional impulse response (DDIR) [6] can be written as
a sum of contributions from MPCs, which is given by
h(t,x
T
,x
R
,τ,Ω T
,Ω R
) =
Np
X
n=1
h
n
(t,x
T
,x
R
,τ,Ω T
,Ω R
) (1.1)
=
Np
X
n=1
a
n
(t)δ(τ −τ
n
)δ(Ω T
−Ω T,n
)δ(Ω R
−Ω R,n
),
where the locations of transmitter isx
T
and receiver isx
R
, the direction of depar-
ture (DoD) Ω T
, and the direction of arrival (DoA) Ω R
. The a
n
are the complex
amplitudes of the physical MPCs, i.e., without any antenna effects. The phases
of the a
n
change quickly, while all other parameters, i.e., absolute amplitude |a|,
delay, DoA and DoD vary slowly with the transmit and receive locations (over
many wavelengths). For this reason, the time dependence is written explicitly
only for a
n
(t). The DoD and DoA are spatial angles, which can be described by
3
theazimuth/elevationpair. Whenonlypropagationinthehorizontalplaneoccurs,
thenrepresentationbytheazimuthaloneissufficient. Note, however, thatneglect-
inganonzeroelevationinmeasurements notonlyeliminatesinformationaboutthe
elevation, but also leads to errors in the estimated azimuth. Similarly, neglecting
polarization information leads to errors in the overall estimated parameters [7].
In fact, a lot of popular geometric-based stochastic channel models (GSCMs)
are derived from the DDCM, such as [8, 9] for mmWave and [10, 11] for V2V
channels. Yet, while some of the fundamental model structures are the same for
all different types of channels, there are several other aspects in which traditional
cellular channels differfrommmWave andV2Vchannels. These differences impact
boththemeasurement/evaluationandmodelingstrategies. Thusitisgenerallydif-
ficult forresearchers todirectly apply the modelsparameterized forthe traditional
cellular cmWave channels to the mmWave or V2V situations. Compared to the
cmWave propagation channel, the mmWave channel tends to lead to a large signal
attenuationduetotheineffectiveness ofthediffractionprocesswhentheLOSpath
is unavailable. The signal attenuation can increase significantly when the mobile
station (MS) is in the transition region and moves from the LOS condition to
the non-line-of-sight (NLOS), according to the results reported in [12, 13]. It was
reported in Ref. [14] that the number of dominant specular signals is reduced for
mmWave channels, which leads to a sparse structure either in the multipath delay
or angular domain. This led to a significant body of work on signal processing
algorithms that exploit the channel sparsity and reduce the system complexity.
The V2V channels are also significantly different from the cellular channels.
In cellular scenarios, the base station (BS) is usually static and elevated above
the rooftop level, such that it will be clear from any surrounding objects. More
importantly, many interaction objects (IOs) in the environment that affect the
4
communication link are either static or moving slowly. In contrast the faster
movement of both terminals and the IOs in V2V channels not only contributes
to the reduced coherence time but also results in channel nonstationarity, i.e. the
popular assumption of wide-sense stationary (WSS) channel can no longer hold
[15].
These important features about the mmWave and V2V channels indicate that
an accurate and comprehensive understanding about them would require a joint
design of the channel sounder and the parameter estimation method.
1.2 Scope of the Thesis
The focus of this thesis is on the signal processing aspects in channel sound-
ing. It emphasizes on the development of HRPE algorithms to evaluate V2V and
mmWave MIMO measurement data. Throughout this thesis we also demonstrate
that it is equally important for the success of the HRPE algorithm to design and
develop the hardware platform (channel sounder) and the calibration techniques,
which take into account the features of the propagation channel and the assump-
tions required by the parameter estimation method.
Since we are mostly interested in MIMO channels, the evaluation techniques
havealotincommonwitharraysignalprocessingmethods. Agoodsurveyonarray
signalprocessingisavailablein[16]. Thesimplestwaytoevaluatethemeasurement
dataistheFourier-basedmethod. Thismethodisalsoknownasspectrumanalysis.
RelyingontheDiscreteFourierTransform(DFT),onecantransformtherepresen-
tationofthechannelfromthefrequencydomaintothedelaydomain,fromthetime
domain to the Doppler frequency domain, and from the antenna/space domain to
the angular domain. However the resolution of the Fourier-based method is quite
5
limited and is usually determined by the “aperture” of each domain. For instance,
the delay resolution of the channel impulse, which is obtained by the inverse Fast
Fourier Transform (IFFT) of the channel transfer function, is given by the inverse
of the system bandwidth. Similarly the inverse of the time-span of sounding snap-
shots provides the Doppler resolution. The angular resolution is proportional to
the inverse of the array dimension measured in wavelengths. This method is also
difficult to produce statistically consistent results when multiple correlated signal
sources are present [17].
Another class of parameter estimators is the subspace-based method, such
as MUltiple SIgnal Classification (MUSIC) and Estimation of Signal Parameter
(ESPRIT). This type of methods relies on the covariance matrix of the signal and
its eigenvalue decomposition, and it exploits the fact that the signal space and
the noise space are orthogonal to each other. The resolution of MUSIC is usually
much better than the Fourier-based approach, but the subspace estimates can be
less accurate when the number of independent realizations is small especially in a
high-dimensional problem when dealing with a wideband MIMO channel sounder.
The ESPRIT algorithm relies the rotation-invariant structure of the signal sub-
space introduced by the antenna array, and the estimation does not require the
knowledge of the array response as long as the radiation patterns are identical
for all the antennas. However it is difficult to generalize ESPRIT to deal with
arbitrary array geometries or real-world arrays with nonidealities such as mutual
coupling and element pattern variation.
An important class of parameter estimators is the maximum likelihood (ML)-
based method. The well-known examples are space-alternating generalized
expectation-maximization(SAGE)[18]andRiMAX[19]. Afullparametrizationof
the popular GSCM usually requires wideband MIMO channel measurements, and
6
theML-basedparameterestimatorholdsitsadvantageintermsoftheadaptability
to various forms of signal models which result from different sounder configura-
tions. SAGE builds on the Expectation Maximization (EM) algorithm and it is
a very popular numerical technique to find ML estimates. To provide just two
examples in environments of interest for this thesis, it was used in a MIMO V2V
channel evaluation in Ref. [20], and used in evaluating the mmWave channel in
Ref. [21]. However SAGE separates the estimation of multiple signal sources as
independent problems and iteratively updates the estimates between them, as a
result the algorithm tends to converge slowly especially when signals are not well
separated. Besides SAGE models all the signal sources as distinct SPs. When
dealing with a cluttered environment rich of MPCs, the number of parameters in
theestimationproblemcangrowquicklywhichcouldleadtounstableestimates. A
more advanced ML-based method is RiMAX [19], which is the first joint ML esti-
mator for both SPs and diffuse multipath componentss (DMCs). To improve the
convergence rate, RiMAX uses the Levenberg-Marquardt method [22, 23] in the
nonlinearoptimizationstageforallSPs’parameters. Salmietal. introducesapath
estimator and tracker in dynamic channels based on the extended Kalman filter
(EKF) [24]. However the main data model utilized by RiMAX and EKF assumes
the channel is completely static within one MIMO snapshot, which does not hold
in general for V2V MIMO channel measurements. Besides, both of them have
shortcomings in their initialization methods of SPs. The initialization for RiMAX
in [19] implements a sequential search in multiple parameter domains to reduce
the computational complexity. Its main drawback is the loss of correlation gain,
because samples from various data domains are processed independently. This
leads to a higher probability of producing “ghost” paths due to imperfect signal
cancellation. In contrast the method in Ref. [24] exploits the full correlation gain
7
by jointly initializing all structural parameters for one SP, but its computational
complexitybecomesprohibitivewhenfacinglargemultidimensionalmeasurements.
The inclusion ofthe DMCmodeling inRiMAX hasseveral benefits. Firstofall
it allows a separate characterization between SPs and diffuse paths, because the
former represent the LOS, specular reflection, and the diffraction, while the latter
are forsignalsgenerated from the surface scattering process. Secondly, it improves
the problem condition and actually reduces the problem complexity when the esti-
matoraimstocharacterizethetotal contribution ofscattered pathsinsteadoftheir
individual parameters. It is shown in [25] that neglecting DMC in the parameter
estimator can lead to inaccurate estimates of SPs in some scenarios. Landmann et
al. discuss more in Ref. [7] about the importance to have a complete signal model
while building the parameter estimator. Various models and estimators for DMC
havebeenproposedintheliterature. Thefrequency-correlated modelisconsidered
in the original version of RiMAX [19, 26], when the power delay profile (PDP) of
DMC is assume to have a single exponentially decaying shape. Others extend the
model and includes the correlation of DMC between TX and receiver (RX) anten-
nas[27, 28]. They assume the DMCtofollow amixture ofVon-Misesdistributions
intheangulardomain. Poutanen et al. discover throughmeasurements thatDMC
hasawelldistinguishable structureinthedelayandangulardomain[29]anditcan
contribute between 10% and 95% of the total received power. Richter et al. gener-
alize the model of DMC in [30] where they consider the covariance matrix of DMC
has an unconstrained Kronecker structure and proposes a ML-based estimator to
provide estimates of the covariance matrix.
Foracomprehensive characterizationthatincludespathdirections, thechannel
sounder is required to deploy antenna arrays at both TX and RX [31, 6]. However
thereareonlyafewV2VMIMOmeasurements, suchas[32,33,20]. Althoughmost
8
of these measurements use a MIMO sounder, few papers utilize it to extract the
directional information about MPCs, which could have provided a deeper insight
into the interactions of radio waves with the environment. Ref. [10] introduces
results on V2V measurements conducted in Lund, Sweden. The channel sounder
has a 4×4 MIMO array, and measures at 5.2GHz with a 240MHz bandwidth. Its
path extraction and tracking algorithm focuses on the delay and signal strength of
MPCs, however the angular and Doppler parameters of MPCs are not included in
theanalysis. Ref. [11]proposesaV2Vchannel modelbasedonmeasurements con-
ductedinHelsinki, Finland. Thechannelsounderisequippedwitha30×30MIMO
arrayandmeasuresat5.3GHzwitha60MHzbandwidth. Itsmainlimitationisthe
low measurement snapshot rate, which leads a maximal resolvable Doppler about
7.15Hz and prevents the capability to measure a fast time-varying channel. The
pathextractionmethodmainlyfollowsasequential search andsubtractprocedure,
which issimilar totheone introduced in[10]. Besides, theanalysis doesnotreport
the angular or Doppler characteristics of MPCs. Abbas et al. present the analysis
on V2V measurements with a HRPE algorithm in [20], which jointly estimates all
parameters of MPCs. It uses the SAGE algorithm [34] to extract parameters of
50 MPCs with a full-polarimetric setup. However the complexity of SAGE forces
the author to reduce the bandwidth to 20MHz from 240MHz. Meanwhile only a
subset of the snapshots are evaluated, which leads to an effective snapshot rate of
2Hz. This could compromise the possibility of analyzing the dynamic statistics
of MPCs in the V2V channel. Part of this thesis discusses an accurate yet effi-
cient parameter estimation algorithm derived from the state-of-the-art algorithm
to jointly estimate the parameters of SPs and DMC in the V2V channel.
9
For the mmWave channels, there are two main categories of path extraction
methods to evaluate the measurement results. The first category is the spectrum-
based peak searching method, where some well-known examples are the methods
in Refs. [35, 14, 36]. Despite its simplicity and effectiveness in some scenarios,
there areobvious drawbacks attributingtothe discarding ofthe phase information
between switched beamsorrotatingantennas. Forinstance, theangularresolution
will be limited to the half power beamwidth (HPBW) of the horn antenna that is
independent of the path’s signal-to-noise ratio (SNR), and it will become very dif-
ficult to resolve multiple paths whose directions are within the HPBW of the horn
antenna, and besides the methods have to account for the sidelobes of imperfect
beam pattern and decide whether any secondary peak attributes to the sidelobes
of the main peak or the contribution from a weak MPC. The second category uses
moreadvancedsignalprocessingalgorithms,whichrelyonthemaximumlikelihood
estimator (MLE). For instance SAGE is used in Ref. [21] with rotating horns and
in Ref. [37, 38] along with virtual arrays. Meanwhile RiMAX [19] outperforms the
conventional SAGE [39] in terms of the convergence speed and RiMAX’s inclusion
of DMC [26] in the parameter estimation. Ref. [40] presents the first RiMAX
evaluation results of an indoor mmWave measurement campaign at 60GHz with
vector network analyzer (VNA) and virtual arrays. Only 1.12GHz out of the total
9-GHz measured channel is eventually used to perform the RiMAX evaluation,
mainly to satisfy the array narrowband assumption. Part of this thesis discusses
our proposed novel calibration techniques that enable the HRPE analysis on data
measured with a real-time mmWave channel sounder based on phased-arrays.
10
1.3 Contributions
To perform V2V MIMO channel measurements, we have built a real-time
MIMO channel sounder based on NI-USRPs, which is capable of measuring V2V
channels continuously [41]. Compared to other V2V channel sounders in the lit-
erature [32, 33], our sounder design emphasizes on improving the stability of the
setup and increasing the MIMO snapshot rate. The system stability allows us to
develop an HRPE algorithm for the data analysis that jointly estimates param-
eters of MPCs such as time-of-arrival, direction of departure, direction of arrival
and Doppler shift. The results also rely on accurate system and array calibration.
The maximized MIMO snapshot rate increases the estimable maximal absolute
Doppler shift to 806Hz. It also provides a more continuous representation of the
channel dynamics. The setup enabled multiple successful measurement campaigns
that aim to study car-to-car, truck-to-car and truck-to-truck channels [42, 43].
These measurement campaigns result in a large database forMIMO V2V channels
across various environments such as urban, suburban, campus and highway. The
number of channel impulse responses is over 350 million. A summary of these
measurement campaigns and sample results are available in Ref. [44]. The sys-
tem architecture and design concepts of this sounder facilitate the construction of
several new setups, such as the 16× 16 device-to-device (D2D) MIMO sounder
at 900MHz band available at USC, and another D2D MIMO channel sounder at
Austrian Institute of Technology (AIT) [45] based on this design.
We also propose and implement a novel HRPE algorithm to evaluate V2V
channel measurement data [46]. The algorithm relies on the double-directional
channel model [6], and models the channel impulse response as a sum of contribu-
tions from plane waves. This underlying signal model provides not only insights
into the directional characteristics of the V2V channel, but also a framework that
11
allows a joint estimation of all parameters related to MPCs. Our adopted data
model also captures the phase evolution within one MIMO measurement, which is
themost suitable forthetime-varying V2Vchannel. Wedevelop anHRPE scheme
basedontheframeworkofRiMAXin[19]. Specialattentionispaidtoimprove the
numerical implementation, such as the initialization of the parameters of SPs, the
score function and Fisher Information Matrix (FIM). The efficient initialization
method is an extension of our method in [47]. To the best of our knowledge, this
is the first published work with optimized implementation that jointly estimates
all parameters of MPCs and DMC from V2V channel measurements.
Furthermore we study the antenna switching pattern design problem for the
TDM channel sounder in fast-varying channels [48, 49]. It is conventionally
believed that there is a fundamental tradeoff between the estimation accuracy
on the path directions and the Doppler frequency estimation range (DFER)in the
TDM channel sounding. Given the maximal Doppler shift in the scenario, there
is a upper limit on the number of combined TX and RX antenna pairs that the
channelsoundercanhave. Yinet al. werethefirsttorealizethattheDFERcanbe
extendedbyafactorequaltotheproductofTXandRXantennasinTDMchannel
sounding. it is the choice the SS sequence that leads to the shrinking DFER when
the number of antennas grows in the TDM MIMO sounder [50]. Pedersen et al.
later discovered that it was the choice of the SS patterns that caused this limit
in the TDM channel sounding [51]. Pedersen et al. proposed to use the so-called
normalized sidelobe level (NSL) of the objective function as the metric to evaluate
switching patterns, and further derived a necessary and sufficient condition of the
switching sequences that lead to ambiguities [52], but the method to derive good
switchingsequencesforrealisticarraysisnotclear. Meanwhiletheiranalysisisper-
formed under ideal conditions: firstly the paper assumes that all antenna elements
12
are isotropic radiators, secondly it requires the knowledge of the phase centers of
allantennas. Ourwork[48,49],ontheotherhand,usesthedecompositionthrough
effective aperture distribution functions (EADFs) [53], which can handle the non-
idealities of real-world arrays and include the effects such as mutual coupling and
the variation of element patterns. We propose a spatio-temporal ambiguity func-
tion and investigate its properties and impact on the estimation of directions and
Dopplershifts ofMPCs. We alsodemonstrate thatthisspatio-temporal ambiguity
function is closely related to the correlation function induced from the MLE uti-
lized inthe state-of-the-artRiMAXalgorithm. Inspired byRef. [54], we modelthe
array pattern design problem as an optimization problem and propose an anneal-
ing algorithm to search for an acceptable solution. Besides we propose an HRPE
algorithm that adopts a signal data model that incorporates the optimized array
switching pattern.
Last but not least, we study the possibility of applying the HRPE to mmWave
channel evaluations [55]. We have built a real-time MIMO mmWave channel
sounder at 28GHz based on steerable phased-arrays. We first study the theo-
retical limits on the estimation error of path directions in the context of phased
arrays, and compare the results with those from a virtual array that has the same
topology. The traditional array calibration method does not apply to our RFU,
because the antenna array is integrated with other RF electronics. In order to
obtain the system response and narrowband array response needed in the RiMAX
analysis, we propose anovel two-step calibration scheme andformulate the extrac-
tion and separation of two responses as an optimization problem. We also study
the impact of the array center misalignment and the residual phase noise on the
13
quality of calibration data and subsequent evaluations through HRPE. The cal-
ibration results are further validated through actual MIMO measurements in an
anechoic chamber with added reflectors.
1.4 Structure of the Thesis
The rest of this thesis is organized as follows. Chapter II introduces a real-
time MIMO channel sounder that we have built to measure the time-varying V2V
propagation channel at 5.9GHz. We focus on the sounder design principles and
highlightitsadvantagesagainstotherV2Vchannelsoundersintheliterature. Par-
ticularly we highlight itssystem stability and optimized datastreaming capability.
The firstfeatureallowsustoperformHRPEanalysisonthedataset, whilethesec-
ond feature maximizes the system’s measurement snapshot repetition rate, which
leads to a more fluent characterization of the time-varying channel. Besides we
also present its calibration procedures. The related publication is [I].
In Chapter III, we mainly describe a HRPE which can be applied to the eval-
uation of fast time-varying MIMO channels, such as V2V and mmWave channels.
Thealgorithm,basedonamaximumlikelihoodestimator,jointlyestimatesparam-
eters of MPCs such as time-of-arrival, direction of departure, direction of arrival
and Doppler shift. We also present details on computationally attractive methods
to initialize parameters in the global search stage and evaluate key components
in the local optimization stage. The algorithm is tested with actual V2V channel
measurement data obtained through the sounder in publication [I]. The related
publications are [II,III].
In Chapter IV, we investigate the impact of array switching patterns on the
accuracy of parameter estimation of MPCs for a TDM channel sounder. The
14
commonly-used sequential (uniform) switching pattern poses a fundamental limit
on the number of antennas that a TDM channel sounder can employ in fast time-
varying channels. We thus aim to design non-sequential switching sequences that
relaxthese constraints. We formulatethesequence design problemasanoptimiza-
tionproblemandproposeanalgorithmbasedonsimulated annealingtoobtainthe
optimalsequence. Asaresultwe can extend the DFERbyeliminatingambiguities
inparameterestimation. ResultsareverifiedthroughMonteCarlosimulationsand
actual V2V channel measurements. The related publications are [IV,V].
In Chapter V, we investigate the capability to perform the HRPE algorithm,
suchasRiMAX,onthemmWaveMIMOchannelsounderbasedonsteerablephased
arrays. We firstly study the limits of direction estimation of MPCs with phased
arrays. We then propose our novel two-step calibration procedure which is essen-
tial to applying an HRPE algorithm such as RiMAX to the mmWave channel
measurement data. Results are verified with simulations and test experiments in
a controlled environment such as the anechoic chamber. The related publication
is [VI].
The symbol notation used in this thesis follows the rules below:
• Bold upper case letters, such asB, denote matrices. B() represents a matrix
valued function.
• Bold lower case letters, such as b, denote column vectors. b
j
is the j-th
column of the matrixB. b() stands for a vector valued function.
• Calligraphic upper-case letters denote higher dimensional tensors.
• [B]
ij
denotes the element in the ith row and jth column of the matrixB.
• Superscripts
∗
,
T
and
†
denote complex conjugate, matrix transpose and
Hermitian transpose.
15
• The operators |f(x)| and kbk denote the absolute value of a scalar-valued
function f(x), and the L2-norm of a vector b.
• The operators ⊗, ⊙ and ⋄ denote Kronecker, Schur-Hadamard, and Khatri-
Rao products.
• The operator⊘ represents the element-wise division between either two vec-
tors, matrices or tensors, and the operator ◦ is the outer product of two
vectors.
• The operators ⌊⌋ and ⌈⌉ are the floor and ceiling functions.
1.5 Summary of Publications
The following is the list of original publications that contribute to this thesis
I Wang, R., Bas, C. U., Renaudin, O., Sangodoyin, S., Virk, U. T., and
Molisch, A. F. (2017, May). A real-time MIMO channel sounder for vehicle-
to-vehicle propagation channel at 5.9 GHz. In Communications (ICC), 2017
IEEE International Conference on (pp. 1-6). IEEE.
II Wang, R., Renaudin, O., Bernas, R. M., and Molisch, A. F. (2015, Septem-
ber). Efficiency improvement for path detection and tracking algorithm in a
time-varying channel. In Vehicular Technology Conference (VTC Fall), 2015
IEEE 82nd (pp. 1-5). IEEE.
III Wang, R., Renaudin, O., Bas, C. U., Sangodoyin, S., and Molisch, A.
F. (2017). High-Resolution Parameter Estimation for Time-Varying Double
Directional V2V Channel. IEEE Transactions on Wireless Communications,
16(11), 7264-7275.
16
IV Wang, R., Renaudin, O., Bas, C. U., Sangodoyin, S., and Molisch, A. F.
(2018, May). Antenna Switching Sequence Design for Channel Sounding in a
Fast Time-varying Channel. In Communications (ICC), 2018 IEEE Interna-
tional Conference on IEEE, Accepted.
V Wang. R.,Renaudin, O., Bas, C.U., Sangodoyin, S., andMolisch, A.F., On
channel sounding with switched arraysin fasttime-varying channels, Wireless
Communications, IEEE Transactions on, Under Review.
VI Wang. R., et al. , Enabling Super-resolution Parameter Estimation forMm-
wave Channel Sounding, Wireless Communications, IEEE Transactions on,
To be submitted.
Here is the list of original publications that are not part of this thesis.
J1 C. U. Bas, V. Kristem, R. Wang and A. F. Molisch, Real-time Ultra-
Wideband Frequency Sweeping Channel Sounder for3-18GHz, IEEE Trans-
actions on Communications, Under Review.
J2 C. U. Bas,R. Wang, O., Sangodoyin, T. Choi, S. Hur, K. Whang, J. Park,
J. Zhang, A. F. Molisch, Outdoor to Indoor Propagation Channel Measure-
ments at 28 GHz, IEEE Transactions on Wireless Communications, Under
Review.
J3 V. Kristem, C. U. Bas, R. Wang and A. F. Molisch, Outdoor Wideband
Channel Measurements and Modeling in the 3âĂŞ18 GHz Band, in IEEE
Transactions on Wireless Communications, vol. 17, no. 7, pp. 4620-4633,
July 2018.
J4 Z. Li, S. Han, S. Sangodoyin,R. Wang and A. F. Molisch, Joint Optimiza-
tion of Hybrid Beamforming for Multi-User Massive MIMO Downlink, in
17
IEEE Transactions on Wireless Communications, vol. 17, no. 6, pp. 3600-
3614, June 2018.
J5 A. Karttunen, C. Gustafson, A. F. Molisch, R. Wang, S. Hur, J. Zhang, J.
Park, Path Loss models with distance-dependent weighted fitting and esti-
mation ofcensored path lossdata. IET Microwaves, Antennas Propagation,
10 (14), 1467-1474
J6 A. Ansuman, E. A. Safadi, M. K. Samimi, R. Wang, G. Caire, T. S. Rap-
paport, andA. F.Molisch. Joint SpatialDivision and Multiplexing forWave
Channels. IEEE Journal on Selected Areas in CommunicationsÂă32, no. 6
(2014): 1239-1255.
C1 R. Wang, S. Sangodoyin, A. F. Molisch, J. Zhang, Y. H. Nam and J. Lee,
Elevation Characteristics of Outdoor-to-Indoor Macrocellular Propagation
Channels, 2014 IEEE 79th Vehicular Technology Conference (VTC Spring),
Seoul, 2014, pp. 1-5.
C2 R. Wang et al., Stationarity region of Mm-Wave channel based on outdoor
microcellularmeasurements at28GHz,MILCOM2017-2017IEEEMilitary
CommunicationsConference(MILCOM),Baltimore,MD,2017,pp. 782-787.
C3 R. Wang, O. Renaudin, C. U. Bas, S. Sangodoyin and A. F. Molisch,
Vehicle-to-vehicle propagation channel for truck-to-truck and mixed passen-
gerfreightconvoy, 2017IEEE 28thAnnual InternationalSymposium onPer-
sonal, Indoor, and Mobile Radio Communications (PIMRC), Montreal, QC,
2017, pp. 1-5.
18
C4 R. Wang, O. Renaudin, C. U. Bas, S. Sangodoyin and A. F. Molisch,
Double-Directional Channel Characterization of Truck-to-Truck Communi-
cation in Urban Environment, 2017 IEEE 86th Vehicular Technology Con-
ference (VTC-Fall), Toronto, ON, 2017, pp. 1-5.
C5 Bas, C. U., R. Wang, D. Psychoudakis, T. Henige, R. Monroe, J. Park,
J. Zhang, and A. F. Molisch. A Real-Time Millimeter-Wave Phased Array
MIMO Channel Sounder. In Vehicular Technology Conference (VTC-Fall),
2017 IEEE 86th, pp. 1-6. IEEE, 2017.
C6 Bas, C.U.,Wang, R., Sangodoyin, S., Hur, S., Whang, K., Park, J., Zhang,
J. and Molisch, A.F., 2018, March. 28 GHz propagation channel measure-
mentsfor5Gmicrocellularenvironments. InAppliedComputationalElectro-
magnetics Society Symposium (ACES), 2018 International (pp. 1-2). IEEE.
C7 Bas, C.U., Wang, R., Sangodoyin, S., Hur, S., Whang, K., Park, J.,
Zhang, J. and Molisch, A.F.. Dynamic Double Directional Propagation
Channel Measurements at 28 GHz. In Vehicular Technology Conference
(VTC-Spring), 2018 IEEE 87th, (Invited Paper).
C8 Bas, C.U.,Wang, R., Sangodoyin, S., Hur, S., Whang, K., Park, J., Zhang,
J.andMolisch, A.F.,2017,December. 28GHzMicrocellMeasurement Cam-
paignforResidentialEnvironment. InGLOBECOM2017-2017IEEE Global
Communications Conference (pp. 1-6). IEEE.
C9 Bas, C.U., Wang, R., Choi, T., Hur, S., Whang, K., Park, J., Zhang, J.
and Molisch, A.F., 2017. Outdoor to Indoor Penetration Loss at 28 GHz for
Fixed Wireless Access. In Communications (ICC), 2018 IEEE International
Conference on IEEE, Accepted.
19
C10 Bas, C. U., Kristem, V., Wang, R., Molisch, A. F. (2017, October). Real-
time ultra-wideband frequency sweeping channel sounder for 3âĂŞ18 GHz.
In Military Communications Conference (MILCOM), MILCOM 2017-2017
IEEE (pp. 775-781). IEEE.
C11 Kristem, V., Bas, C. U., Wang, R., Molisch, A. F. (2017, December).
Outdoor Macro-Cellular Channel Measurements and Modeling in the 3-18
GHz Band. In Globecom Workshops (GC Wkshps), 2017 IEEE (pp. 1-7).
IEEE.
C12 He, R., Molisch, A.F., Tufvesson, F., Wang, R., Zhang, T., Li, Z., Zhong,
Z. and Ai, B., 2016, September. Measurement-Based Analysis of Relay-
ing Performance for Vehicle-to-Vehicle Communications with Large Vehicle
Obstructions. In Vehicular Technology Conference (VTC-Fall), 2016 IEEE
84th (pp. 1-6). IEEE.
C13 Karttunen, A., Molisch, A. F., Wang, R., Hur, S., Zhang, J., Park, J.
(2016, May). Distance dependence ofpath loss models with weighted fitting.
InCommunications(ICC),2016IEEEInternationalConferenceon(pp. 1-6).
IEEE.
20
Chapter 2
V2V Channel Sounder at 5.9 GHz
In this chapter we introduce a real-time multiple input and multiple output
(MIMO) channel sounder to measure the vehicle-to-vehicle (V2V) propagation
channel at 5.9 GHz. Compared to the existing V2V channel sounder, our design
emphasizes onimproving thestabilityofthesetupandincreasing theMIMOsnap-
shot rate. The system stability allows us to develop a high resolution parameter
extractionalgorithminthedataanalysisandjointlyestimatesparametersofmulti-
pathcomponentssuchastime-of-arrival,directionofdeparture, directionofarrival
and Doppler shift. The second emphasis increases the maximal absolute Doppler
shift to 806 Hz, which the system can estimate without ambiguity. The increased
snapshot rate also provides a more fluent representation of the channel dynamics.
We verify the design of the channel sounder with actual V2V channel measure-
ments. Results suggest that 80 percents of the sample snapshots have a diffuse
powerratiolessthan20percentsandtheextracteddominantspecularpathsmatch
well with the environment and dynamics of the measurement.
2.1 Introduction and Motivation
Motivated by the benefits of intelligent transport systems (ITSs), research on
V2V communication technology has proliferated in recent years. It is envisioned
thatallvehicles ontheroadcan gatherdataabouttrafficandroadconditions, and
exchange these data among themselves or with the road infrastructure [56, 57].
21
Through exchanging the information, the vehicles may improve their braking aid
and lane assist systems, which leads a lower accident rate. The vehicles can also
keep a shorter distance between each other, which helps reduce the fuel consump-
tionand trafficjams. However reliable andlow-latency communications are avital
prerequisite for these applications.
Channelmeasurementandmodelingareprerequisitesfordesigningandimprov-
ingcommunicationsystems[3]. MostoftheV2Vmeasurements,e.g.[58,59,60,61],
focus on canonical channel metrics, such as path-loss, signal fading (large or small
scale fading), delay spread and Doppler spread, which characterize the channel
with some “bulk” features. However the safety applications in V2V communica-
tion demand a high level of robustness in the system design, and thus a more
accurate model of the propagation channel it operates in. A complete and more
accurate characterization of the radio channel requires a MIMO channel sounder
equipped with antenna arrays at both TX and RX.
Some V2V measurements do use a MIMO sounder, but very few papers uti-
lize it to extract the directional information about MPCs, which can provide a
deeper insight into the radio wave interaction with the environment. [10] intro-
duces results on V2V measurements conducted in Lund, Swedon. The setup has a
4×4MIMOarray, andabandwidthof240MHzaround5.2GHz. Itsdataanalysis
combines the CLEAN algorithm with a path tracking algorithm, and focuses on
thetime ofarrivalandsignal strength ofMPCs. However theangularandDoppler
parameters of MPCs are not included in the data processing. [11] proposes a V2V
channel model based on measurements conducted in Helsinki Finland. The chan-
nel sounder is equipped with a30×30 MIMO arrayand measures at5.3 GHzwith
a 60 MHz bandwidth. Its main limitation is the low MIMO snapshot rate, which
leads a maximal Doppler about 7.15 Hz and limits the scenarios that the sounder
22
can measure. Besides, the analysis does not report the angular or Doppler char-
acteristics of MPCs. [20] reports a HRPE analysis on a MIMO V2V measurement
witha4×4MIMOarray. ItusestheSAGEalgorithm[39]toextractparametersof
50 MPCs with a full-polarimetric setup. However the complexity of SAGE forces
the author to reduce the bandwidth to 20 MHz from 240 MHz. Meanwhile only a
subset of the snapshots are evaluated, which leads to an effective snapshot rate of
2 Hz. This could compromise the possibility of analyzing the dynamic statistics of
MPCs in the V2V channel.
The main contributions of this chapter are the following:
• we construct a real-time MIMO channel sounder that is capable to measure
a V2V wireless channel with realistic channel dynamics, and produce data
that can be analyzed by an HRPE algorithm;
• we validate the setup with sample results from actual V2V measurement
data.
The remainder of the chapter is organized as follows. In Section II we present
our V2V channel sounder. Section III presents some sample results based on our
V2V measurement data. In Section IV we draw the conclusions.
2.2 Sounder Setup
In this section, we introduce our real-time MIMO channel sounder designed for
measuring V2Vchannels. Oursetup includes apairofNI-USRPRIOsasthe main
RF transceivers, and a pair of 8-element switched antenna arrays. The diagram of
the real-time MIMO channel sounder is given in Fig. 2.1. A list of major pieces of
equipment in this setup is provided in Tab. 2.1.
23
Figure 2.1: Diagram of the real-time MIMO channel sounder at the transmitter
(left) and receiver (right) sides, respectively
Item Manufacture Model No.
USRP National Instrument USRP RIO 2953R
GPSDO Precision Test Systems GPS10eR
PA WENTEQ ABP1500-03-3730
Host controllers National Instrument PXIe-1078 / PXIe-1082
Tx RF switch RF Lambda RFSP8TR0408G
Rx RF switch Taylor Microwave SB8-A01R
Table 2.1: Hardware list of the V2V MIMO channel sounder
2.2.1 Baseband Channel Sounder
The transmitted signal centers around 5.9 GHz with a bandwidth of 15 MHz.
Themeasuredoutputpowerafterthepoweramplifier(PA)isabout26dBm. Since
NI-USRP RIO uses a direct up-conversion (DUC) architecture, we implement the
intermediate frequency (IF) processing for both the transmitter and receiver, so
24
that we minimize negative effects of the direct current (DC) offset problem that
usuallybotherswirelesstransceiverswithDUCarchitectures. Wesetthefrequency
ofthelocaloscillator(LO)atTXandRXtobe5.888GHz,whichis12MHzsmaller
than the center frequency of the sounding signal.
We choose to use a multitone waveform as the sounding signal, similar to an
orthogonal frequency division multiplexing (OFDM) signal. Compared to other
sounding signals such as m-sequence, the multitone signal has the adavantage to
equally distribute power over the bandwidth of interest. Following [62], we opti-
mizethecrest factorofoursoundingwaveform while maintainingequalpower over
frequency tones, and obtain the final waveform with a crest factor of 1.04. The
MIMO sounding signal consists of 64 (8×8) copies of this sounding signal, and
its total length T
0
is 640 µs . Several guard periods are needed in-between these
sounding signals to allow the settling time of TX or RX switches. the maximum
resolvable Doppler shift ν
max
is equal to 1/(2T
0
) and approximately 806 Hz, which
yieldsamaximum relativespeedofapproximately148km/h. Inotherwords, mea-
surement data will suffer from Doppler ambiguities (or, equivalently, the temporal
behavior of the corresponding fading process will be undersampled) whenever the
relative speed islarger than 148km/h, e.g. in highway environments forscatterers
traveling with high velocities in a direction opposite to the one of the TX and RX
vehicles. A burst tranmission of the sounding signals consists of 30 concatenated
MIMO snapshots. Tab. 3.2 provides a detailed list of parameters of the channel
sounder.
As we know the frequency and phase synchronization is crucial for MIMO
channelmeasurements[3]. BecausetheTXandRXarephysicallyseparatedduring
the actual V2V measurements, sharing the reference clock through an RF cable is
not feasible, instead our setup relies on two high quality GPS-disciplined rubidium
25
Parameter Value
Carrier frequency 5.9 GHz
Bandwidth 15 MHz
Transmit power 26 dBm
Sampling rate 20 MS/s
Length of the sounding signal τ
max
4 µs
MIMO signal duration T
0
620µs
Number of frequency tones 61
Number of Tx antennas 8
Number of Rx antennas 8
Number of MIMO per burst 30
Rate of bursts 20 Hz
Table 2.2: Parameters of the real-time V2V MIMO channel sounder
clocks for providing the synchronization. There are two types of synchronization
required in our system. One is the transmission synchronization, and the other
is the reference clock synchronization. The rubidium clocks provide a pulse per
second(PPS)signalforabsolutetimesynchronization,and10MHzasthereference
clock for the timing modules on two USRPs. USRPs further utilize this 10 MHz
rubidium reference to derive the sampling clocks in the FPGA and the LO. The
USRP-RIO 2953R runs with two 16-bit analog-to-digital converters (ADC) for
both I and Q channels. When the receiver’s baseband sampling rate is 20 MS/s,
the output data rate is around 80 MB/s with a full duty cycle.
To characterize the system frequency response, we perform a back-to-back cal-
ibration by replacing the antenna arrays with an RF cable. Some RF attenuators
are also used to ensure a safe level of received power into the RX USRP. Common
clock reference and triggering signal are shared between TX and RX. We have
made continuous measurements with the setup, so we can average over multiple
26
realizations to further improve the SNR of the system calibration data, which is
given by
Y
REF
(f) =E
n
Y(f)
o
. (2.1)
Y
REF
(f) is our final system calibration result and depends on the gain setting of
theRXUSRP.Itisintended topre-processthemeasured channeltransferfunction
Y
MEAS
(f), which is given by
H(f)=
Y
MEAS
(f)
Y
REF
(f)
·G
ATTEN
(f), (2.2)
where H(f) is the channel transfer function ready for an HRPE analysis, whereas
G
ATTEN
(f) is the transfer function of the attenuator used during the back-to-back
system calibration.
An important evaluation we can perform based upon this system calibration
is the stability of this channel sounder. Significant phase drift within multiple
MIMO snapshots may severely undermine the accuracy of any HRPE algorithm
[3]. Fig. 2.2 shows the relative amplitude and phase responses normalized to the
firstSISOsnapshot. Thestandarddeviationofthephasevariationsis2.35
◦
,which
demonstrates that the system is stable over approximately 6000MIMO snapshots.
2.2.2 Antenna Array
We have built two 8-element vertically polarized uniform circular dipole arrays
(VP-UCDA)forourV2Vmeasurementcampaigns. Similartothearrayintroduced
in [63], all the half-wavelength dipoles are mounted in the middle of a cylindrical
metallic pillar that serves as a ground plate. The metallic pillar not only increases
the directivity of each dipole antenna in the direction that it is facing, but also
27
0 1000 2000 3000 4000 5000 6000 7000 8000 9000 10000
t / ms
-0.02
0
0.02
0.04
0.06
dB
(a)
0 2000 4000 6000 8000 10000
t / ms
-20
-10
0
10
20
deg
(b)
Figure 2.2: The variationsof the frequency response of the USRP channel sounder
over time, when we use a common GPS-disciplined rubidium clock: σ
pn
= 2.35
◦
,
LOfrequency is5.888GHz, (a)therelative amplitude indB,(b)therelative phase
in degrees
reducesthemutualcouplingbetweenantennaelements. Thedipoleantennadesign,
which follows [64], features with a quarter-wave balun feed to balance the flow of
current. The final design of the printed dipole antenna is optimized through the
EM simulation software CST. Fig. 2.3 shows the final design in AutoCAD.
Since we plan to mount these antenna arrays on top of moving vehicles, the
safety of the structure is a major concern. As a result we have built two array
stands dedicated to our application. The stand is made of two plates and one
pillar, and the large bottom metal plate is intended to mimic the characteristics
of the car roof during the calibration in the anechoic chamber. The stand is also
equipped with clamps underneath the bottom plate, so that the entire structure is
able to stick with a cargo basket on top of vehicles; see Fig. 2.4.
28
(a) Top Etch Layer of the dipole antenna (b) Bottom Etch Layer of the dipole
antenna
Figure 2.3: Modeling of the dipole antenna in AutoCAD
Figure 2.4: A photo of TX array mounted on a vehicle
High quality array calibration is important for any HRPE algorithm. The
characterization of the response of the antenna array together with the RF switch
needs to be done in an anechoic chamber. Let us treat the array calibration data
s
21
as a 3D complex tensor, and denote it by s
21
(f,ϕ,i), where f is the frequency
index, ϕ is the azimuth angle, and i is the antenna index, respectively. The data
processing from the original s
21
to the array complex beam pattern B(ϕ,i) follows
B(ϕ,i) =E
f
(
s
21
(f,ϕ,i)
G
REF
·H
free
(f)
)
, (2.3)
29
where the expectation operation E{·} performs over 40 MHz around 5.9 GHz
assuming the array response is frequency independent within the measurement
frequency band. G
REF
is the nominal gain of the reference horn antenna used
in the array calibration. H
free
(f) is the free-space LOS channel transfer function
determined by
H
free
(f)=
1
√
4πd
2
e
−j2πfd/c
0
, (2.4)
where d is the distance between the reference antenna and antenna under test in
the calibration, c
0
is the speed of light. We further process the results in Eq. (2.3)
with the estimator in [65], in order to improve the SNR and correct the additional
phase offset due to the axis misalignment. The vertically polarized pattern of each
elementoftheRXandTXantennaarraysisgiveninFigs. 2.5and2.6,respectively.
The 2D ambiguity function is defined as follows
A
b
(φ
1
,φ
2
) =
P
i
B
∗
(φ
1
,i)B(φ
2
,i)
q
P
i
|B(φ
1
,i)|
2
q
P
i
|B(φ
2
,i)|
2
. (2.5)
Fig. 2.7 gives the amplitude of ambiguity function for the Rx antenna array. The
high ridge lies along the diagonal as expected.
2.2.3 Data logging
Two 360
◦
panoramic video cameras are positioned on top of the pillar of the
TX and RX antenna arrays during the measurements, in order to document the
scenariosandroutes(trafficconditions, environment, weather, etc.) as“seen”from
the top of the truck driver cabin or passenger car rooftop, as close as possible to
the antenna array elements.
30
-200 -150 -100 -50 0 50 100 150 200
ϕ
R
/ deg
-50
-45
-40
-35
-30
-25
-20
-15
-10
|B| / dB
Ant1
Ant2
Ant3
Ant4
Ant5
Ant6
Ant7
Ant8
Figure 2.5: Pattern of the elements of the Rx antenna array calibrated with the
RF switch
The global positioning system (GPS) coordinates of the TX and RX vehicles
are also available directly from the channel sounder, recorded 20 times per sec-
ond, hence providing a real-time display of the TX and RX vehicles as well as
their actual instantaneous speeds during the measurements. The GPS data can
be used in combination to the video recordings in the post-processing of the mea-
surement data in order to relate estimated results (channel parameter estimation,
system performance, etc.) to the physical environment surrounding the TX and
RX vehicles, e.g. traffic conditions, number of pedestrians, houses and road side,
etc.
31
-200 -150 -100 -50 0 50 100 150 200
ϕ
T
/ deg
-50
-45
-40
-35
-30
-25
-20
-15
-10
|B| / dB
Ant1
Ant2
Ant3
Ant4
Ant5
Ant6
Ant7
Ant8
Figure 2.6: Pattern of elements of the Tx antenna array calibrated with the RF
switch
Figure 2.7: The amplitude of 2D ambiguity function of the Rx antenna array
32
Chapter 3
HRPE for V2V Channel
Evaluation
In this chapter, we mainly describe a high resolution parameter estimation
algorithm which can be applied to the evaluation on V2V or mmWave MIMO
channel measurements. The algorithm, which is based on a maximum likelihood
estimator, jointly estimates parameters of MPCs such as time-of-arrival, direc-
tion of departure, direction of arrival and Doppler shift. It serves to provide a
comprehensive understanding of the V2V propagation channel. We propose com-
putationally attractive methods to initialize parameters in the global search and
evaluate key components in the local optimization. We also apply the algorithm
to actual V2V channel measurement data. The results suggest an overall good
performance of the estimator, where 80 percents of the snapshots have a diffuse
powerratiolessthan20percentsandthedominantspecularpathsmatchwellwith
the environment and dynamics of the measurement.
3.1 Introduction
Motivated by the benefits of intelligent transport systems (ITSs), research on
V2V communication technology has proliferated in recent years. It is envisioned
thatallvehicles ontheroadcan gatherdataabouttrafficandroadconditions, and
exchange these data among themselves or with the road infrastructure. Through
33
exchanging the information, the vehicles may improve their braking aid and lane
assist systems, which leads a lower accident rate. The vehicles can also keep a
shorter distance between each other, which helps reduce the fuel consumption and
traffic jams. Reliable and low-latency communications are a vital prerequisite for
these applications.
Channelmeasurementandmodelingareprerequisitesfordesigningandimprov-
ing communication systems [3]. Most of the V2V measurements, e.g.[32, 58, 59,
60, 61], focus on canonical channel metrics, such as path-loss, signal fading (large
or small scale fading), delay spread and Doppler spread, which characterize the
channel with some “bulk” features. However the safety applications in V2V com-
munication demand a high level of robustness in the system design, and thus a
more accurate model of the propagation channel it operates in. There is a rich
literature on V2V channel modeling as well, see the survey papers [56, 57] and ref-
erences therein. Most of the V2V channel models fall into these three categories,
deterministic models, stochastic models, GSCMs. All these models require the
support of measurements to either calibrate or parameterize the model.
A complete and more accurate characterization of the radio channel requires a
MIMO channel sounder equipped with antenna arrays at both TX and RX. Some
V2VmeasurementsdouseaMIMOsounder,butfewpapersutilizeittoextractthe
directional information about MPCs, which could have provided a deeper insight
into the interactions of radio waves with the environment. [10] introduces results
on V2V measurements conducted in Lund, Sweden. The channel sounder has a
4× 4 MIMO array, and measures at 5.2 GHz with a 240 MHz bandwidth. Its
path extraction and tracking algorithm focuses on the delay and signal strength of
MPCs, however the angular and Doppler parameters of MPCs are not included in
theanalysis. [11]proposesaV2Vchannelmodelbasedonmeasurementsconducted
34
in Helsinki, Finland. The channel sounder isequipped with a 30×30 MIMO array
and measures at 5.3 GHz with a 60 MHz bandwidth. Its main limitation is the
low measurement snapshot rate, which leads a maximal resolvable Doppler about
7.15 Hz and prevents the capability to measure a fast time-varying channel. The
pathextractionmethodmainlyfollowsasequential search andsubtractprocedure,
which issimilar totheone introduced in[10]. Besides, theanalysis doesnotreport
the angular or Doppler characteristics of MPCs. [20] reports the analysis on V2V
measurements with a HRPE algorithm, which jointly estimates all parameters of
MPCs. It uses the SAGE algorithm [34] to extract parameters of 50 MPCs with
a full-polarimetric setup. However the complexity of SAGE forces the author to
reduce the bandwidth to 20 MHz from 240 MHz. Meanwhile only a subset of the
snapshots are evaluated, which leads to an effective snapshot rate of 2 Hz. This
could compromise the possibility of analyzing the dynamic statistics of MPCs in
the V2V channel. To the best of our knowledge, there are no efficient HRPE
schemes published to jointly estimate the parameters of SPs and DMC in the V2V
channel, and our work attempts to close the gap.
A full parametrization of the channel model has to rely on channel measure-
ments, whose evaluations in turn require efficient and robust methods for a multi-
dimensionalparameterestimation. MostoftheHRPEalgorithmsrelyontheMLE,
which greatly enhances the resolution and accuracy by fitting an appropriate data
model to the measurement data. Over the past two decades, a variety of HRPEs
has been developed for various applications. ESPRIT is a subspace method but it
relies on a special antenna array structure and does not apply to a general case.
SAGEbuilds onthe EMalgorithm, which tendstoshow aslow convergence andin
its conventional version neglects the contributions from DMC. A highly accurate
method is RIMAX [19], which estimates the parameters of both SP and DMC.
35
To improve the convergence rate, RIMAX uses the Levenberg-Marquardt method
[22, 23] in the nonlinear optimization stage for all SPs’ parameters. [24] intro-
duces a MPC tracker based on the EKF. However the main data model utilized
by RIMAX and EKF assumes the channel is completely static within one MIMO
snapshot, which does not hold in general for V2V MIMO channel measurements.
Besides,bothofthemhaveshortcomingsintheirinitializationmethodsofSPs. The
initializationforRIMAXin [19]implements asequential search inmultiple param-
eter domains to reduce the computational complexity. Its main drawback is the
loss of correlation gain, because samples from various data domains are processed
independently. This leads to a higher probability of producing “ghost” paths due
to imperfect signal cancellation. In contrast [24] exploits the full correlation gain
by jointly initializing all structural for one SP, but its computational complexity
becomes prohibitive when we are facing large multidimensional measurements.
Our work relies on the double-directional channel model [6], and models the
channel impulse response as a sum of contributions from plane waves. This under-
lying signal model provides not only insights into the directional characteristics of
the V2Vchannel, but alsoaframework thatallowsa jointestimation ofall param-
etersrelatedtoMPCs. Ouradopteddatamodelalsocapturesthephaseenvolution
within one MIMO measurement, which is the most suitable for the time-varying
V2Vchannel. We develop anHRPE scheme based onthe framework ofRIMAX in
[19]. Special attention is paid to improve the numerical implementation, such as
the initialization of the parameters of SPs, the score function and FIM. The effi-
cientinitializationmethodisanextensionofourconference paper[47]. Tovalidate
ourproposedestimationalgorithm,webuildadedicatedreal-timechannelsounder
suitable forV2Vchannels, andconduct extensive measurement campaigns. To the
36
best of our knowledge, this is the first published work with optimized implemen-
tation that jointly estimates all parameters of MPCs and DMC from V2V channel
measurements.
The main contributions of this work are the following:
• we find a suitable signal data model for V2V MIMO channel measurements,
which enables evaluations with an HRPE algorithm based on the framework
of RIMAX;
• we propose computationally attractive methods to initialize parameters of
SPs,andtoevaluatethescorefunctionandFIMinthenonlinearoptimization
stage;
• we validate the data model and the HRPE algorithm with a self-built V2V
channel sounder and actual V2V measurement data.
The remainder of the chapter is organized as follows. Section II introduces
the general data model of the estimator. Section III introduces a simplified model
catering to our setup for V2V channel measurement and efficient methods for the
numerical implementation. In Section IV we present some sample results based on
our V2V measurement data. In Section V we draw the conclusions.
3.2 Signal Data Model
3.2.1 Double Directional Channel Model
[6]proposesachannelmodelthatcapturesthespatialandtemporalinformation
ofthewirelessMIMOchannel. Thechannelresponseismodeledasasuperposition
of P plane waves or SPs, whose features are delay τ
p
, DoD Ω T,p
, DoA Ω R,p
, and
37
Dopplershiftν
p
. WecandividetheDoDintotheazimuthalDoDϕ
T,p
andelevation
DoD θ
T,p
, and similarly DoA into the azimuth ϕ
R,p
and elevation θ
R,p
. The time-
varying double directional transfer function is given by
H(f,t) =
P
X
p=1
b
R
(Ω R,p
)
T
Γ
p
b
T
(Ω T,p
)·e
−j2π(fτp−νpt)
, (3.1)
whereb
R
andb
T
arepolarimetric antenna arrayresponses forthe receiver and the
transmitter. Γ
p
is the polarimetric path weight matrix and further defined by
Γ
p
=



γ
HH,p
γ
VH,p
γ
HV,p
γ
VV,p


 (3.2)
The subscript of various γ
p
denotes the signal polarization, for example HH is
horizontal-to-horizontal and VH is vertical-to-horizontal.
3.2.2 MIMO Channel Sounding Data Model
Most of the state-of-the-art channel sounders use switched arrays to ensure the
highest phase stability during MIMO channel measurements, which is essential
for evaluating MIMO measurements with any HRPE algorithm [31]. As a result,
this type of channel sounder needs a time division scheduling scheme to collect
data between all antenna pairs. Let us consider a series of MIMO snapshots with
M
T
TX antennas, M
R
RX antennas, and hence altogether M
T
M
R
single-input
single-output (SISO) channels. If we transmit with a multi-tone sounding signal,
one SISO channel consists of M
f
frequency points. The dimension of one MIMO
snapshot is M
f
×M
R
×M
T
.
We assume the channel sounder applies a sequential switching scheme illus-
trated in Fig. 3.1, and the same sounding sequence of length τ
max
is repeated for
38
each TX-RX antenna pair, hence each MIMO snapshot is T
0
= 2M
R
M
T
τ
max
long.
The maximal absolute resolvable Doppler shift without ambiguities is 1/(2T
0
).
However the input to our proposed estimation scheme consists of T MIMO snap-
shots compared to one MIMO snapshot in [24]. A large value of T increases the
observation aperture in the time domain and effectively improves the estimation
accuracy of Doppler shifts, which are important in typical V2V channels. Another
importantassumption ofourworkisthatallstructural parameters, including time
delay of arrival (TDoA)τ, DoDΩ T
, DoA Ω R
, and Doppler shift ν of SPs, are con-
stant within such T MIMO snapshots. Their timespan is 18.6 ms in our channel
sounder setup [41]. Moreover we assume the complex path weights of SPs remain
static within one SISO snapshot. However over T MIMO snapshots, we only con-
strain the amplitudes of path weights remain the same, while their phases may
experience variations captured by the Doppler shift and the switching schedule of
antenna arrays.
Similar to the data model in [19] and [24], we use a vector model for the input
T MIMO snapshots, and denote it asy∈C
M×1
, where M =M
f
×M
R
×M
T
×T.
It includes contributions from SP s(θ
s
), DMC and measurement noise.
y =s(θ
s
)+n
dmc
+n
0
, (3.3)
where the vectorθ
s
represents the parametersofP SPs. It consists ofpolarimetric
pathweightsγ andthestructuralparametersμthatincludeτ,(ϕ
T
,θ
T
),(ϕ
R
,θ
R
),
andν. ϕ
T
,ϕ
R
andν arenormalizedtobetween−π andπ,θ
T
andθ
R
arebetween
0 and π, while τ is normalized to between 0 and 2π.
39
Before breaking down thedetailsabouts(θ
s
), we first introduce thephase shift
matrixA(μ
i
)∈C
M
i
×P
[19], which is given by
A(μ
i
)=







e
−j⌊
M
i
2
⌋μ
i,1
··· e
−j⌊
M
i
2
⌋μ
i,P
.
.
.
.
.
.
e
j(⌈
M
i
2
⌉−1)μ
i,1
··· e
j(⌈
M
i
2
⌉−1)μ
i,P







. (3.4)
μ
i
is a structural parameter vector that represents either τ, ϕ
T
, θ
T
, ϕ
R
, θ
R
or
ν. Together with the system and array calibrations, such as the system frequency
response G
f
and the EADFs G
TH
, G
TV
, G
RH
, and G
RV
introduced in [19] and
[66], we obtain the basis matrices that form important pieces in our data model.
B
f
=G
f
·A(−τ) (3.5)
˜
B
TH
=

G
TH
·

A(θ
T
)⋄A(ϕ
T
)

⊙A
t,T
(ν) (3.6)
˜
B
TV
=

G
TV
·

A(θ
T
)⋄A(ϕ
T
)

⊙A
t,T
(ν) (3.7)
˜
B
RH
=

G
RH
·

A(θ
R
)⋄A(ϕ
R
)

⊙A
t,R
(ν) (3.8)
˜
B
RV
=

G
RV
·

A(θ
R
)⋄A(ϕ
R
)

⊙A
t,R
(ν) (3.9)
B
t
=A(ν) (3.10)
Finally the signal data model for the responses of SPs is given by
s(θ
s
)=B
t
⋄
˜
B
TH
⋄
˜
B
RH
⋄B
f
·γ
HH
+B
t
⋄
˜
B
TH
⋄
˜
B
RV
⋄B
f
·γ
HV
+B
t
⋄
˜
B
TV
⋄
˜
B
RH
⋄B
f
·γ
VH
+B
t
⋄
˜
B
TV
⋄
˜
B
RV
⋄B
f
·γ
VV
. (3.11)
40
Our model differs from the main model of [20, 19] in that we explicitly include
the Doppler shift as a parameter in the MIMO channel signal model, which is
reflectedinEqs. (3.6)-(5.12). WhentheMIMOchannelsounderusesthesequential
switchingscheme,A
t,T
andA
t,R
capturethephaserotationbetweenmeasuredsub-
channels due to the Doppler shifts. This phase rotation becomes non-negligible
when the Doppler shift or the time span is large. For example a path with the
Doppler shift of 100Hz can create a phase difference of 720
◦
over 20ms.
A
t,T
(ν)=







e
j
t
T,1
T
0
ν
1
··· e
j
t
T,1
T
0
ν
P
.
.
.
.
.
.
.
.
.
e
j
t
T,M
T
T
0
ν
1
··· e
j
t
T,M
T
T
0
ν
P







(3.12)
A
t,R
(ν)=







e
j
t
R,1
T
0
ν
1
··· e
j
t
R,1
T
0
ν
P
.
.
.
.
.
.
.
.
.
e
j
t
R,M
R
T
0
ν
1
··· e
j
t
R,M
R
T
0
ν
P







(3.13)
Heret
T,i
andt
R,j
arethestartingtimeofthesub-channelmeasurementcorrespond-
ing to ith Tx antenna and jth Rx antenna.
We also consider the influence of diffuse scattering in our measurements. It is
well known electromagnetic waves are always partially scattered apart in addition
to the specular reflection. Their total power may be non-negligible, even though
each is weak. Besides, including DMC in the signal data model improves the
robustnessoftheestimator[7]. WeassumethatDMCfollowsazero-meancomplex
Gaussian process, and its covariance matrix has a Kronecker structure, which is
given by
R
dmc
=I
t
⊗R
T
⊗R
R
⊗R
f
. (3.14)
41
Figure 3.1: An example of the sequential switching scheme of a channel sounder
equipped with a 3×3 MIMO array
This model allows us to characterize the correlation of the DMC process in the
frequency and angular domains. Furthermore we assume that DMC and noise are
independent, the covariance matrix of the stochastic part in the data model, i.e.
n
dmc
+n
0
, is modeled by
R
dan
=R
dmc
+σ
2
n
I. (3.15)
We denote R
dan
asR for brevity in the rest of chapter.
3.3 Estimation Algorithm
We select the framework of RIMAX [19], because it provides an MLE that is
both robust and efficient. RIMAX implements a joint optimization of all parame-
ters of SPs and it converges considerably faster compared to the SAGE algorithm
used in [20] to analyze the V2V channel measurement data. Besides, RIMAX
includes DMC as a part of the stochastic components in the channel response,
which provides a complete data model and produces stable estimates. We first
42
reviewsomekeypointsabouttheframeworkofRIMAX,andthenfocusontheeffi-
cient implementation of parameter initialization and optimization for SPs regard-
ing with our new data model discussed in Section 3.2.
The general expression of the MLE with the observation vector y and param-
eters θ is given by
ˆ
θ = argmax
θ
n
P
r
(y,θ)
o
(3.16)
In the application of wireless channel sounding, we typically assume that we have
no prior information about θ. We can also separate θ into θ
s
for SPs and θ
d
for
DMC to arrive at



ˆ
θ
s
ˆ
θ
d


 = argmax
θ
{P
r
(y|θ)}. (3.17)
Given the system model in Eq. (5.7), the distribution of y is CN(s(θ
s
),R), and
the conditional probability is determined by
P
r
(y|θ)=
1
π
M
det(R)
e
−[y−s(θs)]
†
R
-1
[y−s(θs)]
. (3.18)
There is no closed-form solution of Eq. (3.17), becauses(θ
s
) is a highly nonlinear
function of μ in θ
s
. RIMAX updates θ
s
and θ
d
separately and iteratively. With
a fixed estimate of R, RIMAX improves θ
s
with the Levenberg-Marquardt (LM)
method [22, 23]. Its input is an initial value ofθ
s
, which we obtain either from the
global initialization algorithm or from the previous snapshot. The main update
equation ofθ
s
within the LM method is given by
ˆ
θ
i+1
s
=
ˆ
θ
i
s
+
h
J(
ˆ
θ
i
s
,R)+ζI⊙J(
ˆ
θ
i
s
,R)
i
-1
q(y|
ˆ
θ
i
s
,R). (3.19)
43
Hereζ istheparametertocontroltheupdatestepsize. Thisiterativeoptimization
step requires the evaluation of the score function q(y|θ
s
,R) and FIM J(θ
s
,R),
for which we will provide more details later in this section.
This chapter mainly focuses on the method to estimate θ
s
. θ
d
is estimated
based on Tab. 6-5 and Tab. 6-6 in [19] for a simplified DMC model that is
explained in Section 3.3.1, while [30] introduces an MLE for the general model
described in Eqs. (3.14) and (3.15).
As a summary, our proposed algorithm performs a joint parameter estimation
forSPsandDMCintheV2Vchannels. Fig. 3.2givesaflowchart ofthealgorithm.
There are two maindifferences when we compare ittothe originalRIMAX scheme
in [19]. The first difference is the increased complexity of the signal data model
of SPs due to the inclusion of phase shift between antennas, which is necessary
for time-varying V2V channels, but we manage to accommodate it with optimized
numerical implementations. The second difference is that we place the detection
on new paths at the end of the snapshot instead of at the beginning, because for a
typical fast time-varying V2V channel it is preferable to further polish the entire
parametervectorθ
s
beforeevaluatingtheresidualresponsey−s(θ
s
)anddetecting
the new paths. Our scheme has shown an overall better stability of the estimates
of SPs and reasonable values of the diffuse power ratio (dPR), which is illustrated
by the evaluation results in Section 3.4.2.
3.3.1 Reduced Data Model
In this section we describe a simplified data model that suits our measurement
setup, which is described with details in [41]. The simplified model focuses on
the vertical-to-vertical polarization component, because both Tx and Rx deploy
vertically polarized antenna arrays. Besides, the angular modeling is limited to
44
Load data y,
EADFs and ˜ μ
L
first
snapshot?
Initialize θ
d
and θ
s
Inherit θ
d
and θ
s
from last snapshot
Start local
maximization
Improve θ
d
with
LM algorithm
Improve θ
s
with
LM algorithm
Convergence?
Path
reliability test
Detect new paths
End and store
estimates
No
Yes
No
Yes
Passed
Failed
Figure 3.2: The flowchart of the HRPE scheme to evaluate V2V MIMO channel
measurements
the azimuthal domain because the 8-element uniform circular array (UCA) has an
ambiguity in the elevation domain. It is assumed that these assumptions do not
significantlydeterioratetheestimationintheV2Vapplication,especiallywhenthe
heightdifferencebetweentwoarraysmountedontopofthevehiclesissmall. InSec.
3.4.2,thediffusepowerratio(dPR)andthegenerallygoodagreementsbetweenthe
extracted SPs with the environment indicate that the above-mentioned assump-
tionsmaybejustifiedinourmeasuredscenarios. However these assumptionsdon’t
45
hold in general for all channel sounding applications, see [7]. The reduced data
model of the response of SPs is then given by
s(θ
s
) =B
t
⋄
˜
B
TV
⋄
˜
B
RV
⋄B
f
·γ
VV
, (3.20)
where θ
s
= [μ,γ
VV
].
We have also selected a simplified DMC model that originates from [67]. The
model assumes the DMC process is only correlated in frequency, and its PDP
follows a single exponential decay model. The main reason is that the entire
estimator with the simplified DMC model produces more stable results over a
continuous route of measurement points. As a result, the covariance matrixR
dmc
from Eq. (3.14) is reduced to
R
dmc
=I
t
⊗I
T
⊗I
R
⊗R
f
(3.21)
R
τ
=F
†
R
f
F, (3.22)
where F is the discrete Fourier transform (DFT) matrix and R
τ
is a diagonal
matrix.
3.3.2 Initialization Method
Accurate initial estimates of the parameters improve the estimation perfor-
mance for most MLEs that implement a local optimization algorithm. Our initial-
ization algorithm is applied to the beginning of the first snapshot, as well as to the
end of each snapshot to detect new paths. Therefore it forms an essential part of
the whole estimator, and could dominate the overall complexity ofthe algorithmif
an inferior method is used. To initialize the parameters of one SP, a global search
46
on a multi-dimensional parameter grid can be computationally expensive. There-
fore [19, 27] avoid the global search on the large grid and resort to a sequential
search in separate dimensions. The method reduces the computational efforts, but
it sarcrifices the correlation gain of a joint multi-dimensional search, which can
lead to inaccurate initial estimates or a misdetection of weak SPs. In this part, we
proposeanefficient initialization methodfortheparametersofSPs, which exploits
the full correlation gain and has a significantly reduced runtime. The new method
is an extension of what we introduced in our conference paper [47].
Although the model has a new data dimension (the time domain) in Eqs. (4.2)
and(3.20),thesamecorrelationfunctionforasingleSPwithstructuralparameters
μ in [47] can still be applied.
C(μ,y) =(y
†
R
-1
B)(B
†
R
-1
B)
-1
(B
†
R
-1
y). (3.23)
[19] proves that locating peaks of the correlation function in Eq. (3.23) is indeed
closelyrelatedtosolvingtheMLE,givenaninitialestimateofR. However thereis
noclosed-form solutiontolocatethese correlationpeaks. Ourmethodsequentially
detects, estimates and substracts SPs in a descending order of their SNRs. It
continues until the maximal correlation value C
max
falls below a certain threshold
ǫ
D
1
or the number of detected paths has exceeded a limit. The pseudocode for the
initialization part is similar to that in [47] and provided in Algorithm 1.
1
An empirical value of 100 is chosen for evaluating the V2V measurement data in Sec. 3.4.2.
47
Algorithm 1 Algorithm for initializing parameters of SPs
1: Loady and the large search grid ˜ µ L
;
2: μ= [];y
res
=y;
3: Compute C(˜ µ L
,y
res
), and find its peak C
max
and μ
max
;
4: while C
max
≥ǫ
D
and need more SPs do
5: while stopzoom == 0do
6: Construct a small grid ˜ µ s
aroundμ
max
;
7: Compute C(˜ µ s
,y
res
), and update μ
max
;
8: if All changes are small then
9: stopzoom = 1;
10: else
11: stopzoom = 0;
12: end if
13: end while
14: μ =[μ;μ
max
]; Compute γ
VV
from Eq. (3.24);
15: Construct θ
s
=[μ;γ
VV
]; y
res
=y−s(θ
s
);
16: Compute C(˜ µ L
,y
res
), and find its peak C
max
andμ
max
;
17: end while
Given the observation vector y and structural parameters μ, the best linear
unbiased estimator (BLUE) ofγ
VV
for the data model in Eq. (4.2) is
ˆ γ
VV
=(B
†
R
-1
B)
-1
B
†
R
-1
y (3.24)
A 4-dimensional (4D) search grid ˜ µ L
is needed for the initial path detection.
A rule of thumb for sufficient oversampling in every parameter domain is that
N
f
= 2M
f
, N
T
= 2M
T
, N
R
= 2M
R
and N
t
= 2T [24]. Similar to the approach
48
in our previous work [47], our method exploits the data structure of y and uses
tensor products to greatly accelerate the computation [68]. Let us introduce the
4D correlation tensor C that shares the same dimension with ˜ µ . With a given
Doppler shift ν
nt
(n
t
th Doppler point), we then introduce two 3-dimensional (3D)
tensors T
nt
1
andT
nt
2
that have the dimension of N
f
×N
R
×N
T
. We can compute
the n
t
th 3D child tensor of C as
C(:,:,:,n
t
) =

T
nt
1

2
⊘T
nt
2
, (3.25)
where ||
2
is the element-wise absolute square, and ⊘ is the element-wise division
between two tensors. Appendix A.1 reveals the detailed procedures on how to
compute the two tensors T
nt
1
andT
nt
2
.
3.3.3 Score function and FIM
The score function and FIM are important elements in the nonlinear optimiza-
tion solution given by Eq. (4.26). The score function is known as the first order
partial derivative of the log-likelihood function, which is given by
q(y|θ
s
,R) =
∂
∂θ
s
ln(P
r
(y|θ)) (3.26)
=2R
n
D
†
(θ
s
)R
-1
(y−s(θ
s
))}, (3.27)
where R{} takes the real value of the expression. On the other hand the FIM is
the negative covariance matrix of the score function, which is defined as
J(θ
s
,R)= 2R
n
D(θ
s
)
†
R
-1
D(θ
s
)
o
. (3.28)
49
Both the score function and FIM requires the value of the Jacobian matrix
D(θ
s
), which is defined as
D(θ
s
)=
∂
∂θ
s
s(θ
s
)∈C
M×L
. (3.29)
Here L = 6P is the length ofθ
s
, i.e. the total number of parameters related with
P SPs. As suggested in [19] and [24], it is not feasible to directly evaluate the
score function in Eq. (3.27) or the inverse of FIM in Eq. (3.28), mainly because
of the large measurement dimension M. Our method utilizes the special structure
of the problem to speed up the computation. The key to our method is that we
may reorganize the Jacobian matrix D(θ
s
) as a sum of Khatri-Rao products of
four matrices.
D(θ
s
) =
1
3

D
1
4
⋄D
1
3
⋄D
1
2
⋄D
1
+D
1
4
⋄D
2
3
⋄D
2
2
⋄D
1
+D
2
4
⋄D
2
3
⋄D
1
2
⋄D
1

(3.30)
=
1
3
(D1+D2+D3), (3.31)
The detailed structures of all these D
j
i
matrices are given in Tab. A.1. To recon-
struct a D
j
i
with the table, one can concatenate the sub-matrices related to D
j
i
along the row direction. Each element in the table has P columns, where P is the
number of SPs.
Withthenew expression oftheJacobianmatrixD(θ
s
), thescorefunctionfrom
Eq. (3.27) becomes
q(y|θ
s
,R) =
2
3
R
n
(D1+D2+D3)
†
R
-1
(y−s(θ
s
))
o
, (3.32)
50
which consists of three small similar terms. In the following we show results for
D1
†
R
-1
(y−s(θ
s
)), and the other terms behave similarly. If we use the eigenvalue
decomposition ofR in Eqs. (A.1)-(A.3), the small term becomes
D1
†
R
-1
(y−s(θ
s
)) =D1
†
UΛ
-1
U
†
(y−s(θ
s
)) (3.33)
=D1
′†
Λ
-1
U
†
(y−s(θ
s
)) (3.34)
=D1
′†
y
′
res
(3.35)
where the projected matrixD1
′
is defined as
D1
′
=U
†
D1 (3.36)
=(I
t
⊗U
T
⊗U
R
⊗U
f
)
†
D1 (3.37)
=(I
t
D
1
4
)⋄(U
†
T
D
1
3
)⋄(U
†
R
D
1
2
)⋄(U
†
f
D
1
) (3.38)
=D
1′
4
⋄D
1′
3
⋄D
1′
2
⋄D
′
1
. (3.39)
An efficient method to compute y
′
res
is provided in Eq. (A.7) through tensor
operations. As a result, we can follow the numerical method summarized in Tab.
5-4 of [19] to compute Eq. (3.35).
To find an alternative expression for the FIM, we substitute Eq. (3.31) into
Eq. (3.28) and have
J =
2
9
R
n
(D1+D2+D3)
†
R
-1
(D1+D2+D3)
o
(3.40)
=
2
9
R
n
3
X
i=1
3
X
j=1
Di
†
R
-1
Dj
o
. (3.41)
51
Here we have 9 elements that follow the structure Dij = Di
†
R
-1
Dj, and Dji =
Dij
†
. We can evaluate each of them following the approach similar to Eq. (42) in
[69]. For instance, with an eigenvalue decomposition onR we can rewrite D12 as
D12=D1
†
R
-1
D2 (3.42)
=D1
†
UΛ
-1
U
†
D2 (3.43)
=D1
′†
Λ
-1
D2
′
, (3.44)
Similar to the definition ofD1
′
in Eq. (3.39),D2
′
is given by
D2
′
=U
†
D2 (3.45)
=D
1′
4
⋄D
2′
3
⋄D
2′
2
⋄D
′
1
. (3.46)
Next according to the PARATREE model proposed in [69], we may decompose
a 4D tensor reshaped from the diagonal vector of Λ
-1
, then find an approximate
expression forΛ
-1
given by
Λ
-1
≈
X
r
f
X
r
R
X
r
T
Λ
(t)
r
f
,r
R
,r
T
⊗Λ
(T)
r
f
,r
R
,r
T
⊗Λ
(R)
r
f
,r
R
⊗Λ
(f)
r
f
. (3.47)
Appendix A.2 provides more details on this approximation expression. After com-
bining Eqs. (3.39), (3.46), and (3.47) into Eq. (3.44), we finally arrive at an
approximate and simpler expression to evaluateD12.
D12≈
X
r
f
(D
′†
1
Λ
(f)
r
f
D
′
1
)⊙
X
r
R
(D
1′†
2
Λ
(R)
r
f
,r
R
D
2′
2
)⊙
X
r
T
(D
1′†
3
Λ
(T)
r
f
,r
R
,r
T
D
2′
3
)⊙(D
1′†
4
Λ
(t)
r
f
,r
R
,r
T
D
1′
4
) (3.48)
52
The other eight elements in Eq. (3.41) can also use this method to speed up the
evaluation.
3.4 Validation with Data
3.4.1 Simulation with Synthetic Channel Responses
We have conducted some simulations based on synthetic two-path channel
responses from Eq. 3.1, where we assume there are only two SPs affected by white
measurement noise. The simulations don’t consider the effect of DMC, because
our method shares the same DMC estimator with that in [19]. With the knowl-
edge of the true parameters, we can compare root mean squared errors (RMSEs)
of estimates between two estimators. The first is our method, and the second one
is the main method in [19] which neglects the phase variation between antenna
switching within one MIMO snapshot.
Table 3.1: RMSE comparison based on synthetic two path channel with one low
and one high Doppler shift (10,400) Hz, our esimator / [19]
Path 1 Path 2
True RMSE True RMSE
τ(ns) 1835.4 1.55e-02/1.55e-02 1100.9 5.93e-02/5.90e-02
ϕ
T
(deg) -94.8 1.14e-02/1.17e-01 -42.3 4.35e-02/4.83
ϕ
R
(deg) 2.3 1.10e-02/1.31e-02 178.3 6.46e-02/4.49e-01
ν(Hz) 10.0 1.20e-02/1.20e-02 400.0 5.53e-02/5.54e-02
|γ|
2
(dB) -6.3 3.07e-03/3.03e-03 -18.7 1.51e-02/8.28e-02
The simulation results are listed in Tab. 3.1. We can observe that the RMSEs
from the two estimators are similar, except for ϕ
T
, ϕ
R
and|γ|
2
where our method
almost always presents the smallest RMSE. The second estimator gives a high
RMSE for ϕ
T
and ϕ
R
whenever the Doppler shift is not small. The reason of
large RMSE for angular estimates in the second estimator is that it neglects the
53
phase variation between antenna switching. Its RMSE of ϕ
T
is higher than that
of ϕ
R
when the Doppler shift is large, as illustrated by Path 2 in Tab. 3.1. This
effect results from the switching pattern applied in this work, as shown in Fig. 3.1
that the phase offset will be larger between adjacent TX antenna switching than
RX due to the bigger time gap. Therefore in the second estimator, neglecting the
phase variation between antenna switching has a worse impact on ϕ
T
than ϕ
R
.
3.4.2 Evaluation with V2V MIMO Measurement Data
In this section, we apply our HRPE algorithm to the V2V measurement data
collected by a real-time MIMO channel sounder that we have constructed. The
sounder includes a pair of NI-USRP RIOs as the main RF transceivers, two GPS-
disciplinedrubidiumreferencesasthesynchronizationunitsandapairof8-element
UCAs. The design of our channel sounder focuses on the system stability and the
real-time streaming capability. With careful system and array calibrations, the
setup allows us to analyze the measurement data with our HRPE algorithm. The
increased measurement snapshot rate expands the resolvable range of Doppler
shift to±806 Hz without ambiguities, meanwhile the increased number of samples
providesasmootherpictureoftheevolutionofparametersofSPs. Tab3.2presents
the key parameters of the setup. More details can be found in [41].
Fig. 3.3 provides three screenshots of the video taken by the 360
◦
camera
mounted on top of the Rx array. Fig. 3.4 presents the route and map of the
sample measurements.
We split this continuous measurement route into four parts. Some sample
average power delay profiles (APDPs) are presented in Fig. 3.6, where “Data” is
the APDP of y, “SP” is s(
ˆ
θ
s
), “Res” is y−s(
ˆ
θ
s
) and “DMC” is the diagonal of
R
τ
. The TX and RX vehicles start with static positions. A sample APDP for this
54
Parameter Value
Carrier frequency 5.9 GHz
Bandwidth 15 MHz
Transmit power 26 dBm
Sampling rate 20 MS/s
Length of the sounding signal τ
max
4 µs
MIMO signal duration T
0
620µs
Number of frequency tones 61
Number of Tx antennas 8
Number of Rx antennas 8
Number of MIMO per burst 30
Rate of bursts 20 Hz
Table 3.2: Parameters of the real-time V2V MIMO channel sounder
(a) A black SUV passes by Rx
(b) Rx approaches Tx
(c) Tx turns around the corner
Figure 3.3: The screenshots of the video taken from the 360
◦
camera at the RX
SUV
scenario is shown in Fig. 3.6(a). We define the driving direction of the vehicle as
0
◦
in azimuth. As shown in Fig. 3.3, the red TX SUV is in front of the RX SUV
55
Figure 3.4: The route and map of the V2V measurements on USC campus, near
the GPS location of (34
◦
1
′
13.08” N, 118
◦
17
′
16.80” W), where the blue arrow is
the route of the Rx vehicle, and the red arrow is the route of the Tx vehicle
and facing away. The angular power spectrum (APS) between 10s and 30s in Fig.
3.7(d) verifies the azimuth DoA of the strong MPCs. Meanwhile the RX vehicle
is in the azimuth direction of 180
◦
, or equivalently −180
◦
, from the perspective
of the TX vehicle, and the results between 10s and 30s in Fig. 3.7(c) verify the
azimuth DoD of dominant MPCs.
In the second scenario, a third black SUV passes the Rx SUV from behind to
its front, stops behind the TX SUV for a while, and partially blocks the LOS, as
shown in Fig. 3.3(a). A sample APDP for this scenario is given in Fig. 3.6(b).
When compared to Fig. 3.6(a), it shows the blockage by the third SUV results
in an approximate 5 dB attenuation of the received power. A string of dots with
a high Doppler shift between t = 10s and 20s in Fig. 3.7(b) corresponds to the
MPCs from the passing black SUV. The Doppler shift quickly falls to zero when
56
the third SUV moves into the space between the Tx and Rx SUVs, which also
explains the waterfall shape of the string of dots.
In the third scenario, the black SUV changes lane and passes the TX around
t = 40s, which explains a string of dots with large negative Doppler between t =
40s and 45s in Fig. 3.7(b). Meanwhile around t = 45s, the RX car pulls out
from the right curb into the middle of the road. This subtle swing of the driving
direction is also tracked and reflected by the results around 40s in Fig. 3.7(d),
when the direction of dominant SPs evolves from 0
◦
to approximately −30
◦
and
back to 0
◦
. The RX car starts moving towards the TX, when the TX car remains
static until the RX car almost reaches the rear of the TX around 60s. Results
between 40s and 60s in Figs. 3.7(a) and 3.7(b) match this activity with decreased
delays, positive Doppler shifts and increased power of the dominant SPs.
In the fourth scenario, the RX stops behind the stop sign, while the TX SUV
starts turning right at the corner around t = 60s before disappearing from the
sight of RX at the intersection around t = 70s. Fig. 3.3(c) provides a picture of
this scenario. The scatter plot in Fig. 3.7(a) indicates an increase of the delays of
strong SPs during this period, while Fig. 3.7(b) shows that the strong SPs have
negative Doppler shifts, because the distance between TX and RX increases. The
azimuth DoA and DoDof the strong SPs also evolve during this process, as shown
in Figs. 3.7(c) and 3.7(d). Fig. 3.6(c) gives a sample APDP for this scenario.
Between t = 70s and t = 80s the link between TX and RX is fully obstructed by
the the building. Figs. 3.7(c) and 3.7(d) provide the angles of the main SPs in
this NLOS scenario and indicate they are mostly likely the diffraction around the
corner and the reflection from the wall of the building. Fig. 3.6(d) gives a sample
APDP for this NLOS scenario.
57
To evaluate the performance of the estimator and the goodness-of-fit of the
model, we examine the dPR, and it is the power ratio due to the contribution of
DMC.
dPR =
ky−s(
ˆ
θ
s
)k
2
−Mσ
2
n
kyk
2
. (3.49)
Fig. 3.5 gives the cumulative distribution function (CDF) of dPR from the eval-
uation of 1400 MIMO burst snapshots on this route, and indicates about 80% of
the path extraction results have a dPR less than 0.2. The values of dPR are com-
parable to the ones in[29, 70, 71]. It is noteworthy that the average dPR is higher
when the distance between TX and RX is less than 20m. An example of such
case is given in Fig. 3.6(c). We conjecture it is because the modeling mismatch
gets worse at a shorter distance between TX and RX, because of the assumption
that SPs travel in the horizontal plane and the data model may omit the elevation
angles. However theperformance ofthe estimatorissatisfying because most ofthe
dominant SPs can be traced and mapped to the environment. Another plausible
interpretation is that the number of scatterers illuminated by the TX increases
when the distance is small, which leads to a relatively larger dPR.
ζ
0 0.2 0.4 0.6 0.8 1
P
r
(dPR≤ζ)
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Figure 3.5: The CDF of the dPR based on 1400 snapshots from the continuous
V2V measurement on USC campus
58
τ/μs
0 1 2 3
dB
-110
-100
-90
-80
-70
-60
(a) t = 11.000s, dPR 5.2%
Data
SP
Res
DMC
τ/μs
0 1 2 3
dB
-110
-100
-90
-80
-70
-60
(b) t = 31.500s, dPR 12.6%
Data
SP
Res
DMC
τ/μs
0 1 2 3
dB
-100
-90
-80
-70
-60
-50
(c) t = 67.950s, dPR 27.7%
Data
SP
Res
DMC
τ/μs
0 1 2 3
dB
-120
-110
-100
-90
-80
-70
-60
(d) t = 75.450s, dPR 21.9%
Data
SP
Res
DMC
Figure 3.6: Sample APDPs of (a) scenario I: Tx and Rx are both static with LOS
(b) scenario II: Tx and Rx are both static with LOS obstructed a third SUV (c)
scenario III: Rx drives toward Tx with LOS (d) scenario IV: Tx drives around the
corner with LOS obstructed by the building.
From the extracted SPs we can conveniently compute some important channel
metrics such asroot-meansquared (RMS) delayspread, Dopplerspread and angu-
lar spread. Figs. 3.8(a) and 3.8(b) present the RMS delay and Doppler spread
based on the extracted SPs. They are basically the square root of the second
central moment of the power delay profile and Doppler spectrum. Both the delay
spread and Doppler spread are important channel metrics in 802.11p [72]. Mean-
while Figs. 3.9(a)and 3.9(b)show the angularspread σ
ϕ
for DoDand DoA,which
is computed by
µ ϕ
=
P
P
p=1
e
jϕp
|γ
VV,p
|
2
P
P
p=1
|γ
VV,p
|
2
(3.50)
σ
ϕ
=
v
u
u
t
P
P
p=1
|e
jϕp
−µ ϕ
|
2
|γ
VV,p
|
2
P
P
p=1
|γ
VV,p
|
2
. (3.51)
59
(a) time varying PDP (b) time varying Doppler spectrum
(c) time varying APS at Tx (d) time varying APS at Rx
Figure 3.7: Color weighted scattered plots of extracted SPs, we show the SPs
whose |γ
VV
|
2
are greater then -80dB
We observe in Fig. 3.9(a) that the angular spread of DoD increases in the second
and fourth scenarios when the LOS is obstructed, while it decreases in the third
scenario when a strong LOS is present. A smart antenna system can exploit some
multiplexing gain in the second and fourth scenarios and some beamforming gain
in the third one.
3.5 Summary and Discussion
In this paper we introduced a robust and efficient HRPE algorithm to evaluate
the time-varying V2V channel. The new scheme provides a joint estimation of the
delay, DoD, DoA and Doppler shift of MPCs with a high accuracy. We have also
60
t / s
10 20 30 40 50 60 70 80
σ
τ
/ ns
20
40
60
80
100
120
140
t / s
10 20 30 40 50 60 70 80
σ
ν
/ Hz
0
10
20
30
40
Figure 3.8: The time-varying RMS delay and Doppler spread of SPs from the
sample continuous route
t / s
10 20 30 40 50 60 70 80
σ
φ
T
/ deg
0
0.2
0.4
0.6
0.8
1
(a)
t / s
10 20 30 40 50 60 70 80
σ
φ
R
/ deg
0
0.2
0.4
0.6
0.8
1
(b)
SUV Passing
by
LOS (close)
Around the
Corner
Figure 3.9: The time-varying angular spread of (a) azimuth DoD, (b) azimuth
DoA
optimizedthenumericalimplementationofthealgorithmtoallowafastevaluation
for large number of points on a continuous measurement route. We have validated
our estimation scheme with actual V2V channel measurement data. Results have
indicated an overall good performance of the estimator, which is based on the
61
reasonable values of dPR and the observation that strong SPs match well with the
environment and the dynamics of the measurement. These new results will enable
a deeper insight into the propagation mechanism of the V2V channel and provide
the necessary tools to develop a new or better V2V channel model.
62
Chapter 4
Switching Sequences for Fast
Time-varying Channels
TDM channel sounders, in which a single RF chain is connected sequentially
via an electronic switch to different elements of an array, are widely used for the
measurement of double-directional/MIMO propagation channels. This chapter
investigates the impact of array switching patterns on the accuracy of parameter
estimation of MPCs for a TDM channel sounder. The commonly-used sequential
(uniform) switching pattern poses a fundamental limit on the number of antennas
that a TDM channel sounder can employ in fast time-varying channels. We thus
aim to design improved patterns that relax these constraints. To characterize the
performance, we introduce a novel spatio-temporal ambiguity function, which can
handle the non-idealities of real-word arrays. We formulate the sequence design
problem asan optimization problem and propose an algorithmbased on simulated
annealingtoobtaintheoptimalsequence. Asaresultwecanextendtheestimation
range of Doppler shifts by eliminating ambiguities in parameter estimation. We
show through Monte Carlo simulations that the root mean square errors of both
directionofdepartureandDopplerarereducedsignificantlywiththenewswitching
sequence. Results are also verified with actual V2V channel measurements.
63
4.1 Introduction
Realistic radio channel models are essential for development and improvement
of communication transceivers and protocols [3]. Realistic models in turn rely on
accurate channel measurements. Variousimportant wireless systems areoperating
in fast-varying channels, such as mobile millimeter wave (mmWave) [73], V2V [57]
and high speed railway systems [74]. The need to capture the fast time variations
of such channels creates new challenges for the measurement hardware as well as
the signal processing techniques.
Since most modern wireless systems use multiple antennas, channel measure-
ments also have to be done with MIMO channel sounders, also known as double-
directional channel sounders. There are three types of implementation of such
sounders: (i) full MIMO, where each antenna element is connected to a different
RF chain [75], (ii) virtual array, where a single antenna is moved mechanically to
emulatethepresenceofmultipleantennas[76],and(iii)switched array,alsoknown
as TDM sounding, where different physical antenna elements are connected via an
electronic switch to a single RF chain [77]. The last configuration is the most
popular in particular for outdoor measurements, as it offers the best compromise
between cost and measurement duration.
From measurements of MIMO impulse responses or transfer functions, it is
possible to obtain the parameters (DoD, DoA, delay, and complex amplitude) of
theMPCsbymeansofHRPEalgorithms. MostHRPEalgorithmsarebasedonthe
assumption thatthedurationofoneMIMOsnapshot (themeasurement ofimpulse
responsesfromeverytransmittoeveryreceiveantennaelement)isshorterthanthe
coherence time of the channel. Equivalently, this means the MIMO cycle rate, i.e.
the inverse of duration between two adjacent MIMO snapshots, should be greater
than or equal to half of the maximal absolute Doppler shift, in order to avoid
64
ambiguities in estimating Doppler shiftsof MPCs. Since in a switched sounder the
MIMO snapshot duration increases with the number of antenna elements, there
seems to be an inherent conflict between the desire for high accuracy of the DoA
and DoDestimates (which demands a larger number of antenna elements) and the
admissible maximum Doppler frequency.
Yin et al. were the first to realize that the Doppler shift estimation range
(DSER) can be extended by a factor equal to the product of TX and RX antennas
in TDM channel sounding [50]. They studied the problem in the context of the
ISI-SAGE algorithm [18]. Pedersen et al. later discovered that it was the choice
of SS patterns that caused this limit in the TDM channel sounding [51]. non-SS
patternscanpotentiallysignificantly extend theDSERbyeliminating theambigu-
ities. However the analysis is performed under ideal conditions: firstly the paper
assumes that all antenna elements are isotropic radiators, secondly it requires the
knowledge of the phase centers of all antennas. Both assumptions are difficult to
fulfill for realistic arrays used in channel sounding, given the unavoidable mutual
coupling between antennasandthe presence ofametallic support frame. Pedersen
et al. proposed to use the so-called NSL of the objective function as the metric to
evaluate switching patterns, and further derived a necessary and sufficient condi-
tion of the switching sequences that lead to ambiguities [52], but the method to
derivegoodswitchingsequencesforrealisticarraysisnotclear. Ourworkattempts
to close these gaps.
Our work adopts an algebraic system model that uses decompositions through
EADFs[53],whichprovidesareliableandelegantapproachforsignalprocessingon
real-world arrays. Therefore our analysis no longer requires the isotropic radiation
pattern or prior knowledge about the antenna phase centers. Based on the Type-
I ambiguity function for an arbitrary array [78], we propose a spatio-temporal
65
ambiguity function and investigate its properties and impact on the estimation
of directions and Doppler shifts of MPCs. We also demonstrate that this spatio-
temporal ambiguity function is closely related to the correlation function induced
from the MLE utilized in the state-of-the-art RiMAX algorithm. Inspired by Ref.
[54], we model the array pattern design problem as an optimization problem and
propose an annealing algorithm to search for an acceptable solution. Besides we
proposeanHRPEalgorithmthatadoptsasignaldatamodelthatincorporatesthe
optimized array switching pattern.
The main contributions of this work are the following:
• wemodeltheselectionofarrayswitchingschemeasanoptimizationproblem,
and introduce the spatio-temporal ambiguity function that also incorporates
realistic arrays with the aid of EADFs;
• we integrate the optimized array switching pattern into an HRPE algorithm
and compare the parameter estimation variance with the Cramer-Rao lower
bound (CRLB) through Monte-Carlo simulations;
• we also modify the switching pattern on a real-time MIMO channel sounder
and use actual V2V measurement data to show the effectiveness of the opti-
mized switching pattern.
The remainder of the chapter is organized as follows. Section II introduces the
general spatio-temporal ambiguity function and the associated signal data model
used in the TDM-based channel sounding. In Section III we simplify the prob-
lem by only allowing the TX to have cycle-dependent switching patterns, and
present the formulation of the optimization problem and its solution based on a
SAalgorithm. InSectionIVwe introduceanHRPEalgorithmthatgeneralizes the
evaluation of TDM MIMO channel measurement and incorporates the optimized
66
switching pattern. Section V validates the switching sequence and correspond-
ing HRPE algorithm via extensive Monte Carlo simulations and measured V2V
channel responses. In Section VI we draw the conclusions.
The symbol notation used in this chapter follows the rules below.
• Bold upper case letters, such asB, denote matrices. Bold lower case letters,
such as b, denote column vectors. For instance b
j
is the j-th column of the
matrixB.
• Calligraphic upper-case letters denote high-dimensional tensors.
• [B]
ij
denotes the element in the ith row and jth column of the matrixB.
• Superscripts
T
and
†
denote matrix transpose and Hermitian transpose.
• The operators |f(x)| and kbk denote the absolute value of a scalar-valued
function f(x) and the Euclidean norm of a vector b.
• The operators ⊗, ⊙ and ⋄ denote Kronecker, Schur-Hadamard, and Khatri-
Rao products respectively.
4.2 Signal Model and Ambiguity Function
4.2.1 Signal Data Model
This work mainly studies the antenna switching sequence in the TDM channel
sounding problem. We consider T MIMO measurement snapshots in one observa-
tion window, each with M
f
frequency points, M
R
receive antennas, and M
T
trans-
mit antennas. The adjacent MIMO snapshots are separated by T
0
. We assume
that all scatterers are placed in the far field of both TX and RX arrays, which
67
also implies that MPCs are modeled as plane waves. Besides TX and RX arrays
are vertically polarized by assumption and have frequency-independent responses
within the operating bandwidth. Such T MIMO snapshots can span larger than
the coherence time of the channel, but we assume that the structural parameters
of the channel, also known as the large-scale parameters, such as path delay, DoA,
DoD and Doppler shift, remain constant during this period.
1
A vectorized data model for the observation of T MIMO snapshots is given
in (5.7). It includes contributions from deterministic SPs s(θ
sp
), DMC n
dmc
and
measurement noise n
0
.
y =s(θ
sp
)+n
dmc
+n
0
(4.1)
The data model of an observation vector for a number of P SPs is determined by
s(θ
sp
) =B(μ,η
T
,η
R
)·γ
vv
, (4.2)
where the SP parameter vector θ
sp
includes the structural parameters μ and the
path weights γ
vv
. The basis matrix B has dimension M × P, and M is the
total number of observations given by M
f
M
R
M
T
T. The i-th column stands for
the vectorized basis channel vector for the i-th SP. The switching sequences η
T
and η
R
for TX and RX respectively not only affect the sequence of samples from
different antennas, but also impact the phase variation because of the nature of
TDM channel sounding.
The complexity of the basis matrix grows as the irregularity of switching
sequence increases. Most papers assume that SS sequences are used at both TX
and RX, hence the periodicity can be exploited to greatly simplifyB. An example
1
This does not mean that the channel is assumedto be static, instead for eachpath the phase
varies over time due to the presence of Doppler shift.
68
ofSSsequencesisshowninFig. 4.1. References[24]and[47]adoptedadatamodel
of SPs where the channel is assumed to be completely static within one MIMO
snapshot, as a result the basis matrix is a Khatri-Rao product of three smaller
matrices. To evaluate fast time-varying channel, such as the V2V communication
channel, a more sophisticated signal model is used in Ref. [46], which considers
the phase variation due to Doppler shifts between switched antennas, but again
the model is only applicable for SS sequences.
4.2.2 Spatio-temporal Ambiguity Function
The Type-I ambiguity function for an antenna array can reflect its ability to
differentiate signals in the angular domain [78]. Generalizing the definition to
include the full structural parametersμfrom Eq. (4.2), we have a spatio-temporal
ambiguity function
X
tot
(μ,μ
′
) =
b
†
(μ)b(μ
′
)
kb(μ)k·kb(μ
′
)k
, (4.3)
whereb is one column of the basis matrixB in Eq. (4.2). The ambiguity function
also depends on the switching sequences η
T
and η
R
, but we drop them from the
notation for brevity.
Moreimportantlythisambiguityfunctioniscloselyrelatedwiththecorrelation
functionthatistightlyconnectedwiththeobjectivefunctionoftheMLEdeveloped
inSection 4.4.1, hence studying the propertiesofthisambiguity function iscritical
for the performance of the MLE.
Here we introduce some properties of the ambiguity function.
Property 1. X
tot
(μ,μ)= 1, and 0≤

X
tot
(μ,μ
′
)

≤ 1.
69
It is straightforward to prove that X
tot
(μ,μ) = 1. We can use the Cauchy-
Schwartz inequality to prove the inequality

X
tot
(μ,μ
′
)

≤ 1, and the equality is
obtained when μ=μ
′
.
Property 2 (Separability of the Ambiguity Function). We can prove that the
ambiguityfunctioninEq. (4.3)isaproductoftwocomponentambiguityfunctions
of delay τ and κ. The latter consists of every parameter in μ except τ.
X
tot
(μ,μ
′
) =X
τ
(τ,τ
′
)X(κ,κ
′
) (4.4)
The proof relies on the following property of Khatri-Rao products between two
vectors. In fact, the Khatri-Rao product between two vectors is equivalent to the
Kronecker product.
(a⋄b)
†
·(a
′
⋄b
′
)= (a
†
a
′
)·(b
†
b
′
) (4.5)
This property holds whena anda
′
have the same length, so dob andb
′
. Detailed
derivations are provided in Appendix B.1.
Property3. Theestimationproblemhasambiguitieswhenthefollowingcondition
holds.

X(κ,κ
′
)

= 1,∃κ
′
6=κ (4.6)
This spatio-temporal array ambiguity function is also closely related to the
ambiguity function well studied in MIMO radar. The MIMO radar ambiguity
function introduced in Ref. [54] allowsTX tosend different waveforms on different
antennas, while our problem considers a repeated sounding waveform for all TX
antennas. The Doppler-(bi)direction ambiguity function introduced in Ref. [52]
is quite similar to ours. However their ambiguity function assumes that the array
hasidentical elementswithoutanymutualcouplingeffects, ourambiguityfunction
70
can handle arbitrary array structures, which suits better for developing non-SS
sequences for MIMO channel sounding.
4.2.3 Simplified Signal Data Model
In this subsection we impose some constraints on the switching sequences in
order to obtain a more tractable problem. As a comparison the basis matrix B
with the SS sequences at both TX and RX is given by [46, (20)]
B(μ)=B
t
⋄
˜
B
TV
⋄
˜
B
RV
⋄B
f
. (4.7)
One straightforward relaxation on the switching patterns from SS sequences is to
allow the TX array to have a cycle-dependent switching pattern,
2
hence the new
basis matrix needs to merge the Khatri-Rao product of two basis matricesB
t
and
˜
B
TV
into
˜
B
TV,T
, in order to reflect such relaxation.
B(μ) =
˜
B
TV,T
⋄
˜
B
RV
⋄B
f
(4.8)
=

˜
B
1
TV
···
˜
B
T
TV

T
⋄
˜
B
RV
⋄B
f
, (4.9)
where
˜
B
j
TV
∈C
M
T
×P
with j = 1,2,...,T represents the spatio-temporal response
of the TX array at the j-th MIMO snapshot, and
˜
B
RV
∈ C
M
R
×P
is for the RX
array with the sequential switching, and B
f
∈ C
M
f
×P
is the basis matrix that
captures the frequency response due to path delay.
Instead of allowing both TX and RX arrayto switch after each sounding wave-
form (fully-scrambled), we focus on the switching sequences where the RX first
switches through all possible antennas while the TX remains connected with the
2
One cycle here means one MIMO snapshot.
71
same antenna. This type of sequences has a relatively simpler basis matrix com-
pared with the fully scrambled case. Compared to (4.8) the fully scrambled case
needs to further merge
˜
B
TV,T
and
˜
B
RV
. Another motivation for studying this
type of switching sequences is the efficiency to operate the channel sounder. Many
high-power TX switches tend to have a longer switching settling time than the
RX switches. This type of switching sequences that we are interested in effectively
limits the number of switching for the TX array, reduces the guard time needed in
thesoundingsignalandimprovestheoverallmeasurement efficiency ofthechannel
sounder.
Since our work focuses on channel sounding in a fast time-varying channel, the
phase variation within one MIMO snapshot is no longer negligible, and the new
TX or RX basis matrices for the t-th MIMO snapshot become weighted versions
of the static array basis matrices. The exact connections are given by
˜
B
t
TV
=B
TV
⊙A
t
T
(4.10)
˜
B
RV
=B
RV
⊙A
R
, (4.11)
where A
R
and A
t
T
are weighting matrices that capture the phase change due to
effects of Doppler and switching sequences. A
t
T
depends on the MIMO snapshot
index t, because TX implements a cycle-dependent switching pattern. Let us use
a M
T
×T matrix η
T
to represent the TX switching pattern, the elements in the
phase weighting matrices A
t
T
and A
R
are given by
[A
t
T
]
m
T
,p
= e
j2πνp[η
T
]m
T
,t
(4.12)
[A
R
]
m
R
,p
= e
j2πνpm
R
t
0
, (4.13)
72
Figure 4.1: An example of the SS switching for the 3×3 MIMO setup
where ν
p
is the Doppler of the p-th path. As shown in the example in Fig. 4.1, we
denote the duration between two switching events as t
1
and t
0
respectively for the
TX and RX array. The TX switching timing matrixη
T
is given by
[η
T
]
m
T
,t
=(t−1)M
T
t
1
+([S
T
]
m
T
,t
−1)t
1
. (4.14)
[S
T
]
m
T
,t
takes an integer value between 1 and M
T
and represents the scheduled
switching index of the m
T
-th TX antenna for the t-th MIMO snapshot. For the
SS sequence at TX, we have [S
T
]
m
T
,t
= m
T
, ∀t = 1,2,...,T. It is not difficult to
see that
˜
B
TV,T
can be broken down intoB
t
⋄
˜
B
TV
in (4.7), which means the basis
matrix in (4.9) is a generalized version of (4.7)
4.3 Switching Pattern Optimization
Based on Property 3, it is reasonable to argue that a good switching sequence
does not lead to an estimation problem that has ambiguities. Although Ref. [52]
focuses on the condition of switching sequences that generate the smallest CRLB,
73
Time Index
0 5 10 15 20 25
Antenna Index
1
2
3
4
5
6
7
8
Uniform
Scrambled
Figure 4.2: Comparison between the uniform and scrambled TX switching
sequences with M
T
= 8 and T = 3
we have found through simulations that CRLB is almost identical for the sequen-
tial switching and the scrambled switching for practical arrays. Suppressing the
sidelobe levels of the ambiguity function within the parameter domain of interest
leads to better switching sequences. This is because CRLB is only relevant for
unbiased estimators [79].
Instead of directly evaluating the ambiguity function in (4.3), we can find a
upper bound for its amplitude, which merely depends on the azimuth DoD, the
Doppler Shift, and the TX switching pattern. The upper bound is given by
|X
tot
(μ,μ
′
)|=|X
τ
(τ,τ
′
)|·|X(κ,κ
′
)| (4.15)
≤|X(κ,κ
′
)| (4.16)
=|X
T
(ϕ
T
,ϕ
′
T
,ν,ν
′
)X
R
(ϕ
R
,ϕ
′
R
,ν,ν
′
)| (4.17)
≤|X
T
(ϕ
T
,ϕ
′
T
,ν,ν
′
)|, (4.18)
74
where the inequalities in (4.16) and (4.18) use Property 1, and Eq. (4.17) uses
Property 2. Furthermore we find that the upper bound only depends on the
Doppler difference ∆ ν = ν
′
−ν, which is given by
X
T
(ϕ
T
,ϕ
′
T
,ν,ν
′
) =
˜
b
†
TV,T
(ϕ
T
,ν)
˜
b
TV,T
(ϕ
′
T
,ν
′
)
k
˜
b
†
TV,T
(ϕ
T
,ν)kk
˜
b
TV,T
(ϕ
′
T
,ν
′
)k
(4.19)
=X
T
(ϕ
T
,ϕ
′
T
,∆ ν). (4.20)
Appendix B.2 provides a detailed derivation of Eq. (4.20), as well as a simpler
form of X
T
(ϕ
T
,ϕ
′
T
,∆ ν).
It is well known that the Doppler shifts and the impinging directions of the
plane waves may contribute to phase changes at the output of the array. The
Doppler shift leads to a phase rotation at the same antenna when it senses at
different time instants, while the propagation direction of the plane wave also con-
tributes to a phase change between two antennas. The periodic structure of the
uniform switching sequence leads to ambiguities in the joint estimation of Doppler
and propagation direction. It is because the estimator may find more than one
plausible combination of Doppler and angle that can produce the phase changes
over different antennas and time instants. For example, Fig. 4.3 plots the ampli-
tudeupperboundgivenin(4.18),whentheTXusestheuniformswitchingpattern.
We observe multiple peaks in addition to the central peak located at (0,0). It also
shows that the non-ambiguous estimation range of Doppler is [−1/2T
0
,1/2T
0
),
when both TX and RX implement uniform switching patterns and ϕ
′
T
= 0. The
value of T
0
in this example is 620µs which is based on the transmitted signal in
Ref. [41].
However the design complexity for a fully scrambled switching array may
becomeprohibitiveespeciallywhenthenumberofantennasbecomeslarge. Instead
75
Figure 4.3: Amplitude of Ambiguity function in dB with Azimuth DoD and
Doppler shift, under uniform switching schemes at both Tx and Rx
we resort to a simplified type of switching sequence introduced in Section 4.2.3,
where only the TX array uses a scrambled switching sequence. The DSER can
potentially grow by a factor of M
T
from 1/T
0
to M
T
/T
0
, compared to the max-
imal boost of M
T
M
R
in the fully scrambled case. Therefore the constraint on
the switching sequence simplifies the problem formulation and the corresponding
parameter extraction algorithm, while it can still provide a boost on the DSER
that is sufficient for many practical purposes.
4.3.1 Problem Formulation
The intuitive objective ofourarrayswitching design problem istofindschemes
that effectively suppress the sidelobes of the spatial-temporal ambiguity function
shown in Fig. 4.3, hence increases the DSER. Refs. [52] and [54] prove that their
76
Figure 4.4: Amplitude of Ambiguity function in dB with Azimuth DoD and
Doppler shift, under uniform Rx switching and scrambled Tx switching scheme
ambiguityfunctionshaveconstantenergy,soapreferableschemeshouldspreadthe
volume under the high sidelobes evenly elsewhere. However the proof again uses
the idealized assumption about antenna arrays, thus it cannot be applied directly
in our case. Here we introduce the function f
p
(η
T
), which is given by
f
p
(η
T
) =
ZZZ
D

X
T
(ϕ
T
,ϕ
′
T
,∆ ν)

p
dϕ
T
dϕ
′
T
d∆ ν, (4.21)
D ={(ϕ
T
,ϕ
′
T
,∆ ν)|ϕ
T
,ϕ
′
T
∈ (−π,π]&∆ ν ∈ [0,ν
up
]}
whereDistheintegrationintervals,andν
up
representsthetargetmaximalDoppler
shiftbelowwhichwewanttoavoidtheambiguity. ThisvalueshouldequalM
T
/2T
0
,
because the periodic sequence is T
0
/M
T
long for the general cycle-dependent η
T
.
Numerical simulations are conducted based on the ambiguity function in Eq.
77
1118 1118.2 1118.4 1118.6 1118.8 1119
x
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
F(x)
Empirical CDF
Figure 4.5: The CDF of the energy of the ambiguity function with various TX
switching schemes η
T
, based on 1000 different configurations
(4.18), with the EADF of the far-field TX array pattern calibrated in the ane-
choic chamber. The array is one used in a V2V MIMO channel sounder [41]. Fig.
4.5 suggests that the energy of the ambiguity function, i.e. f
2
(η
T
), is almost con-
stant regardless of the choices of different η
T
. As a result we can use f
p
(η
T
) with
a higher value of p as the cost function to penalize the TX switching schemes that
lead to high sidelobes. In summary our optimization problem is given by
min
S
T
∈C
f
p
(η
T
) (4.22)
s.t. [η
T
]
m
T
,t
= ([S
T
]
m
T
,t
−1)t
1
+(t−1)M
T
t
1
,
where the elements in the set C are integer-valued matrices with a dimension of
M
T
×T, and every column ofS
T
is a permutation of the vector [1,2,...,M
T
]
T
.
78
4.3.2 Solution and Results
BecauseS
T
takes on discrete values in the feasible set, the simulated annealing
algorithm is known to solve this type of problem [80]. The pseudocode of our
proposed algorithm is given here in Alg. 2. The key parameters related to this
Algorithm 2 The SA-based algorithm to solve the problem (4.22)
1: Initialize η
T
, the temperature T =T
0
, and α = α
0
;
2: while k≤k
max
or f
p
(η
T
)> ǫ
th
do
3: η
′
T
= neighbor(η
T
);
4: if exp
h
(f
p
(η
T
)−f
p
(η
′
T
))/T
i
> random(0,1)then
5: η
T
=η
′
T
6: end if
7: T =αT
8: end while
algorithm are p = 6, the initial temperature T
0
= 100, the cooling rate α
0
= 0.97
and the k
max
= 500. The parameters, particularly α
0
and k
max
, are selected as
a tradeoff between the objective function value and the convergence speed. The
operator random(0,1) outputs a random number uniformly distributed between
0 and 1. The operator neighbor(η
T
) provides a “neighbor” switching pattern by
swapping two elements in the same column ofη
T
.
The SA algorithm implements Markov-Chain Monte Carlo sampling on the
discrete feasible set C. If we denote f
p
(η
T
) as e and f
p
(η
′
T
) as e
′
. The acceptance
probability given by line 4 of Alg. 2 is
P
r
(η
T
,η
′
T
,T)=









exp
h
(e−e
′
)/T
i
, if e
′
≥e
1, otherwise
(4.23)
Fig. 4.6provides the valuesofthe objective function with the iteration number
and decreased temperature in the SA algorithm. We present the amplitude of the
79
100 200 300 400 500
Iteration Number
20
40
60
80
100
120
0
20
40
60
80
100
Figure 4.6: The evolution of f
6
(η
T
) and temperature during the annealing algo-
rithm in Algorithm 2
2Dambiguityfunction(theupperboundgiveninEq. (4.18))withthefinalswitch-
ing sequence in Fig. 4.4, where the high sidelobes clearly disappear in contrast
with Fig. 4.3. Another useful metric is the NSL used in Ref. [51] to measure the
quality of the switching sequence. The NSL here is −13.60dB, while the lowest
NSLamong thethree proposed sequences in Ref. [51] is−11.06dB. Given the fact
thatoursequence design approach isnotlimited toideal uniformlinear arraysand
thus more flexible, the reduction of 2.54dB on NSL over the best sequence in Ref.
[51] is encouraging.
4.4 Parameter Extraction Algorithm
This section introduces an HRPE algorithm to evaluate the channel sounding
data when the TX array employs the optimal switching sequence. It is based on
80
the framework of RiMAX [19], which converges significantly faster than the ISI-
SAGE algorithm thanks to the joint optimization of all SPs’ parameters. For an
introduction to the estimation framework we refer to [46, Sec. III]. We emphasize
the changes made in the parameter initialization algorithm for SPs and the local
optimization algorithm, when compared to our previous work [46].
4.4.1 Path Parameter Initialization Method
The outline of the parameter initialization algorithm of SPs is the same with
that given in [46, Alg. 1]. It is based on the idea of subsequent signal detec-
tion, estimation and subtraction. However the main objective function used in
signal detection and estimation, also known as the correlation function, needs to
be adjusted. It is given by
C(μ,y) =(y
†
R
-1
B)(B
†
R
-1
B)
-1
(B
†
R
-1
y). (4.24)
Accordingto[46,Alg. 1],oneimportantstepistoevaluatethecorrelationfunction
on a N
f
×N
R
×N
T
×N
t
multidimensional search grid ˜ μ
L
. With the increased
DSER N
t
, the number grid points in the time domain, requires an increase from
2T to2M
T
T. Ifwe compute thecorrelationfunction given in(4.24)forallpossible
points in ˜ μ
L
, we can organize the results into a 4D tensor that shares the same
dimension with ˜ μ
L
. Its n
t
-th child tensor in the last domain (time domain) can
be expressed by
C(:,:,:,n
t
)=T
nt
1
⊙T
nt
1
⊘T
nt
2
, (4.25)
where ⊘ is the element-wise division between two tensors. Similar to our method
in Ref. [47], we can exploit the data structure of B and ˜ μ
L
, and apply tensors
81
products to greatly accelerate the computation [68]. Appendix B.4 reveals the
detailed procedures on how to compute the two tensors T
nt
1
and T
nt
2
.
4.4.2 Parameter Joint Optimization Method
With an estimate of R and an initial value of θ
s
, we further improve θ
s
with
the Levenberg-Marquardt (LM) method. The update equation of θ
s
for the i-th
iteration is given by
ˆ
θ
i+1
s
=
ˆ
θ
i
s
+
h
J(
ˆ
θ
i
s
,R)+ζI⊙J(
ˆ
θ
i
s
,R)
i
-1
q(y|
ˆ
θ
i
s
,R). (4.26)
This iterative optimization step requires the evaluation of the score function
q(y|θ
s
,R) and FIM J(θ
s
,R). Computationally attractive methods are provided
in [46, Sec. III-C]. The key to apply this method is to rewrite the Jacobian matrix
D(θ
s
) as a sum of Khatri-Rao products and apply the property in (4.5), but an
update is needed because of the new signal data model given in (4.2) and (4.9).
Details about the Jacobian matrix can be found in Appendix B.5. The expression
is given by
D(θ
s
)=
1
2
(D
1
3
⋄D
1
2
⋄D
1
+D
2
3
⋄D
2
2
⋄D
1
), (4.27)
where the details of each small D matrix are summarized in Tab. A.1. To recon-
structonesmallDmatrixfromthistable,onecanconcatenatethematricesrelated
to D along the row direction. Each element in the table has P columns, where P
is the number of SPs.
82
Table 4.1: Components to compute the Jacobian matrix in Eq. (4.27)
τ ϕ
T
ϕ
R
ν |γ
vv
| φ
vv
D
1
D
f
⋄γ
T
vv
B
f
⋄γ
T
vv
B
f
⋄γ
T
vv
B
f
⋄γ
T
vv
B
f
⋄e
jφ
T
vv
jB
f
⋄γ
T
vv
D
1
2
˜
B
RV
˜
B
RV
D
ϕ
R
,V
⊙A
R
˜
B
RV
˜
B
RV
˜
B
RV
D
2
2
˜
B
RV
˜
B
RV
D
ϕ
R
,V
⊙A
R
B
RV
⊙D
ν,R
˜
B
RV
˜
B
RV
D
1
3
˜
B
TV,T
˜
D
ϕ
T
,T
˜
B
TV,T
˜
D
ν,T
˜
B
TV,T
˜
B
TV,T
D
2
3
˜
B
TV,T
˜
D
ϕ
T
,T
˜
B
TV,T
˜
B
TV,T
˜
B
TV,T
˜
B
TV,T
4.5 Validation with Data
In this section we use both Monte Carlo simulations and actual V2V mea-
surement data to validate the choice of the switching sequence from Section 4.3
and the performance of HRPE algorithm described in Section 4.4, which we call
RiMAX-RS for brevity. We compare it with the HRPE algorithm in Ref. [46] and
will use RiMAX-4D to represent it.
4.5.1 Simulation with Synthetic Channel Responses
First we simulate the single-path channel, whose parameters are listed in Tab.
4.2. To cover several cases of interest, the Doppler shift is larger than 1/2T
0
≈
806Hz in snapshot 1and smaller than 1/2T
0
in snapshot 2, where T
0
equals 620µs
basedontheconfigurationsinRef. [41]. WecomparetheRMSEswiththesquared
root of the CRLB as a function of SNR ρ for two switching sequences, which are
the SS sequence η
T,u
and our optimized non-SS TX sequence η
T,f
. The SNR is
evaluated according to
ρ =
ks(θ
s
)k
2
σ
2
n
(4.28)
83
At each SNR value, the path weight γ is scaled accordingly and 1000 realizations
of the channel are generated and subsequently estimated with RiMAX-RS. The
theoretical CRLB can be determined based on FIM and given by
σ
2
θs
diag(J
-1
(θ
s
)), (4.29)
where is the generalized inequality for vectors. Figs. 4.7 and 4.8 provide such
a comparison forη
T,f
, which demonstrates its good performance in both channels
with high or low Doppler. On the other hand, Fig. 4.9 shows the poor estimation
accuracyinthehighDopplercaseforη
T,u
,althoughthemeansquarederror(MSE)
can achieve the CRLB in the low Doppler scenario as expected in Fig. 4.10.
We also show inFig. 4.11the delay-Doppler spectrum ofsnapshot 1with three
different TX switching sequences, which are η
T,u
, η
T,d
(known as the “dense”
sequential sequence with MIMO snapshot duration reduced toT
0
/8), andη
T,f
. As
a result, the spectrum in Fig. 4.11(a) displays multiple peaks in the same delay
bin but at different Doppler shifts, while Fig. 4.11(c) shows thatη
T,f
successfully
eliminates all the peaks except for one at the desired location. Notice that η
T,f
alsohelpsdistributethepowerunderthoseunwantedpeaksequallyacrossDoppler.
Fig. 4.11(b) shows that with η
T,d
, we can also eliminate the repeated main peaks
(andachieve slightlylowersidelobeenergy); however, atthepriceoftheseparation
time between adjacent MIMO snapshots is reduced to T
0
/8 in η
T,u
, which would
imply that the number of antenna elements would have to be reduced such that
M
T
M
R
decreases by a factor of 8.
Besides we simulate a two-path channel, which is the simplest version of the
multipathchannel. Tab. 4.3providesacomparisonbetweenthetrueandestimated
parameters where we apply η
T,f
and RiMAX-RS. Although both paths’ Doppler
84
Table 4.2: Path parameters of one-path channel with high Doppler in snapshot 1
and small Doppler in snapshot 2
Snapshot τ (ns) ϕ
T
(deg) ϕ
R
(deg) ν (Hz)
1 601.1 11.5 59.6 4032.3
2 1117.3 21.3 160.0 80.6
0 20 40 60 80
(dB)
10
-3
10
-2
10
-1
10
0
10
1
10
2
CRLB
RMSE
0 20 40 60 80
(dB)
10
-3
10
-2
10
-1
10
0
10
1
10
2
10
3
10
4
CRLB
RMSE
Figure 4.7: Comparison about RMSEs of ϕ
T
and ν based on Monte-Carlo simu-
lations with the theoretical CRLB from (4.29) for snapshot 1 in Tab. 4.2 for the
optimized non-SS sequence η
T,f
and RiMAX-RS
shifts are larger than 1/2T
0
with a small difference, the simulation suggests that
the estimated parameters are close to the true values.
4.5.2 Evaluation with Measurement Data
Themeasurement campaignusesareal-timeMIMOchannelsounderdeveloped
at the University of Southern California (USC). The sounder is equipped with
a pair of software defined radios (National Instruments USRP-RIO) as the main
85
0 20 40 60 80
(dB)
10
-3
10
-2
10
-1
10
0
10
1
10
2
CRLB
RMSE
0 20 40 60 80
(dB)
10
-3
10
-2
10
-1
10
0
10
1
10
2
10
3
CRLB
RMSE
Figure 4.8: Comparison about RMSEs of ϕ
T
and ν based on Monte-Carlo simu-
lations with the theoretical CRLB from (4.29) for snapshot 2 in Tab. 4.2 for the
optimized non-SSη
T,f
and RiMAX-RS
Table 4.3: Parameter estimation for a two-path channel withη
T,f
, true/estimate
Path ID τ (ns) ϕ
T
(deg) ϕ
R
(deg) ν (Hz) |γ|
2
(dB)
1 646.2/646.2 67.81/67.79 -59.33/-59.33 3225.8/3225.8 -13.13/-13.13
2 1203.7/1203.7 -60.15/-60.15 -123.79/-123.78 3217.7/3217.8 -18.82/-18.82
transceivers, twoGPS-disciplined rubidiumreferences asthesynchronization units
and a pair of 8-element UCAs. The sounding signal is centered at 5.9GHz with
a bandwidth of 15MHz. The maximum transmit power is 26dBm. The main
advantage of this sounder setup, compared to another V2V sounder introduced
in Ref. [20], is the fast MIMO snapshot repetition rate, which provides a more
accurate representation of the channel dynamics though at the price of reduced
bandwidth. More details about the sounder setup can be found in Ref. [41].
86
0 20 40 60 80
(dB)
10
-3
10
-2
10
-1
10
0
10
1
10
2
CRLB
RMSE
0 20 40 60 80
(dB)
10
-3
10
-2
10
-1
10
0
10
1
10
2
10
3
10
4
(Hz)
CRLB
RMSE
Figure 4.9: Comparison about RMSEs of ϕ
T
and ν based on Monte-Carlo simula-
tions with the theoretical CRLB from (4.29) for snapshot 1 in Tab. 4.2 for the SS
sequence η
T,u
and RiMAX-4D
We presented first MIMO measurement results on truck-to-car (T2C) propa-
gation channel at 5.9GHz in Ref. [42]. The TX unit in the channel sounder was
programmed to measure with switching sequences alternating between η
T,u
and
η
T,f
. Therefore the odd MIMO burst snapshots in the data files were measured
withη
T,u
, while the even ones were withη
T,f
. The adjacent snapshots were 50ms
apart, and it is expected that most of large scale parameters remain the same over
that timescale. For the truck involved in the T2C channel measurement, we use
the 5m studio trucks as our test vehicles. Fig. 4.12 shows a picture of the truck
and the installation of the array on top of the driving cabin. Each truck has a
load capacity about 2722kg and up to 27m
3
of cargo space. The sufficient space
in the driving cabin allowed us to place the equipment rack of the sounder inside.
The platform that holds TX or RX antenna arrays is tightly clamped on metallic
87
0 20 40 60 80
(dB)
10
-3
10
-2
10
-1
10
0
10
1
10
2
CRLB
RMSE
0 20 40 60 80
(dB)
10
-3
10
-2
10
-1
10
0
10
1
10
2
(Hz)
CRLB
RMSE
Figure 4.10: Comparison about RMSEs of ϕ
T
and ν based on Monte-Carlo simu-
lations with the theoretical CRLB from Eq. (4.29) for snapshot 2 in Tab. 4.2 for
the SS sequence η
T,u
and RiMAX-4D
cross-bars installed on top of the driving cabin, in order to ensure the safety of the
array and reduce the vibration while we drive the truck, see Fig. 4.12(b).
To demonstrate the capability of the optimized switching sequence and the
associated HRPE algorithm, we analyze several snapshots of the T2C channel
measurement on I-110 North freeway near Downtown Los Angeles, California,
USA. Both the truck (TX) and SUV (RX) drive in the same direction with an
approximate speed of 10m/s estimated based on the recorded GPS locations. An
incoming large truck driving in the opposite direction creates a reflected signal
path associated with a large Doppler shift. Fig. 4.13 shows the delay and Doppler
spectrum of extracted paths from RiMAX-RS, where we can observe the existence
of a weak MPC with a large Doppler shift that is outside [−806Hz,806Hz), and
it is most likely a reflection from the incoming large truck based on the delay, the
88
0
-5000
10
0.5
/ Hz / ns
10
-6 1 0
20
(a) Regular Switching
1.5
5000
30
2
0
-5000
10
0.5
/ ns / Hz
10
-6
1 0
20
(b) Dense Regular Switching
1.5
5000
30
2
0
-5000
10
0.5
/ ns / Hz
10
-6 1 0
20
(c) Irregular Switching
1.5
5000
30
2
Figure 4.11: Delay-Doppler spectrum of snapshot 1 with the large Doppler under
different TX switching sequences, (a)η
T,u
(b)η
T,d
(c) η
T,f
89
(a) The front & side view (b) Array on top of the cabin
Figure 4.12: The truck used in the T2C and T2T channel measurements
large positive Doppler, and angular estimates. The incoming truck can also be
observed on a video recording at a time corresponding to the snapshot. As a com-
parison, Fig. 4.14 shows the spectrum for a snapshot that is captured 50ms ahead
and processed with RiMAX-4D suitable for the SS sequence in [46]. Most of the
dominant signals including the line-of-sight path and the reflection from the TX
truck’s trailer are present in both plots, and their Doppler shifts are around 50Hz
and delays are between 400ns and 450ns. However there is no component in Fig.
4.14tomatchtheMPCwithlargepositiveDopplergreaterthan1kHzinFig. 4.13,
instead there exists an MPC with a similar delay but a negative Doppler around
−416Hz. The difference between two Doppler shifts is about 1568Hz, which is
close to 1/T
0
≈ 1613Hz given by the length of the DSER. These experimental
results also agree with our simulation results in Fig. 4.11 that the conventional SS
sequence leads to multiple correlation peaks that are 1/T
0
apart.
4.6 Summary and Discussion
In this chapter we revisit the array switching design problem in the context of
channel sounding for fast-time varying channels, where the popular SS sequences
are shown to limit the DSER. To study the problem in the context of real-word
90
400 500 600 700 800
/ ns
-1500
-1000
-500
0
500
1000
1500
/ Hz
-80
-75
-70
-65
-60
-55
-50
Figure 4.13: The delay and Doppler spectrum at 896.45s with each point repre-
senting a path extracted by RiMAX-RS from Sec. 4.4 and color-coded based on
the path power
switched arrays, we use EADFs for the array modeling which directly incorporates
the array nonidealities. We limit our current study to the set of non-SS sequences
which the TX array is allowed to implement the non-SS sequence while the RX
still uses the SS sequence. This design constraint facilitates to reduce the problem
complexity and potentially increase the operation efficiency of the TDM channel
sounder, while still providing a considerable booston the DSER.Future works can
possibly look into the trade-off analysis between the operation efficiency and the
increased gain on the DSER.
We formulate the switching sequence design problem as an optimization prob-
lem that intends to reduce the high sidelobes in the spatio-temporal ambiguity
function, and the proposed SA algorithm with carefully selected parameters can
provide a non-SS sequence with a sufficiently low NSL. We integrate the switching
91
400 500 600 700 800
/ ns
-1500
-1000
-500
0
500
1000
1500
/ Hz
-80
-75
-70
-65
-60
-55
-50
Figure 4.14: The delay and Doppler spectrum at 896.40s with each point repre-
senting a path extracted by RiMAX-4D in Ref. [46] and color-coded based on the
path power
sequence into a state-of-the-art parameter extraction algorithm. Both the switch-
ing sequence and the corresponding parameter estimation algorithm are verified
throughextensiveMonte-CarlosimulationsandactualsampledatafromT2Cchan-
nel measurements on highway, which shows the prospect of this design method to
facilitate V2V MIMO measurements with more antennas and mm-wave MIMO
measurements in dynamic environments.
92
Chapter 5
HRPE for MmWave Channel
Evaluation
This chapter investigates the capability of millimeter-wave (mmWave) channel
sounders with phased arrays to perform super-resolution parameter estimation,
i.e., determine the parameters of multipath components (MPC), such as direc-
tion of arrival and delay, with resolution better than the Fourier resolution of
the setup. We analyze the question both generally, and with respect to a par-
ticular novel multi-beam mmWave channel sounder that is capable of performing
multiple-input-multiple-output (MIMO) measurements in dynamic environments.
We firstly study the limits of direction estimation of MPCs with phased arrays
and establish its connection with virtual arrays, which are commonly used in the
mmWave MIMO setups. Secondly, we propose a novel two-step calibration proce-
dure that provides higher-accuracy calibration data that are required for Rimax
or SAGE. Thirdly, we investigate the impact of center misalignment and residual
phase noiseontheperformance oftheparameterestimator. Finallywe experimen-
tally verify the calibration results and demonstrate the capability of our sounder
to perform super-resolution parameter estimation.
93
5.1 Introduction
Communication in the millimeter-wave (mmWave) band will constitute an
essential part of 5G communications systems, both for mobile access, as well as
fixed wireless access and backhaul [1, 81]. The design and deployment of such sys-
tems requires a thorough understanding of the propagation channel, since impor-
tant design questions, such as beamforming capability, coverage, and equalizer
length, all critically depend on the propagation conditions.
Understanding of mmWave is challenging because the physical propagation
mechanisms differ significantly from those at lower frequencies. The free-space
pathloss increases with frequency (assuming constant antenna gain) [82], and
diffraction processes become less efficient. Meanwhile, due tothe large bandwidth,
and the relatively large antenna arrays used at (mmWave frequencies, the per-
centage of delay/angle bins that carry significant energy is low [14], which can be
interpreted as a sparse structure either in the multipath delay or angular domain.
Due to these different propagation conditions, new channel models are required.
Several standardization groups, e.g., 3GPP [8] and METIS2020 [83], have estab-
lished such models. However, it must be stressed that these models were finalized
under time pressure and have the purpose of comparing different systems under
comparable circumstances; theyare not suitable fordeeper understanding ofprop-
agation effects or absolute performance predictions. Several other standardization
groups,suchasIC1004[84]andNIST5GmmWavechannelmodel[9],arecurrently
developing more detailed models.
Any improved understanding of mmWave channels, as well as the new channel
models, rely on thorough and extensive measurements. Since the late 1980s, hun-
dreds of paper have been published describing mmWave measurements in different
environments, see [85] for an overview and references. Since mmWave systems
94
use (massive) MIMO, directionally resolved channel measurements are of special
interest. ForoutdoorlongTXandRXdistancemeasurements, apopularmmWave
channelsoundingmethodisbasedontheuseofrotatinghornantennas[86,87,88],
i.e., the horn antennas are mechanically pointed into different directions at differ-
enttimes. Dependingonthehornbeamwidth,themechanicalrotationofbothTX
and RX antennas could potentially take up to hours. The phase between TX and
RX LOs is extremely difficult to preserve over such a long period of time, so that
only noncoherent evaluations, are possible. Data can be evaluated by spectrum
peak searching methods, e.g., [35, 36]. Despite the simplicity and effectiveness in
some scenarios, there are obvious drawbacks attributed to the discarding of the
phase information, e.g., difficulty to resolve multiple paths whose directions are
within the HPBW of the horn antenna, and accounting for the sidelobes of imper-
fect beam patterns and deciding whether any secondary peak is attributed to the
sidelobes of the main peak or the contribution from a weak MPC.
In contrast, phase-coherent measurements arepossible with (i) setups based on
a VNAs combined with virtual arrays (mechanical movement of a single antenna)
for short-distance measurements, or (ii) real-time mmWave multi-beam channel
sounders, which may be based on switched horn arrays [89] or phased-arrays with
electronically switched beams [35]. Data can be evaluated by advanced signal
processing algorithms, which rely on the MLE. For instance SAGE [39] is used
in Ref. [37, 38] along with rectangular virtual arrays and Ref. [21] with rotating
horns, while Rimax[40]wasappliedinanindoormmWave measurement campaign
at 60GHz with VNA and virtual arrays.
We have recently constructed a real-time mmWave multi-beam channel
sounder, which is based on phased arrays and electronically switched beams [35].
95
The phased array is integrated with other RF elements, such as converters, fil-
ters and PAs, in one box, which we refer to as RFU for the rest of the chapter.
More details about the RFU can be found in Ref. [90]. The setup achieves a
measurable path loss of 159dB without any averaging or spreading gain, which is
sufficient to handle challenging outdoor NLOS scenarios. More importantly com-
pared to the sounder equipped with rotating horn antennas, our setup is capable
to switch between beams in less than 2µs with a control interface implemented in
field-programmable gate array (FPGA). As a result, the duration of one MIMO
snapshotisreducedfromhourstomilliseconds. Thisuniquefeaturehasthreemain
benefits. Firstlyitallowsthecollectionoftensofthousandsofmeasurement points
inasinglemeasurement campaign, secondlyitmakesthesoundersuitableformea-
surement campaigns in dynamic environments, thirdly given the small phase drift
indicated in [35, Fig. 9] the data evaluation with HRPE algorithms such as Rimax
is feasible.
An essential prerequisite for applying HRPE algorithms on MIMO measure-
ment data is sounder calibration. Calibration usually consists of two steps: the
first is the cable-through calibration, also known as the back-to-back calibration.
It mainly serves the purpose to characterize the TX and RX system frequency
response within the operating bandwidth at different gain settings, such that the
effectsofthechannelsounderonthechannelimpulseresponsecanbecompensated.
The second step is the antenna array calibration, which is critical for any HRPE
algorithm that intends to estimate the directions of MPCs. Another benefit ofcal-
ibrating antennas is to produce the antenna de-embedded characterization of the
propagation channel [6], which is highly desirable for mmWave system simulations
where directional horn antennas with different gain and HPBW, or arrays with
different shapes and sizes, might be tested [91]. The rotating-horn-based sounder
96
from NYU performs back-to-back calibration with two horn antennas pointing
towards each other in the anechoic chamber [92]; the procedures are relatively
simple because the analysis mostly considers the noncoherent response (the PDP
and the APS). The Keysight sounder in [93] uses power dividers and two sepa-
rate back-to-back experiments to obtain the system frequency variation, however
limited details are available on the calibration of inter-channel phase imbalance
or antenna patterns. Papazian et al. present their complete two-step calibration
for their mmWave sounder in Ref. [94], where they first perform the back-to-back
calibration after disconnecting the antennas, then use the NIST near-field probe
to obtain the phase centers and individual complex antenna patterns.
Unfortunatelythesemethodscannotbeappliedtosounderswheretheantennas
are integrated with other RF elements on the same printed circuit board (PCB) in
our RFUs. We consider the important case that the array and the RF electronics
cannot be separated from each other - a situation that occurs not only in our
sounder, but is of general interest for mm-wave and higher frequencies, as system
integration and efficiency considerations often requires joint packaging of these
components.
Inthischapter,wepresentthecalibrationproceduresandtheverificationexper-
iments for MIMO mmWave channel sounders based on phased arrays with inte-
grated RFelectronics. We first study the theoretical limits on the estimation error
of path directions in the context of the phased array, and compare the results
with those from a virtual array with the same topology. To accommodate the
integrated antenna/electronics design and still obtain the system response and
frequency-independent array response needed in HRPE analysis, we propose a
novel two-setup calibration scheme and formulate the extraction and separation of
system frequency response and the beam pattern as an optimization problem.
97
The main contributions are thus as follows,
• we analyze the direction finding problem in the context of phased arrays and
compare the CRLBs of path directions with those in the virtual array setup;
• we introduce detailed calibration procedures for a beam-switching channel
sounder with integrated RFUs, which contains the baseline RFU calibration
and the multi-gain RFU calibration;
• to comply with the data model of Rimax, we formulate the extraction of
the frequency-independent array response as an optimization problem and
propose an optimal solution based on the low rank approximation (LRA)
method;
• we analyze the impact of several imperfections of channel sounder and cali-
bration method on the performance of Rimax evaluation;
• we perform the verification experiments in a static chamber environment
with artificially added reflectors.
This chapter is organized as follows. In section II we introduce the signal mod-
els of phased arrays and the data model used in the path parameter estimation.
Section III provides the overview on the Rimax algorithm as well as the theoret-
ical analysis on the CRLBs of path directions with phased arrays. In section IV
we introduce our proposed calibration procedures that enable the super-resolution
parameterestimationformmWavechannelsounderwithintegratedantennaarrays.
Section V discusses about the impact of two practical issues, array center mis-
alignment and the residual phase noise, on the performance of Rimax evaluation.
Section VI presents the results on the verification experiments with a mmWave
MIMO sounder in an anechoic chamber. In section VI we draw the conclusions.
98
The symbol notation used in this paper follows the rules below:
• Bold upper case letters, such asB, denote matrices. B() represents a matrix
valued function.
• Bold symbols I denote identity matrices. I
b
means that its number of rows
equals the length ofb.
• Bold lower case letters, such as b, denote column vectors. b
j
is the jth
column of the matrixB. b() stands for a vector valued function.
• Calligraphic upper-case letters denote higher dimensional tensors.
• [B]
ij
denotes the element in the ith row and jth column of the matrixB.
• Superscripts
∗
,
T
and
†
denote complex conjugate, matrix transpose and
Hermitian transpose, respectively.
• The operators |f(x)| and kbk denote the absolute value of a scalar-valued
function f(x), and the L2-norm of a vector b.
• The operators ⊗, ⊙ and ⋄ denote Kronecker, Schur-Hadamard, and Khatri-
Rao products.
• The operator⊘ represents the element-wise division between either two vec-
tors, matrices or tensors, and the operator ◦ is the outer product of two
vectors.
• The operators ⌊⌋ and ⌈⌉ are the floor and ceiling functions.
99
5.2 Signal Model
5.2.1 Phased Arrays with Steerable Beams
We consider a uniform rectangular array (URA) in this chapter, which is a
popular array configuration for mmWave phased arrays. We assume without loss
of generality that the arraylies in the y-z plane. Boresight of the array orientation
is thus the azimuth angle ϕ = 0
◦
and the elevation angle θ = 90
◦
, see Fig. 5.1.
The array has N
y
elements along the y-axis and N
z
elements along the z-axis; we
denote the total number of antenna elements N = N
x
N
y
. We assume that the
antenna elements are separated by λ/2, i.e. half a wavelength, in both directions.
Figure 5.1: The structure of URA with N
y
elements along the y-axis and N
z
elements along the z-axis
We label antennas based on their y-z coordinates, for example we choose the
bottom left antenna in Fig. 5.1 as the (1,1) element. If the center of the URA is
100
aligned with the origin of the cartesian coordinates, the coordinates of the (n
y
,n
z
)
element are
y
ny,nz
= (n
y
−
N
y
+1
2
)
λ
2
(5.1)
z
ny,nz
= (n
z
−
N
z
+1
2
)
λ
2
, (5.2)
where n
y
∈ {x|x ∈ Z,1 ≤ x ≤ N
y
} and n
z
∈ {x|x ∈ Z,1 ≤ x ≤ N
z
}. We also
assumethatallantennaelementssharethesameindividualpatternthatisgivenby
A
0
(ϕ,θ) with ϕ∈ [−π,π) and θ∈ [0,π]. LetX(ϕ,θ) be a matrix-valued function,
i.e., X :R
2
→C
Ny×Nz
. It provides the received signal at the URA when a plane
wave with unit gain arrives from the direction (ϕ,θ). The function is determined
by
[X(ϕ,θ)]
ny,nz
=e
jk
T
p
ny,nz
A
0
(ϕ,θ), (5.3)
where p
ny,nz
is the antenna position vector,
p
ny,nz
=

0 y
ny,nz
z
ny,nz

T
(5.4)
and the wave vector k is given by
k =
2π
λ

cosϕsinθ sinϕsinθ cosθ

T
. (5.5)
To allow the phased array to steer beams in different directions, we add nar-
rowband phase shifters to each antenna element. The complex phase weighting
101
matrix W shares the same dimension with X(ϕ,θ), and [W]
ny,nz
is the corre-
sponding complex weight for the (n
y
,n
z
)-th antenna element. The virtual beam
pattern when the URA is excited with W can be mathematically determined by
b
W
(ϕ,θ)= tr

W
T
X(ϕ,θ))= vec(W)
T
·vec(X(ϕ,θ)), (5.6)
where vec() is the vectorization operator. The phase shifters translate the sig-
nals from the antenna ports to the beam ports that are associated with different
weighting matrices. The same concept is adopted in mmWave hybrid beamform-
ing, which is a promising mmWave transceiver structure with a large array [95].
We assume that measurements at different beam ports occur at different times,
but are all within the coherence time of the channel.
5.2.2 Rimax Data Model
Similar to the data model in Refs. [19, 24], we use a vector model for the input
T MIMO snapshots, and denote it asy∈C
M×1
, where M =M
f
×M
R
×M
T
×T.
It includes contributions from SP s(Ω
s
), DMC n
dmc
and measurement noise n
0
:
y =s(Ω
s
)+n
dmc
+n
0
, (5.7)
wherethevectorΩ
s
representstheparametersofP SPs. Itconsistsofpolarimetric
pathweightsγ andthestructuralparametersμthatincludeτ,(ϕ
T
,θ
T
),(ϕ
R
,θ
R
),
andν. ϕ
T
,ϕ
R
andν arenormalizedtobetween−π andπ,θ
T
andθ
R
arebetween
0 and π, whileτ is normalized to between 0 and 2π. The DMC follows a Gaussian
random process with frequency correlation, and its PDP has an exponentially
decaying shape [26]. The measurement noise is i.i.d. and follows the zero-mean
complex Gaussian distribution.
102
AnimportantassumptionoftheoriginalRimaxisthearraynarrowbandmodel,
which means the array response is considered constant over the frequency band
that the channel sounder measures. Although an extension of Rimax to include
the wideband array response between 2GHz and 10GHz is included in Ref. [96],
narrowband approach has been used almost exclusively in practice; we thus leave
wideband calibration for future work. Section 5.4 discusses how to find the best
frequency-independent array pattern from the over-the-air calibration data.
We also assume that the single polarized model is applicable, because the
antennas in our phased-array are designed to be vertically polarized. The cross
polarization discrimination (XPD) of the beam patterns is over 20dB in the main
directions. A detailed discussion of the impact ofignoring the additional polarized
components in parameter estimation is available in Ref. [7].
We assume that a common frequency response attributing to the system hard-
wareissharedbetween antennapairsinaMIMOchannel sounder. Itisdenoted as
adiagonalmatrixG
f
∈C
M
f
×M
f
. Meanwhile weadoptthearraymodelingthrough
EADFs[53],whichprovidesareliableandelegantapproachforsignalprocessingon
real-worldarrays. TheEADFsareobtainedthroughperformingatwo-dimensional
DFT on the complex array pattern either from simulations or array calibration in
an anechoic chamber. We denote the EADFs for TX and RX RFUs as G
TV
and
G
RV
respectively. Before breaking down thedetailsabouts(θ
s
), we first introduce
the phase shift matrixA(μ
i
)∈C
M
i
×P
[19], which is given by
A(μ
i
) =







e
j(−
M
i
−1
2
)μ
i,1
··· e
j(−
M
i
−1
2
)μ
i,P
.
.
.
.
.
.
e
j(
M
i
−1
2
)μ
i,1
··· e
j(
M
i
−1
2
)μ
i,P







. (5.8)
103
μ
i
is a structural parameter vector that represents eitherτ,ϕ
T
,θ
T
,ϕ
R
,θ
R
orν.
These quantities will be essential for HRPE evaluations.
Based on the system response and the EADFs introduced above, we obtain the
basis matrices:
B
f
=G
f
·A(−τ) (5.9)
˜
B
TV
=

G
TV
·

A(θ
T
)⋄A(ϕ
T
)

(5.10)
˜
B
RV
=

G
RV
·

A(θ
R
)⋄A(ϕ
R
)

(5.11)
B
t
=A(ν) (5.12)
With phased arrays we use the beam ports instead of the antenna ports, hence
M
T
and M
R
become the number of beam ports and reflect how many different
beamforming matrices W are applied to TX and RX arrays respectively. Finally
the signal data model for the responses of SPs is given by
s(Ω
s
)=B
t
⋄B
TV
⋄B
RV
⋄B
f
·γ
VV
(5.13)
If the measurement environment is static such as the anechoic chamber, and one
measurement snapshot y contains only one MIMO snapshot, i.e. M
T
×M
R
pairs
of sweeping-beam measurements, the signal model of SPs can be simplified to
s(Ω
s
)=B
TV
⋄B
RV
⋄B
f
·γ
VV
(5.14)
104
5.3 Limits in Angular Estimation
This section investigates the CRLB of azimuth angle of an MPC when the
channel sounder is equipped with phased arrays. We also compare this estimation
bound with that from the switched array that shares the same structure.
For the TDM switched arrays of the URA introduced in Section 5.2.1, the
received signal of an MPC with the direction (ϕ,θ) and the complex weight γ
under the narrowband model is given by
y
0
=vec

X(ϕ,θ)

γ +n
0
. (5.15)
Let usdenote vec

X(ϕ,θ)

γ ass forbrevity. Forthe phased arraysthe vectorized
N
b
-beam measurement signal can be modeled by
y
b
=







vec(W
1
)
T
y
1
.
.
.
vec(W
N
b
)
T
y
N
b







N
b
×1
=







vec(W
1
)
T
.
.
.
vec(W
N
b
)
T







s+







vec(W
1
)
T
n
1
.
.
.
vec(W
N
b
)
T
n
N
b







=Φs+ ˜ n. (5.16)
It is essential that the noise realizations in different y
i
are uncorrelated; this is
fulfilled because we assume that the N
b
-beam measurements are conducted at
different timeinstances. Becausen
i
isfromazero-meancomplexGaussianprocess
105
∀i = 1,...,N
b
, we know that ˜ n is also zero-mean. Furthermore they are mutually
independent, thus its covariance R
˜ n
is determined by
[R
˜ n
]
ij
=E{˜ n
i
˜ n
∗
j
}=









kW
i
k
2
F
, if i =j;
0, otherwise.
(5.17)
If all phase shifters are ideal with unit gain and follow the structure [W
i
]
ny,nz
=
e
jφ
i,ny,nz
, the noise process in (5.16) has a white spectrum and its covariance
becomes
R
˜ n
= N
y
N
z
σ
2
n
I =Nσ
2
n
I. (5.18)
Letusdefine aparameter vectorα, [ϕ,θ,γ]
T
associated with thenarrowband
signal, then the FIM J(α) can be determined by
J(α), 2Re
n
D
†
(α)R
-1
˜ n
D(α)
o
=
2
N
y
N
z
σ
2
n
Re{D
†
(α)D(α)}
=
2
N
y
N
z
σ
2
n
Re
n
(
∂s
∂α
T
)
†
Φ
†
Φ(
∂s
∂α
T
)
o
. (5.19)
Here D(α) is the Jacobian matrix of s with respect to α. Meanwhile it is well-
known that the CRLB [79], which is the smallest possible estimation variance of
any unbiased estimator, is connected to the FIM by
var(α
i
)≥[J(α)
-1
]
ii
, (5.20)
where i =1,2,3.
In the following we consider the CRLB of the azimuth angle of a single path
of the switched array with that of a phased array. Consider first a switched array
106
with dimension of the URA being 8×2 and assume that θ and γ are known and
θ =90
◦
. The angular estimation limits based on the signal model in (5.15) can be
computed according to the method in [19, Chapter 4],
J(ϕ) =
2
σ
2
n
Re
n
(
∂s
∂ϕ
)
†
(
∂s
∂ϕ
)
o
=8π
2
ρcos
2
ϕ·kΞxk
2
. (5.21)
Here x,s/γ ∈C
N
, and the SNR ρ,|γ|
2
/σ
2
n
. Furthermore the diagonal matrix
Ξ is defined as
Ξ,
1
λ











y
1,1
0 ··· 0
0 y
2,1
··· 0
.
.
.
.
.
.
.
.
.
.
.
.
0 ··· 0 y
Ny,Nz











, (5.22)
whose diagonal elements are the y-coordinates of antenna elements as determined
by (5.1). Meanwhile for the phased arrays withΦ or the set of{W
i
}, we can find
the expression of the CRLB of ϕ from (5.19):
var(ϕ)
-1
=J(ϕ) =
8π
2
ρcos
2
ϕ
N
y
N
z
·kΦΞxk
2
. (5.23)
Fig. 5.2 provides the CRLB comparison for ϕ when ρ = 10dB and θ = 90
◦
.
1
We present two cases for the phased array, both of which implement the conjugate
beamformers whose associated virtual beams have their peak direction lie in the
azimuthplaneandareuniformlyspacedforϕ∈ [−60
◦
,60
◦
]. Thefirstcaseincludes
16 beams with an equivalent beam spacing of 8
◦
in azimuth, while the second
consists of only 8 beams with a larger beam angular spacing. It is noteworthy
1
Because sin and cos functions naturally take angle in radians, the σ
ϕ
presented in Fig. 2 is
the inverse of (5.23) or (5.21) multiplied by 180/π
107
that in the first case, where the number of beams equals the number of antennas,
the CRLB of ϕ of the phased array is lower than that of the switched array for
the values of ϕ of interest, which can be attributed to the beamforming into the
right elevation angle. In the meantime the phased array can obtain similar CRLB
values with only 8 beams in the azimuth plane, although the insufficient beam
number creates non-uniform coverage of different azimuth angles, which leads to
the oscillation of the CRLB over ϕ in Fig. 5.2. The optimal structure of Φ that
minimizes the CRLB is out of the scope of this thesis and left for future work.
-60 -40 -20 0 20 40 60
0.3
0.4
0.5
0.6
0.7
0.8
0.9
/ deg
CRLB comparison with =10 dB and =90
o
Switched Array
Phased Array with 16 beams
Phased Array with 8 beams
Figure 5.2: The squared root of CRLB of ϕ with the 8×2 URA when ρ = 10dB
and θ =90
◦
5.4 Calibration Procedures
In this section, we describe the channel sounder calibration for a phased-array
sounder. We consider the important case that the array and the RF electronics
cannot be separated from each other - a situation that occurs not only in our
sounder, but is of general interest for mm-wave and higher frequencies, as system
108
integration and efficiency considerations often requires joint packaging of these
components. Wehighlightournoveltwo-stepcalibrationmethodforthiscase. The
simplified diagramofthetime-domainsetup isgiven inFig. 5.3(a), andtheoverall
calibration procedures can also be categorized into a back-to-back calibration in
Fig. 5.3(b) and the RFU/antenna calibration in Fig. 5.3(c).
(a) Over-the-air verification/measurements
(b) Through-calbe back-to-back calibration
(c) Over-the-air RFU calibration
Figure 5.3: The summary of diagrams that are relevant for the RFU calibration
and verification
Conventional array calibration is based on the assumption that the antenna
or beam ports can be connected to RF signals at the operating frequency, and
typically the array to be calibrated would consist only of passive components. In
contrast, we consider here the situation that the RFU has an embedded mixer
and an LO at 26GHz, so that the input and output frequencies are different.
The intermediate frequency (IF) frequency being between 1.65GHz and 2.05GHz.
109
This motivates us to use both TX and RX RFUs in the antenna calibration, so
that the generator and receiver of the calibration signal in a VNA can operate on
the same frequency (even though it is lower than the operating frequency of the
array). Although the measured frequency response based on the setup in Fig. 5.4
varies significantly over the 400MHz bandwidth, we still attempt to find the best
narrowband fit of the RFU pattern to the calibration data.
In this subsection we introduce the main procedures of RFU calibration. Here
we limit our objective to calibrate the frequency and beam pattern responses of
one TX and one RX RFU. The procedures consist of two main steps
1. the baseline calibration of one TX and one RX RFU, i.e., calibration of the
gain pattern and frequency response for one setting of the amplifier gains at
TX and RX;
2. the PNA-assisted multi-gain calibration of TX and RX RFUs.
Tosimplifythediscussion,weonlyconsidercalibrationasafunctionofazimuth;
elevation can treated similarly. The measurement setup is illustrated in Fig. 5.4.
The same Rubidium reference is shared between two RFUs and the VNA (KT-
8720ES). The main procedures are given as follows. In the first experiment we
treat the TX RFU as the probe and the RX RFU as the device under test (DUT),
so we fix both the orientation and the beam setting of the TX RFU while placing
the RX RFU on a mechanical rotation stage and turning it to different azimuthal
orientationsandmeasuring with theVNA.The VNAoutputsS21responses atdif-
ferent RX phase shifter settings (i.e., beams) and different azimuth angles. These
S21 values are denoted as Y
1
(n,ϕ
R
,f), where the beam index n = 1,2,...,19.
The calibration sequence for these three dimensions basically follows embedded
FOR loops, and the loop indices from inside to outside are f → n → ϕ
R
. This
110
sequence is motivated by the fact that scanning through frequencies is faster than
scanning throughbeams(which requires phase shifter switching, which takes afew
microseconds), which in turn is faster than scanning through observation angles,
which requires mechanical rotation and thus a few seconds. Shortening the mea-
surement time is not only a matter of convenience, but also reduces the sensitivity
to inevitable phase noise, see Section 5.5. Similarly we swap the positions and the
rolesofTXandRXRFUs, the second experiment generatesthedataY
2
(m,ϕ
T
,f),
and the TX beam index m = 1,2,...,19.
The goal of the baseline calibration is to estimate the frequency-independent
beam patterns B
T
(m,ϕ
T
) and B
R
(n,ϕ
R
) from Y
1
and Y
2
. We can build a joint
estimator by minimizing the sum of squared errors, and the corresponding opti-
mization problem is stated as
min
b
T
,b
R
,g
f
,k
X
|B
T0
B
R
(n,ϕ
R
)G
0
(f)−Y
1
(n,ϕ
R
,f)|
2
+|kB
R0
B
T
(m,ϕ
T
)G
0
(f)−Y
2
(m,ϕ
T
,f)|
2
, (5.24)
where k is a complex scalar that attempts to model the gain and reference phase
offset between two experiments illustrated in Fig. 5.4. The gain difference is due
to the small boresight misalignment and the phase offset is because of different
phase values in local oscillators at the start of the two experiments.
To simplify the problem formulation, we use vector notation. For example
b
R
, vec(B
R
), b
T
, vec(B
T
) and g
f
is the frequency vector related to G
0
(f).
We also need to transform the two data sets, Y
1
, reshape(Y
1
(n,ϕ
R
,f),[],N
f
)
111
and Y
2
, reshape(Y
2
(m,ϕ
T
,f),[],N
f
). Here reshape() is a standard MATLAB
function. The original problem in (5.24) can then be rewritten as
min
b
R
,b
T
,g
f
,k
kB
T0
b
R
g
T
f
−Y
1
k
2
F
+kkB
R0
b
T
g
T
f
−Y
2
k
2
F
. (5.25)
If we combine the two data sets into one, we can haveY= [Y
1
;Y
2
]. The problem
in 5.25 is equivalent to the following one
min
u,v
kuv
†
−Yk
2
F
, (5.26)
which is a typical LRA problem and it can be quickly solved through singular
value decomposition (SVD) of Y [97]. If we denote its optimal solutions asu
◦
and
v
◦
, we easily find the mapping from u
◦
to b
T
, b
R
and k, as well as the mapping
from v
◦
to g
f
. The optimal solution to (5.26) can be found through Alg. 3. Here
Algorithm 3 The SVD-based algorithm to solve the problem (5.26)
1: StackY
1
and Y
2
in the rows, Y=[Y
1
;Y
2
]
2: Perform the SVD on Y = UΣV
†
and find the largest singular value σ
1
and
its related singular vectors u
1
and v
1
3: Initialize g
f
= σ
1
v
∗
1
; Divide u
1
into two halves with equal length u
1
=
[u
1,1
;u
1,2
]
4: Finallyb
R
= au
1,1
; a is selected such that the center element ofb
R
is 1
5: ˜ u
1,2
=au
1,2
, the finalg
f
=
1
a
g
f
6: Finally k equals the center element of ˜ u
1,2
, the final b
R
= ˜ u
1,2
/k
we normalize the center points of the frequency-independent patterns such that
B
T0
= 1 and B
R0
= 1.
After implementing and testing the algorithm on actual calibration data with
100MHzbandwidth, wehave obtainedquitegoodpatternextraction results
2
. The
2
In general the larger bandwidth the calibration data contains, the worse the extraction
becomes, although a largerbandwidth could help improving the estimation accuracy. The deter-
mination of the optimum bandwidth is out of the scope of this thesis.
112
Figure 5.4: The setup diagram of the baseline calibration that involves one TX
and one RX RFU
sorted singular values of Y are shown in Fig. 5.5, where the ratio between the
largest and the second largest singular value is 16.7. For the ideal solution there
should only be one nonzero eigenvalue. With the solutions to (5.26), we compare
Y
1
(n,ϕ,f) against
ˆ
Y
1
in Fig. 5.6 while fixing f = 1.85GHz. Similarly the com-
parison results of Y
2
can be found in Figs. 5.7 when we fix n = 8. The complex
scalar k is 0.078−1.045i, and its amplitude is about 0.40dB.
We now turn to the second step of the calibration. In order to calibrate the
RFU responses at different gain settings, we need to have a calibration setup that
can handle the gain variations at both TX and RX RFUs without saturating any
device. The requirement is quite difficult to fulfill in the baseline calibration setup
113
0 2 4 6 8 10
Index
10
-1
10
0
10
1
10
2
10
3
Singular Value
Sorted Singular Values of Y
Figure 5.5: The distribution of the ten largest singular values of Y = [Y
1
;Y
2
] in
Alg. 3, where a dominant singular value exists.
highlighted in Fig. 5.4, therefore we propose to perform a second multi-gain cal-
ibration procedure via using a PNA for TX and RX separately. The setup is
illustrated in Fig. 5.8. It uses the measurement class known as the frequency
converter application (Option 083)in the Keysight PNA series. We use the config-
urationknown as“SMC+phase" which provides goodmeasurement resultsonthe
conversion loss and the group delay of the DUT. To simplify the task of measuring
the conversion loss and the phase response of a mixer, Dunsmore [98] introduces a
new calibrationmethodonaVNA by using aphase reference, such acomb genera-
tor traceable to NIST, to calibrate the input and output phase response of a VNA
independently, which eliminates the need for either reference or calibration mixer
in the test system. However we cannot measure with the vector mixer calibration
(VMC) configuration, which could have provided the reference phase apart from
the group delay, because finding reciprocal calibration mixers that match the IF
114
-100 0 100
2
4
6
8
10
12
14
16
18
Beam index
|Y
1
| / dB
-40
-35
-30
-25
-20
-15
-10
-100 0 100
2
4
6
8
10
12
14
16
18
Beam index
| Y
1
| / dB
-40
-35
-30
-25
-20
-15
-10
Figure 5.6: The comparison between Y
1
(n,ϕ
R
,f) and reconstructed
ˆ
Y
1
when the
IF f =1.85GHz
and the mmWave RF frequency is very difficult. The quality of the calibration
mixers ultimately limits the performance of the VMC method [99]. Although we
do not have a direct control of the LO into the RFU, we can share the 10MHz
reference clock of the PNA with the RFU, so that the phase response from PNA
is also stable.
The objectives of this multi-gain calibration are two folds, i) validating the
frequency-independent patterns in the baseline calibration
3
; ii) calibrating and
estimating the frequency responses of the TX and RX RFUs separately. As shown
in [100, Fig. 8], the variable gain controller affects the signal power at IF for both
TX and RX RFUs. The calibration procedures for the setup in Fig. 5.8 are given
as follows. We align the standard gain horn antenna with the boresight of the RX
3
We expect the frequency-independent EADFs implicitly required by Rimax to hold well at
different RFU gain settings.
115
-100 0 100
1.8
1.82
1.84
1.86
1.88
1.9
f / GHz
|Y
2
| / dB
-40
-35
-30
-25
-20
-15
-10
-100 0 100
1.8
1.82
1.84
1.86
1.88
1.9
f / GHz
| Y
2
| / dB
-40
-35
-30
-25
-20
-15
-10
Figure 5.7: The comparison between Y
2
(n,ϕ
R
,f) and reconstructed
ˆ
Y
2
when the
beam index n =8
RFU, then measure the S21 for the RX RFU with different beam configurations,
different gain settings before we rotate the RFU to the next azimuth angle. The
same steps are repeated for the TX RFU, except that we reverse the input and
output signals when the horn antenna becomes the receiving antenna. These two
steps produce two data sets, which we denote as Y
R
(g,n,ϕ,f) and Y
T
(g,n,ϕ,f),
where the 4-tuple (g,n,ϕ,f) represents the RFU IF gain setting, the beam index,
theazimuthangleandfrequency index. Similarlytothebaselinecalibrationsetup,
the calibration procedures here closely follow embedded FOR loops. and the loop
indicesfrominsidetotheoutsidearef →n→g→ϕ. Animportantfeatureabout
the two data sets is that they are only phase coherent within the same frequency
sweep, because of the random initial phase of each sweep in the “SMC+phase”
calibration configuration.
116
Figure 5.8: The setup diagram of the PNA-assisted multi-gain calibration that
involves one RFU and one horn antenna
First we remove the responses of the standard gain horn and the LOS channel
fromY
R
orY
T
. Toestimate different frequency responses andverifythemagnitude
of RFU patterns extracted in the baseline calibration, we formulate the following
optimization problem,
min
b,g
kbg
T
−Y
R
k
2
F
(5.27)
s.t.|b|=b
a
=|b
R
|.
Thisproblemformulationtriestofindthebestrank-1approximationtoY
R
subject
to the constraint that the magnitude of the optimal b is equal to the amplitude
pattern b
a
extracted from the baseline calibration. Comparing to the problem in
(5.26), the vector equality constraint prevents us from applying Alg. 3. However
wecansubstitutebwithb
a
⊙e
jφ
intheobjectivefunction,andproposeaniterative
algorithm based on the alternating projection method in Alg. 4. The algorithm
117
attemptstosolvetwo smallersub-problemsiterativelyuntil thesolutionconverges.
The two matrices in the subproblems are given by A
b
= I
g
⊗ (b
a
⊙ e
jφ
) and
A
g
=g⊗I
b
Algorithm 4 The iterative optimization algorithm to solve the problem (5.27)
1: Find initial estimates for the pattern phase vector φ and frequency response
g;
2: while k(b
a
⊙e
jφ
)g
T
−Yk
2
F
has yet converged do
3: Fix φ, and use the least-square method to solve the sub-problem 1:
min
g
kA
b
g−vec{Y}k
2
;
4: Fixg, and use the Levenberg-Marquardt method [23] to solve the nonlinear
optimization sub-problem 2:
min
φ
kA
g
(b
a
⊙e
jφ
)−vec{Y}k
2
.
5: end while
Firstly we process Y
R/T
when the RFU gain setting equals that in the baseline
calibration, so that we can obtain the frequency response from TX and RX RFUs
separately, because only one RFU is involved in the PNA-aided gain calibration.
However the product of these two frequency responses should in principle be close
to g
f
extracted from Alg. 3, which serves as a part of the verifications. Secondly
we repeat the steps in Alg. 4 for different TX and RX RFU gain settings. Among
them we select the set of gain settings whose mismatch errors, evaluated according
totheobjectivefunctionin(5.27), arerelatively small, andweconsider using these
gain settings in future measurements.
5.5 Calibration Practical Limitations
This section investigates two important practical issues in the mmWave cali-
bration procedure. The first is the misalignment between the calibration axis and
the center of the antenna array. The second is the phase stability measurement of
two RFUs in the anechoic chamber, and we observe that the residual phase noise
118
consists of a slow-walking process anda fast-varying one. These issues areapplica-
ble to mmWave channel sounders whose antenna array cannot be separated from
other RF electronics. We study the impact of these issues on the performance of
Rimax evaluation with simulations.
5.5.1 Center Misalignment
It is important to align the rotation axis with the phase center of an antenna
in the antenna calibration in the anechoic chamber. Different methods have been
proposed to calculate the alignment offset based on the phase response of the
calibration data [101, 102].
We again consider the URA shown in Fig. 5.1. Let us assume that the origin
of the Cartesian coordinates is aligned with the center of the URA. The probe
horn antenna is placed at p
t
= [5,0,0]
T
. The antenna position vector with the
ideal alignment is given by (5.4) for the n
y
-th and the n
z
-th element. We assume
that this is the initial position with ϕ = 0
◦
and θ = 90
◦
of the array pattern
calibration. If we denote the offset vector at the initial position as ∆ p, the actual
initial position is given by ˜ p
ny,nz
=p
ny,nz
+∆ p.
To measure the array at ϕ
0
and θ
0
we can compute the new antenna position
with the rotation matrix, which is given by
˜ p
ny,nz
(ϕ
0
,θ
0
) =R
y
(θ
0
−90)R
z
(ϕ
0
)˜ p
ny,nz
. (5.28)
where R
y
and R
z
are the standard 3×3 transformation matrices that represent
rotationalong the y axis with the right-hand rule and the z axis with the left-hand
rule respectively [103]. The distance between the probe and the rotated antenna is
119
givenbyd
ny,nz
(ϕ
0
,θ
0
) =k˜ p
ny,nz
(ϕ
0
,θ
0
)−p
t
k
2
. Thereforethesimulated“distorted”
calibration response is given by
˜
b
ny,nz
(ϕ
0
,θ
0
) =A
ny,nz
(ϕ
0
,θ
0
)e
−j2πfc
dny,nz
c
0
, (5.29)
wheref
c
isthecarrierfrequency,c
0
isthespeedoflightintheair,andA
ny,nz
(ϕ
0
,θ
0
)
is the element pattern. For simplicity we assume in the simulations that antennas
are isotropic radiators, i.e.A
ny,nz
(ϕ
0
,θ
0
) =1. Similarly we could acquire the ideal
patternb
ny,nz
(ϕ
0
,θ
0
)viasetting∆ pas0. Examples areshowninFig. 5.10. Asthe
offset ∆ p increases we can observe that the high power coefficients in EADFs tend
to be more spread-out when compared to the ideal case in Fig. 5.10(a). This will
decrease the effectiveness of mode gating, where we could truncate the coefficients
in EADF in order to reduce the calibration noise.
The array ambiguity function is used to check the performance of the array to
differentiate signals from different directions [104]. It is usually defined as
A
b
(ϕ
1
,θ
1
,ϕ
2
,θ
2
) =
b
†
(ϕ
1
,θ
1
)b(ϕ
2
,θ
2
)
kb
†
(ϕ
1
,θ
1
)k·kb(ϕ
2
,θ
2
)k
. (5.30)
Wecanreplace thesecondbwith
˜
binthe aboveequation andexamine the“cross”
ambiguity function. Fig. 5.9 shows that it presents a high ridge in the off-diagonal
direction, which means that if the actual response is b while the calibrated array
response is
˜
b, the estimator is very likely to provide the correct result, as shown
through the following simulation results.
We provide simulation results with Rimax evaluation based on synthetic chan-
nel responses, in order to study the impact of center misalignment and phase noise
during the calibration. A similar approach togenerate synthetic channel responses
is also presented in Ref. [47]. Tab. 5.1 summarizes the simulation settings. We
120
-80 -40 0 40 80
-80
-40
0
40
80
Cross Ambiguity when =90
o
-25
-20
-15
-10
-5
Figure 5.9: The cross ambiguity function between b and
˜
b when θ
1
=θ
2
=90
◦
Table 5.1: Simulation settings to generate synthetic double-directional channel
responses
Simulation Parameter Value
Carrier frequency f
c
28GHz
Bandwidth B 100MHz
Number of frequency points M
f
601
TX array 8×2 URA
RX array 8×2 URA
have simulated channel responses when the ideal calibrated array responses b is
used. One example is shown in Fig. 5.10(a). In the Rimax evaluation, we then
use the EADFs extracted from the “distorted” patterns when the center offset
∆ p = 3λ1, which is the upper limit on the center misalignment considering our
efforts to align the probe with the RFU. Fig. 5.10(d) provides the EADF ampli-
tude pattern for one of the corner elements. In Fig. 5.11 we compare the APDPs
of the synthetic channel response, the reconstructed channel response based on
121
-30 -15 0 15 30
Elevation Mode
-30
-15
0
15
30
Azimuth Mode
-40
-35
-30
-25
-20
-15
-10
(a) Δp =0
-30 -15 0 15 30
Elevation Mode
-30
-15
0
15
30
Azimuth Mode
-40
-35
-30
-25
-20
-15
-10
(b) Δp = λ1
-30 -15 0 15 30
Elevation Mode
-30
-15
0
15
30
Azimuth Mode
-40
-35
-30
-25
-20
-15
-10
(c) Δp= 2λ1
-30 -15 0 15 30
Elevation Mode
-30
-15
0
15
30
Azimuth Mode
-40
-35
-30
-25
-20
-15
-10
(d) Δp= 3λ1
Figure 5.10: Amplitude of EADF of (n
y
= 1,n
z
= 1) in the 8× 2 URA with
different center offset values ∆ p
Rimax estimates and the residual channel response due to center misalignment.
Thepeakreductionisaround30dBforeachpath. Wedefinepeakreductionasthe
power difference between APDPpeaksoftheoriginalsignalandtheresidual signal
after the parameter estimation. Tab. 5.2 provides the comparison of parameters
for each path. Because of the imperfect amplitude estimate of path 1, the residual
peak of path 1 is still higher than that of path 10 in this simulation, and hence the
estimator fails to pick up path 10.
122
0 200 400 600 800 1000 1200
delay [ns]
-60
-50
-40
-30
-20
-10
0
dB
Data
SP est
Residual
X: 100
Y: -28.46
X: 100
Y: 6.051
Figure 5.11: The APDP comparison between the synthetic channel response, the
reconstructed channel response from Rimax estimates and the residual channel
response based on center-misaligned RFU patterns
Table 5.2: Compare the estimated path parameters with the center misaligned
array responses, format true/estimated
Path ID τ(ns) ϕ
T
(deg) θ
T
(deg) ϕ
R
(deg) θ
R
(deg) P (dBm)
1 100.00/100.00 18.45/18.67 109.53/108.60 -31.21/-30.70 104.56/104.36 -20.00/-20.00
2 165.17/165.17 -0.70/-0.33 90.00/89.51 46.38/45.97 53.39/53.91 -22.83/ -22.85
3 311.45/311.45 33.49/33.60 88.39/87.37 -56.56/-55.77 55.72/55.47 -29.18/-29.20
4 383.01/382.99 25.80/26.07 122.38/121.98 -1.21/-0.84 91.73/91.36 -32.29/-32.31
5 430.16/430.16 48.45/48.48 98.79/98.66 -39.85/-39.14 57.74/57.64 -34.34/-34.35
6 468.86/468.86 46.91/46.89 99.41/98.95 57.44/56.81 115.45/114.16 -36.02/-36.02
7 561.61/561.60 -19.90/-19.43 118.76/118.68 25.52/25.53 115.40/114.16 -40.05/-40.04
8 661.73/661.74 23.85/23.97 114.44/113.21 0.06/0.47 107.80/105.80 -44.40/-44.39
9 662.94/662.94 -36.26/-35.76 96.14/95.58 -3.47/-3.03 61.99/61.88 -44.45/-44.52
10
∗
990.65/100.00 -56.34/3.59 64.63/-19.28 -52.85/-30.48 102.77/102.41 -58.68/-52.66
5.5.2 Phase Noise
Because the LO signals in two RFUs are generated separately, although the
TX and RX RFUs share the same 10MHz reference clock, there still remain some
small phase variations. To study the potential impact of phase noise on the pat-
tern calibration in Section 5.4, we measured the system phase response for about
15 minutes. We denote the time-varying S21 measurement as S
21
(f,t), and the
123
0 200 400 600 800 1000
t / s
-30
-20
-10
0
10
20
30
40
/ deg
Relative Phase evolution comparison
Figure 5.12: Phase response of two RFUs measured in the anechoic chamber over
a 20min timespan
normalized phase response as φ
rel
(f,t) =
180
π
arg(S
21
(f,t)/S
21
(f,0)). As shown in
Fig. 5.12, the relative phase averaged over frequency
¯
φ
rel
(t) =E
f
{φ
rel
(f,t)} shows
a combination of fast and slow variations over time.
Theslow-varying phaseresponsecanbeobserved byperformingamovingaver-
ageon
¯
φ
rel
(t), andFig. 5.13presents theautocorrelationoftwo typesofvariations.
The local fast phase variation can be closely modeled by a i.i.d. Gaussian process
with the mean −0.01
◦
and the standard deviation 4.8
◦
. Its fitting to a normal
distribution passes the two-sample KS test with a 5% confidence level.
To model its influence on the measured response, we consider the calibration
scheme highlighted in Section 5.4 and assume that there is an i.i.d. Gaussian
variation between switched beams at the same panel rotation angle, while another
124
0 100 200 300 400 500 600 700 800 900
t / s
-0.5
0
0.5
1
correlation
Autocorrelation of two types of phase variation
Slow Variation
Local Fast Variation
Figure 5.13: The autocorrelation of “slow” and “fast” phase variations based on
the phase stability test
slow-varying process is kept constant within the same angle and only changes
between orientations.
We have also conducted simulations to study the effects of residual phase noise
(PN) in the calibrated pattern on Rimax evaluation. The simulated slow-varying
phase is presented in Fig. 5.15. However the actual measurement data obtained in
real-timearecorruptedwiththefastphasevariationbutfreefromtheslow-varying
PN, because each MIMO measurement with the time domain setup indicated in
Fig. 5.3(a) takes about 1.4ms, which is insignificant when compared with the
correlation time of the slow-varying phase term shown in Fig. 5.13. Fig. 5.16
showsthecomparisonofAPDPsafterweperformaRimaxevaluationwiththePN-
corrupted RFU patterns. The peak reduction can be only 18dB in this scenario.
125
-20 -15 -10 -5 0 5 10 15 20
/ deg
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
F(x)
Empirical CDF
Data
Normal fit
mean = -0.01
o
std = 4.8
o
Pass KS test at 5
Figure 5.14: The statistics of the local “fast” variation of phase noise shown in
Fig. 5.12
In summary, the limiting factor on the performance of Rimax evaluation with
our mmWave channel sounder is the PN corruption on the calibrated RFU pat-
terns,wherethepeakreductionislimitedtoabout20dBforeachpath,anditcould
lead to the existence of “ghost” paths as well as failures to detect relatively weaker
paths at larger delays. However it may be sufficient in most of practical environ-
ments, suchasindoorofficeandoutdoormacrocell, wherethepower ofDMCcould
contribute between 10% and 90% of the total power [70, 29], therefore the residual
signals would be below the spectrum of DMC and have a limited influence on the
estimation results especially when the Rimax-based estimator considers DMC as
a part of the channel response [26].
126
0 200 400 600 800 1000 1200 1400 1600
t / s
-10
-5
0
5
10
15
/ deg
Figure 5.15: The simulated slow-varying phase noise that follows the model in
Section 5.5.2
5.6 Measurement
In this section, we first summarize the steps to apply the calibration results
from Section 5.4 to pre-process the measurement data before running an HRPE
algorithm. Secondly, we conduct MIMO channel measurements in an anechoic
chamber and present the Rimax evaluation results for two types of test channels.
The first is the two-path channel that consists of a direct LOS signal and a strong
reflection fromametallic plane. Thesecond isthetwo-poletest channel where two
standing poles provide potential weaker reflections and the position of one pole is
adjusted to create different angular separation between the two reflection signals.
127
0 500 1000 1500
delay [ns]
-70
-60
-50
-40
-30
-20
-10
0
10
20
dB
PN slow-plus-fast BW 100MHz
Data
SP est
Residual
X: 150
Y: 11.83
X: 150
Y: -7.346
Figure 5.16: The APDP comparison between the synthetic channel response, the
reconstructed channel response from Rimax estimates and the residual channel
response based on PN-corrupted RFU patterns
5.6.1 Preprocessing with Calibration Results
We provide the recipe to apply various calibrated frequency responses to the
measurement data before inputting them into Rimax. The measurement data is
generally produced by the time-domain setup given in Fig. 5.3(a), and we denote
them as Y
data
(f,m,n). We calibrate the through cable between 1.65GHz and
2.05GHz, represented by the green line in Fig. 5.3(b), with a VNA and obtain its
response H
cable
(f). On the other hand, the data generated by the setup in Fig.
5.3(b) is denoted as Y
IF
(f). The common frequency response of two RFUs, which
isproduced by thetwo-step calibrationandextracted according to Alg. 3 andAlg.
4, is denoted as G
0
(f).
128
As a result, the frequency compensation to the original data Y
data
is given by
˜
Y(f,m,n) =
Y
data
(f,m,n)
G
0
(f)Y
IF
(f)/H
cable
(f)
. (5.31)
5.6.2 Two-path Experiment
For an experimental verification of calibration and Rimax evaluations, we cre-
ated a two-path test channel is created in an anechoic chamber with our MIMO
mmWave channel sounder [35]. The setup is illustrated in Fig. 5.17 and 5.18.
Although the reflector is a plane, since we enforce the reflector to be parallel to
the wall of the chamber, the actual specular reflection point should have the same
distance to both TX and RX, which is a = b = 3.15m, and the distance between
TX and RX is d
LOS
= 5.65m. Based on trigonometry, the azimuth DoD of the
reflected path is around 26.26
◦
and the azimuth DoA is about−26.26
◦
. The extra
path delay of the reflection, when compared with LOS, is about 2.17ns, which is
about one quarter of the inverse of the 100MHz bandwidth. The channel is basi-
cally composed of one dominant LOS path and a reflection from a metallic plane.
The metallic plane is elevated to the same height of TX and RX RFUs, hence we
can assume all significant signals travel in the azimuth plane.
We have analyzed in total 470 MIMO snapshots with Rimax after following
steps in Section. 5.4. The statistics, such as mean and standard deviation, of
path parameters is listed in Tab. 5.3. Path 1 is the LOS path, and Path 2 is
the reflected path, andboth of their pathdirections match well with the geometry.
The theoretical LOSpower is−26.03dB, which iscalculated by1/4πd
2
. The extra
delay of Path 2 is 2.25ns, which equals 67.5cm in path length and is close to the
65cmcomputedbasedonthegeometry. Path3isthedouble-reflectionbetweentwo
RFUs when they are aligned and facing each other. The extra run length is about
129
Figure 5.17: The picture of TX RFU and the inserted reflector in an anechoic
chamber
Figure5.18: Thetwo-pathtest channel withmmWaveRFUsintheanechoiccham-
ber
10.55mwhichisclosetotwo timesofd
LOS
. Thepeakreductionisabout12dBand
the unresolved residual signal power is less than 8% of the original signal, which is
130
somewhat close to the peak reduction level in the PN-corrupted-data simulation.
Path 4 is the “ghost” path as its power is more than 10 dB lower than that of Path
2 and it cannot be physically mapped to any reflector in the chamber. Based on
the results of this verification measurement and the simulations in Section 5.5, the
paths with similar delay values (less than 1ns) whose power is more than 10dB
weaker than the strongest one should be considered as the “ghost” path.
Table5.3: ListofRimaxestimatesforthethetwo-pathtestchannelintheanechoic
chamber, mean / standard deviation
Path ID τ (ns) ϕ
T
(deg) ϕ
R
(deg) P (dB)
1 20.30/0.06 -1.43/0.06 -0.14/0.03 -23.80/0.06
2 22.55/0.07 26.27/0.07 -25.55/0.19 -28.83/0.23
3 55.47/0.08 -1.18/0.07 -0.25/0.02 -40.94/0.11
4 22.40/7.58 6.54/17.30 13.57/30.92 -46.74/3.36
0 100 200 300 400 500
delay / s
-60
-55
-50
-45
-40
-35
-30
-25
-20
-15
-10
dB
Final APDP after extracting 4 paths with Rimax
Data
SP Est
Res
Figure5.19: The APDPcomparison between the two-pathtest channel, the recon-
structed two-path test channel and the residual response
131
5.6.3 Two-pole Experiment
We further investigate the capability of the setup to differentiate signals with a
limited angular separation, and compare the results with the geometry as well as
theFourier-based beamforming APS. The measurement setup canbefoundinFig.
5.20. We setup two standing posts with their upper parts wrapped with reflective
aluminum foil. The diameters of these posts are small so that they are weaker
reflectors compared to the metallic plane in the previous two-path experiment.
The metallic plane is inserted between TX and RX so that the direct LOS is
attenuated.
Figure 5.20: The picture of the two-pole experiment in an anechoic chamber, the
two poles are marked with yellow stars.
We name the post, which is closer to the TX RFU in Fig. 5.20, as post A, and
the other as post B. We move post A to five different positions so that the angular
separation of reflected signals from two posts gradually decrease. The complete
parameter list of six paths is provided in Tab. 5.4. Path 2 and 5 are from post
A and B respectively. The angular difference is about 17
◦
for the azimuth DoD
and 10
◦
for the azimuth DoA. The estimates of two diffraction signals around the
132
Table5.4: ListofRimaxestimatesforthethetwopolemeasurementinananechoic
chamber when post A is at position 1
Path type τ (ns) ϕ
T
(deg) ϕ
R
(deg) P (dB)
Diffraction 18.81 -4.62 3.32 -37.99
Post A 20.68 40.10 -17.44 -40.29
Diffraction 18.26 6.33 -5.90 -46.99
Unidentified 20.80 37.33 34.54 -43.81
Post B 21.23 23.75 -29.37 -44.03
Unidentified 18.79 -3.98 -27.59 -49.67
metallic plane are consistent when post A moves among the five locations. The
summary of the estimated reflections from post A and post B is available in Tab.
5.5. We can observe that the estimated azimuth DoDs of both reflections have an
approximateoffsetof5
◦
whencomparedtotheirtruevaluesbasedonthegeometry.
However the estimated angular difference is very close to the true value, and thus
the estimated angular spreads will also be close. Besides, we also see that the
calibrated setup with Rimax is unable to see two reflected signals from position 4,
where the azimuth DoA separation is about 5
◦
. This angular separation limit is
certainly affected by the imperfect estimation of other strong signals, such as the
two diffractions, with reasons mentioned in Section 5.5. But if we compare these
results with the Fourier-based beamforming results of position 1 and position 4 in
Fig. 5.21, we could see in Fig. 5.21(a) that even for position 1 when two posts
havethelargestseparation, theAPSappearstofailtoproducetwo distinguishable
peaks around the region that matches the directions of the posts.
5.7 Summary and Discussion
In this chapter, we presented the calibration procedures and the verification
experiments for our MIMO mmWave channel sounder based on phased arrays, in
133
Table 5.5: Rimax estimates ofthe reflections fromtwo poles inthe two-poleexper-
iment in an anechoic chamber, estimate/true
Position Post A ϕ
T
(deg) Post A ϕ
R
(deg) Post B ϕ
T
(deg) Post B ϕ
R
(deg)
1 40.10/45.89 -17.44/-17.12 23.75/28.64 -29.37/-27.57
2 37.11/42.44 -20.38/-19.22 24.94/28.64 -28.72/-27.57
3 34.48/39.53 -22.81/-21.66 21.99/28.64 -28.94/-27.57
4 30.41/36.50 -27.05/-22.98 NA/28.64 NA/-27.57
5 NA/31.35 NA/-22.77 24.92/28.64 -28.60/-27.57
order to demonstrate its capability to perform anHRPE evaluation such as Rimax
orSAGE.TheintegrateddesignofantennaarraywithotherRFelectronicsrequires
us to rethink about the traditional sounder calibration procedures. As a result, we
proposed a novel two-setup calibration scheme and formulated the extraction and
separationofsystem frequency responseandthebeampatternastwooptimization
problems.
We also investigated two practical issues that might impact mmWave array
calibration, such as the array center misalignment and the phase noise variation.
We also conclude through simulations that the fast-varying phase noise could to
18dB,asimilarlevelthatweobservedinthechamberverificationexperiment. The
two-path channel measurement in an anechoic chamber shows that the sounder
with the calibration results can perform HRPE evaluation, and the estimates of
strongpathsmatchwellwiththeenvironment. Thetwo-polechannelmeasurement
suggests that the calibration enables the sounder to differentiate signals from two
reflectors which are only about 5
◦
apart in azimuth DoA.
Inthefuture,wewillstudyifthereisanoptimalsetofbeamstoestimatepropa-
gationpaths. Besideswewillinvestigatetheoptimalbandwidthinthenarrowband
pattern extraction algorithm. We will also investigate how to apply the wideband
Rimax to the data evaluation and if the calibration procedures can therefore be
simplified.
134
-60 -40 -20 0 20 40 60
-60
-40
-20
0
20
40
60
Joint Angular Spectrum based on BF
-40
-35
-30
-25
-20
-15
X: 3
Y: -4
Z: -14.98
X: -17
Y: 40
Z: -14.44
X: -11
Y: 6
Z: -15.52
(a) Position 1
-60 -40 -20 0 20 40 60
-60
-40
-20
0
20
40
60
Joint Angular Spectrum based on BF
-40
-35
-30
-25
-20
-15
X: -23
Y: 30
Z: -13.82
X: 3
Y: -4
Z: -15.31
X: -9
Y: 4
Z: -17.54
(b) Position 4
Figure 5.21: The two-dimensional beamforming APS of the measurements for
position 1 and position 4 of the two-pole measurement
135
Chapter 6
Conclusions
6.1 Summary
In this thesis we presented several works that aims at developing useful tools
for studying V2V and mmWave channels. Both V2V and mmWave channels could
be highly dynamic in nature, which propels us to rethink the design of MIMO
channel sounders and the associated parameter estimation algorithms.
In Chapter 2, we present the framework to develop a real-time MIMO chan-
nel sounder for V2V channels at 5.9GHz. The design is focused on improving
the system stability, maximizing the measurement snapshot rate and enabling
synchronous continuous channel measurements. Together with system and array
calibration in Section 2.2.1 and the HRPE algorithm introduced in Chapter 3,
we can jointly estimate parameters of MPCs from the measurement data, which
includes path delay, azimuth DoD, azimuth DoA, Doppler shift and path power.
With this V2V channel sounder, we were able to conduct extensive MIMO V2V
channel measurement campaignsinvariousenvironments suchasurban, suburban,
campus, and highway.
In Chapter 3, we introduce a novel HRPE algorithm suitable to evaluate time-
varying channels such as V2V and mmWave. The algorithm is developed based
on the state-of-the-art RiMAX algorithm, which jointly estimates the parameters
of SPs and the statistics of DMC. Because of the potential high Doppler spreads,
136
the signal model of the HRPE algorithm has to take into account the phase rota-
tion between antenna switchings in both TX and RX switched arrays. Special
attention is paid to reducing the numerical complexity of the HRPE algorithm by
simplifying the implementation of the path parameter initialization and the eval-
uation of key elements in the local optimization stage. Simulations with synthetic
multipath channel responses are conducted to demonstrate the performance of the
HRPE algorithm. Furthermore evaluation with actual V2V measurement data
has suggested that most of the dominant SPs can be traced to reflectors or events
recorded on camera during the measurement.
In Chapter 4, we exploit the possibility to extend the DSER of MIMO TDM
channel sounding, which is a limit introduced by the conventional SS sequences at
both TX and RX arrays. This requires a joint design of the switching control in
the hardware and the HRPE algorithm. We formulate the switching design prob-
lem as an optimization problem that intends to reduce the high sidelobes in the
spatio-temporalambiguity function, andtheproposedSA algorithmwithcarefully
selected parameters can provide a non-SSsequence with a sufficiently low NSL. We
integrate the new sequence into a state-of-the-art parameter extraction algorithm.
Both the switching sequence and the corresponding parameter estimation algo-
rithm are verified through extensive Monte-Carlo simulations and actual sample
datafromT2Cchannelmeasurements onhighway, whichshowstheprospectofthe
method to facilitate V2V MIMO measurements with more antennas and mmWave
MIMO measurements in dynamic environments.
In Chapter 5, we present the calibrationprocedures andthe verification experi-
mentsforourMIMOmmWavechannelsounderbasedonphasedarrays. Compared
to the VNA-based sounder that also applies RiMAX for the data evaluation, our
sounder is proven to measure in dynamic environments which are more relevant
137
for the actual scenarios of mmWave deployment. We first study the theoretical
limits onthe estimation error of path directions in the context of the phased array,
and compare the results with those from a virtual array with the same topol-
ogy. To accommodate the design of the RFU and still obtain the system response
and frequency-independent array response needed in Rimax analysis, we propose
a novel two-setup calibration scheme and formulate the extraction and separation
of system frequency response and the beam pattern as an optimization problem.
We performed the verification experiments in an anechoic chamber with added
reflectors and show that the calibrated setup can produce path estimates whose
accuracy is much than the Fourier resolution.
6.2 Future Work
Subsequent work will be additional improvements on the path extraction algo-
rithm and the statistical channel modeling based on the extraction results. First
one could expand the evaluation based on the HRPE algorithm developed in the
publication [III] to more data points. These results, though based on a snapshot-
by-snapshot analysis, canbeuseful inbuilding statistical modelsofsomekey chan-
nel parameters, such as path-loss, rms delay spread, angular spread and Doppler
spread. Particular results involving the truck-to-car and truck-to-truck propaga-
tion channel are new and appealing to the scientific society, because there is a
dearth of publications on the related topics except for our recent papers [42, 43].
Secondly,researchersarealsointerestedinthestationarityregionorthedynam-
ics of dominant SP in the V2V channel. To characterize the lifetime of SPs and
incorporate them into a GSCM V2V model will have great values. One alterna-
tive yet promising method is the EKF-based path tracking algorithm proposed in
138
Ref. [24]. The method is expected to have a smaller estimation variance com-
pared to the conventional ML-based evaluation, largely because of the Kalman
filtering approach. Furthermore it directly outputs a tractable evolution of one
SP. It also eliminates the requirement to run an additional path association algo-
rithm from adjacent snapshots evaluated by 4D-RiMAX in Publication [III]. More
details about the path association algorithm can be found in Ref. [44]. However
the algorithm in Ref. [24] cannot be applied directly in its original form, and
proper adjustments have to be made to evaluate V2V or mmWave measurement
data. It is also interesting to compare the results from this EKF-based estimator
and with those from the Rimax-plus-tracking approach.
Thirdly, since the increased bandwidth in the mmWave band and the multi-
band operation via carrier aggregation (CA) below 6GHz, related channel mea-
surements and modeling activities will very likely be extended to cover a larger
bandwidth, where the narrowband array assumption can be violated in either the
original Rimax [19] or 4D-Rimax [105]. It will be interesting to apply the wide-
band Rimax to evaluate the channel measurement data, which basically removes
the narrowband assumption made about the arrays. One can extend the single-
input multiple-output (SIMO) wideband implementation outlined in Ref. [96] into
the full MIMO version.
Last but not least, it is also worthwhile to investigate the modeling of DMC
in mmWave and study its impact on mmWave system impact, such as mmWave
initial access, channel estimation etc.
139
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150
Appendix A
Supplemental Materials to
Chapter 3
A.1 3D tensors T
1
and T
2
in 4D-RiMAX
The two 3D tensors T
1
andT
2
are important elements in computing the corre-
lation grid in Eq. (3.25). First we introduce their relationship with the correlation
function defined in Eq. (3.23). We assume that an estimator has provided the
eigenvalues and eigenvectors of the covariance matrix R for DMC and measure-
ment noise, which follows the data model in Eqs. (3.14) and (3.15).
R=UΛU
†
(A.1)
U =I
t
⊗U
T
⊗U
R
⊗U
f
(A.2)
Λ =I
t
⊗Λ
T
⊗Λ
R
⊗Λ
f
+σ
2
n
I (A.3)
ByutilizingUandΛwe canpre-process theobservationvectoryandthebasis
matrixB,
y
′
=Λ
-1
U
†
y (A.4)
B
′
(μ)=U
†
B(μ). (A.5)
151
An efficient implementation of Eq. (A.4) is through tensor operations, which is
given by
Y = reshape

y,[M
f
M
R
M
T
T]

(A.6)
y
′
= vec
(

(Y×
1
U
†
f
)×
2
U
†
R

×
3
U
†
T

×
4
I
t
)
⊘λ. (A.7)
Here “reshape()” and “vec()” are standard MATLAB commands to transform
between a vector and a tensor. λ is the diagonal vector of Λ. The ×
n
repre-
sents the n-mode product between a matrix and a tensor [106]. Similarly we may
form a 4D data tensor Y
′
from y
′
according to Eq. (A.6).
After substituting Eqs. (A.4) and (A.5) into Eq. (3.23), we have an alternative
expression for the correlation function
C(μ,y
′
)= (y
′†
B
′
)(B
′†
Λ
-1
B
′
)
-1
(B
′†
y
′
) (A.8)
Each point on the large search grid ˜ μ
L
represents the structural parameters for
onepotential SP. The correlationgridC, based on Eq. (A.8) and ˜ μ
L
, has thesame
dimension with ˜ μ
L
. From Eq. (A.8), we may observe that the 4D tensors T
1
and
T
2
correspond to the two terms B
′†
y
′
andB
′†
Λ
-1
B
′
when evaluated over ˜ μ
L
. The
computation method described here mainly focuses on an efficient implementation
of the two tensors.
We need to exploit the tensor structure of the data model to simplify the
implementation. Compared to the problem in [47], the new structural parameter,
i.e. Doppler shift ν, is involved in several basis matrices, such as B
t
,
˜
B
TV
and
˜
B
RV
. However if we fix the Doppler shift in the search grid ˜ μ
L
, the problem at
hand will be very similar to the one studied in [47]. With a given Doppler shift ν
nt
152
(the n
t
th Doppler point), let us denote T
nt
1
and T
nt
2
as the 3D child tensors from
T
1
and T
2
. The dimension of the two new 3D tensors is N
f
×N
R
×N
T
.
T
nt
1
=

(Y
′
×
1
B
′†
f
)×
2
B
n
′
t
†
RV

×
3
B
nt′†
TV

×
4
b
nt′†
t
(A.9)
The new B matrices are determined by
B
′
f
=U
†
f
B
f
(A.10)
B
nt′
RV
=U
†
R

B
RV
⊙

A
t,R
(ν
nt
) ··· A
t,R
(ν
nt
)

| {z }
N
R

(A.11)
B
nt′
TV
=U
†
T

B
TV
⊙

A
t,T
(ν
nt
) ··· A
t,T
(ν
nt
)

| {z }
N
T

(A.12)
b
nt′
t
= U
†
t
h
e
−j⌊
T
2
⌋νn
t
e
j(−⌊
T
2
⌋+1)νn
t
··· e
j(⌈
T
2
⌉−2)νn
t
e
j(⌈
T
2
⌉−1)νn
t
i
T
. (A.13)
Similarly in terms ofT
nt
2
, first we reshape the diagonal elements ofΛ
-1
to form
a 4D tensor L
i
, then we perform several n-mode products between L
i
and four
BB
′
matrices to acquire T
nt
2
.
L
i
= reshape

diag{Λ
-1
},[M
f
M
R
M
T
T]

(A.14)
T
nt
2
=

(L
i
×
1
(BB
′
f
)
T
)×
2
(BB
nt′
RV
)
T

×
3
(BB
nt′
TV
)
T

×
4
(bb
nt′
t
)
T
(A.15)
153
where these special BB
′
matrices are given by
bb
nt′
t
=b
nt′
t
⊙(b
nt′
t
)
∗
(A.16)
BB
′
f
=B
′
f
⊙(B
′
f
)
∗
(A.17)
BB
nt′
TV
=B
nt′
TV
⊙(B
nt′
TV
)
∗
(A.18)
BB
nt′
RV
=B
nt′
RV
⊙(B
nt′
RV
)
∗
(A.19)
A.2 Jacobian Matrix and FIM in 4D-RiMAX
All relevant partial derivatives in the Jacobian matrix in Eq. (3.29) can be
found as follows
∂
∂τ
s(θ
sp
) =B
t
⋄
˜
B
TV
⋄
˜
B
RV
⋄D
f
⋄γ
T
vv
(A.20)
∂
∂ϕ
T
s(θ
sp
) =B
t
⋄(D
ϕ
T
,V
⊙A
t,T
)⋄
˜
B
RV
⋄B
f
⋄γ
T
vv
(A.21)
∂
∂ϕ
R
s(θ
sp
) =B
t
⋄
˜
B
TV
⋄(D
ϕ
R
,V
⊙A
t,R
)⋄B
f
⋄γ
T
vv
(A.22)
∂
∂γ
vv,r
s(θ
sp
) =B
t
⋄
˜
B
TV
⋄
˜
B
RV
⋄B
f
(A.23)
∂
∂γ
vv,i
s(θ
sp
) = j·B
t
⋄
˜
B
TV
⋄
˜
B
RV
⋄B
f
. (A.24)
Particularly since Doppler shifts are parameters in multiple basis matrices, their
partial derivatives are given by
∂
∂ν
s(θ
sp
) =B
t
⋄(B
TV
⊙D
ν,T
)⋄
˜
B
RV
⋄B
f
⋄γ
T
vv
+B
t
⋄
˜
B
TV
⋄(B
RV
⊙D
ν,R
)⋄B
f
⋄γ
T
vv
+D
t
⋄
˜
B
TV
⋄
˜
B
RV
⋄B
f
⋄γ
T
vv
(A.25)
154
Table A.1: Parts needed to reconstruct the Jacobian matrix in (3.30)
θ
sp
τ ϕ
T
ϕ
R
ν γ
VV,r
γ
VV,i
D
1
D
f
⋄γ
T
vv
B
f
⋄γ
T
vv
B
f
⋄γ
T
vv
B
f
⋄γ
T
vv
B
f
jB
f
D
1
2
˜
B
RV
˜
B
RV
D
ϕ
R
,V
⊙A
t,R
˜
B
RV
˜
B
RV
˜
B
RV
D
2
2
˜
B
RV
˜
B
RV
D
ϕ
R
,V
⊙A
t,R
B
RV
⊙D
ν,R
˜
B
RV
˜
B
RV
D
1
3
˜
B
TV
D
ϕ
T
,V
⊙A
t,T
˜
B
TV
3(B
TV
⊙D
ν,T
)
˜
B
TV
˜
B
TV
D
2
3
˜
B
TV
D
ϕ
T
,V
⊙A
t,T
˜
B
TV
3
˜
B
TV
˜
B
TV
˜
B
TV
D
1
4
B
t
B
t
B
t
B
t
B
t
B
t
D
2
4
B
t
B
t
B
t
D
t
B
t
B
t
As a summary, Tab. A.1 provides details about the parts needed to reconstruct
D(θ
s
) in Eq. (3.30). If we take one column of Tab. A.1 and plug into Eq. (3.30),
it matches the partial derivative mentioned above.
In order to present details about all the newD matrices, we need to introduce
a new type of derivative of the matrixA(μ
i
)∈C
M
i
×P
with respect toμ
i
∈R
P×1
,
and we denote the operation by D().
D(A,μ
i
)=

∂
∂μ
i,1
a
1
···
∂
∂μ
i,P
a
P

(A.26)
155
Hence the new smallD matrices in the above-mentioned partial derivatives can be
expressed as
D
f
=D(B
f
,τ) (A.27)
D
ϕ
T
,V
=G
TV
·D(A,ϕ
T
) (A.28)
D
ϕ
R
,V
=G
RV
·D(A,ϕ
R
) (A.29)
D
t
=D(B
t
,ν) (A.30)
D
ν,T
=D(A
t,T
,ν) (A.31)
D
ν,R
=D(A
t,R
,ν). (A.32)
TofindanalternativeexpressionofFIMinSection3.3.3,wewillfirstdecompose
L
i
into a sum of outer-products of vectors based on the PARATREE model [69],
Λ
-1
= diag

vec(L
i
)

(A.33)
≈ diag

vec

X
r
f
X
r
R
X
r
T
l
r
f
f
◦l
r
f
,r
R
R
◦l
r
f
,r
R
,r
T
T
◦l
r
f
,r
R
,r
T
t

(A.34)
=
X
r
f
X
r
R
X
r
T
diag{l
r
f
,r
R
,r
T
t
}⊗diag{l
r
f
,r
R
,r
T
T
}⊗diag{l
r
f
,r
R
R
}⊗diag{l
r
f
f
} (A.35)
=
X
r
f
X
r
R
X
r
T
Λ
(t)
r
f
,r
R
,r
T
⊗Λ
(T)
r
f
,r
R
,r
T
⊗Λ
(R)
r
f
,r
R
⊗Λ
(f)
r
f
(A.36)
Here Eq. (A.35) uses the property given by in [69, (7)].
156
Appendix B
Supplemental Materials to
Chapter 4
B.1 Derivation of Separability of the Ambiguity
Function
If we start with the basis vector based on the basis matrix in Eq. (4.8) and
apply the property in Eq. (4.5) and, we can further express the inner product
between two basis vectors by
b
†
(μ)b(μ
′
) =
h
˜
b
†
TV,T
(Ω T
,ν)
˜
b
TV,T
(Ω ′
T
,ν
′
)
i
·
h
˜
b
†
RV
(Ω R
,ν)
˜
b
RV
(Ω ′
R
,ν
′
)
i
·
h
b
†
f
(τ)b
f
(τ
′
)
i
, (B.1)
where we see that the inner product is factored into three inner products of small
basis vectors from different data domains, i.e. frequency (f), RX array (RV), and
TX-plus-time (TV,T).
Similarly we can also express the denominator of the ambiguity function in Eq.
(4.3) with vector inner products,
kb(μ)k·kb(μ
′
)k=
r
h
b
†
(μ)b(μ)
i
·
h
b
†
(μ
′
)b(μ
′
)
i
, (B.2)
157
then we can apply the separability feature derived from Eq. (B.1) with μ
′
= μ
and obtain the following
kb(μ)k·kb(μ
′
)k=k
˜
b
TV,T
(Ω T
,ν)k·k
˜
b
RV
(Ω R
,ν)k·
k
˜
b
TV,T
(Ω ′
T
,ν
′
)k·k
˜
b
RV
(Ω ′
R
,ν
′
)k·
kb
f
(τ)k·kb
f
(τ
′
)k (B.3)
From Eqs. (B.1) and (B.3) we show that both the numerator and denominator
in Eq. (4.3) can be factored into a product of two parts. The first one is only
associated with delay τ, and the other contains the rest of parameters in µ except
τ. Thismeansweprovethattheambiguityfunctionisaproductoftwocomponent
ambiguity functions, which is given in Eq. (4.4).
B.2 A Simpler Expression of X
T
In this appendix we provide a simpler expression of the essential ambiguity
function X
T
(ϕ
T
,ϕ
′
T
,ν,ν
′
) given by Eq. (4.20). This simplification is made possi-
ble by exploiting the structure of
˜
b
TV,T
, when TX implements a cycle-dependent
switching pattern and RX uses a sequential switching pattern.
The derivation of Eq. (4.20) requires another property related with the
Hadamard-schur product of two column vectors. It is given by
(a
1
⊙b
1
)
†
(a
2
⊙b
2
) =(a
1
⊙a
∗
2
)
†
(b
∗
1
⊙b
2
), (B.4)
158
wherea
1
,a
2
,b
1
andb
2
have the same length. Using this property we can express
the numerator of Eq. (4.19) by
T
X
t=1
˜
b
t
TV,T
(ϕ
T
,ν)
†
˜
b
t
TV,T
(ϕ
′
T
,ν
′
)
=
T
X
t=1
[b
TV
(ϕ
T
)⊙e
j2πνη
t
T
]
†
[b
TV
(ϕ
′
T
)⊙e
j2πν
′
η
t
T
]
=

b
TV
(ϕ
T
)⊙b
∗
TV
(ϕ
′
T
)

†
T
X
t=1
e
j2π(ν
′
−ν)η
t
T
(B.5)
Besides the Euclidean norm in the denominator of Eq. (4.19) can be efficiently
evaluated by substituting ϕ
′
T
with ϕ and ν
′
with ν.
k
˜
b
TV,T
(ϕ
T
,ν)k =
q
˜
b
TV,T
(ϕ
T
,ν)
†˜
b
TV,T
(ϕ
T
,ν)
=
√
T ·kb
TV
(ϕ
T
)k (B.6)
B.3 Ambiguity function and Correlation func-
tion
This appendix provides the relationship between the ambiguity function X
tot
and the correlation function C(μ,y) in Eq. (3.23). Assuming there is one SP, we
replace B with b in the derivation. Furthermore with large SNR the observation
159
vector y can be replaced by its mean or b(μ
′
)γ
vv
. Finally we also assume that
R= σ
2
n
I. After incorporating three assumptions we have
C(μ,y(μ
′
)) =
1
σ
2
n
(y
†
b)(b
†
b)
-1
(b
†
y)
=
1
σ
2
n
[b
†
(μ
′
)b(μ)γ
∗
vv
][b
†
(μ)b(μ
′
)γ
vv
]
b
†
(μ)b(μ)
=
|γ
vv
|
2
σ
2
n
|b
†
(μ)b(μ
′
)|
2
kb(μ)k
2
≈ρ
vv
|b
†
(μ)b(μ
′
)|
2
kb(μ)k·kb(μ
′
)k
=ρ
vv
|X
tot
|
2
, (B.7)
where the approximation assumes that kb(μ)k has similar values for different μ.
This is more or less fulfilled for antenna arrays such as UCA designed to cover
signals from all possible azimuth angles.
The derivation in Eq. (B.7) shows that our proposed ambiguity function is
closely related with the correlation function and hence the MLE developed in
Sect. 4.4.
B.4 3D Tensors T
1
and T
2
in RiMAX-RS
Similar to the parameter initialization method in [46, Appendix A], we first
construct a 4D tensor Y
′
from y, then reorganize it by combining the data in the
last two dimensions, i.e. TX array and time, into one dimension.
Y
′
3D
= reshape(Y
′
,[M
f
M
R
M
T
T]), (B.8)
160
where reshape() is a standard MATLAB function. The dimension of the 3Dtensor
Y
′
3D
is M
f
×M
R
×M
T
T. As a result we can compute T
nt
1
as
T
nt
1
=
h
(Y
′
3D
×
1
B
′†
f
)×
2
B
nt′†
RV
i
×
3
B
nt′†
TV,T
. (B.9)
The expressions ofB
′
f
andB
nt′
RV
can be readily found in [46, (61-62)], whileB
nt′
TV,T
is a new component and given as follows.
B
nt′
TV,T
= (U
t
⊗U
T
)
†

B
TV,T
⊙A
t,N
T
(ν
nt
)

, (B.10)
whereB
TV,T
∈C
M
T
T×N
T
isverticallystackedwithT copiesofB
TV
,andthematrix-
valued functionA
t,N
T
(ν
nt
)∈C
M
T
T×N
T
is constructed by horizontally stacking N
T
copies of the column vector a
T
(ν
nt
). This column vector is built by concatenating
the columns ofA
t
T
with t =1,2,...,T, which are related with the n
t
-th grid point
of Doppler shift and given in (4.12).
ForanotherkeyelementT
nt
2
usedinEq. (4.25),weagaincanfollowthemethod
outlined in [46, Appendix A] with some minor adjustments. First we need to build
a 3D tensor L
i
based on the eigenvalues of R, which is given by
L
i
= reshape

diag(Λ
-1
,[M
f
M
R
M
T
T])

. (B.11)
Then the child-tensor T
nt
2
can be computed with tensor products according to
T
nt
2
=
h
(L
i
×
1
(BB
′
f
)
T
)×
2
(BB
nt′
RV
)
T
i
×
3
(BB
nt′
TV,T
)
T
. (B.12)
161
B.5 Jacobian Matrix in RiMAX-RS
The Jacobian matrixD(θ
s
) is defined as
D(θ
s
) =
∂
∂θ
T
s
s(θ
s
), (B.13)
with each column related to one partial derivative. More details about different
columns in D(θ
s
) are given by
∂
∂τ
T
s(θ
sp
)=
˜
B
TV,T
⋄
˜
B
RV
⋄D
f
⋄γ
T
vv
(B.14)
∂
∂ϕ
T
T
s(θ
sp
)=







D
ϕ
T
,V
⊙A
1
T
.
.
.
D
ϕ
T
,V
⊙A
T
T







⋄
˜
B
RV
⋄B
f
⋄γ
T
vv
(B.15)
∂
∂ϕ
T
R
s(θ
sp
)=
˜
B
TV,T
⋄(D
ϕ
R
,V
⊙A
R
)⋄B
f
⋄γ
T
vv
(B.16)
∂
∂γ
T
vv,r
s(θ
sp
)=
˜
B
TV,T
⋄
˜
B
RV
⋄B
f
(B.17)
∂
∂γ
T
vv,i
s(θ
sp
)= j
˜
B
TV,T
⋄
˜
B
RV
⋄B
f
, (B.18)
wherej standsfortheunitimaginarynumber here. Particularlythepartialderiva-
tive with respect to the Doppler shift is determined by
∂
∂ν
T
s(θ) =















B
TV
⊙D
1
T
···
B
TV
⊙D
t
T
···
B
TV
⊙D
T
T















⋄
˜
B
RV
⋄B
f
⋄γ
T
vv
+
˜
B
TV,T
⋄(B
RV
⊙D
ν,R
)⋄B
f
⋄γ
T
vv
. (B.19)
162
Most of theD matrices in this appendix and Tab. A.1 are given in [46, (78)-(83)]
except for D
j
T
with j = 1,2,...,T, which is determined by D
j
T
=D(A
j
T
,ν). The
operator D() is defined based on [46, (77)].
163

Abstract (if available)

Abstract Connected Vehicles and millimeter wave (mmWave) communications are both key parts of the upcoming 5G wireless communication systems. In order to study the performance of these new systems, improved channel models that capture the unique characteristics of vehicle-to-vehicle (V2V) and mmWave channels are of great importance. This thesis introduces several contributions in the field of signal processing tools for evaluating channel measurement data of V2V and mmWave. ❧ Firstly, we focus on the differences between the V2V channel and the typical cellular channel, and between the mmWave channel and the centimeter wave (cmWave) channel. To perform large scale V2V multiple-input multiple-output (MIMO) channel measurements, a real-time MIMO channel sounder based on NI-USRPs is designed, constructed and calibrated. Our sounder design emphasizes improvements of the stability of the setup and increasing the MIMO snapshot rate. A novel high resolution parameter estimation (HRPE) algorithm to evaluate fast time-varying channels is proposed, which is an extension of the state-of-the-art RiMAX algorithm. In order to improve the estimation accuracy, the signal model of specular paths (SPs) includes the phase rotation due to the Doppler effect. ❧ Secondly, we study the fundamental limit on the number of antennas that a time-division multiplex (TDM) channel sounder can deploy in fast time-varying channels, which is posed by the commonly-used sequential switching (SS) sequence. A novel spatio-temporal ambiguity function is introduced to characterize the performance of different non-sequential switching (non-SS) sequences that takes into account the non-idealities of real-world arrays. An optimization problem is formulated to search for a better switching pattern and an algorithm based on simulated annealing (SA) is proposed as the solution. As a result, we can extend the estimation range of Doppler shifts by eliminating ambiguities in parameter estimation. Both simulation and measurement results show that the improved performance of the new switching sequence over the conventional SS sequence. ❧ Finally, we investigate the capability of mmWave channel sounder with phased arrays to perform the super-resolution parameter estimation. Our novel multi-beam mmWave channel sounder is capable to conduct MIMO measurements in dynamic environments. We also study the limits of direction finding of multipath components (MPCs) with phased arrays and establish its connection with virtual arrays. A novel two-step calibration procedure is proposed for our mmWave channel sounder, in order to extract the system frequency responses and the frequency-independent beam patterns. We verify and validate the RiMAX algorithm and calibration results with synthetic channel responses and sample outdoor mmWave measurement data.

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Asset Metadata

Creator Wang, Rui (author)

Core Title Signal processing for channel sounding: parameter estimation and calibration

School Viterbi School of Engineering

Degree Doctor of Philosophy

Degree Program Electrical Engineering

Publication Date 11/29/2018

Defense Date 06/11/2018

Publisher University of Southern California (original), University of Southern California. Libraries (digital)

Tag 5G,channel sounding,millimeter wave communications,MIMO,multi-dimensional parameter estimation,OAI-PMH Harvest,RiMAX,vehicle-to-vehicle communications

Format application/pdf (imt)

Language English

Contributor Electronically uploaded by the author (provenance)

Advisor Molisch, Andreas (committee chair), Moghaddam, Mahta (committee member), Razaviyanyn, Meisam (committee member)

Creator Email rwangsc18@gmail.com,wang78@usc.edu

Permanent Link (DOI) https://doi.org/10.25549/usctheses-c89-110076

Unique identifier UC11676750

Identifier etd-WangRui-6986.pdf (filename),usctheses-c89-110076 (legacy record id)

Legacy Identifier etd-WangRui-6986.pdf

Dmrecord 110076

Document Type Dissertation

Format application/pdf (imt)

Rights Wang, Rui

Type texts

Source University of Southern California (contributing entity), University of Southern California Dissertations and Theses (collection)

Access Conditions The author retains rights to his/her dissertation, thesis or other graduate work according to U.S. copyright law. Electronic access is being provided by the USC Libraries in agreement with the a...

Repository Name University of Southern California Digital Library

Repository Location USC Digital Library, University of Southern California, University Park Campus MC 2810, 3434 South Grand Avenue, 2nd Floor, Los Angeles, California 90089-2810, USA