Available via license: CC BY-NC-ND 4.0
Content may be subject to copyright.
Received 29 November 2020; revised 28 December 2020; accepted 3 January 2021. Date of publication 13 January 2021; date of current version 8 February 2021.
Digital Object Identifier 10.1109/OJAP.2021.3051478
Embedded Propagation Graph Model for Reflection
and Scattering and Its Millimeter-Wave
Measurement-Based Evaluation
YUAN LIU 1,2, XUEFENG YIN 1,3 (Member, IEEE), XIAOKANG YE 1(Student Member, IEEE),
YONGYU HE 1(Student Member, IEEE), AND JUYUL LEE 4(Senior Member, IEEE)
1College of Electronics and Information Engineering, Tongji University, Shanghai 200092, China
2Guangdong Communications & Networks Institute, Guangzhou, China
3National Computer and Information Technology Practical Education Demonstration Center, Tongji University, Shanghai 200092, China
4Telecommunication and Media Research Laboratory, Electronics and Telecommunications Research Institute, Daejeon 34129, South Korea
CORRESPONDING AUTHOR: X. YIN (e-mail: yinxuefeng@tongji.edu.cn)
This work was supported in part by the China National Science Foundation General Project under Grant 61971313, and in part by the Institute for
Information & Communications Technology Planning & Evaluation (IITP) grant funded by the Korean Government (MSIT,
Development of Time-Space Based Spectrum Engineering Technologies for the Preemptive Using of Frequency).
ABSTRACT Propagation graph (PG) is a stochastic channel simulation method for scattering propagation.
In this article, an embedded-PG (EPG) approach, an extension of conventional PG, is proposed to simulate
reflection and scattering multi-path behaviors in wireless channels. In this method, multiple propagation
paths are categorized into scattering-path, reflecting-path, and scattering-reflecting mixed paths among
reflectors and scatterers. The matrix recursive formula of conventional PG modeling is used to calculate
scattering-path, a recursive mathematical transformation is applied to adapt reflecting-path into the recursive
formula, and an embedded graph method is used to decompose mixed-path into scattering effects and
reflection effects. The proposed simulation approach is validated by comparison with conventional PG and
measurement in 39 GHz millimeter-wave (mm-wave) time-variant corridor scenario. Power delay profiles
(PDPs) and spatial consistency of multiple paths observed in concatenated-PDPs (CPDPs) obtained by
EPG are more consistent with measurement than conventional PG, differences of mean delay and delay
spread between simulations and measurement in typical snapshots are within 3 ns and 1.5 ns, respectively.
INDEX TERMS Channel modeling, channel simulation, millimeter-wave channel, reflection-embedded
propagation graph, time-variant channel.
I. INTRODUCTION
DESIGN of algorithm and performance optimization for
next generation of wireless communication systems
become research focus both in the academia and indus-
try, owing to the rapid increase in the need of channel
bandwidth and data rate [1]–[3]. Channel modeling of
millimeter-wave (mm-wave) is essential for 5G and beyond
wireless communication systems [4]. There are two typ-
ical approaches to characterizing parameters of channel
modeling, i.e., measurement-based and simulation-based [5].
However, measurement-based channel models concentrate on
statistical behaviors, which require abundant experimental
data [6]. Meanwhile, challenges in carrying out field mea-
surements increase drastically in mm-wave bands, so as the
deployment costs. As a result, an accurate and efficient
simulation-based method is of great necessity for mm-wave
and higher-frequency band channels.
Geometry-based channel simulation tools have a signifi-
cant advantage on predicting the specific propagating path in
space and spatial information over other simulation methods,
e.g., room electro-magnetics. With those information, it is
possible to study the the spatial consistency and space-time
randomness of channels, multi-path clustering and so on.
The most widely used geometry-based channel simulation
This work is licensed under a Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 License.
For more information, see https://creativecommons.org/licenses/by-nc-nd/4.0/
VOLUME 2, 2021 191
LIU et al.: EPG MODEL FOR REFLECTION AND SCATTERING AND ITS MILLIMETER-WAVE MEASUREMENT-BASED EVALUATION
TABLE 1. Summary of state-of-the-art geometry-based channel simulation algorithms.
methods are Ray-tracing (RT) [7], [8] and Propagation-
graph (PG) model proposed in a recent decade [9], [10].
RT is a deterministic simulation tool, which makes use of
image method to calculate geometrical optic rays of waves
propagate along paths involving reflection [11], [12]. It is
worthy mentioning that some commercial softwares of RT
also include other basic propagation mechanisms, such as
scattering and diffraction [13]–[15]. However, the shortages
of RT are: (1) there is a high time and resource consumption
for high order reflection calculations; (2) it is hard to include
reverberating effects of different propagation mechanisms.
The PG is a stochastic simulation method, which mainly
makes use of the topology structure of directed graph [16] to
calculate the effect of wave propagation along paths involv-
ing scattering in matrices with a significant advantage in
the reduction of computational compexity [17], [18]. For
this reason, the PG model has been widely applied in chan-
nel simulations of variant scenarios after being originally
proposed. For indoor and outdoor scenarios, the PG the-
ory was used to predict closed room reverberating effects
and exponential power decay [19], [20], human block-
age and doppler frequency of ultra wide band (UWB)
channel [21], and high speed train channels [22]. For multi-
room channel predictions, PG and Rays were combined
in [23], and an iterative transfer matrix computation method
was used for acceleration in [24]. Furthermore, PG even
showed interesting results in analyzing characteristics and
channel capacity of Multiple-input-multiple-output (MIMO)
systems [25].
Due to the efficiency and flexibility of PG, researchers
contributed themselves to modifying the model by including
reflection [18] and diffraction effects [26], which are also
considered as the three basic propagation mechanisms along
with scattering [27]–[29]. The most commonly used method
is hybrid model, which combines PG and RT [30]–[32].
The main difference of [30], [31], and [32] is the calcula-
tion of scattering coefficients. Time efficiency of these two
simulation algorithms were studied and compared in [30],
which revealed that time consumption of PG grows lin-
early as bouncing order of wave increases, while that of
RT grows exponentially, hence PG model owns obviously
better performance on time consumption than RT when the
reflection order is larger than three. A Summary of literature
review is listed in Table 1.
To overcome the shortages of conventional PG mentioned
in Table 1, a so-called embedded propagation graph (EPG)
is proposed that can be used to predict wireless propagation
channels with the capability of calculating both reflecting
and scattering components. It may provide powerful tools
for channel researches, e.g., multi-path clustering analysis
in wireless channel, high frequency channel predictions, and
other applications like radio based localizations.
Since EPG is an extension of conventional PG model, on
the one hand, it inherits good time consumption of matrices
calculation from PG; on the other hand, the accuracy is
improved by considering reflection effects and reverberating
effects of scattering and reflection. The technical novelties
of the proposed algorithm can be concluded as:
1) Use PG model to calculate reflection effects in radio
channel by applying a linear mathematic transformation to
adapt the multi-bounce reflection into the recursive matrices
formula.
2) Use embedded PG model to compute the reverberat-
ing paths between reflection and scattering by exploiting an
embedded method to decompose the reverberating paths into
scattering-path and reflecting-path.
3) Evaluate the newly proposed EPG in a time-varying
scenario using mm-wave frequency band.
The remaining parts of this article are arranged as follows:
Section II introduces the methodology of EPG. Section III
192 VOLUME 2, 2021
FIGURE 1. Vertices illustration for conventional PG model.
describes the procedure of EPG. The simulation results of
EPG and its validation. Finally, conclusive remarks are given
in Section IV.
II. METHODOLOGY
A. REVISIT OF SCATTERING PROPAGATION GRAPH
THEORY AND ITS MODIFICATION
A wireless communication system generally contains trans-
mitters (Txs), receivers (Rxs), and environment that
wave propagating through. The conventional PG proposed
in [19], [20] is based on the assumption that the environ-
ment is discretized as scattering points. Scattering points,
Txs, and Rxs are regarded as vertices Vs,VT, and VRin a
propagation graph, respectively, as Fig. 1 shows. The paths
among these vertices are defined as edges εd,εTs,εRs, and
εss, which represent edges of VTto VR,VTto Vs,Vsto VR,
and Vsto Vs, respectively.
The transfer coefficient of a propagation edge connecting
two vertices in the graph can be expressed as
Ae(f)=| ge(f)|·exp(−j2πfτe+jφ),(1)
where fis the carrier frequency of signal, φcan be regarded
as a random variable following uniform distribution on the
interval [0, 2π), |ge(f)|is the edge gain, which can be
defined as
|ge(f)|2=
⎧
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎨
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎩
c
4πτef2
,e∈εd
1
4πfμ(εTs)
·τe−2
S(εTs),e∈εTs
1
4πfμ(εRs)
·τe−2
S(εRs),e∈εRs
g2
odi(e)2,e∈εss
(2)
where odi(e)denotes the out degree of the corresponding
scatterers, and for any edges belong to ε
μ(ε) =1
|ε|
e=ε
τeand S(ε) =
e∈ε
τe−2,(3)
FIGURE 2. Transfer matrices model for conventional PG computation.
in which
τe=de
cwith de=|rv−rv|,(4)
where cis the speed of light, |.|denotes two dimensional
norm, rvand rvdenote position vectors of vertices of any
edges.
Then, define D(f),T(f),R(f), and B(f)denote the transfer
matrices from vertices VTto VR,fromVTto Vs,fromVs
to VR, and from Vsto Vs, respectively, the transfer matrices
model is illustrated by Fig. 2. The channel transfer function
can be calculated as [20]
H(f)=D(f)+R(f)I+B(f)+··· +Bn(f)T(f)
=D(f)+R(f)I−B(f)−1T(f), (5)
where I is identity matrix. The PG model can evaluate the
effects of infinite-bounce among scatterers by calculating a
matrix inverse.
The term odi(e)in Eq. (2) ensures that the total out-
bounding power does not exceed the input power, however, it
also results in the transmitting power is uniformly distributed
to other scatterers. For this reason, a semi-deterministic
approach is proposed to modify the edge gain in [30], [31] as
|ge(f)|2=
⎧
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎨
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎩
c
4πdef2
,e∈εd
dS·cos(θi)
4πd2
e
,e∈εTs
S2·cos(θs)
πd2
e
·c2
4πf2,e∈εRs
S2·dS·cos(θi2)cos(θs1)
πd2
e
,e∈εss
(6)
where Srepresents the scattering loss, dS denotes the area of
the small tile at scatterers, θirepresents the angle between
incident direction of a wave and the normal vector of a
scattering surface, θsrepresents the angle between scattering
direction of a wave and the normal vector of a scattering
surface. It is verified by measurement that the modified edges
gains obtain a better accuracy than conventional PG model.
Thus, the modified edge gains is applied for the proposed
EPG in this article.
VOLUME 2, 2021 193
LIU et al.: EPG MODEL FOR REFLECTION AND SCATTERING AND ITS MILLIMETER-WAVE MEASUREMENT-BASED EVALUATION
FIGURE 3. Transfer matrices model for EPG computation.
B. CALCULATION OF REFLECTION PROPAGATION
GRAPH
Define a new Reflection Propagation Graph (RPG) which
contains only Txs, Rxs and reflectors. To discretize them
into vertices as VT,VRand Vr, then the transfer matrices
model can be illustrated as Fig. 3. The paths among these
vertices are defined as εd,εTr ,εRr and εrr, which represent
edges from VTto VR,VTto Vr,Vrto VR, and Vrto Vr,
respectively.
The Line-of-sight (LoS) paths of RPG can be calculated
as the same method of PG. For simplification, only introduce
Non-LoS (NLoS) paths in the following discussion.
1) ANALYSIS OF EDGE GAINS FOR REFLECTION AND
SCATTERING
For multiple scattering propagation, the edge gain contains
a product of distances multiplication. For example, the Friis
equation in [34] describes received power gain after bouncing
sequentially with n scatterers named No. 1,2,...,(n+1),
i.e., the radio wave propagating through nedges can be
calculated as
Pn+1=P1·Sn·λ
4π(d1·d2···· ·dn)2
,(7)
where Pn+1represents the received power, P1is the trans-
mitted power, dndenotes the distance between the NO.nto
the NO.(n+1)scatterer. Thus, the scattering matrices B(f)n
can be used to calculate the overall transfer matrix of signal
scattering ntimes among the scatterers.
However, for reflection, the received power after reflect-
ing through (n+1)reflectors contains a factor of distances
addition [34], i.e., radio wave propagating through nedges
can be calculated as
Pn+1=P1·Rnλ
4π(d1+d2+··· +dn)2
,(8)
where Ris polarimetric reflection loss [35]. Equation (8)
shows that the power gain for multiple-reflection is depen-
dent on the additive distances of all the paths. Since
distance-factor can not be multiplied sequentially, the con-
tinuous matrices multiplication of PG are hard to be directly
used for calculating the effect of n-bounce reflection. To
adapt the reflection in the framework of conventional PG
iterative matrices operation, we have to calculate distance
factors separately.
2) FACTORIZATION OF EDGE GAINS OF RPG
The propagation gain for any edges can be factorized into
gain-factor and distance-factor, i.e.,
|ge(f)|2=ggain(f)
2·gpath
2,(9)
where gain-factor ggain(f)can be calculated based on
Eq. (6) as
ggain(f)
2=⎧
⎪
⎪
⎪
⎪
⎪
⎨
⎪
⎪
⎪
⎪
⎪
⎩
dS·cos(θi)
4π,e∈εTr
S2·cos(θs)
π·c2
4πf2,e∈εRr
S2·dS·cos(θi2)cos(θs1)
π,e∈εrr
(10)
and gpath contains the edge distance-factor, we proposed a
recursive mathematical transformation to calculate it.
3) A MATHEMATICAL TRANSFORMATION FOR
N-BOUNCE REFLECTION
Continuous addition of distance Dnis defined as
Dn=1
d1+d2+···+dn,n≥2
D1=1
d1,n=1(11)
Applying the following manipulation, we can obtain a
sequential approach for calculating Dnbased on Dn−1as:
Dn=1
d1+d2+··· +dn−1·1
1+dn
d1+d2+···+dn−1
=Dn−1·1
1+Dn−1dn
=Dn−1·1
d
n
(12)
with d
ndefined as the so-called equivalent distance
d
n1+Dn−1·dn,n≥2.(13)
Then using the mathematical transformation of Eq. (11),
once the distances between vertices in the propagation graph
are determined, the new distance d
ncan be calculated
sequentially by Dn−1and dn.
4) CALCULATION OF DISTANCE-FACTOR GPAT H OF RPG
(1) For one-bounce reflection, the propagation paths are:
Tx →reflectors →Rx. The distances of Tx to reflectors, as
well as reflectors to Rx are known. Use dTr to denote the
distance matrix of Tx and reflectors, and dRr to denote the
distance matrix of Rx and reflectors. Apply (11) and (13),
we obtain
D1=I
dTr ,
d
2=I+D1·dRr.(14)
194 VOLUME 2, 2021
Then gpath can be calculated as
gpath=⎧
⎨
⎩
D1,e∈εTr
I
d
2
.e∈εRr (15)
(2) For multiple reflection, the number of the propagation
edges n≥3. Tx and Rx are at two terminals, and the
propagated waves would interact with reflectors for (n−1)
times. Use distance matrix drr to denote distances among
reflectors. We adapt drr,ito denote the distance of the ith to
the (i+1)th reflecting paths, i∈[1,n−2]. Hence the edge
distance d2······dn−1are sequentially equal to the distance
among reflectors drr,1,drr,1···drr,(n−2), respectively. Apply
the transformation (11) and (13), the n-time reflection can
be represented as
⎧
⎪
⎪
⎪
⎪
⎪
⎪
⎨
⎪
⎪
⎪
⎪
⎪
⎪
⎩
D1=I
dTr ,
···
d
n−1=I+Dn−2·dn−1=I+Dn−2·drr,(n−2),
Dn−1=Dn−2·I
d
n−1
,
d
n=I+Dn−1·dn=I+Dn−1·dRr.
(16)
Then, gpath can be calculated as
gpath=⎧
⎪
⎪
⎪
⎪
⎨
⎪
⎪
⎪
⎪
⎩
I
d
n
.e∈εRr
D1,e∈εTr
I
d
n−1
.e∈εrr
(17)
5) CALCULATION OF TRANSFER MATRICES AND
CHANNEL TRANSFER FUNCTION OF RPG
We use D(f)∈C1×1,Tr(f)∈C1×N,Rr(f)∈CN×1and r(f)
∈CN×Nto denote the transfer matrices between the vertices
VTand VR,VTand Vr,Vrand VR,Vrand Vr, respectively.
Moreover, Rrn(f)∈CN×1is introduced to denote the trans-
fer matrix of reflectors to Rx after bouncing among reflectors
for ntimes, rn(f)∈CN×Nto denote the transfer function
among reflectors after bouncing n times. Notice that rn(f)
is calculated sequentially based on the distance factor d
n−1
in Eq. (17).
For the transfer matrix, we also consider it in two parts
as shown in (9), i.e., gain-factor and distance-factor.
Use Trgain (f)to denote the gain-factor transfer matrix of
Tr(f),Trdis to denote the distance-factor transfer matrix of
Tr(f), then Tr(f)can be calculated as
Tr(f)=Trgain (f)Trdis ,(18)
where means Hadamard product of matrices, Trgain (f)can
be generated as (10), Trdis can be generated as (15) and (17).
Similarly, we use rgain,n(f)to denote the gain-factor trans-
fer matrix of rn(f),rdis,nto denote the distance-factor transfer
matrix of rn(f), then rn(f)can be represented as
rn(f)=rgain,n(f)rdis,n.(19)
Algorithm 1 Modeling Algorithm of Pure Reflection
Propagation Graph
Input: Geometrical information of scenario, electromag-
netic properties of major reflecting materials, radiation
pattern of antennas.
Output: Channel transfer function HRPG(f).
Step 1: Divide surfaces into uniform small vertices, and
obtain locations of reflecting vertices, Txs and Rxs.
Step 2: Generate distance matrices dTr,dRr, and drr
based on locations in Step 1. Identify reflection paths
based on snell’ law. Then, calculate the iterative distance
matrices Drr,nand drr,n.
Step 3: Generate gain-factor matrices Trgain(f),
rgain,n(f), and Rrgain,n(f).
Step 4: Generate distance-factor matrices Trdis(f),
rdis,n(f), and Rrdis,n(f).
Step 5: Calculate transfer matrices Tr(f),rn(f), and
Rrn(f). Obtain channel transfer function HRPG(f).
The first part rgain,n(f)can be obtained as
rgain,n(f)=rgain,(n−1)(f)rgain(f)
=rgainn(f), (20)
where entries in rgain(f)can be generated as (10).
Use Rrgain,n(f)to denote the gain-factor transfer matrix
of Rrn(f), and Rrdis,nto denote the distance-factor transfer
matrix of Rrn(f). Thus, Rrn(f)can be represented as
Rrn(f)=Rrgain,n(f)Rrdis,n,(21)
where the entries in Rrgain(f)can be generated as Eq. (10),
Rrdis can be generated based on drr,n−1according to Eq. (17)
and Eq. (24).
Finally, the NLoS part channel transfer function of RPG
can be obtained as
HRPG(f)=Tr(f)Rr1(f)+
∞
n=2Tr(f)rn−1(f)Rrn(f).(22)
The implementation of RPG model can be summarized as
the following flowchart.
In Step2, to generate the distance-factor matrices rdis,n,it
is necessary to consider the following aspects:
1) The rdis,nis a sequentially result of the initial input
Trdis ,rdis,1,rdis,2,..., and rdis,n−1.
2) Define Drr,n∈CN×Nto denote the intermediate
variable in (14) and (16).
3) Use matrix CrrCN×Nto filter out the extra 1elements
resulted from term (I+Dn−1)in Eq. (16).
Crr describes the relationship of every two reflectors, in
which 1means existing reflection path between the two
reflectors and otherwise not. Define d∈CN×Nto denote the
Euclidean space distance of every two reflectors, which can
be obtained from digital map. So that the distance matrix of
reflectors with connectivity drr ∈CN×Ncan be calculated as
drr =dCheckrr.(23)
VOLUME 2, 2021 195
LIU et al.: EPG MODEL FOR REFLECTION AND SCATTERING AND ITS MILLIMETER-WAVE MEASUREMENT-BASED EVALUATION
FIGURE 4. Diagram of PG with both Reflectors and Scatterers for Transfer function
computation.
Apply the formulation of (16) on distances matrices Drr,n
and drr, we obtain
⎧
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎨
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎩
Drr,0=Trdis T;TrdisT···TrdisT,
drr,1=I+Drr,0drrCheckrr ,
···
drr,n−1=I+Drr,n−2·drrCheckrr n−1,
Drr,n−1=Drr,n−2·I
drr,n−1
,
drr,n=I+Drr,n−1·drrCheckrr n,
(24)
where [.]Tdenotes transpose of matrices.
Then rdis,ncan be calculated as
rdis,n=I
drr,n
.(25)
C. CALCULATION OF GRAPH WITH REFLECTORS AND
SCATTERERS USING EPG
1) TRANSFER FUNCTION MODEL OF FULL
PROPAGATION GRAPH
The EPG is expanded based on adding reflectors into the
conventional scattering graph. Similarly to the conventional
PG, we define the full propagation graph as shown in Fig. 4,
which contains M1 transmitters, M2 receivers, N1 scatter-
ers, and N2 reflectors. Then the transfer matrices can be
defined as
D(f)∈CM1×M2:transmitters →receivers
Ts(f)∈CM1×N2:transmitters →scatterers
s(f)∈CN2×N2:transmitters →reflectors
Rs(f)∈CN2×M2:scatterers →receivers
Rrn(f)∈CN1×M2:reflectors →receivers
r(f)∈CN1×N1:reflectors →reflectors
Tr(f)∈CM1×N1:transmitters →reflectors
sr(f)∈CN2×N1:reflectors →scatterers
rs(f)∈CN1×N2:scatterers →reflectors.
Matrices D(f),Ts(f),s(f)and Rs(f)can be calculated
in similar manner as in Eq. (6). Matrices Rrn(f),rn(f)and
Tr(f)can be calculated as using the RPG approach illustrated
in Eq. (18), (19) and (21), respectively.
Matrix sr(f)and rs(f)contain the so-called mixed paths,
which are reverberating bounces among reflectors and scat-
terers, i.e., the transfer matrix sr(f)contains the signal flows
of the two components as
case 1:Tx →reflectors →scatterers,
case 2:Tx →scatterers →reflectors →scatterers,
and matrix rs(f)contains signal flows of two components as
case 3:Tx →scatterers →reflectors
case 4:Tx →reflectors →scatterers →reflectors.
Signal flows of both case 1and case 3of matrices sr(f)
and rs(f)are directly start from Tx, while the case 2and
case 4contain the so-called reverberating effects between
reflectors and scatterers.
The purpose of the following proposed embedded method
is to decouple the case 2and case 4signal flows from
matrices sr(f)and rs(f).
2) EMBEDDED METHOD
In case 2of matrix sr(f), the scatterers at two terminals can
be regarded as relay base stations. Then the sub-channel of
a full propagation channel with only reflectors and scatterers
can be illustrated in Fig. 5 (a), in which scatterers at two
sides are transmitters and receivers, s(f)and rn(f)are defined
in Fig. 4, sr1(f)and rs1(f)represent the transfer matrix
from reflectors to scatterers and from scatterers to reflectors,
respectively.
Then the effects of case 2can be embedded into the the
embedded-scatterers transfer matrix ss(f)as
ss(f)=s(f)+
∞
n=2rs1(f)rn−1(f)sr1,n(f),(26)
where rn(f)represents bouncing among reflectors for ntimes
and can be calculated by the manner of Eq. (19), sr1,n(f)
can be computed in the same way illustrated in Eq. (21).
In case 2of matrix rs(f), the reflectors at two terminals
illustrated in Fig. 5 (b) also can be regarded as relay base
stations. Applying the same formulation in Eq. (26), the
effects of case 2can be embedded into the the embedded-
reflection transfer matrix rr(f)as
rr(f)=r(f)+
∞
n=2sr2(f)sn−1(f)rs2(f),(27)
where sr2(f)and rs2(f)represent the transfer matrix from
reflectors to scatterers and from scatterers to reflectors in
Fig. 5 (b), respectively.
After embedded the reverberating effects of reflecting
and scattering into matrices ss(f)and rr(f), we only need
to consider components in case 1of matrices sr(f)and
rs(f). Then, matrix sr(f)can be calculated using the RPG
approach denoted in Eq. (18), (19) and (21). The edge gains
scatterers →reflectors in matrix rs(f)can be calculated
using Eq. (21).
196 VOLUME 2, 2021
FIGURE 5. Model for reflecting and scattering transfer function matrix.
3) CALCULATION OF CHANNEL TRANSFER FUNCTION
FOR EPG
The full propagation graph can be divided into three parts,
i.e., the LoS parts D(f), the NLoS parts Hs(f)and Hr(f).
The NLoS parts, i.e., subgraph for wave of Hs(f)and Hr(f),
are illustrated in Fig. 6 (a) and (b), respectively.
The NLoS portion of channel transfer function Hs(f)can
be calculated as
Hs(f)=Tr(f)
∞
n=2rrn−1(f)srn(f)I−ss(f)−1·Rs(f)
+Ts(f)I−ss(f)−1Rs(f). (28)
The first part denotes the propagating flows: Tx →
reflectors →scatterers →Rx. The second part denotes the
propagating flows: Tx →scatterers →Rx.
The NLoS portion of channel transfer function Hr(f)can
be calculated as
Hr(f)=
∞
n=2Ts(f)I−ss(f)−1sr(f)rn−1(f)Rrn(f)
+
∞
n=2Tr(f)rn−1(f)Rrn(f).(29)
FIGURE 6. Subgraph for NLoS parts.
The first part denotes the propagating flows: Tx →
scatterers →reflectors →Rx. The second part denotes the
propagating flows: Tx →reflectors →Rx.
Then, the total channel transfer function can be calculated
by superimposing these components as
H(f)=D(f)+Hr(f)+Hs(f). (30)
After all, the H(f)includes line of sight propaga-
tion, reflecting-path, scattering-path, and reflecting-scattering
mixed paths.
III. MEASUREMENT-BASED PERFORMANCE
EVALUATION
A. MEASUREMENT CAMPAIGN
In this section, the EPG was evaluated by a millimeter-
wave (mm-wave) field measurement in a corridor scenario.
The channel sounder was constructed using a programmable
network analyzer (PNA), a computer, a storage disk, oscil-
lators, frequency doublers, amplifiers, band-pass filter and a
low noise amplifier. Detailed information about the channel
VOLUME 2, 2021 197
LIU et al.: EPG MODEL FOR REFLECTION AND SCATTERING AND ITS MILLIMETER-WAVE MEASUREMENT-BASED EVALUATION
TABLE 2. The parameters of measurement campaign.
FIGURE 7. The photo for the measurement environment of the corridor.
sounder can be found in [36], [37]. Two 39 GHz omni-
antennas were used in both Tx and Rx sides. Important
configuration parameters are reported in Table 2.
The measurement campaign is performed in a corridor
showed in Fig. 7.
The sketch map of the corridor is depicted in Fig. 8.
The Tx was fixed at one end and the Rx was on the trolley
which gradually departs forward along the route connecting
Site.7 to Site.1 in the sketch map showed by Fig. 8.
B. IMPLEMENTATION PROCEDURES OF EPG
Step 1 (Set Up Digital Map According to the Geometrical
Information of Environment): Divide the objects in the envi-
ronment into scatterers and reflectors according to surface
roughness.
In this corridor scenario, glass windows are discretized as
reflecting cells, whose area dS is 0.01 m2with coefficient
0.2, while walls, floors, ceilings and handrails are discretized
as scattering cells, whose area dS is 0.04 m2with coefficient
0.6. The parameters of mm-wave are justified in [15], [38]
and used in unified PG in [18]. For more accurate simula-
tion, the coefficients need to be calibrated with measurement
results, even machine learning can also be applied to modify
the coefficients [33].
Step 2 (Generate Transfer Matrix): Identify the scattering
paths and reflecting paths of every pair of vertices, e.g., Tx
and reflectors, reflectors and scatterers.
Fig. 9 and Fig. 10 illustrate one-bouncing scattering paths
and one-bouncing reflection paths from Tx to Rx of one
snapshot, respectively.
Then, calculate transfer matrices, i.e., D(f),Tr(f),Ts(f),
s(f),sr(f),Rs(f),r(f),rs(f),Rr(f). It is reported in [18]
when the bouncing order is up to 3, the influence of multi-
path of mm-wave was quite weak, hence, we set n=4as
the highest reflecting order in this simulation.
Step 3 (Calculate the Transfer Matrices Using EPG): Use
Eq. (27) to calculate scattering-embedded transfer reflect-
ing matrices. Use Eq. (26) to calculate reflection-embedded
transfer scattering matrices. Calculate channel transfer func-
tions by Eq. (30) and obtain the corresponding channel
impulse responses (CIRs).
C. COMPARISONS AND DISCUSSION
1) CONCATENATED POWER DELAY PROFILES (CPDPS)
Fig. 11 (a), (b) and (c) illustrate the measured CPDPs, simu-
lated CPDPs by EPG, and simulated CPDPs by conventional
PG in the corridor scenario, respectively. It can be observed
that the moving trajectory in delay for LoS paths are simi-
lar for both two simulations and the measurement, however,
NLoS paths are different by appearance.
Some continuous NLoS paths marked as multi-path com-
ponents 1 (MCP1) and MCP2 in Fig. 11 (a) and (b) can
be observed both in the measurement and the EPG simu-
lation. Delays of NLoS paths increase along with the LoS
path with less power. However, for the conventional PG in
Fig. 11 (c), MPC1 disappears quickly as the distance of Tx
and Rx increases. Besides, obvious MPC2 track can not be
observed in conventional PG.
2) POWER DELAY PROFILES (PDPS)
To elaborate the effects of the proposed EPG, the PDPs gen-
erated by conventional PG, proposed EPG and measurement
are used for comparison. In both simulations of EPG and
conventional PG, the variance of the noise is set to -105 dBm
to emulate the thermal noise of measurement equipment.
Fig. 12 depicts comparison between measurement and
conventional PG, comparison between measurement and
proposed EPG, respectively in three typical snapshots. The
three snapshots are marked as site.7, site.5, and site.2 in the
sketch showed by Fig. 8. The LoS distance of site.7, site.5,
and site.2 can be approximately calculated as 4 meters, 7.5
meters, and 15 meters, respectively.
In Fig. 12 (a), the PDPs generated by both conventional
PG and the proposed EPG are able to reproduce the two
NLoS peaks as the measurement when the distance of Tx
and Rx is not far away.
However, as the distance between Tx and Rx increases in
Fig. 12 (b), the conventional PG can only reproduce part of
the NLoS peaks, while the newly proposed PG still works
well, the two important NLoS peaks match the measurement
with little error.
The contrast becomes more evident in Fig. 12 (c), when
the distance between Tx and Rx is more than 15 meters, it
can be observed that two obvious peaks for NLOS paths of
EPG and measurement in around 55 ns and 57 ns coincides
with each other, however, these two NLoS components are
invisible in the PDP generated using the conventional PG.
3) DELAY AND DELAY SPREAD
The mean delay and delay spread of site.7, site.5, and site.2
mentioned above are listed in Table 3, from which, it is
198 VOLUME 2, 2021
FIGURE 8. A diagram of the environment considered in the measurement, on the upper side of this sketch is glass windows, on the other side is a concrete wall.
FIGURE 9. One-bouncing scattering paths illustration.
FIGURE 10. One-bouncing reflection paths illustration.
observed that delay parameters of measurement and both
two simulations are in good consistency. The difference of
mean delay between measurement and simulation is within
3 ns, and difference of root-mean-square is within 1.5 ns.
4) DISCUSSION OF VALIDATION
The CPDPs obtained using the newly proposed EPG pro-
vide more accurate descriptions of this time-variant channel,
i.e., the reflection effects caused by the glass windows in
this corridor scenario can be observed in both measurement
and EPG simulation as the delay varies. Moreover, spatial
consistency of the moving track is also can be inferred.
The observation of single PDP comparison is in line with
CPDPs, furthermore, single PDP can explain the MPC tracks
revealed in CPDPs more concretely. The scattering compo-
nent becomes weaker and weaker as the distance increases,
TABLE 3. The delay parameters comparison of conventional PG, EPG, and
measurement.
hence the MPC tracks of conventional PG fading quite
fast due to lack of considering reflection effects. On the
contrary, the proposed EPG always keeps consistency with
measurement.
The mean delay and delay spread obtained through con-
ventional PG and EPG are both in good agreement with
measurement, it means the in such a corridor scenario, the
LoS path plays the most vital role in propagating signals.
These observations reveal two postulations, firstly, even
in the mm-wave frequency band the reflection mechanism
still plays a vital role in generating multipaths. Secondly the
improved EPG can simulate channels in better accordance
with the measurements than the conventional PG where dom-
inant reflection paths exist. The reasons why the simulation
results are not strictly identical to the measurement, accord-
ing to our conjecture, include inaccurate estimation of the Rx
location, the exact radiation pattern of the omnidirectional
antenna which is not considered in graph simulations, and
the thermal noise in the measurement equipment, etc.
IV. CONCLUSION
In this article, a novel simulation-based channel modeling
approach based on EPG is proposed and evaluated by field
measurement. Distinguished from the conventional scattering
PG, reflectors are added to generate a reflector-scatterer-
hybrid propagation graph. Such an extended graph can imi-
tates wave propagating along scattering-path, reflecting-path
and scattering-reflecting-mixed-path. An essential specifi-
cally subgraph-embedded method is utilized, which allows
decomposing mixed-path of full graph into scattering-path
and reflecting-path. In addition, a recursive formulation is
implemented and applied to calculating the composite effect
of scattering-path and reflecting-path. The procedures of the
newly proposed EPG are elaborated in this article.
VOLUME 2, 2021 199
LIU et al.: EPG MODEL FOR REFLECTION AND SCATTERING AND ITS MILLIMETER-WAVE MEASUREMENT-BASED EVALUATION
FIGURE 11. CPDPs comparison for measurements, EPG simulation and
conventional PG simulation.
The comparison of CPDPs and single PDPs was per-
formed among the conventional PG, the newly proposed
EPG, and measurement in a mm-wave corridor scenario.
The results demonstrate that the reflection mechanisms can
not be ignored in mm-wave band frequency. With reflection
effects added into the PG, the channel spatial consistency
observed in NLoS scenarios was explained more reason-
ably in the graph simulation. With these evidences, it is
concluded that the newly proposed method is capable of
accordingly reproducing the channel characteristics attributed
to reflecting components and scattering components.
Accurate and efficient channel simulation algorithms may
provide powerful tools for wireless channel and physical
FIGURE 12. PDPs comparison for measurements, proposed EPG, conventional PG
in three typical sites.
layer researches, in an era of Internet of Things with
increasingly frequency.
REFERENCES
[1] T. Obara, T. Okuyama, Y. Aoki, S. Suyama, J. Lee, and Y. Okumura,
“Indoor and outdoor experimental trials in 28-GHz band for 5G
wireless communication systems,” in Proc. IEEE 26th Annu. Int.
Symp. Pers. Indoor Mobile Radio Commun. (PIMRC), Aug. 2015,
pp. 846–850.
[2] M. S. Alouini, “Paving the way towards 5G wireless communication
networks,” in Proc. 2nd Int. Conf. Telecommun. Netw. (TEL-NET),
Aug. 2017, p. 1.
[3] Z. Cao et al., “Advanced integration techniques on broadband
millimeter-wave beam steering for 5G wireless networks and beyond,”
IEEE J. Quantum Electron., vol. 52, no. 1, pp. 1–20, Jan. 2016.
200 VOLUME 2, 2021
[4] J. Huang, C.-X. Wang, R. Feng, J. Sun, W. Zhang, and Y. Yang,
“Multi-frequency mmWave massive MIMO channel measurements and
characterization for 5G wireless communication systems,” IEEE J. Sel.
Areas Commun., vol. 35, no. 7, pp. 1591–1605, Jul. 2017.
[5] T. Abbas, J. Nuckelt, T. Kurner, T. Zemen, C. F. Mecklenbrauker, and
F. Tufvesson, “Simulation and measurement-based vehicle-to-vehicle
channel characterization: Accuracy and constraint analysis,” IEEE
Trans. Antennas Propag., vol. 63, no. 7, pp. 3208–3218, Jul. 2015.
[6] C. Gustafson, K. Haneda, S. Wyne, and F. Tufvesson, “On mm-Wave
multipath clustering and channel modeling,” IEEE Trans. Antennas
Propag., vol. 62, no. 3, pp. 1445–1455, Mar. 2014.
[7] V. Degli-Esposti, D. Guiducci, A. de’Marsi, P. Azzi, and F. Fuschini,
“An advanced field prediction model including diffuse scattering,”
IEEE Trans. Antennas Propag., vol. 52, no. 7, pp. 1717–1728,
Jul. 2004.
[8] A. O. Kaya, L. J. Greenstein, and W. Trappe, “Characterizing indoor
wireless channels via ray tracing combined with stochastic modeling,”
IEEE Trans. Wireless Commun., vol. 8, no. 8, pp. 4165–4175,
Aug. 2009.
[9] T. Pedersen and B. H. Fleury, “A realistic radio channel model based
in stochastic propagation graphs,” in Proc. 5th MATHMOD, 2006,
pp. 324–331.
[10] T. Pedersen, G. Steinbock, and B. H. Fleury, “Modeling of
outdoor-to-indoor radio channels via propagation graphs,” in Proc.
XXXIth URSI Gen. Assembly Sci. Symp. (URSI GASS), Aug. 2014,
pp. 1–4.
[11] G. E. Athanasiadou, A. R. Nix, and J. P. McGeehan, “A microcellular
ray-tracing propagation model and evaluation of its narrow-band and
wide-band predictions,” IEEE J. Sel. Areas Commun., vol. 18, no. 3,
pp. 322–335, Mar. 2000.
[12] B. Choudhury and R. M. Jha, “A refined ray tracing approach for
wireless communications inside underground mines and metrorail tun-
nels,” in Proc. IEEE Appl. Electromagn. Conf. (AEMC), Dec. 2011,
pp. 1–4.
[13] G.-Y. Wang, Y.-J. Liu, and S.-D. Li, “Simulation and analysis of
indoor millimeter-wave propagation based on the ray tracing method,”
in Proc. IEEE Int. Conf. Comput. Electromagn. (ICCEM), Feb. 2016,
pp. 350–352.
[14] P. Koivumaki, S. L. H. Nguyen, K. Haneda, and G. Steinböck, “A
study of polarimetric diffuse scattering at 28 GHz for a shopping
center facade,” in Proc. IEEE 29th Annu. Int. Symp. Pers. Indoor
Mobile Radio Commun. (PIMRC), Sep. 2018, pp. 182–187.
[15] E. M. Vitucci, J. Chen, V. Degli-Esposti, J. S. Lu, H. L. Bertoni,
and X. Yin, “Analyzing radio scattering caused by various building
elements using millimeter-wave scale model measurements and ray
tracing,” IEEE Trans. Antennas Propag., vol. 67, no. 1, pp. 665–669,
Jan. 2019.
[16] T. G. Lewis, Graphs. Hoboken, NJ, USA: Wiley, 2009. [Online].
Available: https://ieeexplore.ieee.org/document/8041132
[17] J. Zhang, C. Tao, L. Liu, and R. Sun, “A study on channel modeling
in tunnel scenario based on propagation-graph theory,” in Proc. IEEE
83rd Veh. Technol. Conf. (VTC Spring), May 2016, pp. 1–5.
[18] J. Chen, X. Yin, L. Tian, and M.-D. Kim, “Millimeter-wave chan-
nel modeling based on a unified propagation graph theory,” IEEE
Commun. Lett., vol. 21, no. 2, pp. 246–249, Feb. 2017.
[19] T. Pedersen, G. Steinbock, and B. H. Fleury, “Modeling of reverber-
ant radio channels using propagation graphs,” IEEE Trans. Antennas
Propag., vol. 60, no. 12, pp. 5978–5988, Dec. 2012.
[20] T. Pedersen and B. H. Fleury, “Radio channel modelling using stochas-
tic propagation graphs,” in Proc. IEEE Int. Conf. Commun., Jun. 2007,
pp. 2733–2738.
[21] R. Zhang, X. Lu, Z. Zhong, and C. Lin, “A study on spatial-
temporal dynamics properties of indoor wireless channels,” in Proc.
Wireless Algorithms Syst. Appl. Int. Conf., Chengdu, China, Aug. 2011,
pp. 410–421.
[22] L. Tian, X. Yin, Q. Zuo, J. Zhou, Z. Zhong, and S. X. Lu, “Channel
modeling based on random propagation graphs for high speed railway
scenarios,” in Proc. IEEE 23rd Int. Symp. Pers. Indoor Mobile Radio
Commun. (PIMRC), Sep. 2012, pp. 1746–1750.
[23] Y. Miao, T. Pedersen, M. Gan, E. Vinogradov, and C. Oestges,
“Reverberant room-to-room radio channel prediction by using rays and
graphs,” IEEE Trans. Antennas Propag., vol. 67, no. 1, pp. 484–494,
Jan. 2019.
[24] R. Adeogun, A. Bharti, and T. Pedersen, “An iterative transfer
matrix computation method for propagation graphs in multiroom envi-
ronments,” IEEE Antennas Wireless Propag. Lett., vol. 18, no. 4,
pp. 616–620, Apr. 2019.
[25] O. Souihli and T. Ohtsuki, “Benefits of rich scattering in MIMO chan-
nels: A graph-theoretical perspective,” IEEE Commun. Lett., vol. 17,
no. 1, pp. 23–26, Jan. 2013.
[26] Y. Liu, X. Yin, J. Lee, and M. Tong, “A graph-based simulation
method for propagation channels with multiple-knife-edge diffrac-
tion,” in Proc. IEEE Int. Conf. Comput. Electromagn. (ICCEM), 2020,
pp. 270–272.
[27] A. F. Molisch, Propagation Mechanisms. Hoboken, NJ,
USA: Wiley-IEEE Press, 2011. [Online]. Available:
https://ieeexplore.ieee.org/document/5635436
[28] X. Yin and X. Cheng, “Propagation channel characteriza-
tion, parameter estimation, and modeling for wireless com-
munications,” in Deterministic Channel-Parameter Estimation.
Hoboken, NJ, USA: Wiley-IEEE Press, 2016. [Online]. Available:
https://ieeexplore.ieee.org/document/7656797
[29] T. K. Sarkar, M. Salazar Palma, and M. N. Abdallah, Mechanism of
Wireless Propagation. Hoboken, NJ, USA: Wiley-IEEE Press, 2018.
[Online]. Available: https://ieeexplore.ieee.org/document/8410155
[30] L. Tian, V. Degli-Esposti, E. M. Vitucci, and X. Yin, “Semi-
deterministic radio channel modeling based on graph theory
and ray-tracing,” IEEE Trans. Antennas Propag., vol. 64, no. 6,
pp. 2475–2486, Jun. 2016.
[31] L. Tian, V. Degli-Esposti, E. M. Vitucci, X. Yin, F. Mani, and
S. X. Lu, “Semi-deterministic modeling of diffuse scattering com-
ponent based on propagation graph theory,” in Proc. IEEE 25th Annu.
Int. Symp. Pers. Indoor Mobile Radio Commun. (PIMRC), Sep. 2014,
pp. 155–160.
[32] G. Steinböck et al., “Hybrid model for reverberant indoor radio chan-
nels using rays and graphs,” IEEE Trans. Antennas Propag., vol. 64,
no. 9, pp. 4036–4048, Sep. 2016.
[33] R. O. Adeogun, “Calibration of stochastic radio propagation mod-
els using machine learning,” IEEE Antennas Wireless Propag. Lett.,
vol. 18, no. 12, pp. 2538–2542, Dec. 2019.
[34] A. Goldsmith, Wireless Communications, Path Loss and Shadowing.
Cambridge, U.K.: Cambridge Univ. Press, 2005, pp. 27–63.
[35] D. M. Pozar, Microwave Engineering—Transmission Line Theory,4th
ed. New Delhi, India: Wiley, 2012, pp. 68–114.
[36] Y. Lv, X. Yin, C. Zhang, and H. Wang, “Measurement-based char-
acterization of 39 GHz millimeter-wave dual-polarized channel under
foliage loss impact,” IEEE Access, vol. 7, pp. 151558–151568, 2019.
[37] C. Zhang, X. Yin, X. Cai, and Z. Yu, “Wideband 39 GHz millimeter-
wave channel measurements under diversified vegetation,” in Proc.
IEEE 29th Annu. Int. Symp. Pers. Indoor Mobile Radio Commun.
(PIMRC), Sep. 2018, pp. 1–6.
[38] V. Degli-Esposti et al., “Ray-tracing-based mm-Wave beamforming
assessment,” IEEE Access, vol. 2, pp. 1314–1325, 2014.
YUAN LIU received the bachelor’s degree in electronics engineering from
Central China Normal University, Wuhan, China, in 2017, and the master’s
degree in electronics engineering from Tongji University, Shanghai, China,
in 2020. From September 2018 to June 2019, he was a Visitor with the
Group of Antennas and Propagation, Aalto University, Helsinki, Finland.
He is currently a Research Engineer with Guangdong Communications &
Networks Institute. His research interests include radio propagation sim-
ulation tools, such as propagation-graph and ray-tracing, statistical signal
processing for channel parameters estimation and characterization, terahertz
channel modeling and terahertz/RF systems for B5G communications.
VOLUME 2, 2021 201
LIU et al.: EPG MODEL FOR REFLECTION AND SCATTERING AND ITS MILLIMETER-WAVE MEASUREMENT-BASED EVALUATION
XUEFENG YIN (Member, IEEE) received the bachelor’s degree in opto-
electronics engineering from the Huazhong University of Science and
Technology, Wuhan, China, in 1995, and the M.Sc. degree in digi-
tal communications and the Ph.D. degree in wireless communications
from Aalborg University, Aalborg, Denmark, in 2002 and 2006, respec-
tively. From 2006 to 2008, he was an Assistant Professor with Aalborg
University. In 2008, he joined the College of Electronics and Information
Engineering, Tongji University, Shanghai, China. He became a Full
Professor, in 2016, and has been the Vice Dean of the College of
Electronics and Information Engineering since then. He has authored
or coauthored more than 100 technical articles and coauthored the
book Propagation Channel Characterization, Parameter Estimation, and
Modeling for Wireless Communications (Wiley, 2016). His research interests
include high-resolution parameter estimation for propagation channels,
measurement-based channel characterization and stochastic modeling for
5G wireless communications, channel simulation based on random graph
models, radar signal processing, and target recognition.
XIAOKANG YE (Student Member, IEEE) received the bachelor’s degree in
electrical engineering from the Shanghai University of Engineering Science,
Shanghai, China, in 2014. He is currently pursuing the Ph.D. degree with
the College of Electronics and Information Engineering, Tongji University.
His research interests include statistical channel characterization, millime-
ter wave channel characterization and modeling, channel fingerprint, and
applications of machine learning based techniques on propagation channel
characterization.
YONGYU HE (Student Member, IEEE) was born in Pingan, China, in
1989. He received the bachelor’s degree in electronics science and tech-
nology and the master’s degree in circuit and systems from the Tongji
University, Shanghai, China, in July 2011 and April 2014, respectively,
where he is currently pursuing the Doctoral degree in physics. His research
interests include wireless propagation channel and more specifically, the
characterization and modeling of millimeter-wave and higher frequency
band channels, and self-interference channels.
JUYUL LEE (Senior Member, IEEE) received the Ph.D. degree in electrical
engineering from the University of Minnesota at Twin Cities, USA, in 2010.
He was with the Agency for Defense Development, Daejeon, South Korea,
from 1998 to 2000. Since 2000, he has been with the Electronics and
Telecommunications Research Institute, Daejeon, where he is currently a
Principal Researcher with the Telecommunications and Media Research
Laboratory. He has contributed to ITU-R recommendations and reports
in Study Group 3 (Propagation), including millimeter-wave propagation
models. He is currently the Chairman of the ITU-R Correspondence Group
3K-6, which is responsible for studying the impact of higher frequencies
(from 6 GHz to 450 GHz) on propagation models and related characteristics.
His current research interests include wireless channel modeling, machine
learning, and information theory.
202 VOLUME 2, 2021
