MS Thesis_Sangjo_Yoo

Thesis for Master’s Degree
Geometry-Based Stochastic Models for MIMO
Fading Channels
Sangjo Yoo
School of Information and Mechatronics
Gwangju Institute of Science and Technology
2012

석 사 학 위 논 문
다중안테나 페이딩 채널 환경을 위한 기하학적
확률 채널 모델링 방법 연구
유 상 조
정 보 기 전 공 학 부

광 주 과 학 기 술 원
2012

With love and aﬀection for my family and friends

MS/SIM
20101213
Sangjo Yoo. Geometry-Based Stochastic Models for MIMO Fading Chan-
nels. School of Information and Mechatronics. 2012. 64p. Advisor: Prof.
Kiseon Kim.
Abstract
A geometry-based stochastic model (GBSM) has been received significant
attention due to a number of advantages. A GBSM can capture the essential charac-
teristics of propagation channels of interest, and it can be easily applied to different
propagation scenarios by considering arbitrary geometries reflecting diverse scattering
regions; therefore, it is applicable to modeling of the diverse propagation environments
ranging from classical cellular networks to contemporary complex communication sys-
tems such as relay networks using multi-element antennas.
In this thesis we introduce geometry-based stochastic channel modeling method-
ologies and analysis of channel statistics for multiple communication environments. By
reviewing three important models that have led to the present-day theories of GB-
SMs from the viewpoints of geometrical and physical representations, we discuss the
requirements that must be considered for the development of a GBSM for multiple
mobile-to-mobile (M2M) wideband multiple-input multiple-output (MIMO) channel
environments. The main objectives of this thesis are: to propose a GBSM based on re-
alistic geometries and tapped-delay-line structure useful for the analysis and simulation
of diverse M2M propagation scenarios, and to propose a method to incorporate mea-
sured or specified PDPs in the proposed GBSM for reliable and less complex link-level
simulations. We believe this thesis is useful for flexible channel model design based on
GBSM for contemporary communication systems.
©2012
Sangjo Yoo
ALL RIGHTS RESERVED
– i –

MS/SIM
20101213
유상조. 다중안테나 페이딩 채널 환경을 위한 기하학적 확률 채널 모델링
방법 연구. 정보기전공학부. 2012. 64p. 지도교수: 김기선.
국 문 요 약
기하학-기반 확률적 채널 모델 (geometry-based stochastic model, GBSM)
은 그 다양한 장점으로 인해 지금까지 학계에서 큰 주목을 받아왔다. GBSM은 전파
채널의 특성을 쉽게 모델링 할 수 있으며, 산란자들의 다양한 위치 및 분포를 임의의
기하학적 도형을 사용하여 모델링이 가능하므로, 고전적인 단순한 셀룰러 네트워크에
서부터, 현대의 복잡한 다중안테나 기반 릴레이 네트워크에 이르기까지의 다양한 전파
환경에 적용 가능하다.
본 논문에서는, GBSM을 활용한 채널 모델링 기법과 이것으로 부터 얻은 채널
응답을 통계적으로 분석하는 방법론을 소개한다. 또한, 최근까지 학계에 발표된 중요
한 GBSM 모델 3 종류를 분석하고 검토하여, 최근 이슈가 되고 있는 이동-대-이동
(mobile-to-mobile, M2M) 광대역 다중 입출력 안테나 (multiple-input multiple-output,
MIMO) 통신 환경에 적합한 GBSM을 개발하기 위해 고려해야할 점들을 논의한다.
위의 논의 사항을 바탕으로, 본 논문에서는 다양한 M2M 광대역 MIMO 통신환경에
적용 가능한 tapped-delay-line 구조를 사용하는 기하학적 다중 링 기반의 채널 모델
(geometrical multi-radii two-rings,GMRTR) 을 제안하고 그 통계적 특성을 분석한다.
또한, 측정된 전력 지연 프로파일을 제안된 TDL 기반 GBSM에 적용하는 방법을 제안
한다. 제안된 방법은 TDL 기반 GBSM을 이용한 링크 레벨 시뮬레이션에서의 유연성
및 신뢰성을 높이고 연산 복잡도를 낮출 수 있는 장점이 있다.
©2012
유 상 조
ALL RIGHTS RESERVED
– ii –

Contents
Abstract (English) i
Abstract (Korean) ii
Acknowledgements iii
List of Contents vi
List of Tables viii
List of Figures ix
1 Introduction 1
1.1 Research Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 Research Objectives . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
1.3 Thesis Organization . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
2 Overview of MIMO Channel Models 6
2.1 MIMO Channel Model Classiﬁcation . . . . . . . . . . . . . . . . . . . 6
2.1.1 Physical Channel Model . . . . . . . . . . . . . . . . . . . . . . 7
2.1.2 Analytic Channel Model . . . . . . . . . . . . . . . . . . . . . . 10
2.2 Geometry-Based Stochastic Model . . . . . . . . . . . . . . . . . . . . 11
2.2.1 Geometrical Single-Radii One-Ring Model for Base-to-Mobile
Narrowband MIMO Channels . . . . . . . . . . . . . . . . . . . 12
2.2.2 Geometrical Single-Radii Two-Ring Model for Mobile-to-Mobile
Narrowband MIMO Channels . . . . . . . . . . . . . . . . . . . 14
2.2.3 Geometrical Single-Radii Two-Ring Model for Mobile-to-Mobile
Wideband MIMO Channels . . . . . . . . . . . . . . . . . . . . 16
2.3 Conclusive Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
3 Proposed Geometrical Multi-Radii Two-Ring Model for Mobile-to-
Mobile Wideband MIMO Channel 21
3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
– vi –

3.2 Proposed Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
3.3 Spece-Time-Frequency Correlation Function of the GMRTR Model . . 30
3.4 Correlation Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
3.5 Conclusive Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42
4 Incorporating geometry-based stochastic channel model in tapped-
delay-line for mobile-to-mobile wideband MIMO channels 45
4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46
4.2 Problem Description . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47
4.3 A Methodolgy to Obtain a ∆τ–Spaced Power Delay Proﬁle . . . . . . . 50
4.4 Applications and Simulation Results . . . . . . . . . . . . . . . . . . . 52
5 Conclusions and Further Works 54
5.1 Summary of Thesis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54
5.2 Further Works . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56
Abbreviations 58
References 60
– vii –

List of Figures
2.1 Typical mobile wireless communication environment where there is mul-
tipath propagation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
2.2 Doppler eﬀect caused by movement of the mobile station. . . . . . . . . 8
2.3 Ray-based physical channel model. . . . . . . . . . . . . . . . . . . . . 9
2.4 Lee’ s geometry-based model [26] providing spatial correlation between
two base stations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
2.5 GSROR model for B2M narrowband MIMO channels . . . . . . . . . . 13
2.6 GSRTR model for M2M narrowband MIMO channels . . . . . . . . . . 15
2.7 GSRTR model for M2M wideband MIMO channels . . . . . . . . . . . 17
2.8 A M2M channel environment. . . . . . . . . . . . . . . . . . . . . . . . 18
3.1 GMRTR model for NηT
× NηR
MIMO antennas with the description of
SBT, SBR and DB components. . . . . . . . . . . . . . . . . . . . . . . 26
3.2 GMRTR model for NηT
× NηR
MIMO antennas with the description of
LOS components. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
3.3 The ACF of the GMRTR model with αT = π, αR = 0. . . . . . . . . . 34
3.4 The ACF of the GMRTR model w.r.t. αT ∈ [0, 2π] with αR = π/4 and
αR ∈ [0, 2π] with αT = π/4 at ∆t = 0.05sec for RT,1 = RR,1 = 10m. . . 35
3.5 The ACF of the GMRTR model w.r.t. αT ∈ [0, 2π] with αR = π/4
and αR ∈ [0, 2π] with αT = π/4 at ∆t = 0.05sec for RT,1 = 40m and
RR,1 = 10m. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36
– ix –

3.6 The ACF of the GMRTR model w.r.t. αT ∈ [0, 2π] with αR = π and
αR ∈ [0, 2π] with αT = π at ∆t = 0.05sec for RT,1 = RR,1 = 40m. . . . . 37
3.7 The ACF comparison for the scenario A and B. . . . . . . . . . . . . . 38
3.8 The CCF of the SBT component. . . . . . . . . . . . . . . . . . . . . . 39
3.9 The CCF of the SBR component. . . . . . . . . . . . . . . . . . . . . . 40
3.10 The CCF of the DB component. . . . . . . . . . . . . . . . . . . . . . . 40
3.11 The SF-CF comparison for the scenario A, B and C. . . . . . . . . . . 42
4.1 Reference PDP and fractional path powers of ∆τ–spaced PDP obtained
from simulation results. . . . . . . . . . . . . . . . . . . . . . . . . . . 53
4.2 Absolute value of reference FCF and FCF obtained from ∆τ-spaced PDP. 53
– x –

Chapter 1
Introduction
In this chapter, the overall scope of this thesis is briefly introduced. First, we provide
the research backgrounds of the thesis in section 1.1. Next, the research objectives are
given in section 1.2. Finally, the overview and organization of this thesis are given in
section 1.3.
1.1 Research Background
The mobile wireless channel is a physical transmission medium that lies between a
(mobile) transmitter and (mobile) receiver, and it critically affects the quality of the
signal at the receiver and eventually limits the performance of communication systems.
A well designed channel model provides channel information, which closely illustrates
behavior of the channel; by exploiting this information, it is possible to mitigate the
negative effects of channels. Therefore, to design reliable communication systems, it is
no longer questionable; developing a realistic one that efficiently and accurately models
channel environments of interest is necessary.
In deed, a lot of efforts has been devoted to modeling and measurements of mobile
wireless channels for last decades. From the extensive measurement of the envelope of
the received signals via flat fading channels, Rician and Rayleigh processes were proven
– 1 –

to be suitable models for the stochastic behavior of the received signal’s envelops under
line-of-sight (LOS) and no LOS (NLOS) environment, respectively [1]. Under the NLOS
environment, Clarke [2] was among the first researcher to derive the well-known math-
ematical reference Doppler spectrum model based on Rice’s sum-of-sinusoids (SOS)
method, and Jakes [3] suggested its implementable channel simulator. The aforemen-
tioned two classical channel models were simple compared to the contemporary channel
models, yet they provided sufficient statistics for the analogue narrowband transmission
systems they considered [4]. However, as communication systems have evolved to meet
the needs of high link quality, high data transmission rate, and diverse communica-
tion topologies, developing the channel models sufficiently reflecting the corresponding
channel environments and providing more accurate statistics, has been a big issue in
the literature.
Among all the other advanced channel models, a geometry-based stochastic model
(GBSM) has been received significant attention due to a number of advantages. A
GBSM can capture the essential characteristics of propagation channels of interest, be-
cause it uses reasonable geometries and physical parameters, such as distance between
transmitter (Tx) antenna, receiver (Rx) antenna, antenna spacing, antenna tilt angles,
mobile speeds, carrier frequency, and so on. Also, the modeling method can be easily
applied to different scenarios by considering geometry of scattering region, such as one-
ring, two-ring, or ellipse [5]; therefore, it is possible to model the diverse propagation
environments ranging from classical base-to-mobile (B2M) single-input single-output
(SISO) cellular networks [3] to mobile-to-mobile (M2M) SISO channels [6, 7], B2M
– 2 –

MIMO channels [10]-[13], M2M MIMO channels [14]-[20], and it is even applicable to
high-altitude platforms (HAP) MIMO [21] and MIMO relay systems [22, 23] whose
channel environments are even more complex than conventional one-to-one communi-
cation systems.
Yet, main difficulties remain in developing wideband channel models, especially for
M2M MIMO communication environments satisfying the following requirements:
• Req. 1: modeling the environments accurately using reasonable geometries,
• Req. 2: finding one that can be parameterized for many environments,
• Req. 3: using reliable channel parameters extracted from extensive measurements,
• Req. 4: moderate/low link-level simulation complexity.
In order to satisfy the Req. 1 and 2, acceptable and reasonable geometries should be
used to represent effective scatterers placed around both MSs. For wideband channel
modeling, GBSMs are typically incorporated in the tapped-delay-line (TDL) structure,
so that the models can be used for diverse M2M frequency selective fading environ-
ments. Especially for the Req. 3, the propagation delays and the corresponding path
gains should be reliable for the concerning propagation scenarios. Reliable channel pa-
rameters can be obtained from the measured or specified power delay profiles (PDPs)
such as [24], and this makes the TDL-based GBSM more reliable than those based
on pure GBSMs, e.g., [18, 19], as pointed out in [25]. Moreover, a uniformly spaced
PDP with a tunable interval obtained from a empirical PDP can reduce the link-level
simulation complexity by reducing the upsampling rate of transmission symbols to be
– 3 –

used in the simulation and can increase the flexibility of the simulation by adjusting
the discrete PDP’s interval.
To sum up, a TDL-based GBSM using reasonable geometry and uniformly spaced
PDP extracted from measured or specified PDPs will be the best GBSM satisfying
all the aforementioned criteria. However, existing GBSMs proposed for M2M MIMO
channels are based on the inappropriate geometry to consider the near-far scatterers,
e.g., [17], and also not able to adapt measured or specified PDPs [18, 20]. Even though
the GBSM proposed in [17] can adapt measured or specified PDPs, but the PDP is
not uniformly spaced. These unsolved problems have motivated us to develop a new
GBSM to fill the aforementioned gaps. The corresponding research objectives will be
presented in the following section.
1.2 Research Objectives
As it follows from the title, in this thesis we introduces geometry-based stochastic
channel modeling methodologies and analysis of channel statistics for MIMO com-
munication environments. By reviewing three important models that have led to the
present-day theories of GBSMs from the viewpoints of geometrical and physical rep-
resentations, we discuss the factors that must be considered to satisfy the aforemen-
tioned requirements for developing a GBSM applicable to multiple M2M wideband
MIMO propagation environments. The objectives of this thesis are: (1) to propose a
GBSM based on realistic geometries and TDL structure useful for the simulation of
diverse M2M wideband MIMO propagation scenarios, and (2) to propose a method to
– 4 –

incorporate measured or specified PDPs in the proposed GBSM in such a way that
the PDP of the proposed GBSM is uniformly spaced with an adjustable spacing factor
∆τ, which enables the simulation of TDL-based GBSM to be more flexible and less
complex.
1.3 Thesis Organization
This thesis is outlined as follows. Overview of existing MIMO channel models are
given in chapter2. Specifically, physical channel models and analytic channel models are
introduced in a comparative manner, as well as three important GBSMs. In chapter
3, a GBSM based on geometrical multi-radii two-ring (GMRTR model) is proposed
for multiple M2M wideband MIMO environments. To investigate the fading statistics
of the proposed models, analytical channel correlation functions (CFs) are derived
and analyzed. In chapter 4, a simple yet effective method to incorporate measured or
specified PDPs in the GMRTR model is proposed. We show that the resulting discrete
propagation delays and the corresponding channel gains form a uniformly spaced PDP
with a tunable spacing factor, which enhance the flexibility and reduce the complexity
of the simulation of the TDL-based GMRTR model. Finally, conclusions of this thesis
and suggestions for the further works are given in chapter 5.
– 5 –

Chapter 2
Overview of MIMO Channel Models
This chapter introduces an overview of existing MIMO channel models. In section
2.1, physical channel models and analytic channel models are introduced in a com-
parative manner. In section 2.2, we review three important models that have led to
the present-day theories of GBSMs from the viewpoints of geometrical and physical
representations. Each channel model’s contributions to the GBSM theory and their
limitations are also discussed. In section 2.3, based on our discussions in section 2.2,
we point out some requirements for geometry-based stochastic channel modeling for
multiple M2M wideband environments as conclusive remarks.
2.1 MIMO Channel Model Classification
A number of MIMO channel models have been introduced in the literature, and
they can be classified in many different ways depending on the statistical behavior
and impulse response characteristics of channels. One useful way to distinguish these
channel models is based on the modeling approaches, either physical channel model or
analytic channel model. In the following two subsections, we introduce the two type
of channel models and compare them in terms of their underlying modeling principles,
pros, and cons.
– 6 –

2.1.1 Physical Channel Model
In typical mobile wireless communications, signals transmitted from a base station
(BS) go through the reflection, diffraction, shadowing and scattering until its reception
at mobile station (MS) due to the obstables located between the two site as described
in Fig. 2.1. As the result, the transmitted signals, which is electromagnetic waves,
take different propagation paths and propagate while experiencing random phase shift,
Doppler frequency shift, and amplitude attenuation for each path. The received signals
are then represented by superposition of these signals coming via all these multipaths,
and the received signal strength suffers severe fluctuation due to the phase difference
Shadowing
Base Station
Scattering
)(tv
Mobile
Station
...
Scattering
frequency-shifted, and
phase-shifted replicas
Reflection
LOS component
Figure 2.1: Typical mobile wireless communication environment where there is multi-
path propagation.
– 7 –

Mobile station
x
y
nI
Direction of motion
n-th incoming wave’s
angle of arrival
n-th incoming wave
Mobility causes
Doppler spread
Figure 2.2: Doppler effect caused by movement of the mobile station.
of each received signal. This multipath propagation causes signal time-spreading in
time domain and frequency selective fading in frequency domain. This effect distorts
transmitted signals in a such way that the signals spreads over time and eventually
causes inter-symbol interference (ISI) and distortion of the channel frequency response.
Besides, mobile wireless communication frequently faces the Doppler spread due to
the movement of the MS as described in Fig. 2.2. Doppler effect broaden transmitted
signal’s bandwidth up to maximum Doppler frequency, which is determined by the
mobile speed, angle of motion, and the angle of incoming wave (frequently referred to
angle of arrival). Since the mobile speed varies with time, the corresponding Doppler
shift is also time dependent. This makes the channel time-variant, meaning that the
channel impulse response varies with time. Due to multipath propagation and the
Doppler effect, received signal strength suffers severe stochastic fluctuation, which is
called fading.
Fading channel with MIMO Tx and Rx antennas can be modeled using physi-
cal channel modeling principle. In the physical channel modeling, each propagation
– 8 –

Reflection,
refraction,
scattering,
diffraction
Path N
Path 1
Tx Antennas
Rx Antennas
Cluster 1
Cluster N
Figure 2.3: Ray-based physical channel model.
path is represented by sum of the “rays” which are reﬂected, refracted, scattered, and
diﬀracted by the objects between the Tx and Rx, and the objects are represented by
physical clusters. In other words, one fading cluster generates an independent Rayleigh
fading path, and the angles of incoming rays with respect to the receiver determine the
correlations between MIMO sub-channels (or antennas). The illustration of physical
channel model is given in Fig. 2.3. Physical channel model uses physical parameters,
such as angle of arrival (AOA), angle of departure (AOD), and antenna tilt angles,
which determine the physical characteristics of each propagation path. Currently, the
representative physical channel models are ray tracing model, GBSM, spatial channel
model (SCM), WINNER, and GBSMs are our particular interest of this thesis. More
details about GBSMs will be discussed in section 2.2.
– 9 –

2.1.2 Analytic Channel Model
Another class of channel models is analytic channel model. Analytic channel model
does not consider the physical reality of the propagation of rays but only statistical
characteristics of channel matrix. The modeling method directly calculates channel
matrix, and the fading characteristics of MIMO sub-channels are typically assumed to
be independent identically distributed (i.i.d) when the scatterers placed around Tx and
Rx are very large, i.e., rich isotropic scattering environment. To represent time-variant
behavior of the channel, the random numbers generated from a complex random gen-
erator are processed via Doppler filter, in order to obtain the corresponding channel
matrices. The correlation between MIMO antenna elements can be also modeled by
channel matrices. In this case, the random numbers are generated in such a way that
specific correlation exists between the numbers via mathematical methods. The chan-
nel models using this method is referred as the correlation-based channel model. In
this chapter, we briefly introduce one of well-known correlation-based channel models,
METRA model.
METRA model is the result of IST METRA project, called “WP2 Channel Char-
acterization,” and this model generates MIMO matrices for given channel parameters,
such as antenna correlation. The key of the METRA model is the use of correlation
matrices, meaning that the MIMO channel matrices do not reflect the MIMO channel
environment physically, but it is simply generated based on the given antenna cor-
relation information; thus, it simply provides MIMO channel matrices for link-level
simulation. The antenna correlation information is typically given from empirical data
– 10 –

or mathematical models, which are function of spatial parameters, such as AOA and
power azimuth spectrum (PAS).
The one advantage of METRA model is that it is possible to control the correla-
tion between MIMO antennas directly. However, this is also disadvantageous to itself
because the channel matrices do not reflect neither MIMO system configuration (e.g.,
antenna spacing and antenna tilt angles) nor real physical channel environment. For
this reason, the channel matrices generated by METRA model is not statistically reli-
able with the consideration of the real channel environment of interest.
2.2 Geometry-Based Stochastic Model
As discussed in the section 2.1 and 2.2, physical channel models are more suitable
for the simulation of real channel environments of interest. For this reason, the re-
cent MIMO channel models developed for mobile ad-hoc wireless networks, intelligent
transportation systems, and relay-based cellular networks are mostly physical chan-
nel models, and especially GBSMs are the most actively researched areas in channel
modeling. In fact, its origin stems from spatial channel models developed for receiver
diversity system using two BSs, and one-ring geometry is used to represent the effec-
tive scatterers placed around the MS. Then, the spatial correlation between two BSs
are given as a function of physical parameters. As can be seen in Fig 2.4, a GBSM
can capture the essential characteristics of propagation channels of interest, because
it uses geometries and physical parameters describing the physical reality of channels,
such as distance between Tx antenna, Rx antenna, antenna spacing, antenna tilt an-
– 11 –

y
x
RD
D
BWT
Effective
scatterers
MS
d
E
Ring of scatterers
E
BWT
D
R - Ring radius
- Distance between Tx and Rx antennas
- Angular spread
- Angle of motionD
- Rx antenna tile angle
Figure 2.4: Lee’ s geometry-based model [26] providing spatial correlation between two
base stations.
gles, mobile speeds, carrier frequency, and so on. Moreover, the modeling method can
be easily applied to different channel scenarios by considering geometry of scattering
region, such as one-ring, two-ring, or ellipse [5]; therefore, it is possible to model the
diverse propagation environments ranging from classical B2M SISO cellular networks
to contemporary MIMO M2M channel environments.
In the following sections, three important models that have led to the present-day
theories of GBSMs are reviewed.
2.2.1 Geometrical Single-Radii One-Ring Model for Base-to-Mobile Nar-
rowband MIMO Channels
Geometrical single-radii one-ring model (GSROR) model is the first GBSM for
B2M MIMO communication environments proposed by Shiu [8], and its simulation
model was proposed by Pätzold [9]. This model considers a fixed BS and a MS, and
it calculates the MIMO channel matrix using physical parameters, such as antenna
– 12 –

RD
RV
x
y
TG
T
T
AK
T
A1
TE
,T n,,
nS
RD
R
A1
RE
R
R
AK,R n,,
RG,maxT
TO RO
Figure 2.5: GSROR model for B2M narrowband MIMO channels
spacing, antenna tilt angles, distribution of scatterers around MS, AOAs, AODs, and
so on. In B2M channel environments, the BS antenna is highly elevated compared to
the surrounding objects, while the MS antenna is not elevated. Since only MS antenna
elements are subjected to the local scattering objects, a ring with radius R is used
to represent the local scatterers around MS as can be seen in Fig. 2.5. In this model,
it is assumed that an infinite number of scatterers are uniformly distributed on the
circumference over the interval [−π, π), so the transmitted signals from BS is captured
by a number of scatterers lying on the ring at first and received at the MS. Since
the mobility of MS and the scatterers cause Doppler shift and random phase shifts,
respectively, a received signal can be represented by a superposition of an infinite
number of randomly phase shifted, Doppler shifted replicas of a transmitted signal. In
typical one-ring based GBSMs, it is assumed that the BS (MS) has ηT (ηR) antennas
with the inter antenna spacing δT (δR) and tilt angle βT (βR). The MS moves with
– 13 –

velocity VR in the direction determined by the angle of motion αR. The nth scatterer
located on the ring placed around the MS is denoted by Sn, where n ∈ {1, 2, ..., N}. The
AODs and the AOAs are denoted by ϕT,n and ϕR,n, respectively. The AOAs are i.i.d
random variables following a uniform distribution over [−π, π). The distance between
MSs, denoted by D, is very large, and the ring radius is also larger than the antenna
spacings such that the relations, max {(ηT ), (ηR)} ≪ min{R} and max {R} ≪ D hold.
As mentioned before, the GSROR model was proposed for conventional B2M nar-
rowband MIMO channel environments. Despite of its simpleness, it has been an im-
portant framework for the GBSMs developed for more complex channel environments.
2.2.2 Geometrical Single-Radii Two-Ring Model for Mobile-to-Mobile Nar-
rowband MIMO Channels
By extending the GSROR model introduced in the previous section, Pätzold [9] pro-
posed a geometrical single-radii two-ring model (GSRTR) model for M2M narrowband
MIMO channel environments as shown in the Fig. 2.6. In B2M channel environments,
the one-ring geometry is used to represent the eﬀective scatterers around MS due to
the low elevated MS antennas. On the other hand, in M2M environments, both Tx and
Rx are mobile stations and move with diﬀerent velocities and directions. Since both
MSs’ antennas are not elevated, these are subjected to the local scattering. To represent
such local scatterers placed around each MS, Pätzold introduced two-ring geometry,
and he derived the channel impulse response by assuming that only double-bounced
rays carry the most of the received signal power due to dense isotropic scattering envi-
– 14 –

RRTR D
RV
x
y
,T mS
TG
TV
TD
T
T
AK
T
A1
TE ,T m,,
,R nS
RD
R
A1
RE
R
R
AK,R n,,
RG
,m nd
,T mdK
1,md
,R ndK
1,nd
Figure 2.6: GSRTR model for M2M narrowband MIMO channels
ronment. As can be seen in Fig. 2.6, two rings are placed around each MS with radius
RT and RR to represent the local scatterers around mobile transmitter and receiver,
denoted by MST and MSR, respectively. MST (MSR), has ηT (ηR) number of anten-
nas with the inter antenna spacing δT (δR) and tilt angle βT (βR). It is assumed that
MST (MSR) moves with velocity VT (VR) in the direction determined by the angle
of motion αT (αR). The m (n)th scatterer located on the ring placed around Tx is
denoted by ST,m (SR,n), where m ∈ {1, 2, ..., M} (n ∈ {1, 2, ..., N}). It is also assumed
that M (N) is infinite, and the scatterers are uniformly distributed over [−π, π). The
AOD and AOA, denoted by ϕT,m and ϕR,n, respectively, are i.i.d. random variables
following a uniform distribution over [−π, π). The distance between MSs, denoted by
D, is very large, and the ring radii are also larger than the antenna spacings such that
the relations max {ηT , ηR} ≪ min{RT , RR} and max {RT , RR} ≪ D hold. Pätzold not
only proposed the first GBSM for M2M MIMO channels, but also he firstly derived the
space-time correlation function (ST-CF) of the GBSM for M2M MIMO environments.
– 15 –

The ST-CF provides the joint statistics between the spatial correlation and temporal
correlation, which is useful statistics for designing M2M communication systems with
the consideration of time-selective channel environment with MIMO antennas. He later
extended his work to the GBSM for non-isotropic case in [16].
2.2.3 Geometrical Single-Radii Two-Ring Model for Mobile-to-Mobile Wide-
band MIMO Channels
The growing need for higher data rate M2M communication have stimulated the
studies in wideband (frequency selective or time dispersive) channel modeling. Re-
searchers have tried to implement TDL structure to wideband channel modeling since
the most of time-dispersive and stochastic time-variant features of wireless channel can
be easily described. In this way, Y. Ma and M. Pätzold presented two GSRTR model
[17] considering two diﬀerent scattering scenarios: double bounced (DB) scattering and
single bounced (SB) scattering, respectively. These two channel models were obtained
from [10] by partitioning each ring at both MSs into several disjoint intervals, which
correspond to diﬀerent relative propagation delays as can be seen in Fig. 2.7. The over-
all time-variant impulse response consisting of i ∈ {0, 1, ..., I − 1} discrete propagation
paths between pth Tx antenna and qth Rx antenna can be represented by using TDL
structure as blow:
hpq(t, τ) =
I−1∑
i=0
˜αihpq,i(t)δ(τ − ˜τi), (2.1)
where δ(·), ˜αi, and ˜τi denote Dirac delta function, discrete path gains, and relative
propagation delays, respectively. The discrete path gains and relative propagation de-
– 16 –

RRTR D
RV
x
y
TV
,R nS
RD
R
A1
RE
R
R
AK,R n,,
RG
,m nd
,R ndK
1,nd
,T mS
TG
TD
T
T
AK
T
A1
TE
,T m,,
,T mdK
1,md
0W0
1W1
2W2
3W3
4W4
IWI
SBT:
SBR:
DB:
Figure 2.7: GSRTR model for M2M wideband MIMO channels
lays determine the frequency selectivity of the channel, and these two sets of parameters
in this model were obtained from the specification of the 18-path HiperLAN/2 model
C [27]. hpq,i(t), which denotes the time-variant complex channel gain of ith path gener-
ated by superposition of scattered rays, can be obtained by applying wave propagation
law based on the model geometry.
In [17], Y. Ma successfully modeled M2M wideband MIMO channel environments
using GSRTR based on the TDL structure, and the model is able to adapt measured
or specified PDPs using the 18-path HiperLAN/2 model C. Since the model can use
reliable channel parameters obtained from the specification, reliable frequency selec-
tive channel simulations are possible using this model. However, besides its usefulness,
the model geometry is inappropriate to consider the near and far effective scatterers,
as described in the Fig. 2.8. This causes the model’s space-time-frequency correlation
– 17 –

Effective
scatterer
Moving direction
Single bounced ray around transmitter (SBT)
Single bounced ray around receiver (SBR)
Double bounced ray (DB)
Line-of-site (LOS)
Figure 2.8: A M2M channel environment.
error by modeling the near and far scatterers on the same rings. Moreover, as pointed
out in [19], all LOS, single bounced ray around transmitter (SBT), single bounced ray
around receiver (SBR), and DB components contribute the total channel statistics with
different proportions. Since the model only considers two scattering scenarios, either
double bounced scattering or single bounced scattering, the model provides not suffi-
cient channel statistics for M2M environments. For the link-level simulation complexity
aspect, using non-uniformly spaced PDP, i.e., [27] increases the link-level simulation
complexity due to the small greatest common divisor (GCD) of the propagation delays.
To cope with all these disadvantages, we propose a new GBSM based on geometrical
multi-radii two-rings in section 3, and we also propose a method to incorporate mea-
sured or specified PDPs in the proposed GBSM in such a way that the PDP of the
– 18 –

proposed GBSM is uniformly spaced with an adjustable spacing factor ∆τ, which en-
ables the TDL-based GBSM proposed in section 3 to be more flexible and less complex
for the link-level simulation.
2.3 Conclusive Remarks
In this chapter, we provided an overview of existing MIMO channel models. In chap-
ter 2.1, we classified the existing MIMO channel models into two categories: physical
channel model and analytic channel model. The physical channel model generates the
MIMO channel matrix using the law of propagation with physical channel parameters
while analytic channel model directly generates the MIMO channel matrix for given
MIMO sub-channel correlation information. Since the analytic channel model does not
reflect the physical channel environment, the channel matrices generated by the ana-
lytic channel model cannot statistically represent the channel environment. For those
reasons, physical channel models have been considered as more suitable models for the
simulation of real channel environments of interest. In fact, the GBSM is classified into
the physical channel model category, and it has been received significant attention due
to a number of the aforementioned advantages.
In chapter 2.2, we reviewed three important models that have led to the present-day
theories of GBSMs from the viewpoints of geometrical and physical representations.
From the three models, we have observed that the ring geometry is used to represent
the effective scatterers: for B2M narrowband MIMO channel environments and M2M
narrowband MIMO channel environments, one-ring and two-ring geometries are used,
– 19 –

respectively. For the M2M wideband MIMO channels, Y. Ma proposed the wideband
GSRTR model using same geometry as in [10], but intersecting the two-rings into sev-
eral disjoint parts to represent different propagation delays. Even though the model
successfully incorporated the GSRTR model into the TDL structure, but the model
geometry itself has a limitation in the representation of the near and far effective scat-
terers, which causes the model’s space-time-frequency correlation error. Also, the model
only considers limited scattering scenarios, either DB or SB rays; however, to provide
more accurate M2M MIMO channel statistics, a GBSM should be able to consider all
of the scattering components, as well as LOS components. Also, the specified PDP [27]
used in the model is not uniformly spaced; thus its small GCD increases the link-level
simulation complexity. To remedy these problems, a GBSM based on the new suitable
geometry is proposed in chapter 3, and a method to produce a uniformly spaced PDP
is proposed in chapter 4.
– 20 –

Chapter 3
Proposed Geometrical Multi-Radii Two-Ring
Model for Mobile-to-Mobile Wideband MIMO
Channel
In this chapter, a new GBSM based on geometrical multi-radii two-rings (GMRTR
model) is proposed for multiple M2M wideband MIMO channel environments. The
multi-radii two-ring geometry is more realistic and generic to represent near and far
scatterers placed around MSs than the single-radii two-ring geometry of conventional
models for frequency selective propagation environments. Also, the proposed model can
be applied to diverse propagation and scattering scenarios by considering a line-of-site
component, single bounced scattering components, and double bounced scattering com-
ponents, where these contribute their energies to the total received signal power with
diﬀerent proportions. The propagation channel is assumed to be wide-sense stationary
uncorrelated scattering (WSSUS), and TDL structure is used in such a way that the
proposed model can adapt a measured or speciﬁed PDPs for reliable wideband channel
simulations. Detailed modeling procedures are also presented in this chapter. For the
investigation of the proposed model’s statistical properties, the space-time-frequency
– 21 –

correlation function (STF-CF) is derived for isotropic two-dimensional (2-D) scattering
scenario, and the auto-correlation function (ACF), cross-correlation function (CCF),
and space-frequency correlation function (SF-CF) are derived from the STF-CF and
investigated for the statistical analysis of the proposed model.
3.1 Introduction
M2M communication channels have received great attention due to their vast ap-
plications in mobile ad-hoc wireless networks [30], intelligent transportation systems
[31], and relay-based cellular networks [32]. To successfully design and test such com-
munication systems, it is necessary to develop a realistic channel model which is useful
for diverse M2M channel environments by utilizing suitable geometries.
Early studies for narrowband single-input and single-output M2M Rayleigh fading
channels have been done by Akki and Haber based on scattering geometry around
MSs [6][7], and the channel models were further developed in [14] by using double-ring
scattering model. In [10][11], MIMO antennas were additionally considered under DB
scattering scenario, and these were further extended to more generalized model which
includes LOS, SB, and DB components [15, 20]. In addition, the growing needs of higher
data rate M2M communication have stimulated the studies in wideband (frequency
selective or time dispersive) channel modeling. Researchers have tried to implement
TDL structure to wideband channel modeling since the most of time-dispersive and
stochastic time-variant features of wireless channel can be easily described. In this
way, Y. Ma and M. Pätzold presented two GSRTR model [17] based on DB and SB
– 22 –

scatterings, respectively. These two channel models in [17] were obtained from [10]
by partitioning each ring at both MSs into several disjoint intervals corresponding
to relative propagation delays. Also, three-dimensional wideband M2M channel was
presented in [18] based on concentric-cylinders, and the author in the reference [19]
proposed 2-D channel model based on single-radii two-rings with confocal ellipses.
However, the GBSM proposed in [17] is based on single-radii two-ring geometry,
which is inappropriate to consider the near and far scatterers as described in Fig. 2.8.
Moreover, as pointed out in [19], all LOS, SBT, SBR, and DB components contribute
the total channel statistics differently. Since the model [17] only considers two scattering
scenarios, either double bounced scattering or single bounced scattering, the model
provides not sufficient channel statistics for M2M environments. Also, the pure GBSMs
proposed in [18, 19] not able to adapt measured or specified PDPs; therefore, the
models are less reliable than the models using reliable model parameters extracted
from empirical data, such as measured or specified PDPs (e.g. [24, 27]) as pointed out
in [25].
To fill the aforementioned gaps, we propose a new GBSM based on geometrical
multi-radii two-rings (GMRTR model) for multiple M2M wideband MIMO channel
environments. Since the multi-radii two-ring geometry is more general than the single-
radii two-ring used in the conventional wideband GBSM in [17], the proposed model is
more useful for diverse frequency selective M2M environments. Moreover, the proposed
model considers a line-of-site component, single bounced scattering components, and
double bounced scattering components for diverse propagation and scattering scenarios.
– 23 –

Also, the proposed model is designed based on the TDL structure in such a way that
the model can adapt a measured or specified PDPs for reliable channel simulations.
A method to incorporate measured or specified PDPs in the GMRTR model will be
discussed in chapter 4. In chapter 3, however, it is task to present the GMRTR model
and the detailed modeling procedures, as well as the analysis of the proposed model’s
correlation properties by investigating the ACF, CCF, and SF-CF obtained from the
STF-CF.
The remainder of this paper is organized as follows. Chapter 3.2 describes the
GMRTR model, and the corresponding time-variant complex channel gain is derived
with detailed procedures. In chapter 3.3 , the STF-CF of the proposed model is derived.
In chapter 3.4, the ACF, CCF, and SF-CF of the proposed models are obtained from
the STF-CF and their statistical properties are analyzed. Finally, conclusions are drawn
in chapter 3.4.
3.2 Proposed Model
We consider a M2M wideband MIMO communication environment under the WS-
SUS 2-D isotropic scattering scenario. Fig. 3.1 shows the GMRTR model with SBT,
SBR, and DB components, and Fig. 3.2 shows LOS components. As can be seen in
Fig. 3.1, it is assumed that K and L number of concentric rings with different radii
RT,k and RR,l for k ∈ {0, 1, ..., K −1} and l ∈ {0, 1, ..., L−1} are placed around mobile
Tx and mobile Rx, denoted by MST and MSR, respectively. The mobile transmitter
MST (receiver MSR), has ηT (ηR) antennas with the inter antenna spacing δT (δR) and
– 24 –

tilt angle βT (βR). It is assumed that MST (MSR) moves with velocity VT (VR) in the
direction determined by the angle of motion αT (αR). The m (n)th scatterer located on
the k (l)th ring at MST (MSR) is denoted by ST,k,m (SR,l,n), where m ∈ {1, 2, ..., Mk}
(n ∈ {1, 2, ..., Nl}). The propagation scenario considered here is macro- and micro-cell
scenarios, where the distance between MSs, denoted by D, is much larger than the ring
radii and inter antenna spacings, such that the following conditions
max {δT , δR} ≪ min{RT,k, RR,l}, (3.1)
max {RT,k, RR,l} ≪ D, (3.2)
hold. It is assumed that Mk (Nl) is inﬁnite, and the scatterers are uniformly distributed
over [−π, π) on the k (l) th ring. The angles ϕT,K,m and ϕT,L,m denote the AODs
determined by ST,K,m and SR,L,n, respectively. Also, the angles ϕR,L,n and ϕR,K,m denote
the AOAs determined by SR,L,n and SR,K,m, respectively. For the isotropic scattering
scenario, ϕT,K,m and ϕR,L,n are assumed to be i.i.d. random variables following a uniform
distribution over [−π, π) due to the relation 3.2. The other angles, ϕT,L,n and ϕR,K,m,
are due to the single-bounced scatterings and therefore subject to the distribution of the
ϕT,K,m and ϕR,L,n. In other words, the distribution of ϕT,K,m and ϕR,L,n are dependent
on the ϕT,L,n and ϕR,K,m. Their relations are important to derive the time-variant
channel impulse response of the GMRTR model, and it will be discussed later.
The parameters in Fig. 3.1, d1,K,m (dηT ,K,m), d,K,m,1 (dK,m,ηR
), dK,m, d1,L,n (dηT ,L,n),
dL,n, dK,m,L,n, and dL,n,1 (dL,n,ηR
) denote the distance AT
1 (AT
ηT
)–ST,K,m, ST,K,m–AR
η1
(AR
ηR
), ST,K,m–OR, AT
1 (AT
ηT
)–SR,L,n, OT –SR,L,n, ST,K,m–SR,L,n, and SR,L,n–AR
1 (AR
ηR
),
respectively. In Fig. 3.2, the parameters d1,1, dηT ,ηR
, dηT ,1, d1,ηR
, and d1 (dηT
) denote the
– 25 –

Single bounced ray
around transmitter (SBT)
Single bounced ray
around receiver (SBR)
Double bounced ray (DB)
,1RR
,2RR
,R LR
,1TR
,2TR
,T KR D
RV
x
y
. . .
, ,T K mS
TG
TV
TD
T
T
AK
T
A1
TE mKT ,,I
, ,R L nS
RD
R
A1
RE
R
R
AKnLR ,,I
. . .
RG
, ,T L nI , ,R K mI
, ,T K mdK
1, ,K md
, , ,K m L nd
, , RL nd K
, ,1L nd
,K md,L nd
1, ,L nd
, ,1K md
, , RK md K
, ,T L ndK
TO
RO
Figure 3.1: GMRTR model for NηT
×NηR
MIMO antennas with the description of SBT,
SBR and DB components.
distance AT
1 –AR
1 , AT
ηT
–AR
ηR
, AT
ηT
–AR
1 , AT
1 –AR
ηR
, and AT
1 (AT
ηT
)–OR, respectively. These
distance parameters determine the phases and propagation delays of the received waves
traveled via diﬀerent paths generated by following scattering scenarios: LOS, SBT,
SBR, and DB rays. Hence, the normalized time-variant channel impulse response (CIR)
between the pth Tx antenna element (AT
p ) and the qth Rx antenna element (AR
q ) is a
superposition of the LOS, SBT, SBR, and DB components and can be expressed as
follows:
hpq(t, τ) = hLOS
pq (t, τ) + hSBT
pq (t, τ) + hSBR
pq (t, τ) + hDB
pq (t, τ) (3.3)
– 26 –

RV
x
y
. . .
TG
TV
TD
T
T
AK
T
A1
TE
RD
R
A1
RE
R
R
AK
. . .
RG
LOS
I
,T R
dK K
1,1d
,1T
dK
1, R
d K
T
dK
1d
TO RO
D
Figure 3.2: GMRTR model for NηT
×NηR
MIMO antennas with the description of LOS
components.
where
hLOS
pq (t, τ) =
√
Kpq
Kpq+1
ej[2π(fLOS
R +fLOS
T )t]
ej 2π
λ
DLOS
δ(τ − τLOS
) (3.4)
hSBT
pq (t, τ) =
√
ζSBT
Kpq+1
K∑
k=1
˜αSBT
k lim
Mk→∞
1
Mk
Mk∑
m=1
ej[θT,k,m+2π(fT,k,m+fR,k,m)t]
·e−j 2π
λ
DSBT
δ(τ − τSBT
k,m )
(3.5)
hSBR
pq (t, τ) =
√
ζSBR
Kpq+1
L∑
l=1
˜αSBR
l lim
Nl→∞
1
Nl
Nl∑
n=1
ej[θR,l,n+2π(fT,l,n+fR,l,n)t]
·e−j 2π
λ
DSBR
δ(τ − τSBR
l,n )
(3.6)
hDB
pq (t, τ) =
√
ζDB
Kpq+1
K∑
k=1
L∑
l=1
˜αDB
kl lim
Mk→∞
Nk→∞
1√
MkNk
Mk∑
m=1
Nl∑
n=1
ej[θT,k, m|R,l,n+2π(fT,k,m+fR,l,n)t]
·e−j 2π
λ
DDB
δ(τ − τDB
k,m,l,n)
(3.7)
In (3.4)–(3.7), Kpq denotes the Rician K factor of the MIMO sub-channel, i.e., AT
p –AR
q ,
and the parameters ζSBT , ζSBR, and ζDB denote the power contribution ratios of each
component w.r.t. the total received signal power satisfying the boundary condition:
– 27 –

ζSBT + ζSBR + ζDB = 1. ˜αSBT
k , ˜αSBR
l , and ˜αDB
kl denote fractional path gains specifying
how much the scatterers placed on the circumference of a specific ring contribute in the
total received power, and these parameters satisfy the following boundary conditions:
∑
k
(
αSBT
k
)2
=
∑
l
(
αSBR
l
)2
=
∑
k,l
(
αDB
k,l
)2
= 1. (3.8)
Also, fLOS
T , fLOS
R , fT,k,m, fT,l,n, fR,k,m, and fR,l,n denote Doppler shifts and defined as
fLOS
T = fTmax cos αT , fLOS
R = −fRmax cos αR, fT,k,m = fTmax cos(ϕT,k,m − αT ), fT,l,n ≈
fTmax cos
[
RR,l
D
sin ϕR,l,n − αT
]
, fR,k,m ≈ −fRmax cos
[
RT,k
D
sin (ϕT,k,m) + αR
]
, and fR,l,n =
fRmax cos(ϕR,l,n − αR), where fTmax and fRmax denote maximum Doppler frequencies by
movement of each MS. λ denotes the wavelength of the carrier frequency. The random
phase shifts, denoted by θT,k,m, θR,l,n are i.i.d. random variables uniformly distributed
over [−π, π). The joint phases, θT,k,m|R,l,n, can be expressed by a sum of two i.i.d random
phases, i.e. θT,k,m|R,l,n = θT,k,m + θR,l,n due to the independent locations of ST,k,m and
SR,n,l with the condition (3.2) [10]. Finally, DLOS
, DSBT
, DSBR
, and DDB
denote the
distance traveled by transmitted waves from AT
p to AR
q in each propagation scenario,
and these are defined as DLOS
= dp,q, DSBT
= dp,k,m + dk,m,q, DSBR
= dp,l,n + dl,n,q,
and DDB
= dp,k,m + dk,m,l,n + dl,n,q.
By observing the geometries in Fig. 3.1 and 3.2, and by using the relation (3.1),
(3.2), trigonometric identity, and the approximation, i.e.,
√
1 + x ≈ 1 + x/2 for x ≪ 1,
the distances dp,q, dp,k,m, dk,m,q, dp,l,n, dl,n,q, and dk,m,l,n can be expressed as follows:
dpq ≈ D + εR,q cos βR − εT,p cos βT , (3.9)
dp,k,m ≈ RT,k − εT,p cos(βT − ϕT,k,m), (3.10)
– 28 –

dk,m,q ≈ D + εR,q cos βR − RT,k cos ϕT,k,m, (3.11)
dp,l,n ≈ D − εT,p cos βT + RR,l cos ϕR,l,n, (3.12)
dl,n,q ≈ RR,l − εR,q cos(ϕR,l,n − βR), (3.13)
dk,m,l,n ≈ D − RT,k cos(ϕT,k,m) + RR,l cos(ϕR,l,n), (3.14)
where εT,p = (ηT −2p+1)δT
2
and εR,q = (ηR−2q+1)δR
2
. Using (3.9)–(3.14), the equations
(3.4)–(3.7) become, respectively
hLOS
pq (t, τ) =
√
Kpq
Kpq+1
ej[2π(fLOS
R +fLOS
T )t]
·e−j 2π
λ
(D−εT,p cos βT +εR,q cos βR)
δ(τ − τLOS
),
(3.15)
hSBT
pq (t, τ) =
√
ζSBT
Kpq+1
K∑
k=1
˜αSBT
k lim
Mk→∞
1
Mk
Mk∑
m=1
·ej 2π
λ [RT,k cos ϕT,k,m+εT,p cos(βT −ϕT,k,m)+θSBT
k,q ]δ(τ − τSBT
k,m ),
(3.16)
hSBR
pq (t, τ) =
√
ζSBR
Kpq+1
L∑
l=1
˜αSBR
l lim
Nl→∞
1
Nl
Nl∑
n=1
·ej 2π
λ [−RR,l cos ϕR,l,n+εR,q cos(ϕR,l,n−βR)+θSBR
l,p ]δ(τ − τSBR
l,n ),
(3.17)
hDB
pq (t, τ) =
√
ζDB
Kpq+1
K∑
k=1
L∑
l=1
˜αDB
kl lim
Mk→∞
Nk→∞
1√
MkNk
Mk∑
m=1
Nl∑
n=1
·ej 2π
λ [εT,p cos(βT −ϕT,k,m)+εR,q cos(ϕR,l,n−βR)+RT,k cos ϕT,k,m−RR,l cos ϕR,l,n+θDB
k,l ]δ(τ − τDB
k,m,l,n),
(3.18)
where τLOS
, τSBT
k,m , τSBR
l,n , and τLOS
k,m,l,n denote the propagation delays, and using the
relations, (3.1) and (3.2), these are approximately obtained as τLOS = DLOS
c0
≈ D
c0
,
τSBT
k,m = DSBT
c0
≈ D
c0
+
RT,k(1+cos ϕT,k,m)
c0
, τSBR
l,n = DSBR
c0
≈ D
c0
+
RR,l(1+cos ϕR,l,n)
c0
, and τDB
k,m,l,n =
DDB
c0
≈ D
c0
+
RT,k(1−cos ϕT,k,m)
c0
+
RR,l(1+cos ϕR,l,n)
c0
.
– 29 –

3.3 Spece-Time-Frequency Correlation Function of the GMRTR Model
Since the time-variant CIR (3.3) is wide-sense stationary (WSS) zero mean complex
Gaussian random process, its ﬁrst and second order statistics completely characterize
such MIMO channels, and especially both auto- and cross-correlation properties w.r.t.
time, space, and frequency only depend on the time separation, antenna spacing, and
frequency separation, respectively. In fact, for a WSSUS MIMO channel model, its
ACF, CCF, and FCF are typically analyzed to investigate its statistical properties. The
ACF provides information of channel’s fading rapidity w.r.t. mobile speeds or maxi-
mum Doppler frequencies, and the CCF provides correlation between two arbitrary
MIMO sub-channels w.r.t. the Tx and Rx antenna spacing. Also, the FCF provides
the frequency coherence of information. However, for higher data rate and link quality,
contemporary communication systems exploits the multiple dimensions of space, time,
and frequency, or their joint domain, such as MIMO-OFDM systems. For the design
and optimization of such systems, the conventional correlation functions w.r.t. a sin-
gle domain, i.e., ACF, CCF, and FCF, can not provide joint statistics (between any
combination of time, space, and frequency) required for the design of such systems.
Among all the other joint statistics, the STF-CF provides not only joint correlation
between time, space, and frequency domains but also its sub-correlation properties,
such as space-time CF (ST-CF), time-frequency CF (TF-CF), SF-CF, ACF, CCF, and
FCF as special cases. In this section, we derive the STF-CF of the GMRTR model by
assuming that the number of scatterers on the circumference of the rings placed around
each MS is inﬁnite, i.e. Mk, Nl → ∞ ∀k, l and uniformly distributed over [−π, π). The
– 30 –

STF-CF between the two normalized time-variant CIR is deﬁned as follows:
RHpqHp′q′
(δT , δR, ∆t, ∆f) = E
[
Hpq(t + ∆t, f + ∆f)H∗
p′q′ (t, f)
]
(3.19)
where E [·] and (·)∗
denote expectation operation and complex conjugation, respec-
tively, and Hpq(t, f) denotes the time-variant transfer function, which can be obtained
by taking the Fourier transform of the time-variant CIR (3.3) [1] as below:
Hpq(t, f) = Fτ {hpq(t, τ)} = HLOS
pq (t, f)+HSBT
pq (t, f)+HSBR
pq (t, f)+HDB
pq (t, f). (3.20)
In (3.20), HLOS
pq (t, f), HSBT
pq (t, f), HSBR
pq (t, f), and HDB
pq (t, f) denote LOS, SBT, SBR,
and DB components of the time-variant transfer function. The expressions of these
components are given as below:
HLOS
pq (t, f) =
√
Kpq
Kpq+1
ej[2π(fLOS
R +fLOS
T )t]
·e−j 2π
λ
(D−εT,p cos βT +εR,q cos βR)
e−j2πfτLOS
,
(3.21)
HSBT
pq (t, f) =
√
ζSBT
Kpq+1
K∑
k=1
˜αSBT
k lim
Mk→∞
1
Mk
Mk∑
m=1
·ej 2π
λ [RT,k cos ϕT,k,m+εT,p cos(βT −ϕT,k,m)+θSBT
k,q ]e−j2πfτSBT
k,m ,
(3.22)
HSBR
pq (t, f) =
√
ζSBR
Kpq+1
L∑
l=1
˜αSBR
l lim
Nl→∞
1
Nl
Nl∑
n=1
·ej 2π
λ [−RR,l cos ϕR,l,n+εR,q cos(ϕR,l,n−βR)+θSBR
l,p ]e−j2πfτSBR
l,n ,
(3.23)
HDB
pq (t, f) =
√
ζDB
Kpq+1
K∑
k=1
L∑
l=1
˜αDB
kl lim
Mk→∞
Nk→∞
1√
MkNk
Mk∑
m=1
Nl∑
n=1
·ej 2π
λ [εT,p cos(βT −ϕT,k,m)+εR,q cos(ϕR,l,n−βR)+RT,k cos ϕT,k,m−RR,l cos ϕR,l,n+θDB
k,l ]e−j2πfτDB
k,m,l,n .
(3.24)
Since HLOS
pq (t, f), HSBT
pq (t, f), HSBR
pq (t, f), and HDB
pq (t, f) are independent zero-mean
– 31 –

Gaussian complex random processes, (3.19) can be expressed as below:
RHpqHp′q′
(δT , δR, ∆t, ∆f) = RLOS
HpqHp′q′
(δT , δR, ∆t, ∆f) + RSBT
HpqHp′q′
(δT , δR, ∆t, ∆f)
+RSBR
HpqHp′q′
δT , δR, ∆t, ∆f) + RDB
HpqHp′q′
(δT , δR, ∆t, ∆f),
(3.25)
where RLOS
HpqHp′q′
(δT , δR, ∆t, ∆f), RSBT
HpqHp′q′
(δT , δR, ∆t, ∆f), RSBR
HpqHp′q′
(δT , δR, ∆t, ∆f),
RDB
HpqHp′q′
(δT , δR, ∆t, ∆f) denote the STF-CF of the LOS, SBT, SBR, DB, components,
respectively, and these are deﬁned as follows:
RLOS
HpqHp′q′
(δT , δR, ∆t, ∆f) = E
[
HLOS
pq (t + ∆t, f + ∆f)
(
HLOS
p′q′ (t, f)
)∗]
(3.26)
RSBT
HpqHp′q′
(δT , δR, ∆t, ∆f) = E
[
HSBT
pq (t + ∆t, f + ∆f)
(
HSBT
p′q′ (t, f)
)∗]
(3.27)
RSBR
HpqHp′q′
(δT , δR, ∆t, ∆f) = E
[
HSBR
pq (t + ∆t, f + ∆f)
(
HSBR
p′q′ (t, f)
)∗]
(3.28)
RDB
HpqHp′q′
(δT , δR, ∆t, ∆f) = E
[
HDB
pq (t + ∆t, f + ∆f)
(
HDB
p′q′ (t, f)
)∗]
(3.29)
By substituting (3.21-3.24) into (3.26-3.29), and evaluating the expectation operations
w.r.t. i.i.d zero-mean random phases, i.e. θT,k,m, θR,l,n, and θT,k,m|R,l,n, only the terms
for k = k′
, l = l′
are 1; otherwise, zero. Moreover, since we assume that Mk, Nl → ∞
∀k, l and the scatterers are uniformly distributed over [−π, π), the probability mass
functions (pmfs) of discrete AOAs (ϕR,l,n) and AODs (ϕT,k,m) can be replaced with
probability density functions (pdfs), i.e. fϕ(ϕR,l) and fϕ(ϕT,k), respectively, where ϕR,l
and ϕT,k are continuous random variables. Also, using trigonometric transformations
and the equality
∫ π
−π
ea sin θ+b cos θ
dθ = 2πI0
(√
a2 + b2
)
, (3.26-3.29) can be obtained as
RLOS
TpqTp′q′
(δT , δR, ∆t, ∆f) =
√
KpqKp′q′
(Kpq+1)(Kp′q′ +1)
ej2π[fTmax cos αT −fRmax cos αR]∆t
e
−j2π∆f D
c0 ej 2π
λ
{(p′−p)δT cos βT +(q−q′)δR cos βR}
(3.30)
– 32 –

RSBT
TpqTp′q′
(δT , δR, ∆t, ∆f) = ζSBT√
(Kpq+1)(Kp′q′ +1)
K∑
k=1
(
˜αSBT
k
)2
ej 2π
λ
δR(q−q′) cos βR
e
−j2π∆f
(D+RT,k
c0
)
e−j2π∆tfRmax cos αR
· I
(k)
0
(√
V 2
SBT + W2
SBT
) (3.31)
RSBR
TpqTp′q′
(δT , δR, ∆t, ∆f) = ζSBR√
(Kpq+1)(Kp′q′ +1)
L∑
l=1
(
˜αSBR
l
)2
ej 2π
λ
δT (p′−p) cos βT
e
−j2π∆f
(D+RR,l
c0
)
· ej2π∆tfTmax cos αT
· I
(l)
0
(√
X2
SBR + Y 2
) (3.32)
RDB
TpqTp′q′
(δT , δR, ∆t, ∆f) = ζDB√
(Kpq+1)(Kp′q′ +1)
K∑
k=1
L∑
l=1
(
˜αDB
kl
)2
·I
(k)
0
(√
V 2
DB + W2
DB
)
I
(l)
0
(√
X2
DB + Y 2
)
· e
−j2π∆f
D+RT,k+RR,l
c0 ,
(3.33)
where VSBT = j2π∆tfTmax sin αT + j2π∆tfRmax
RT,k
D
sin αR + j2πδT
λ
(p′
− p) sin βT ,
WSBT = j2π∆tfTmax cos αT + j2πδT
λ
(p′
− p) cos βT − j2π∆f
RT,k
c0
,
XSBR = j2π∆tfTmax
RR,l
D
sin αT + j2π∆tfRmax sin αR + j2πδR
λ
(q′
− q) sin βR,
Y = j2π∆tfRmax cos αR + j2πδR
λ
(q′
− q) cos βR − j2π∆f
RR,l
c0
,
VDB = j2π∆tfTmax sin αT + j2πδT
λ
(p′
− p) sin βT ,
WDB = j2π∆tfTmax cos αT + j2πδT
λ
(p′
− p) cos βT + j2π∆f
RT,k
c0
,
XDB = j2π∆tfRmax sin αR +j2πδR
λ
(q′
−q) sin βR. Moreover, I0(·) denotes zero-order
modiﬁed Bessel function of the ﬁrst kind.
3.4 Correlation Properties
In this section, based on the STF-CF derived in the previous section, the ACF,
CCF, ST-CF, and SF-CF is visualized and analyzed for the investigation of the pro-
posed model’s statistical properties. From the STF-CF, the ACF, CCF, and SF-CF are
respectively obtained as follows :
RTpq (∆t) = RTpqTpq (δT = 0, δR = 0, ∆t, ∆f = 0), (3.34)
– 33 –

RTpqTp′q′ (δT , δR) = RTpqTp′q′ (δT , δR, ∆t = 0, ∆f = 0), (3.35)
RTpqTp′q′ (δT , δR, ∆f) = RTpqTp′q′
(δT , δR, ∆t = 0, ∆f). (3.36)
The following parameters are used for the correlation function analysis: carrier fre-
quency fc = 2.4GHz, fTmax = fRmax = 111Hz, D = 400m, βT = 3π/4, and βT = π/4.
Moreover, for the investigation of the individual CFs of SBT, SBR, and DB compo-
nents, we set the power related parameters as Kpq = Kp′q′ = 0, ζSBT = ζSBR = ζDB = 1,
and some ˜αSBT
k , ˜αSBR
l , and ˜αDB
kl satisfying 3.8 for given K and L. At ﬁrst, the absolute
values of the ACFs of the SBT, SBR, and DB components for αT = π, and αR = 0
(meaning that MST and MSR are moving in opposite directions) for K = L = 1
0 0.5 1 1.5 2 2.5 3 3.5 4
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Normalized time delay [tau*fDmax
]
AbsolutevaluesoftheACFs
SBT, R
T
=R
R
=10
SBR, RT
=RR
=10
DB, R
T
=R
R
=10
SBT, RT
=20, RR
=40
SBR, R
T
=20, R
R
=40
DB, R
T
=20, R
R
=40
Figure 3.3: The ACF of the GMRTR model with αT = π, αR = 0.
– 34 –

0 1 2 3 4 5 6
0
0.02
0.04
0.06
0.08
0.1
0.12
0.14
0.16
0.18
0.2
Angle of motion [0, 2pi]
AbsolutevalueoftheACFs
SBT, =[0, 2pi]
SBR, =[0, 2pi]
DB, =[0, 2pi]
SBT, =[0, 2pi]
SBR, =[0, 2pi]
DB, =[0, 2pi]
αT
T
R
T
R
R
α
α
α
α
α
Figure 3.4: The ACF of the GMRTR model w.r.t. αT ∈ [0, 2π] with αR = π/4 and
αR ∈ [0, 2π] with αT = π/4 at ∆t = 0.05sec for RT,1 = RR,1 = 10m.
are visualized in Fig. 3.3. The figure shows that the SBT, SBR and DB components
are unchanged despite of the different ring radii, and this result is same for the case
that MST and MSR are moving towards each other, i.e., αT = 0, αR = π, as well
as when MST and MSR are moving in same direction, i.e., αT = αR = 0(π). For the
further investigation of the effect of different ring radii on the ACFs, we evaluated the
absolute values of the ACFs of SBT, SBR, and DB components w.r.t. the two cases:
(1) RT,1 = RR,1 = 10m, and (2) RT,1 = 40m and RR,1 = 10m for αT ∈ [0, 2π] with
αR = π/4; and αR ∈ [0, 2π] with αT = π/4 at fixed time lag ∆t = 0.05sec. The results
– 35 –

are visualized in Fig. 3.4 for the case (1) and Fig. 3.5 for the case(2), respectively.
The result of the case (1) shows that the ACFs of the SBT and SBR components for
both cases are different for all αT and αR. The maximum differences in the correla-
tions occur at αT = αR = π/2 and αT = αR = 3π/2. On the other hand, the ACF of
the DB component was not changed for neither varying αT and fixed αR; nor varying
αR and fixed αT . This indicates that the absolute value of DB component does not
depend on the angle of motion. This phenomena also was pointed out in [20] by ob-
serving Doppler spectrum. Moreover, as can be observed in Fig. 2.5, the SBT, SBR,
and DB components differently affects the overall ACF value of the GMRTR model
0 1 2 3 4 5 6
0
0.02
0.04
0.06
0.08
0.1
0.12
0.14
0.16
0.18
0.2
AbsolutevaluesoftheACFs
SBT, =[0, 2pi]
SBR, =[0, 2pi]
DB, =[0, 2pi]
SBT, =[0, 2pi]
SBR, =[0, 2pi]
DB, =[0, 2pi]
α
α
α
α
α
T
T
T
R
R
Rα
Figure 3.5: The ACF of the GMRTR model w.r.t. αT ∈ [0, 2π] with αR = π/4 and
αR ∈ [0, 2π] with αT = π/4 at ∆t = 0.05sec for RT,1 = 40m and RR,1 = 10m.
– 36 –

0 1 2 3 4 5 6
0
0.02
0.04
0.06
0.08
0.1
0.12
0.14
0.16
0.18
0.2
SBT, =[0, 2pi], =pi
SBR, =[0, 2pi], =pi
DB, =[0, 2pi], =pi
SBT, =pi, =[0, 2pi]
SBR, =pi, =[0, 2pi]
DB, =pi, =[0, 2pi]
α
α
α
α
α
α
α
α
α
α
α
α
T
T
T
T
T
R
R
R
R
R
R
T
Figure 3.6: The ACF of the GMRTR model w.r.t. αT ∈ [0, 2π] with αR = π and
αR ∈ [0, 2π] with αT = π at ∆t = 0.05sec for RT,1 = RR,1 = 40m.
for RT,1 = RR,1. To investigate the effect of angle of motion on the ACFs of SBT, SBR,
and DB components for different ring radii, we once again evaluated the absolute value
of the ACFs with same parameters used in Fig. 3.4, and the results are depicted in Fig.
3.5. The result shows that the absolute ACF values of the SBT and SBR components
are significantly different due to RT,1 ̸= RR,1 while the absolute ACF value of the DB
– 37 –

0 0.5 1 1.5 2 2.5 3 3.5 4
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Normalized time delay [tau*fDmax
]
AbsolutevalueoftheACFs
SBT, scenario A
SBR, scenario A
DB, scenario A
SBT, scenario B
SBR, scenario B
DB, scenario B
Figure 3.7: The ACF comparison for the scenario A and B.
component remains the same. Finally, Fig. 3.6 shows that the absolute ACF values of
the SBT and SBR components vary with angle of motion parameters for same radii,
RT,1 = RR,1 = 40m. In this result, the effect of angle of motion becomes more obvi-
ous than those in Fig. 3.4 and 3.5. As can be seen in Fig. 3.6, the angle of motion
of Tx (Rx) does not affect the absolute value of the ACF of SBT (SBR) component
while the angle of motion of Tx (Rx) affects the absolute value of the ACF of SBR
(SBT) component. This is because no matter what an angle of motion of a mobile
station surrounded by isotropic scatterers is given, the Doppler shifts caused by any
scatterer on a circumference are the same. To observe the effect of multi-radii rings
– 38 –

on the ACFs, we evaluated the ACFs of SBT, SBR, and DB components for the two
different scenarios, i.e., scenario A (K = L = 1, (˜αSBT
1 )2
= (˜αSBR
1 )2
= (˜αDB
11 )2
= 1,
RT = 20, and RR = 40) and for the scenario B (K = 2, L = 3, (˜αSBT
)2
= {0.7, 0.3},
(˜αSBR
)2
= {0.5, 0.3, 0.2}, (˜αDB
)2
= {0.35, 0.25, 0.15, 0.12, 0.08, 0.05}, RT = {10, 20}m,
and RR = {15, 30, 40}m), and the results are given in Fig. 3.7. In this figure, the differ-
ences between the ACFs of the two scenarios gets bigger as the normalized time delay
increases except the ACFs of the DB component for the two scenarios. This is because
the ACF of the DB component does not depend on the ring radii while the others do.
To observe the correlation between two MIMO sub-channels, i.e., p = q = 1 and
p′
= q′
= 2, the absolute values of CCFs for the SBT, SBR, and DB components are
evaluated for Kpq = Kp′q′ = 0, ζSBT = ζSBR = ζDB = 1, K = L = 1, and (˜αSBT
1 )2
=
(˜αSBR
1 )2
= (˜αDB
11 )2
= 1 because the CCFs do not depend on the ring radii, thus the
index k and l can be eliminated. The results are visualized in Fig. 3.8-3.10, respectively.
The evaluation results show that only the spacing of antennas surrounded by isotropic
dense scatterers effectively de-correlates the two MIMO sub-channels. For example, the
0
1
2
3 0
1
2
3
0
0.5
1
AbsolutevalueoftheCCF
δ δT R/λ /λ
Figure 3.8: The CCF of the SBT component.
– 39 –

0
1
2
3
0
1
2
3
0
0.5
1
δ /λ δ /λR T
Figure 3.9: The CCF of the SBR component.
absolute value of the CCF of the SBT component is zero when δT /λ = 0.3; however,
δR/λ does not contribute to reducing the correlation between the sub-channels.
To investigate frequency correlation properties of the GMRTR model w.r.t. the
antenna spacing (the correlation between two MIMO sub-channels), we evaluated the
ST-CFs of SBT, SBR, and DB components for the following three scenarios:
0
1
2
3 0
1
2
30
0.5
1
δ /λ
δ /λ
T
R
Figure 3.10: The CCF of the DB component.
– 40 –

• Scenario A: K = L = 1, (˜αSBT
1 )2
= (˜αSBR
1 )2
= (˜αDB
11 )2
= 1, RT = 10, and
RR = 15, p = q = p′
= q′
= 1, and δT /λ = δR/λ = 0.
• Scenario B: K = 2, L = 3, (˜αSBT
)2
= {0.7, 0.3}, (˜αSBR
)2
= {0.5, 0.3, 0.2},
(˜αDB
)2
= {0.35, 0.25, 0.15, 0.12, 0.08, 0.05}, RT = {10, 20}m, RR = {15, 30, 40}m,
p = q = p′
= q′
= 1, and δT /λ = δR/λ = 0.
• Scenario C: same as Scenario B except for p = q = 1, p′
= q′
= 2, δT /λ = δR/λ =
3.
As can be seen in Fig. 4.2, the consideration of near and far scatterers represented by
additional concentric radii rings result in significant difference in the absolute values
of FCFs of the SBT, SBR, and DB components. This indicates that the single-radii
two-ring geometry may result in correlation error, when it is applied to the modeling
of the frequency selective channels where scatterers are distributed from the Tx (Rx)
antennas at difference distances. Moreover, the multi-radii two-ring geometry is more
general than the single-radii two-ring geometry, it is more suitable to represent diverse
frequency-selective fading environments. Also, the scenario C shows that the frequency
dependency between the two MIMO sub-channels can be decreased by increasing the
Tx and Rx antenna spacings. In fact, from the observation of the CCFs of the SBT, SBR
and DB components, δT /λ = δR/λ = 0.3 was enough to obtain independence between
the two MIMO sub-channels. However, to to reduce the frequency dependency between
them, at least δT /λ = δR/λ = 3 is required as show in the Fig. 4.2.
– 41 –

0 1 2 3 4 5 6 7 8 9 10
x 10
6
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Frequency seperation [MHz]
AbsolutevalueofFCFs
SBT, scenario A
SBR, scenario A
DB, scenario A
SBT, scenario B
SBR, scenario B
DB, scenario B
SBT, scenario C
SBR, scenario C
DB, scenario C
Figure 3.11: The SF-CF comparison for the scenario A, B and C.
3.5 Conclusive Remarks
In this chpater, we introduced the GMRTR model applicable or multiple M2M
wideband MIMO channel environments. Based on the model geometry and the law of
propagation, we derived the time-variant CIR consisting of LOS, SBT, SBR, and LOS
components by considering multiple scattering scenarios. As can be seen in Fig. 3.1 and
(3.3), the model is more general than multi-radii one-ring models, and many existing
CIRs of the conventional models [8]-[17] are its special cases; thus the proposed model
satisﬁes the Req. 2.
From the CIR, the STF-CF of the proposed model was derived for the investigation
– 42 –

of the model correlation properties over time, space, and frequency, respectively, and
also over their joint domains under the 2D isotropic rich scattering environment. From
the observation of the ACFs of the SBT, SBR, and DB components, we have found
some important properties of the ACFs:
• When the angle of motion, i.e., αT and αR are either 0(2π) or π, the absolute
values of the ACFs of SBT and SBR components do not depend on the ring radii,
i.e., RT,k, RR,l; otherwise, the ring radii significantly affect the absolute values of
the ACFs.
• When αT and αR are either 0 (2π) or π, the absolute values of the ACFs of SBT
and SBR components are the same.
• αT (αR) does not affect the absolute values of the ACF of the SBT (SBR) com-
ponents whereas αT (αR) affects the absolute value of the ACf of the SBR (SBT).
• None of αT , αR, RT,k, nor RR,l do not affect the absolute value of the ACF of the
DB component.
Also, the observation of the SF-CF tells us that modeling near and far scatterers
by single-radii two-rings results in the frequency correlation error. In fact, the multi-
radii two-ring geometry is more general than the single-radii two-ring geometry, so it is
advantageous to model the near and far scatterers more accurately. Thus, the proposed
model is satisfiable for the Req. 1. Besides of these observations, the proposed model’s
STF-CF provides various joint second-order statistical properties which are useful for
– 43 –

designing contemporary communication systems exploiting joint dimensions over time,
space, and frequency.
– 44 –

Chapter 4
Incorporating geometry-based stochastic chan-
nel model in tapped-delay-line for mobile-
to-mobile wideband MIMO channels
In this chapter, a simple and effective method to find appropriate tap delays and the
corresponding multipath channel gains of TDL-based GBSMs (e.g. (3.3)) is proposed.
The underlying motivation is that, by properly extracting the tap delays and channel
gains required for TDL-based GBSMs from empirical (or specified) PDP, reliable and
site-specific frequency selective channel simulations are possible. The proposed method
successfully generates the parameters which form a uniformly spaced PDP with a
spacing factor. The spacing factor is also adjustable; therefore, the proposed method
can enhance the flexibility and reduce the complexity of TDL-based GBSMs. This
chapter is organized as follows: the background of this problem is addressed in 4.1, and
the problem description is introduced in 4.2. In 4.3, the proposed method is addressed,
and its application and the simulation result are given in 4.4. Some conclusive remarks
are also given in the last section as well.
– 45 –

4.1 Introduction
TDL-based stochastic channel models have been widely used as propagation models
owing to their mathematical tractability and ease of implementation. The advantage of
this model type is that reliable frequency- and time-selective channel simulations are
possible by adapting empirical (or specified) PDPs (e.g. [24]) and stochastic processes
reflecting the mobility-geometry-induced Doppler spread of the channels. Thus far,
several papers [17]-[19] have reported the modeling of such stochastic processes for
wideband M2M MIMO channels, and GBSMs have been considered as versatile due
to their consideration of a large number of parameters that reflect channel geometries
and communication system specifications [25]. By combining the TDL structure with
GBSM models, it is possible to take into account multiple M2M environments as well
as MIMO system parameters such as the number of antenna elements and antenna tilt
angles. However, the main difficulty lies in determining the delay taps and coefficients of
TDL, i.e., discrete propagation delays and multipath channel gains, which are obtained
by sampling or interpolating a specified PDP in order to satisfy GBSM constraints
consisting of ring radii, AOA, and AODs. In fact, a few papers [12, 17] dealing with
this problem have been introduced in the literature, but the resulting parameters form
non-uniformly spaced PDPs; this is not desirable from the viewpoints of complexity
and flexibility of TDL-based simulations. In this section, using numerical optimization
and an initial guess determination method, we propose a simple yet effective method
to find the delay taps and coefficients that satisfy all of the GBSMs’ constraints. The
proposed method yields a uniformly spaced PDP that satisfies the constraints; the
– 46 –

yielded PDP has a tunable spacing factor, which enhances the flexibility and reduces
the complexity of TDL-based GBSMs.
4.2 Problem Description
The stochastic time-variant impulse response between the pth Tx and the qth Rx
antenna in an M2M wideband MIMO channel by incorporating a GBSM in TDL having
K · L number of taps can be expressed as follows:
hpq(t, τ) =
K−1∑
k=0
L−1∑
l=0
˜αklhpq,kl(t)δ(τ − ˜τkl), (4.1)
where δ(·), ˜αkl, and ˜τkl (k ∈ {0, 1, ..., K − 1} and l ∈ {0, 1, ..., L − 1}) denote the
Dirac delta function, discrete path gains, and relative propagation delays, respectively.
hpq,kl(t) denotes the time-variant complex channel gain, which is obtained by summing
the scattered plane waves impinging on the effective scatterers on kth and lth rings
placed at Tx and Rx, respectively. hpq,kl(t) can be modelled using the corresponding
propagation geometry and observing MIMO system configurations as shown in Fig. 3.1
and 3.2. Generalized descriptions of this model can be found in (3.3). In this chapter,
however, we consider a propagation scenario that line-of-site path does not exist, but
only DB scattering components are received via isotropic uniform linear arrays of Tx
and Rx antennas under the WSSUS conditions for mathematical tractability. Then,
(4.1) becomes (3.7). The propagation delays caused by DB scattering is given by
τk,m,l,n ≈
D
c0
+
RT,k(1 − cos ϕT,k,m)
c0
+
RR,l(1 + cos ϕR,l,n)
c0
, (4.2)
where c0, D, RT,k, RR,l, ϕT,k,m, and ϕR,l,n denote the speed of light, distance between
Tx and Rx, radii of rings placed at Tx, radii of rings placed at Rx, AOD, and AOA,
– 47 –

respectively. The distance parameters satisfy the relations, i.e., (3.1) and (3.2). For
readability, we re-write the two equations as below:
max {δT , δR} ≪ min{RT,k, RR,l}, (4.3)
max {RT,k, RR,l} ≪ D, (4.4)
where δT and δR denote Tx and Rx antenna spacing, respectively.
As can be seen in (4.2), the propagation delays are subject to the AOAs, AODs
and ring radii, and these are constraints for obtaining a set of propagation delays from
specified PDPs. To relax these geometrical constraints, we assume that the scatterers
lying on the same ring cause equal propagation delay. Then, (4.2) can be further reduced
to τkl = (D+RT,k +RR,l)/c0. Because τ00 is common to all paths, we can obtain relative
propagation delays as follows:
˜τkl = τkl − τ00 = (RT,k − RT,0 + RR,l − RR,0)/c0. (4.5)
Note that this assumption is typically made in some GBSMs to obtain the statistical
independence between the frequency- and time-selectivity at a expense of TF-CF error;
however, the assumption enables us to find two sets of radii, {RT,k} and {RR,l} from
some given set {˜τkl} by only linear operations without trigonometric functions which
are typically found in M2M wideband MIMO channel models [18, 19] whose relative
delays and Doppler spreads are correlated.
Typically, ˜αkl and ˜τkl of (4.1) are chosen from a specified (reference) PDP denoted
by Sτ (τ), and sampling or interpolating the reference PDP is a well-known approach
to determining these parameters for an arbitrary number of taps. In this case, the
– 48 –

resulting parameters form a new discrete PDP [29]-eq. (10) having K · L number of
paths. However, the new PDP should be constrained by (4.5). Moreover, frequency
selectivity of (4.1) should be approximated to that of the reference PDP. In fact, this
problem is equivalent to a constrained numerical optimization problem for generating a
new discrete PDP out of the reference PDP under the design constraints stated above.
In the literature, unconstrained cases have been introduced in [29], and the a (L2NM)
has outperformed the others. The minimization problem addressed in [29] is defined as
follows:
min
˜αkl,˜τkl
[∫ vmax
0
Rτ (v) − ˜Rτ (v; ˜αkl, ˜τkl)
2
· dv
]1/2
, (4.6)
where vmax denotes the maximum frequency separation that determines the optimiza-
tion interval, Rτ (v) denotes the reference frequency correlation function (FCF) ob-
tained by taking the Fourier transform of Sτ (τ), and ˜Rτ (v; ˜αkl, ˜τkl) denotes the FCF
of (4.1) obtained from the definition ˜Rτ (v; ˜αkl, ˜τkl) = E[Hpq(t, f + v)H∗
pq(t, f)] stated
below
˜Rτ (v; ˜αkl, ˜τkl) =
K−1∑
k=0
L−1∑
l=0
˜α2
kle−j2πv˜τkl
, (4.7)
where E [·] denotes the expectation operator, and Hpq(t, f) denotes the time-variant
transfer function, which is the Fourier transform of (4.1).
Although the solution of (4.6) yields a new discrete PDP whose FCF is approxi-
mated to Rτ (v), it is not possible to find RT,k and RR,l such that they satisfy (4.5) from
the resulting set of relative propagation delays {˜τkl} due to the irregular spacing be-
tween two adjacent delay elements for K·L K+L cases. Moreover, the non-uniformly
– 49 –

spaced delay set {˜τkl} yields a smaller greatest common divisor (GCD), i.e., gcd({˜τkl}),
than that of uniformly spaced delay sets in general. For simulating TDL-based mod-
els, it is necessary to sample or interpolate the reference PDP with a proper spacing
factor ∆τ = gcd({τkl}). A smaller ∆τ value results in higher simulation complexity
owing to the increased number of taps and upsampling rate m, i.e., m = Ts/∆τ, where
Ts denotes symbol duration. Therefore, the determination of uniformly spaced {˜τkl}
that satisfies (4.5) with tunable ∆τ (∆τ-spaced PDP) enhances model flexibility and
reduces complexity. One method for obtaining such {˜τkl} is addressed in the following
section.
4.3 A Methodolgy to Obtain a ∆τ–Spaced Power Delay Profile
Because solving (4.6) yields non-uniformly spaced {˜τkl}, we change (4.6) into a
problem with fewer unknowns by pre-defining uniformly spaced {˜τkl}. To maintain
the same frequency selectivity, we consider the use of root mean square delay spread
(RMS-DS), which characterizes temporal dispersive wireless channels, as an equality
constraint. Then, for the unknown vector ˜α = (˜α00, ..., ˜αK−1,L−1) ∈ RK·L
, the new
optimization problem can be expressed as follows:
min
˜α∈RK·L
[∫ vmax
0
Rτ (v) − ˜Rτ (v; ˜α, ˜τkl)
2
· dv
]1/2
subject to τrms = ˜τrms,
where τrms and ˜τrms denote the RMS-DS of Sτ (τ) and (4.1), respectively, and ˜τrms
is defined by [29]-eq. (16). Eq. (4.8) is a non-linearly constrained non-linear opti-
mization problem, and a local minimum ˜α∗
can be found using active-set algorithms
with a proper initial guess ˜α0. Because the optimization performance is subject to
– 50 –

α0 = {αkl} = Sτ (τ )|τ=˜τkl∀k,l, it is important to find {τkl} satisfying (4.5). If we as-
sume {˜τkl} = { ˜τkl| ˜τkl := κ∆τ, κ = 0, 1, ..., K · L − 1} with a spacing factor ∆τ =
τT
max/(K · L − 1), where τT
max denotes the maximum relative propagation delay of a
truncated reference PDP having reduced interval [0, τT
max] ∈ [0, τmax], we can convert
(4.5) into linear system equations via a small modification given below:
A · x = (0, 1, ...K · L − 1)T
· ∆τc0 + ε, (4.8)
where A is a (K · L) by (K + L) matrix defined as follows:
AT
=




CT
1 CT
2 · · · CT
K
I I · · · I




T
. (4.9)
In (4.9), Ck ∈ RL×K
and its elements in only the kth column are one; otherwise,
zero. I denotes a L by L identity matrix. x is an unknown vector defined by x =
(RT,0, ..., RT,K−1, RR,0, ..., RR,L−1)T
. In addition, ε is an offset value defined by ε =
RT,0 + RR,0. Although (4.9) is an overdetermined system, choosing suitable K, L, ε,
and ∆τ values reduces rank(A) to K + L or less. For determining K and L for the
given system bandwidth B, we can use the relation, i.e., B = vmax ≤ 1
2·∆τ
= K·L−1
2·τT
max
.
Solving (4.8) for x using QR factorization and back-substitution gives {RT,k} and
{RR,l}. If these radii satisfy the GBSM’s constraints, i.e., (4.3) and (4.4), we can use
the ∆τ–spaced {˜τkl} to obtain ˜α0. It is noteworthy that ∆τ is adjustable, as long as
rank(A) ≤ K + L is satisfied.
– 51 –

4.4 Applications and Simulation Results
The proposed method was applied to obtain the ∆τ–spaced PDP for the TDL-based
GBSM by using the continuous Typical Urban profile [24]
Sτ (τ) =



1/ (1 − e−7
) · e−τ/µs
, 0 ≤ τ ≤ τmax = 7µs
0, else
, (4.10)
where τmax denotes the maximum relative propagation delay. The corresponding RMS-
DS is given by τrms = 0.977µs. The Fourier transform of (4.10) yields the reference
FCF as follows:
Rτ (v) =
1
1 − e−τmax
·
1 − e−τmax(1+j2πv)
1 + j2πv
. (4.11)
In this application, we use K = 3, L = 4, ε = 400m, and ∆τ = 0.4µs in order to solve
(4.8) for obtaining x. In this case, the result is x = (355.2, 835.2, 1315.2, 44.77, 164.77
, 284.77, 404.77)T
m. Assuming that x satisfies (4.3) and (4.4), we substitute the 0.4µs–
spaced {˜τkl} into (4.10) to obtain ˜α0.
From the optimization result, we obtained a ∆τ–spaced PDP satisfying (4.5) and its
FCF as can be seen in Figs. 2. and 3, respectively. The results show that the FCF of the
∆τ–spaced PDP is closely approximated to the reference FCF for [0, vmax ≈ 0.8MHz],
thus validating the proposed method. The adjustable ∆τ is noteworthy, because it en-
ables TDL-based GBSMs to be more flexible and less complex by increasing gcd({τkl}).
– 52 –

0 1 2 3 4 5 6 7 8
0
0.2
0.4
0.6
0.8
1
Propagation delay tau [micro sec.]
Powerdelayprofiles[lin.]
Reference PDP
−spaced PDP∆τ
Figure 4.1: Reference PDP and fractional path powers of ∆τ–spaced PDP obtained
from simulation results.
0 0.2 0.4 0.6 0.8 1
0
0.2
0.4
0.6
0.8
1
Frequrncy seperation v [MHz]
AbsolutevalueoftheFCF
Reference FCF
FCF of the −spaced PDP∆τ
Figure 4.2: Absolute value of reference FCF and FCF obtained from ∆τ-spaced PDP.
– 53 –

Chapter 5
Conclusions and Further Works
5.1 Summary of Thesis
In this thesis, we have focused on the development of a GBSM for M2M MIMO
wideband channel environments. As the background research, a brief overview of exist-
ing MIMO channel models were introduced in chapter 2, as well as a brief introduction
to the GBSM theory. We classiﬁed the existing MIMO channel models into two cate-
gories: physical channel model and analytic channel model. The pros and cons of each
model type are comparatively explained, and we conclude physical channel models are
more suitable for the simulation of real channel environments of interest because of its
statistical reliability and representativity based on the realistic geometry and physical
parameters with their underlying the law of propagation. We also reviewed three im-
portant models that have led to the present-day theories of GBSMs from the viewpoints
of geometrical and physical representations. From the observations, we introduced the
four requirement for designing a GBSM applicable for multiple M2M environments,
i.e.:
• Req. 1: modeling the environments accurately using reasonable geometries,
– 54 –

• Req. 2: finding one that can be parameterized for many environments,
• Req. 3: using reliable channel parameters extracted from extensive measurements,
• Req. 4: moderate/low link-level simulation complexity.
In chapter3, the GMRTR model concerning the Req 1 and 2 are proposed. The
proposed model is based on the multi-radii two-ring geometry, and the model is a
generalized version of many existing GBSMs, i.e., [8]-[17]. The proposed model can be
applied to diverse scattering scenarios and frequency selective channel environments,
by jointly considering LOS, NLOS, DB, SBT, and SBR components. For the statistical
analysis of the channel model, we derived the analytical STF-CF, and we investigated
the channel’s ACF, CCF, and SF-CF. From the observation of the ACFs of the SBT,
SBR, and DB components, we have found some important properties of the ACFs,
and they are summarized in section 3.5. From the observation of the SF-CF, we have
shown that modeling near and far scatterers using single-radii two-rings results in the
frequency correlation error when its SF-CF is compared with that of the proposed
model. The multi-radii two-ring geometry is more general than the single-radii two-
ring geometry; thus, it is advantageous for modeling of M2M wideband propagation
environments.
In chapter4, the Req. 3 and 4 were discussed. Especially for the Req. 3, the reliable
channel parameters can be obtained from the measured or specified power delay profiles
such as [24], and this makes the TDL-based GBSM more reliable than those based on
pure GBSMs, e.g., [18]-[20], as pointed out in [25]. There are several ways to extract
– 55 –

the propagation delays and the corresponding path gains from measured or specified
PDPs, such as non-uniform sampling and uniform sampling methods. The non-uniform
sampling case was introduced in the reference [29], but the non-uniformly spaced PDP
increases the link-level simulation complexity of the TDL-based GBSM by increasing
up-sampling rate; thus the Req. 4 is not satisfied. To fill the gap, we proposed a method
to obtain a ∆τ-spaced PDP from measured or specified PDPs in chapter4. The ad-
justable ∆τ enabled the link-level simulation using TDL-based GBSMs to be more
flexible and less complex by adjusting and increasing the greatest common divisor of
the propagation delays.
To sum up, in this thesis, we successfully modeled a GBSM which can cover many
existing propagation environments ranging from classical cellular networks to contem-
porary M2M wideband MIMO communication systems. The proposed model (GMRTR
model) can make use of reliable channel parameters obtained from measured or spec-
ified PDPs using the proposed method introduced in this thesis. We believe that the
proposed model and method are useful for designing contemporary communication sys-
tems exploiting joint dimensions over time, space, and frequency, and also useful for
more flexible and less complex link-level simulations.
5.2 Further Works
Currently, two pioneering works [22, 23] proposing GBSMs for M2M MIMO coop-
erative networks have been proposed in the literature. However, in comparison to the
M2M MIMO channel environments, the detailed joint correlation properties for diverse
– 56 –

cooperative network scenarios have not been well analyzed, and also there many ex-
isting unsolved problems still exist. Due to the various industrial applications of M2M
MIMO cooperative network technologies, the modeling, analysis, simulation, and mea-
surement of various propagation scenarios in cooperative networks will be the most
vivid research area for next few years.
The other future work is the channel model verification. Up to now, a number of
GBSMs have been introduced in the literature, and there also have been various efforts
to verify the validity of those proposed models. Yet, some channeling problems such as
lack of research works specifying every channel parameters required for the verification
of the validity of GBSMs exist. To cope with these problems, the development of a
quantitative methods to check the reliability of GBSMs is needed.
– 57 –

Abbreviations
2-D two-dimension
ACF auto-correlation function
AOA angle of arrival
AOD angle of departure
B2M base-to-mobile
BS base station
CCF cross-correlation function
CF correlation function
CIR channel impulse response
DB double bounced
GBSM geometry-based stochastic model
GCD the greatest common divisor
GMRTR geometrical multi-radii two-rings
GSROR geometrical single-radii one-ring
GSRTR geometrical single-radii two-ring model
HAP high-altitude platforms
i.i.d. independent identically distributed
ISI inter-symbol interference
L2NM L2-norm method
LOS line-of-sight
– 58 –

M2M mobile-to-mobile
MIMO multiple-input multiple-output
MS mobile station
NLOS no line-of-sight
PAS power azimuth spectrum
PDP power delay proﬁle
RMS-DS root mean square delay spread
Rx receiver
SB single bounced
SBR single bounced ray around receiver
SBT single bounced ray around transmitter
SCM spatial channel model
SF-CF space-frequency correlation function
SISO single-input single-output
SOS sum-of-sinusoids
ST-CF space-time correlation function
ST-CF space-time correlation function
STF-CF space-time-frequency correlation function
TDL tapped-delay-line
TF-CF time-frequency correlation function
Tx transmitter
WSS wide-sense stationary
WSSUS wide-sense stationary uncorrelated scattering
– 59 –

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– 64 –

Curriculum Vitae
Name : Sangjo Yoo
Birth Date : Feb. 13, 1985
Birth Place : Pusan
Permanent Address : Changwon-si, Rep. of Korea
Education
2010.9–2012.8 Department of Information and Mechatronics, Gwangju Institute of
Science and Technology (GIST), Gwangju, Rep. of Korea (M.S.)
2008.8–2009.8 Brookhaven College in Texas, USA (visiting student)
2003.3–2010.8 School of Information and Communications, Changwon National Uni-
versity, Changwon-si, Rep. of Korea (B.S.)
Work Experience
2010.9–2012.4 A study on the intelligent RF communication based on MIMO an-
tenna, Agency for Defence Development (ADD), Rep. of Korea.
2010.6-2010.9 Development of an intelligent game board using printed electronics,
Electria, Finland.
Professional Activities
2012.06–present Student Member of the Institute of Electrical and Electronics Engi-
neers (IEEE)
2012.06–present Student member of IEEE Signal Processing Society
2012.01–current Student member of the Korea Information and Communications So-
ciety (KICS)
Honors and Awards

2010.9–2012.7 Brain Korea 21 (BK21) scholarship from Ministry of Education and
Human Resource Development, Rep. of Korea
2010.9–2012.7 A full government scholarship for M.S. degrees from GIST
2003.3–2010.8 Won 12 times for university scholarships such as NURI Star, NURI
Gold, and national scholarships for outstanding grades.
Publications
1. Sangjo Yoo and Kiseon Kim,“Incorporating geometry-based stochastic channel
model in tapped-delay-line for mobile-to-mobile wideband MIMO channels,”
submitted to IET Elec. Letters.
2. Sangjo Yoo, Sujung Yoo, Jeehoon Lee, and Kiseon Kim,“Modeling and charac-
teristics of mobile-to-mobile wideband MIMO channel based on the geometrical
multi-radii two-rings with speciﬁed frequency selectivity,” in Proc. EuCAP 12,
pp. 2030-2034, Prague, Czech Rep., Apr. 2012.
3. Sangjo Yoo, Jeehoon Lee, and Kiseon Kim, “Modeling of sum-of-sinusoid chan-
nel simulation for fading channel,” Proc. KICS Gwangju-Jeonnam Fall Confer-
ence, pp. 82-85., Rep. of Korea, Jun. 2011.

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