# A General 3D Non-Stationary 5G Wireless Channel Model

Shangbin Wu, Cheng-Xiang Wang, el-Hadi M. Aggoune, Mohammed M. Alwakeel, and
Xiao-Hu You, Manuscript received August 29, 2017; revised November 12, 2017;
accepted November 24, 2017. The authors gratefully acknowledge the support of
this work from the EU H2020 ITN 5G Wireless project (No. 641985), the EU H2020
RISE TESTBED project (No. 734325), the EU FP7 QUICK project (No. PIRSES-
GA-2013-612652), the EPSRC TOUCAN project (No. EP/L020009/1), the Natural
Science Foundation of China (No. 61210002), and SNCS Research Center, the
University of Tabuk, Saudi Arabia. Part of this work was presented in Dr. S.
Wu’s PhD thesis, October 2015. The associate editor coordinating the review of
this paper and approving it for publication was L. Dai.S. Wu is with Samsung
R&D Institute UK, Staines-upon-Thames, TW18 4QE, UK (E-mail:
shangbin.wu@samsung.com).C. -X. Wang (corresponding author) is with the
Institute of Sensors, Signals and Systems, School of Engineering and Physical
Sciences, Heriot-Watt University, Edinburgh, EH14 4AS, U.K. He is also with
the Mobile Communications Research Laboratory, Southeast University, Nanjing,
211189, China (E-mail: cheng-xiang.wang@hw.ac.uk).el-H. M. Aggoune and M. M.
Alwakeel are with the Sensor Networks and Cellular Systems (SNCS) Research
Center, University of Tabuk, P. O. Box: 6592-2, 47315/4031 Tabuk, Saudi Arabia
(E-mail: {haggoune.sncs, alwakeel}@ut.edu.sa).X. -H. You is with the Mobile
Communications Research Laboratory, Southeast University, Nanjing, 211189,
China (E-mail: xhyu@seu.edu.cn).

###### Abstract

A novel unified framework of geometry-based stochastic models (GBSMs) for the
fifth generation (5G) wireless communication systems is proposed in this
paper. The proposed general 5G channel model aims at capturing small-scale
fading channel characteristics of key 5G communication scenarios, such as
massive multiple-input multiple-output (MIMO), high-speed train (HST),
vehicle-to-vehicle (V2V), and millimeter wave (mmWave) communication
scenarios. It is a three-dimensional (3D) non-stationary channel model based
on the WINNER II and Saleh-Valenzuela (SV) channel models considering array-
time cluster evolution. Moreover, it can easily be reduced to various
simplified channel models by properly adjusting model parameters. Statistical
properties of the proposed general 5G small-scale fading channel model are
investigated to demonstrate its capability of capturing channel
characteristics of various scenarios, with excellent fitting to some
corresponding channel measurements.

###### Index Terms:

3D non-stationary 5G wireless channel models, massive MIMO systems, mmWave
communications, high-speed train communications, V2V communications.

##  I Introduction

To satisfy the demands of the fifth generation (5G) wireless communication
networks, known as increased data rate, reduced latency, energy, and cost [1],
a number of advanced technologies have been proposed in the literature as
potential 5G technologies. Massive multiple-input multiple-output (MIMO),
i.e., an enhanced MIMO technique with a large number of antennas, is able to
greatly improve communication reliability, spectral efficiency, and energy
efficiency [2]–[5]. Vehicle-to-vehicle (V2V) communications [6], [7] were
proposed in the heterogeneous network architecture of 5G to enable vehicles to
connect mutually without base stations. High-speed train (HST) communication
also attracts attention for the emerging development of high mobility trains
with speed expected to be higher than $500$ km/h. Furthermore, millimeter wave
(mmWave) frequency bands (30–300 GHz) have been proposed to be used for 5G
wireless communications to solve the spectrum crisis problem. The mmWave
frequency bands are capable of providing large bandwidth (in the order of GHz)
and exploiting polarization and massive MIMO [8]–[10]. In order to design and
evaluate 5G systems, a channel model which can capture channel characteristics
of the above-mentioned potential technologies is essential. However,
conventional channel models such as 3GPP spatial channel model (SCM) [11], the
WINNER II [12], WINNER+ [13], 3GPP three dimensional (3D) SCM [14], IMT-A
[15], and COST 2100 channel models [16]–[18] are not able to sufficiently meet
these emerging 5G requirements. Although both large-scale fading and small-
scale fading have large impacts on system performance, small-scale fading of
5G wireless channels is less studied in the literature.

The first geometry-based stochastic model (GBSM) for 5G wireless channels was
proposed by the METIS project [19]. However, the METIS GBSM did not
sufficiently support channel characteristics of massive MIMO, V2V, and mmWave
communications [19, Table 4–1]. Later, multiple industrial partners and
academic institutes formed a special interest group (SIG) and jointly
published a white paper for 5G channel models [20]. This SIG white paper has
covered new channel characteristics from 6 GHz to 100 GHz, such as blockage,
spatial consistency, support of large bandwidth and array, and novel path loss
models. The SIG white paper was then used as the guideline for the 3GPP new
radio (NR) channel model for frequencies from 0.5 to 100 GHz in [21]. Based on
the IMT-A channel model and 3GPP 3D SCM, the IMT-2020 channel model [22] was
recently proposed by the ITU. The IMT-2020 channel model supports frequency
bands up to 100 GHz and can cover many new features, e.g., 3D propagation,
spatial consistency, large bandwidth, large antenna array, etc. Additionally,
another modeling approach known as the map-based model was introduced in the
METIS channel model [19], 3GPP NR channel model [21], and IMT-2020 channel
model [22]. Map-based models aim at computing channel coefficients in a
deterministic manner when the layout of a network is predefined. The
millimetre-wave evolution for backhaul and access (MiWEBA) project [23]
proposed a channel model combining deterministic approach based on network
layout with stochastic components to model mmWave channels. However, since
layouts of networks are not always accessible, GBSMs attract more attentions
from researchers. Therefore, we will focus on GBSMs and small-scale fading in
this paper.

###  I-A Related Work I: GBSMs for Massive MIMO

It was reported in [24] and [25] that massive MIMO channels have some specific
characteristics which were ignored in conventional MIMO (numbers of transmit
and receive antennas are relatively small) channel models [12]–[18]. First,
when the number of antennas is large, the distance $\mathcal{D}$ between the
transmitter (cluster) and the receiver may not be larger than the Rayleigh
distance $2\mathcal{L}^{2}/\lambda$ [26], where $\mathcal{L}$ is the dimension
of the antenna array and $\lambda$ is the carrier wavelength. In this case,
the effect of spherical wavefront is significant and the conventional plane
wavefront assumption in [12]–[15] is not well-justified. The COST 2100 channel
model and the METIS GBSM can support spherical wavefronts after minor
extensions. Second, cluster appearance and disappearance can occur on the
array axis. As a result, each antenna may have its own set of observable
clusters. This has not been considered in most advanced GBSMs such as the SCM-
extension model [27], the COST 2100 channel model [16], the METIS GBSM [19],
the 3GPP NR channel model [21], and the IMT-2020 channel model [22]. A two-
dimensional (2D) confocal ellipse model and a 3D twin-cluster model were
proposed for massive MIMO channels in [28] and [29], respectively, while a
comprehensive survey of massive MIMO channel measurements and models was given
in [30]. Both the ellipse model and twin-cluster model in [28] and [29]
employed the spherical wavefront assumption and adapted the cluster birth-
death process in [31] and [32] to both the time and array axes to characterize
cluster appearance and disappearance. However, the mean power evolution of
clusters and rays, directional antennas, and polarized antennas were ignored
in [28] and [29]. Most importantly, channel models in [28] and [29] were not
designed to accommodate mmWave channels, V2V channels, and arbitrary antenna
array layouts.

###  I-B Related Work II: GBSMs for V2V and HST

In V2V channels, the transmitter, scatterers, and receiver can be all moving.
Doppler frequencies caused by either the transmitter and/or scatterers and/or
receiver should be taken into account. Wideband 2D GBSMs for V2V channels were
proposed and validated via measurements in [33]–[35], where clusters were
categorized into mobile clusters and static clusters. In V2V communications,
transmitters and receivers may be lower than clusters on surrounding
buildings. Therefore, 3D clusters were included in [36]–[38]. A 3D concentric-
cylinder V2V channel model was introduced in [36], while 3D GBSMs combining a
two-sphere model and an elliptic-cylinder model were proposed in [37] and
[38]. Recently, birth-death process was used to model cluster dynamics in V2V
channels in [39]. For HST communications, relevant non-stationary GBSMs can be
found in [40]–[43] and some measurement results were given in [44]. The non-
stationary channel behavior of HST systems is similar to that of V2V channels
with large Doppler frequencies. However, cluster evolution on the time axis
was ignored in the METIS GBSM [19], 3GPP NR channel model [21], IMT-2020
channel model [22], and models in [33]–[39]. Thus, it is difficult to track
the channel with respect to time in a continuous manner.

###  I-C Related Work III: GBSMs for mmWave

As the supported bandwidth for mmWave is large (in the order of GHz [8]–[10]),
GBSMs for mmWave channels need to consider high delay resolution, i.e., rays
within a cluster may be resolvable and the numbers of rays within clusters may
vary. The GBSM of the METIS channel model [19] supports frequency bands up to
70 GHz. However, the resolvable rays within clusters and the varying numbers
of rays within clusters were not included. The quasi deterministic radio
channel generator (QuaDRiGa) channel model [46] was used as the initial model
for mmWave channels in the mmWave based mobile radio access network for 5G
integrated communications (mmMAGIC) project [45]. However, it also ignored the
resolvable rays and varying numbers of rays within clusters. The Saleh-
Valenzuela (SV) channel model [47], which was originally proposed for indoor
multipath propagations, has been used to evaluate system performance in the
IEEE wireless personal area network (PAN) standard [48]–[51] with the
supported bandwidth over $500$ MHz. Since mmWave channels are expected to have
bandwidths over $500$ MHz, applications of the SV channel model to mmWave
channels can be found in [52]–[56]. In a SV channel model, the number of rays
within each cluster was assumed to follow a Poisson distribution. Complex gain
and delay were assigned to each ray. A 3D mmWave channel model with random
numbers of clusters and paths within each cluster was proposed in [57].
However, in the above SV-based mmWave channel models, time evolution and mean
ray power evolution have not sufficiently been studied.

###  I-D Contributions

To address the abovementioned research gaps, this paper has the following
contributions [58].

  1. 1.

A general 3D non-stationary 5G GBSM for terrestrial wireless communication
systems is proposed, having the capability of simulating massive MIMO, V2V,
HST, and mmWave small-scale fading channels [58]. It also considers time
evolution of channels with all the model parameters as time varying. The
proposed general 5G channel model is based on the WINNER II channel model
[12], in order to keep consistency of 4G and 5G channel models, and the SV
channel model in order to support the high delay resolution of mmWave channels
[48]–[51]. Spherical wavefront and array-time cluster evolution are included
to represent massive MIMO channel characteristics. Array-time evolution
includes cluster birth-death process in both the array and time axes and
geometrical relationship updates. The mean power updates of rays are also
embedded in the proposal 5G small-scale fading channel model with the
assumption of the inverse square law.

  2. 2.

The proposed general 5G GBSM can easily be reduced to various simplified
channel models by setting proper channel parameters, which is demonstrated by
fitting some statistical properties to the corresponding channel measurement
data.

The rest of this paper is organized as follows. Section II gives a general
description of the proposed unified framework for 5G small-scale fading
channel models. Statistical properties of the proposed general 3D 5G GBSM are
investigated in Section III. Simulation/numerical results and analysis are
presented in Section IV. Conclusions are drawn in Section V.

##  II A General 3D Non-Stationary 5G Small-Scale Fading Channel Model

Let us consider a MIMO system with $M_{R}$ receive and $M_{T}$ transmit
antennas communicating at carrier frequency $f_{c}$. Let
$\mathrm{Ant}_{q}^{R}$ denote the $q$th receive antenna and
$\mathrm{Ant}_{p}^{T}$ denote the $p$th transmit antenna. Also, let
$\mathrm{Cluster}_{n}$ denote the $n$th cluster. It should be noted that
arbitrary antenna array layouts are assumed in the proposed model. Typical
antenna array layouts include uniform linear arrays, 2D planar arrays, and 3D
cube arrays. Antenna responses can be modified subject to actual antenna
settings. The scattering environment between the transmitter and receiver is
abstracted as effective clusters [12], which characterize the first and last
bounces of the channel. Multi-bounces between the first and last bounces are
abstracted by a virtual link. The proposed general 3D non-stationary 5G GBSM
is illustrated in Fig. 1. It should be noticed that
($x_{\mathrm{G}},y_{\mathrm{G}},z_{\mathrm{G}}$) axes are established as the
global coordinate system (GCS) with origin at the center of the transmit
array. This needs to be distinguished from the local coordinate systems (LCSs)
with origins at the centers of transmit and receive arrays when calculating
antenna pattern in a 3–D space. Spherical wavefront and cluster appearance and
disappearance are assumed in order to support massive MIMO scenarios. In this
case, each antenna may have its own set of observable clusters. Let
$C^{R}_{q}(t)$ and $C^{T}_{p}(t)$ represent the cluster sets of
$\mathrm{Ant}_{q}^{R}$ and $\mathrm{Ant}_{p}^{T}$ at time $t$, respectively.
Then, the total number of clusters $N(t)$ observable by both the transmitter
and receiver at time $t$ can be calculated as
$N(t)=\textrm{card}\left(\bigcup\limits^{M_{T}}_{p=1}\bigcup\limits^{M_{R}}_{q=1}S_{qp}(t)\right)$
where $S_{qp}(t)=C^{R}_{q}(t)\bigcap C^{T}_{p}(t)$,
$\mathrm{card}(\mathcal{S})$ denotes the cardinality of the set $\mathcal{S}$,
$\bigcup$ and $\bigcap$ denote the union and intersection of sets,
respectively. Also, both the transmitter and receiver are assumed to be in
motion to support V2V scenarios, which results in Doppler frequencies at both
sides. In addition, rays within clusters are considered to be resolvable in
order to support the high time resolution in mmWave scenarios. Complex gain
and delay should be assigned to each ray.

Figure 1: A general 3D non-stationary 5G GBSM. TABLE I: Definitions of key 5G channel model parameters. $\psi_{A}^{R}$, $\psi_{E}^{R}$ | azimuth and elevation angles of the receive array broadside
---|---
$\psi_{A}^{T}$, $\psi_{E}^{T}$ | azimuth and elevation angles of the transmit array broadside
$N(t)$ | total number of observable clusters
$M_{n}$ |  number of rays within $\mathrm{Cluster}_{n}$
$C^{R}_{q}(t)$, $C^{T}_{p}(t)$ |  cluster set of $\mathrm{Ant}_{q}^{R}$ and cluster set of $\mathrm{Ant}_{p}^{T}$
$S_{qp}(t)$ |  set of clusters observable for both $\mathrm{Ant}_{q}^{R}$ and $\mathrm{Ant}_{p}^{T}$
$\phi_{n}^{A}(t)$, $\phi_{n}^{E}(t)$ |  azimuth and elevation angles between $\mathrm{Cluster}_{n}$ and the receive array center
$\varphi_{n}^{A}(t)$, $\varphi_{n}^{E}(t)$ |  azimuth and elevation angles between $\mathrm{Cluster}_{n}$ and the transmit array center
$\phi_{n,m_{n}}^{A}(t)$, $\phi_{n,m_{n}}^{E}(t)$ |  azimuth and elevation angles between the $m_{n}$th ray of $\mathrm{Cluster}_{n}$ and the receive array center
$\varphi_{n,m_{n}}^{A}(t)$, $\varphi_{n,m_{n}}^{E}(t)$ |  azimuth and elevation angles between the $m_{n}$th ray of $\mathrm{Cluster}_{n}$ and the transmit array center
$\mathbf{A}_{q}^{R}(t)$, $\mathbf{A}_{p}^{T}(t)$ |  3D position vectors of $\mathrm{Ant}_{q}^{R}$ and $\mathrm{Ant}_{p}^{T}$
$\mathbf{D}_{n}^{R}(t)$, $\mathbf{D}_{n}^{T}(t)$ |  3D distance vectors between $\mathrm{Cluster}_{n}$ and the receiver (transmitter) array center
$\mathbf{D}_{n,m_{n}}^{R}(t)$,$\mathbf{D}_{n,m_{n}}^{T}(t)$ |  3D distance vectors between $\mathrm{Cluster}_{n}$ and the receive (transmit) array center via the $m_{n}$th ray
$\mathbf{D}_{qn,m_{n}}^{R}(t)$,$\mathbf{D}_{pn,m_{n}}^{T}(t)$ |  3D distance vectors between $\mathrm{Cluster}_{n}$ and $\mathrm{Ant}_{q}^{R}$ ($\mathrm{Ant}_{p}^{T}$) via the $m_{n}$th ray
$f_{qn,m_{n}}^{R}(t)$,$f_{pn,m_{n}}^{T}(t)$ |  Doppler frequencies of $\mathrm{Ant}_{q}^{R}$ ($\mathrm{Ant}_{p}^{T}$) via $\mathrm{Cluster}_{n}$ and the $m_{n}$th ray
$\mathbf{D}_{qp}^{\mathrm{LOS}}(t)$ |  3D distance vector of the LOS component between $\mathrm{Ant}_{q}^{R}$ and $\mathrm{Ant}_{p}^{T}$
$f_{pq}^{\mathrm{LOS}}(t)$ |  Doppler frequency of the LOS component between $\mathrm{Ant}_{q}^{R}$ and $\mathrm{Ant}_{p}^{T}$
$\mathbf{v}^{R}$,$\mathbf{v}^{T}$ | 3D velocity vectors of the receive and transmit arrays
$\mathbf{v}_{n}^{R}$, $\mathbf{v}_{n}^{T}$ |  3D velocity vectors of the last bounce and first bounce of $\mathrm{Cluster}_{n}$
$P_{n,m_{n}}(t)$ |  mean power of the $m_{n}$th ray of $\mathrm{Cluster}_{n}$
$\mathbf{D}$ | 3D distance vector between the receive and transmit array centers
$K$ | Rician factor
$\tilde{\lambda}$ | mean number of rays within a cluster
${\kappa}$ | cross polarization power ratio
${\lambda_{G},\lambda_{R}}$ | generation rate and recombination rate of clusters

Let $\mathbf{A}^{R}_{q}(t)$ and $\mathbf{A}^{T}_{p}(t)$ denote the position
vectors of $\mathrm{Ant}_{q}^{R}$ and $\mathrm{Ant}_{p}^{T}$, respectively.
Also, let $\psi_{A}^{R}$ and $\psi_{E}^{R}$ be azimuth and elevation angles of
the receive array broadside, and let $\psi_{A}^{T}$ and $\psi_{E}^{T}$ be
azimuth and elevation angles of the transmit array broadside, respectively.
Let $\mathbf{D}$ denote the initial position vector of the receiver and is
assumed to equal $\left[D,0,0\right]^{\mathrm{T}}$, and $D$ is the initial
distance between the transmitter and receiver centers. The line-of-sight (LOS)
distance vector $\mathbf{D}^{\mathrm{LOS}}_{qp}(t)$ between
$\mathrm{Ant}_{q}^{R}$ and $\mathrm{Ant}_{p}^{T}$ is computed as

$\mathbf{D}^{\mathrm{LOS}}_{qp}(t)=\mathbf{A}^{R}_{q}(t)-\mathbf{A}^{T}_{p}(t).$
(1)

According to the geometrical relationships in Fig. 1 and the key parameters
listed in Table I, distance vectors of $\mathrm{Cluster}_{n}$ at the
transmitter and receiver are calculated as

$\mathbf{D}^{R}_{n}(t)=D^{R}_{n}(t)\begin{bmatrix}\cos\phi^{E}_{n}(t)\cos\phi^{A}_{n}(t)\\\
\cos\phi^{E}_{n}(t)\sin\phi^{A}_{n}(t)\\\
\sin\phi^{E}_{n}(t)\end{bmatrix}^{\mathrm{T}}+\mathbf{D}$ (2)
$\mathbf{D}^{T}_{n}(t)=D^{T}_{n}(t)\begin{bmatrix}\cos\varphi^{E}_{n}(t)\cos\varphi^{A}_{n}(t)\\\
\cos\varphi^{E}_{n}(t)\sin\varphi^{A}_{n}(t)\\\
\sin\varphi^{E}_{n}(t)\end{bmatrix}^{\mathrm{T}}$ (3)

where $D^{R}_{n}(t)$ and $D^{T}_{n}(t)$ are the Frobenius norms of
$\mathbf{D}^{R}_{n}(t)$ and $\mathbf{D}^{T}_{n}(t)$, respectively. Distance
vectors of the $m_{n}$th ray of $\mathrm{Cluster}_{n}$ to the transmitter and
receiver centers are calculated as

$\mathbf{D}^{R}_{n,m_{n}}(t)=D^{R}_{n}(t)\begin{bmatrix}\cos\phi^{E}_{n,m_{n}}(t)\cos\phi^{A}_{n,m_{n}}(t)\\\
\cos\phi^{E}_{n,m_{n}}(t)\sin\phi^{A}_{n,m_{n}}(t)\\\
\sin\phi^{E}_{n,m_{n}}(t)\end{bmatrix}^{\mathrm{T}}+\mathbf{D}$ (4)
$\mathbf{D}^{T}_{n,m_{n}}(t)=D^{T}_{n}(t)\begin{bmatrix}\cos\varphi^{E}_{n,m_{n}}(t)\cos\varphi^{A}_{n,m_{n}}(t)\\\
\cos\varphi^{E}_{n,m_{n}}(t)\sin\varphi^{A}_{n,m_{n}}(t)\\\
\sin\varphi^{E}_{n,m_{n}}(t)\end{bmatrix}^{\mathrm{T}}.$ (5)

Distance vectors between the $m_{n}$th ray of $\mathrm{Cluster}_{n}$ and
antenna elements are calculated as

$\mathbf{D}^{R}_{qn,m_{n}}(t)=\mathbf{D}^{R}_{n,m_{n}}(t)-\mathbf{A}^{R}_{q}(t)$
(6)
$\mathbf{D}^{T}_{pn,m_{n}}(t)=\mathbf{D}^{T}_{n,m_{n}}(t)-\mathbf{A}^{T}_{p}(t).$
(7)

It should be noticed that position vectors are all time dependent. After all
vectors are obtained in the 3D space, the channel impulse response can be
derived.

###  II-A Channel Impulse Response

Based on the WINNER II and SV channel models, the proposed 5G GBSM at time $t$
with delay $\tau$ can be characterized by an $M_{R}\times M_{T}$ matrix
$\mathbf{H}(t,\tau)=\left[h_{qp}(t,\tau)\right]$. The entries of
$\mathbf{H}(t,\tau)$ consist of two components, i.e., the LOS component and
the non-LOS (NLOS) component, and can be written as

$\displaystyle h_{qp}(t,\tau)$
$\displaystyle=\underbrace{\sqrt{\frac{K(t)}{K(t)+1}}h^{\mathrm{LOS}}_{qp}(t)\delta\left(\tau-\tau^{\mathrm{LOS}}(t)\right)}_{\mathrm{LOS}}$
$\displaystyle+\underbrace{\sqrt{\frac{1}{K(t)+1}}\sum\limits_{n=1}^{N(t)}\sum\limits_{m_{n}=1}^{M_{n}(t)}h_{qp,n,m_{n}}(t)\delta\left(\tau-\tau_{n}(t)-\tau_{m_{n}}(t)\right)}_{\mathrm{NLOS}}.$
(8)

In (8), $K(t)$ is the Rician factor, $N(t)$ is the time variant number of
clusters, $M_{n}(t)$ is the number of rays within $\mathrm{Cluster}_{n}$,
$\tau_{n}(t)$ is the delay of $\mathrm{Cluster}_{n}$, and $\tau_{m_{n}}(t)$ is
the relative delay of the $m_{n}$th ray in $\mathrm{Cluster}_{n}$. It is
important to mention that all the parameters of the proposed 5G GBSM are time-
variant, which has the capability to model the time-evolution and high
mobility features of channels and is essentially a non-stationary channel
model. To simplify the model, we assume that the Rician factor and relative
delays are constants during the generation of channel coefficients, i.e.,
$K(t)=K$ and $\tau_{m_{n}}(t)=\tau_{m_{n}}$. These may not hold in certain
scenarios such as the HST cutting scenario where the Rician factor is changing
with time [59]. However, the cutting scenario is not frequently occurring in
5G scenarios and it is of high complexity. Also, the number of rays within a
cluster is assumed to follow a Poisson distribution
$\mathrm{Pois}\left(\tilde{\lambda}\right)$ [54], i.e.,
$M_{n}(t)=M_{n}=\max\left\\{\mathrm{Pois}\left(\tilde{\lambda}\right),1\right\\}$,
where $\tilde{\lambda}$ is both the mean and variance of $M_{n}$ and
$\max\left\\{\cdot\right\\}$ calculates the maximum value. Each ray within a
cluster has its own complex gain and delay to support mmWave channels in the
proposed 5G GBSM.

For the LOS component, if polarized antenna arrays are assumed at both the
receiver and transmitter sides, the complex channel gain
$h^{\mathrm{LOS}}_{qp}(t)$ is presented as (9),

$\displaystyle h^{\mathrm{LOS}}_{qp}(t)=$
$\displaystyle\begin{bmatrix}F_{p,\mathrm{V}}^{T}(\mathbf{D}_{qp}^{\mathrm{LOS}}(t),\mathbf{A}_{p}^{T}(t))\\\
F_{p,\mathrm{H}}^{T}(\mathbf{D}_{qp}^{\mathrm{LOS}}(t),\mathbf{A}_{p}^{T}(t))\end{bmatrix}^{\mathrm{T}}\begin{bmatrix}e^{j\Phi_{\mathrm{LOS}}}&0\\\
0&-e^{j\Phi_{\mathrm{LOS}}}\end{bmatrix}\begin{bmatrix}F_{q,\mathrm{V}}^{R}(\mathbf{D}_{qp}^{\mathrm{LOS}}(t),\mathbf{A}_{q}^{R}(t))\\\
F_{q,\mathrm{H}}^{R}(\mathbf{D}_{qp}^{\mathrm{LOS}}(t),\mathbf{A}_{q}^{R}(t))\end{bmatrix}e^{j2\pi
f_{qp}^{\mathrm{LOS}}(t)t}$ (9)

where $\Phi_{\mathrm{LOS}}$ is uniformly distributed within $(0,2\pi]$ [14].
The superscripts V and H denote vertical polarization and horizontal
polarization, respectively. Functions $F^{T}(\mathbf{a},\mathbf{b})$ and
$F^{R}(\mathbf{a},\mathbf{b})$ are antenna patterns with input vectors
$\mathbf{a}$ and $\mathbf{b}$ in the GCS. The input vectors $\mathbf{a}$ and
$\mathbf{b}$ need to be transformed into the LCS to obtain the antenna
patterns. Detailed calculations are presented in Appendix A [14] and the
antenna pattern functions can be modified according to practical antenna
settings. For large bandwidth support, the antenna patterns can be frequency
dependent. In this case, the antenna patterns can be implemented accordingly
in this model as well. The Doppler frequency $f_{qp}^{\mathrm{LOS}}(t)$
between $\mathrm{Ant}_{q}^{R}$ and $\mathrm{Ant}_{p}^{T}$ of the LOS component
is expressed as
$f_{qp}^{\mathrm{LOS}}(t)=\frac{1}{\lambda}\frac{\left\langle\mathbf{D}_{qp}^{\mathrm{LOS}}(t),\mathbf{v}^{R}-\mathbf{v}^{T}\right\rangle}{\left\|\mathbf{D}_{qp}^{\mathrm{LOS}}(t)\right\|}$
where $\left\langle\cdot,\cdot\right\rangle$ is the inner product operator,
$\|\cdot\|$ calculates the Frobenius norm, and $\lambda$ is the wavelength
with respect to the central carrier frequency. Given the speed of light $c$,
the delay $\tau^{\mathrm{LOS}}(t)$ of the LOS component is computed as
$\tau^{\mathrm{LOS}}(t)=\left\|\mathbf{D}(t)\right\|/c$.

For NLOS components, if $\mathrm{Cluster}_{n}$ is observable to
$\mathrm{Ant}_{q}^{R}$ and $\mathrm{Ant}_{p}^{T}$, i.e.,
$\mathrm{Cluster}_{n}\in S_{qp}(t)$, the complex channel gain is expressed as
(10),

$\displaystyle h_{qp,n,m_{n}}(t)=$
$\displaystyle\begin{bmatrix}F_{p,\mathrm{V}}^{T}(\mathbf{D}_{n,m_{n}}^{T}(t),\mathbf{A}_{p}^{T}(t))\\\
F_{p,\mathrm{H}}^{T}(\mathbf{D}_{n,m_{n}}^{T}(t),\mathbf{A}_{p}^{T}(t))\end{bmatrix}^{\mathrm{T}}\begin{bmatrix}e^{j\Phi_{n,m_{n}}^{\mathrm{VV}}}&\sqrt{\kappa}e^{j\Phi_{n,m_{n}}^{\mathrm{VH}}}\\\
\sqrt{\kappa}e^{j\Phi_{n,m_{n}}^{\mathrm{HV}}}&e^{j\Phi_{n,m_{n}}^{\mathrm{HH}}}\end{bmatrix}\begin{bmatrix}F_{q,\mathrm{V}}^{R}(\mathbf{D}_{n,m_{n}}^{R}(t),\mathbf{A}_{q}^{R}(t))\\\
F_{q,\mathrm{H}}^{R}(\mathbf{D}_{n,m_{n}}^{R}(t),\mathbf{A}_{q}^{R}(t))\end{bmatrix}\times$
$\displaystyle\sqrt{P_{n,m_{n}}(t)}e^{j2\pi f_{qn,m_{n}}^{R}(t)t}e^{j2\pi
f_{pn,m_{n}}^{T}(t)t}$ (10)

where $\kappa$ is the cross polarization power ratio and $P_{n,m_{n}}$ is the
normalized mean power of the $m_{n}$th ray in $\mathrm{Cluster}_{n}$. The
normalized mean power of $\mathrm{Cluster}_{n}$ can be calculated as
$P_{n}=\sum\limits_{m_{n}}P_{n,m_{n}}$. Random phases
$\Phi_{n,m_{n}}^{\mathrm{VV}},\Phi_{n,m_{n}}^{\mathrm{VH}},\Phi_{n,m_{n}}^{\mathrm{HV}},\Phi_{n,m_{n}}^{\mathrm{HH}}$
are uniformly distributed within $(0,2\pi]$ [12].

Conversely, if $\mathrm{Cluster}_{n}$ is not observable, i.e.,
$\mathrm{Cluster}_{n}\notin S_{qp}(t)$, the complex channel gain
$h_{qp,n,m_{n}}(t)=0$. Accordingly, the Doppler frequencies at the receiver
and transmitter are calculated as

$\displaystyle
f_{qn,m_{n}}^{R}(t)=\frac{1}{\lambda}\frac{\left\langle\mathbf{D}_{qn,m_{n}}^{R}(t),\mathbf{v}^{R}-\mathbf{v}_{n}^{R}\right\rangle}{\left\|\mathbf{D}_{qn,m_{n}}^{R}(t)\right\|}$
(11) $\displaystyle
f_{pn,m_{n}}^{T}(t)=\frac{1}{\lambda}\frac{\left\langle\mathbf{D}_{pn,m_{n}}^{T}(t),\mathbf{v}^{T}-\mathbf{v}_{n}^{T}\right\rangle}{\left\|\mathbf{D}_{pn,m_{n}}^{T}(t)\right\|}.$
(12)

Moreover, the delay $\tau_{n}(t)$ of the NLOS component is computed as

$\displaystyle\tau_{n}(t)=\left[\left\|\mathbf{D}_{n}^{R}(t)\right\|+\left\|\mathbf{D}_{n}^{T}(t)\right\|\right]/c+\tilde{\tau}_{n}(t)$
(13)

where $\tilde{\tau}_{n}(t)$ is an exponentially distributed random variable
representing the virtual delay caused by the virtual link between the first
and last bounces of $\mathrm{Cluster}_{n}$ in the scattering environment. The
time evolution of virtual delays will be introduced in Section II-D.

###  II-B Array-Time Cluster Evolution for the General 3D 5G GBSM

The array-time cluster evolution for the proposed unified 5G GBSM framework is
developed based on the birth-death process and the algorithm described in
[29]. However, the proposed algorithm for the unified 5G GBSM framework in
Fig. 2 improves the one in [29] by including mean power evolution and updates
of rays within clusters. Let us assume the generation (birth) and
recombination (death) rates of clusters are $\lambda_{G}$ and $\lambda_{R}$,
respectively. Then, the array-time cluster evolution for the unified 5G GBSM
framework can be described as follows.

Figure 2: Flowchart of the array-time cluster evolution for the proposed
general 3D non-stationary 5G GBSM.

Step 1: An initial set of clusters are generated at time $t$. The generation
procedure of initial clusters will be described in Section II-C.

Step 2: At time $t+\Delta t$, cluster evolution on the time axis is operated.
In principle, each cluster should have its own survival probability according
to the relative motion. However, for simplicity, mean relative velocities of
clusters will be used to calculate survival probabilities of clusters. Mean
relative velocities $\Delta v^{R}$ and $\Delta v^{T}$ are characterized as
$\Delta
v^{R}=\mathrm{E}\left[\left\|\mathbf{v}^{R}-\mathbf{v}^{R}_{n}\right\|\right]$
and $\Delta
v^{T}=\mathrm{E}\left[\left\|\mathbf{v}^{T}-\mathbf{v}^{T}_{n}\right\|\right]$,
respectively. As a result, the survival probability $P_{\mathrm{T}}(\Delta t)$
of a cluster after $\Delta t$ is computed as

$P_{\mathrm{T}}(\Delta t)=e^{-\lambda_{R}\frac{P_{F}(\Delta v^{R}+\Delta
v^{T})\Delta t}{D^{s}_{c}}}$ (14)

where $P_{F}$ is the percentage of moving clusters and $D_{c}^{s}$ is a
scenario-dependent coefficient describing space correlation. Typical values of
$D_{c}^{s}$ such as $10$ m, $30$ m, $50$ m, and $100$ m can be chosen with the
same order of correlation distances in [21]. The survivability of each cluster
at time $t+\Delta t$ is determined by $P_{\mathrm{T}}(\Delta t)$. Geometrical
relationships, delays, and mean powers of survived clusters will be updated
according to Section II-D. Meanwhile, a random number of new clusters are
generated. This random number is generated according to a Poisson distribution
with mean

$\textrm{E}[N_{\textrm{new}}(t+\Delta
t)]={{\lambda_{G}}\over{\lambda_{R}}}(1-P_{\mathrm{T}}(\Delta t)).$ (15)

Rays and geometrical parameters will be assigned to these new clusters as
described in Section II-C, which also presents the array axis evolution for
clusters.

Step 3: When the array-time evolution is finished, the algorithm returns to
Step 2 to enter the next time instant.

###  II-C Generation of New Clusters

For a new cluster, say $\mathrm{Cluster}_{n}$, certain parameters such as the
number of rays within the cluster, virtual delay, mean power, angular
parameters, and relative delays of rays need to be assigned to this cluster.
These parameters are randomly generated according to the distributions listed
in Table II. The virtual delays $\tilde{\tau}_{n}$ of clusters are assumed to
be exponentially distributed, as in the WINNER II channel model [12], and can
be expressed by

$\displaystyle\tilde{\tau}_{n}=-r_{\tau}\sigma_{\tau}\cdot\mathrm{ln}u_{n}$
(16)

where $u_{n}$ is uniformly distributed within $(0,1)$, $r_{\tau}$ is the delay
scalar ($r_{\tau}=2.3$ for NLOS urban outdoor scenario and $r_{\tau}=2.4$ for
NLOS indoor office scenario [12]), and $\sigma_{\tau}$ is a randomly generated
delay spread ($\mathrm{E}\left[\log_{10}\sigma_{\tau}\right]=-6.63$ and
$\mathrm{std}\left[\log_{10}\sigma_{\tau}\right]=0.32$ for NLOS urban outdoor
scenario and $\mathrm{E}\left[\log_{10}\sigma_{\tau}\right]=-7.60$ and
$\mathrm{std}\left[\log_{10}\sigma_{\tau}\right]=0.19$ for NLOS indoor office
scenario [12]). The mean powers $\tilde{P}^{\prime}_{n}$ of clusters are
generated as [12]

$\displaystyle\tilde{P}^{\prime}_{n}=\exp\left(-\tilde{\tau}_{n}\frac{r_{\tau}-1}{r_{\tau}\sigma_{\tau}}\right)10^{-\frac{Z_{n}}{10}}$
(17)

where $Z_{n}$ follows a Gaussian distribution $\mathcal{N}(0,3)$ [12]. Unlike
the virtual delay and mean power of clusters, which are generated as in the
WINNER II channel model, generations of angular parameters, relative delays of
rays, and mean power of rays are not following the WINNER II channel model.
The angular parameters $\phi_{n}^{A}$, $\phi_{n}^{E}$, $\varphi_{n}^{A}$, and
$\varphi_{n}^{E}$ of $\mathrm{Cluster}_{n}$ are assumed to obey wrapped
Gaussian distributions. Angles of arrival (AoAs) of $\mathrm{Cluster}_{n}$ are
generated as

$\displaystyle\phi_{n}^{A}=\mathrm{std}\left[\phi_{n}^{A}\right]Y^{A}_{n}+\psi_{A}^{R}$
(18)
$\displaystyle\phi_{n}^{E}=\mathrm{std}\left[\phi_{n}^{E}\right]Y^{E}_{n}+\psi_{E}^{R}$
(19)

where $Y^{A}_{n},Y^{E}_{n}\sim\mathcal{N}(0,1)$,
$\mathrm{std}\left[\phi_{n}^{A}\right]$ and
$\mathrm{std}\left[\phi_{n}^{E}\right]$ are standard deviations of AoAs and
need to be estimated. The parameter estimation procedure is introduced in
Appendix C. The mean power generation method of a cluster is extended to
compute the mean power of rays within clusters as [54]

$\displaystyle\tilde{P}^{\prime}_{n,m_{n}}=\exp\left(-\tau_{m_{n}}\frac{r_{\tau}-1}{\mathrm{E}\left[\tau_{m_{n}}\right]}\right)10^{-\frac{Z_{n,m_{n}}}{10}}$
(20)

where $Z_{n,m_{n}}$ follows a Gaussian distribution $\mathcal{N}(0,3)$ [12].
The mean relative delay $\mathrm{E}\left[\tau_{m_{n}}\right]$ of rays of
$\mathrm{Cluster}_{n}$ will be given in Section IV. The mean power of rays is
then scaled by the cluster power as
$\tilde{P}_{n,m_{n}}=\tilde{P}^{\prime}_{n}\frac{\tilde{P}^{\prime}_{n,m_{n}}}{\sum\limits_{m_{n}}\tilde{P}^{\prime}_{n,m_{n}}}$.
Then, the angular parameters of $\mathrm{Cluster}_{n}$ via the $m_{n}$th ray
can be calculated by adding the angular offset of the ray, i.e.,

$\displaystyle\left[\phi_{n,m_{n}}^{A}\,\phi_{n,m_{n}}^{E}\,\varphi_{n,m_{n}}^{A}\,\varphi_{n,m_{n}}^{E}\right]^{\mathrm{T}}$
$\displaystyle=\left[\phi_{n}^{A}\,\phi_{n}^{E}\,\varphi_{n}^{A}\,\varphi_{n}^{E}\right]^{\mathrm{T}}+\left[\Delta\phi^{A}\,\Delta\phi^{E}\,\Delta\varphi^{A}\,\Delta\varphi^{E}\right]^{\mathrm{T}}$
(21)

where $\Delta\phi^{A}$, $\Delta\phi^{E}$, $\Delta\varphi^{A}$, and
$\Delta\varphi^{E}$ are angular offsets of the ray and are assumed to follow
Laplace distributions [12] with zero mean and standard deviation of $1$ degree
($0.017$ radian) for simplicity. The standard deviation of angular offsets can
be modified subject to measurements.

TABLE II: Distributions of Parameters of Clusters. Parameters | $M_{n}$ |  $\tilde{\tau}_{n}$, $\tau_{m_{n}}$, $D_{n}^{T},D_{n}^{R}$ |  $\phi_{n}^{A}$, $\phi_{n}^{E}$,$\varphi_{n}^{A}$, $\varphi_{n}^{E}$
---|---|---|---
Distributions | Poisson | Exponential | Wrapped Gaussian

Next, which antennas are able to observe the newly generated cluster should be
determined. To avoid repeated description, only the receiver side is
presented, the transmitter side follows the same procedure. First, the newly
generated cluster is added to the cluster set of a randomly selected receive
antenna $\mathrm{Ant}^{R}_{\tilde{q}}$. Second, we generate a 3D ball with
radius $r\sim\exp\left(\frac{\lambda_{R}}{D^{a}_{c}}\right)$ where $D^{a}_{c}$
is the scenario-dependent coefficient normalizing antenna spacings. Typical
values of $D^{a}_{c}$ such as $30$ m and $50$ m can be chosen with the same
order of correlation distances in [21]. Third, we compute the distances
between antennas $\mathrm{Ant}^{R}_{\tilde{q}}$ and $\mathrm{Ant}^{R}_{{q}}$
for all $q$. Then, we add the newly generated cluster to the cluster sets of
antennas satisfying $\|\mathbf{A}^{R}_{q}-\mathbf{A}^{R}_{\tilde{q}}\|\leq r$.
As a consequence, the probability that both $\mathrm{Ant}^{R}_{\tilde{q}}$ and
$\mathrm{Ant}^{R}_{{q}}$ are able to observe this newly generated cluster will
be
$\exp\left(\frac{\lambda_{R}}{D^{a}_{c}}\|\mathbf{A}^{R}_{q}-\mathbf{A}^{R}_{\tilde{q}}\|\right)$.
The psuedo codes of the new cluster generation algorithm are shown in Fig. 3.

1:Generate $M_{n}\sim\mathrm{Pois}\left(\tilde{\lambda}\right)$ rays for the
cluster;

2:Generate virtual delay/mean power/distance for the cluster;

3:Generate AoAs/Angles of departure (AoDs)/relative delays/relative mean
powers for the rays within the cluster;

4:Generate discrete $\tilde{q}\sim\mathrm{U}\left(1,M_{R}\right)$, generate
$r\sim\exp\left(\frac{\lambda_{R}}{D^{a}_{c}}\right)$, and let $\tilde{i}=1$;

5:while ($\tilde{i}\leq M_{R}$) do

6: if ($\|\mathbf{A}^{R}_{\tilde{i}}-\mathbf{A}^{R}_{\tilde{q}}\|\leq r$) then

7: Add the cluster to $C^{R}_{\tilde{i}}(t)$;

8: end if

9: $\tilde{i}=\tilde{i}+1$;

10:end while

Figure 3: Psuedo codes for the new cluster generation algorithm.

###  II-D Evolution of Survived Clusters

In order to highlight time evolution of the proposed model, geometrical
relationships, virtual delays, and mean powers of survived clusters need to be
updated from $t$ to $t+\Delta t$. To begin with, antenna position vectors are
updated as

$\displaystyle\mathbf{A}_{q}^{R}(t+\Delta
t)=\mathbf{A}_{q}^{R}(t)+\mathbf{v}^{R}\Delta t$ (22)
$\displaystyle\mathbf{A}_{p}^{T}(t+\Delta
t)=\mathbf{A}_{p}^{T}(t)+\mathbf{v}^{T}\Delta t.$ (23)

At the same time, distance vectors of clusters need to be adjusted as

$\displaystyle\mathbf{D}_{n}^{R}(t+\Delta
t)=\mathbf{D}_{n}^{R}(t)+\mathbf{v}_{n}^{R}\Delta t$ (24)
$\displaystyle\mathbf{D}_{n}^{T}(t+\Delta
t)=\mathbf{D}_{n}^{T}(t)+\mathbf{v}_{n}^{T}\Delta t.$ (25)

Other distance vectors in (1)–(7) can be updated accordingly. Delays are
updated as

$\displaystyle\tau_{n}(t+\Delta t)$
$\displaystyle=\left[\left\|\mathbf{D}_{n}^{R}(t+\Delta
t)\right\|+\left\|\mathbf{D}_{n}^{T}(t+\Delta
t)\right\|\right]/c+\tilde{\tau}_{n}(t+\Delta t).$ (26)

The random virtual delays $\tilde{\tau}_{n}(t+\Delta t)$ are modeled as
$\tilde{\tau}_{n}(t+\Delta t)=e^{-\frac{\Delta
t}{\varsigma}}\tilde{\tau}_{n}(t)+(1-e^{-\frac{\Delta t}{\varsigma}})X$ where
$X$ is a random variable independent to $\tilde{\tau}_{n}$ but identically
distributed as $\tilde{\tau}_{n}$ and $\varsigma$ is a scenario-dependent
parameter describing the coherence of virtual links. Typical values of the
coherence of virtual links such as $5$ s, $7$ s, and $30$ s are chosen based
on reasonable assumptions. Thus, the updated delay will carry information of
the delay in the previous time instant.

Another important aspect is the evolution of cluster mean power. Constant
cluster mean powers were assumed in [28] and [29], which were not sufficient
to characterize time evolution of the channel. Therefore, in this paper, with
the assumption that the cluster mean powers satisfy the inverse square law,
the time evolution of cluster mean power can be expressed as (derivations
given in Appendix B)

$\displaystyle\tilde{P}_{n,m_{n}}(t+\Delta
t)=\tilde{P}_{n,m_{n}}(t)\frac{3\tau_{n}(t)-2\tau_{n}(t+\Delta
t)+\tau_{m_{n}}}{\tau_{n}(t)+\tau_{m_{n}}}.$ (27)

The mean power terms $\tilde{P}_{n,m_{n}}$ in the mean power evolution in (27)
are not normalized. They need to be normalized such that
$P_{n,m_{n}}=\tilde{P}_{n,m_{n}}/\sum\limits_{n,m_{n}}\tilde{P}_{n,m_{n}}$
before being plugged into (10). To guarantee smooth power transitions when
clusters appear or disappear, a simple linear power scaling is performed
within 1 ms. That is to say, the power of a disappearing cluster will be
scaled linearly from its instant power to zero within 1 ms, and the power of a
newly generated cluster will be scaled from 0 to its power shown in (17). The
choice of a 1 ms transition period is aligned with the length of one subframe
in LTE [60], which is easier for system-level simulations with the proposed
channel model. Other non-linear transitions can be found in [46].

###  II-E Simplified Channel Models

The proposed general 3D non-stationary 5G GBSM can easily be reduced to
various simplified channel models by adjusting certain model parameters.

  1. 1.

By setting the numbers of antennas ($M_{R}$ and $M_{T}$) as relatively small
numbers, the spherical wavefront effect and cluster evolution on the array
axis will become insignificant. In this case, the general 5G massive MIMO
channel model is reduced to a conventional (non-massive) MIMO channel model.

  2. 2.

By setting the velocity of the transmitter $\mathbf{v}^{T}=\mathbf{0}$, the
general 5G V2V channel model is reduced to a fixed-to-mobile (F2M) channel
model.

  3. 3.

By setting the relative delays of rays as $0$, i.e., $\tau_{m_{n}}=0$, rays
within a cluster will become irresolvable in the delay domain. Consequently,
the general 5G mmWave channel model is simplified as a SCM-like wideband
channel model.

  4. 4.

By setting all elevation angles as zero, i.e.,
$\psi_{E}^{R}=\psi_{E}^{T}=\phi_{n}^{E}(t)=\varphi_{n}^{E}(t)=0$, the impacts
of elevation angles are ignored. Then, the 3D 5G channel model is simplified
to 2D.

By properly adjusting the parameters of the proposed general 5G channel model,
various simplified channel models can be obtained, such as 3D wideband massive
MIMO, 3D wideband HST conventional MIMO, 3D mmWave conventional MIMO, and 2D
wideband V2V conventional MIMO channel models.

##  III Statistical Property Analysis

###  III-A Time-Variant Power Delay Profile (PDP)

The time-variant PDP $\Lambda(t,\tau)$ of the channel can be expressed as [61]

$\displaystyle\Lambda(t,\tau)=\sum\limits_{n}^{N(t)}\sum\limits_{m_{n}}^{M_{n}}P_{n,m_{n}}(t)\delta\left(\tau-\tau_{n}(t)-\tau_{m_{n}}\right).$
(28)

It should be noted that the PDP is from all observable clusters at both the
transmit and receive arrays. The time-variant properties of PDP are caused by
the time-dependent mean powers and delays of rays. These are related to the
geometrical relationship updates of the scattering environment.

###  III-B Stationary Interval

The stationary interval is utilized in [44] to measure the estimated period
within which the channel amplitude response can be regarded as stationary. It
can be used to determine the frequency of channel estimation in HST
communications. The definition of the stationary interval is the maximum time
length within which the autocorrelation function (ACF) of the PDP exceeds the
$80\%$ threshold [44]. It should be noticed that the $80\%$ threshold in [44]
is empirical and can be adjusted according to requirements. Also, the
definition of stationary interval in [44] would fail to work if the ACF of the
PDP is not monotonically decreasing or has multiple crossing points at the
threshold. Therefore, an improved definition of the stationary interval $I(t)$
at time $t$ is proposed as $I(t)=\mathrm{inf}\left\\{\Delta
t|_{R_{\Lambda}(t,\Delta t)\leq 0.8}\right\\}$ where
$\mathrm{inf}\left\\{\cdot\right\\}$ calculates the infimum of a function and
$R_{\Lambda(t,\Delta t)}$ is the normalized ACF of the PDP defined by [44]

$\displaystyle R_{\Lambda}(t,\Delta
t)=\frac{\int\Lambda(t,\tau)\Lambda(t+\Delta
t,\tau)d\tau}{\max\left\\{\int\Lambda^{2}(t,\tau)d\tau,\int\Lambda^{2}(t+\Delta
t,\tau)d\tau\right\\}}.$ (29)

###  III-C Time-Variant Transfer Function

The time-variant transfer function $H_{qp}(\xi,t)$ is the Fourier transform of
the channel impulse response with respect to delay, which can be expressed as
(30),

$\displaystyle H_{qp}(\xi,t)$
$\displaystyle=\int\limits_{-\infty}^{\infty}h_{qp}(t,\tau)e^{-j2\pi\xi\tau}d\tau={\sqrt{\frac{K}{K+1}}h^{\mathrm{LOS}}_{qp}(t)e^{-j2\pi\xi\tau^{\mathrm{LOS}}(t)}}+{\sqrt{\frac{1}{K+1}}\sum\limits_{n=1}^{N(t)}\sum\limits_{m_{n}=1}^{M_{n}(t)}h_{qp,n,m_{n}}(t)e^{-j2\pi\xi\left[\tau_{n}(t)+\tau_{m_{n}}\right]}}$
(30)

where $\xi$ is frequency [61].

###  III-D Space-Time-Frequency Correlation Function

To study the correlation properties, the space-time-frequency correlation
function (STFCF)
$\rho_{qp,q^{\prime}p^{\prime}}(\|\mathbf{A}^{T}_{p}-\mathbf{A}^{T}_{{p^{\prime}}}\|,\|\mathbf{A}^{R}_{q}-\mathbf{A}^{R}_{{q^{\prime}}}\|,\Delta\xi,\Delta
t;\xi,t)$ can be calculated as [61]

$\displaystyle\rho_{qp,q^{\prime}p^{\prime}}(\|\mathbf{A}^{T}_{p}-\mathbf{A}^{T}_{{p^{\prime}}}\|,\|\mathbf{A}^{R}_{q}-\mathbf{A}^{R}_{{q^{\prime}}}\|,\Delta\xi,\Delta
t;\xi,t)$
$\displaystyle=\mathrm{E}\left[H^{*}_{qp}(\xi,t)H_{q^{\prime}p^{\prime}}(\xi+\Delta\xi,t+\Delta
t)\right].$ (31)

The LOS component is computed based on the relative position between the
transmitter and receiver. The parameters of the NLOS components are randomly
generated. Therefore, the LOS and NLOS components are assumed to be
uncorrelated for simplicity. Then, (31) can be written as the sum of the
correlation of the LOS component and the correlation of the NLOS components,
i.e.,

$\displaystyle\rho_{qp,q^{\prime}p^{\prime}}$
$\displaystyle(\|\mathbf{A}^{T}_{p}-\mathbf{A}^{T}_{{p^{\prime}}}\|,\|\mathbf{A}^{R}_{q}-\mathbf{A}^{R}_{{q^{\prime}}}\|,\Delta\xi,\Delta
t;\xi,t)$
$\displaystyle=\rho^{\mathrm{LOS}}_{qp,q^{\prime}p^{\prime}}(\|\mathbf{A}^{T}_{p}-\mathbf{A}^{T}_{{p^{\prime}}}\|,\|\mathbf{A}^{R}_{q}-\mathbf{A}^{R}_{{q^{\prime}}}\|,\Delta\xi,\Delta
t;\xi,t)$
$\displaystyle+\rho^{\mathrm{NLOS}}_{qp,q^{\prime}p^{\prime}}(\|\mathbf{A}^{T}_{p}-\mathbf{A}^{T}_{{p^{\prime}}}\|,\|\mathbf{A}^{R}_{q}-\mathbf{A}^{R}_{{q^{\prime}}}\|,\Delta\xi,\Delta
t;\xi,t).$ (32)

The correlation of the LOS component is calculated as

$\displaystyle\rho^{\mathrm{LOS}}_{qp,q^{\prime}p^{\prime}}($
$\displaystyle\|\mathbf{A}^{T}_{p}-\mathbf{A}^{T}_{{p^{\prime}}}\|,\|\mathbf{A}^{R}_{q}-\mathbf{A}^{R}_{{q^{\prime}}}\|,\Delta\xi,\Delta
t;\xi,t)$
$\displaystyle=\frac{K}{K+1}h^{\mathrm{LOS}*}_{qp}(t)h^{\mathrm{LOS}}_{q^{\prime}p^{\prime}}(t+\Delta
t)e^{j2\pi\sigma_{1}}$ (33)

with $\sigma_{1}=\xi\left[\tau^{\mathrm{LOS}}(t)-\tau^{\mathrm{LOS}}(t+\Delta
t)\right]+\Delta\xi\tau^{\mathrm{LOS}}(t+\Delta t).$ Similarly, the
correlation of the NLOS components is calculated as (34),

$\displaystyle\rho^{\mathrm{NLOS}}_{qp,q^{\prime}p^{\prime}}(\|\mathbf{A}^{T}_{p}-\mathbf{A}^{T}_{{p^{\prime}}}\|,\|\mathbf{A}^{R}_{q}-\mathbf{A}^{R}_{{q^{\prime}}}\|,\Delta\xi,\Delta
t;\xi,t)=\frac{1}{K+1}\mathrm{E}\left[\sum\limits_{n=1}^{N(t)}\sum\limits_{n^{\prime}=1}^{N(t+\Delta
t)}\sum\limits_{m_{n}=1}^{M_{n}}\sum\limits_{m_{n^{\prime}}=1}^{M_{n^{\prime}}}h_{qp,n,m_{n}}^{*}(t)h_{q^{\prime}p^{\prime},n^{\prime},m_{n^{\prime}}}(t+\Delta
t)e^{j2\pi\sigma_{2}}\right]$ (34)

with $\sigma_{2}=\xi(\tau_{n}(t)+\tau_{m_{n}}-\tau_{n^{\prime}}(t+\Delta
t)-\tau_{m_{n^{\prime}}})-\Delta\xi(\tau_{n^{\prime}}(t+\Delta
t)+\tau_{m_{n^{\prime}}}).$ Because cluster evolution is considered in the
proposed general 5G GBSM, the mean number of survived cluster shared by
$h_{qp,n,m_{n}}(t)$ and
$h_{q^{\prime}p^{\prime},n^{\prime},m_{n^{\prime}}}(t+\Delta t)$ can be
calculated as

$\displaystyle\mathrm{E}$
$\displaystyle\left\\{\textrm{card}\left(S_{qp}(t)\bigcap
S_{q^{\prime}p^{\prime}}(t+\Delta t)\right)\right\\}$
$\displaystyle=P_{\mathrm{survival}}\mathrm{E}\left\\{\textrm{card}\left(S_{qp}(t)\right)\right\\}$
(35)

where $P_{\mathrm{survival}}$ is the cluster survival probability when a
cluster evolves from $\mathrm{Ant}_{p}^{T}$, $\mathrm{Ant}_{q}^{R}$, and $t$
to $\mathrm{Ant}_{p^{\prime}}^{T}$, $\mathrm{Ant}_{q^{\prime}}^{R}$, and
$t+\Delta t$, respectively, i.e.,

$P_{\mathrm{survival}}=e^{-\lambda_{R}\left[\frac{\|\mathbf{A}^{T}_{p}-\mathbf{A}^{T}_{{p^{\prime}}}\|+\|\mathbf{A}^{R}_{q}-\mathbf{A}^{R}_{{q^{\prime}}}\|}{D^{a}_{c}}+\frac{P_{F}(\Delta
v^{R}+\Delta v^{T})\Delta t}{D^{s}_{c}}\right]}.$ (36)

Those newly generated clusters from $h_{qp,n,m_{n}}(t)$ to
$h_{q^{\prime}p^{\prime},n^{\prime},m_{n^{\prime}}}(t+\Delta t)$ are
independent to the survived clusters. Therefore, they do not contribute to the
correlation coefficient. Then, the STFCF for NLOS components in (34) reduces
to (37).

$\displaystyle\rho^{\mathrm{NLOS}}_{qp,q^{\prime}p^{\prime}}(\delta_{T},\delta_{R},\Delta\xi,\Delta
t;\xi,t)=\frac{P_{\mathrm{survival}}}{K+1}\mathrm{E}\left[\sum\limits_{n=1}^{N(t)}\sum\limits_{n^{\prime}=1}^{N(t)}\sum\limits_{m_{n}=1}^{M_{n}}\sum\limits_{m_{n^{\prime}}=1}^{M_{n^{\prime}}}h_{qp,n,m_{n}}^{*}(t)h_{q^{\prime}p^{\prime},n^{\prime},m_{n^{\prime}}}(t+\Delta
t)e^{j2\pi\sigma_{2}}\right]$ (37)

As the dimension of the STFCF is high, it is difficult to present the STFCF
visually. However, by setting $\Delta\xi=0$, $q=q^{\prime}$, and
$p=p^{\prime}$, the STFCF is reduced to the time-variant ACF. By setting
$\Delta t=0$, $\Delta\xi=0$ and $p=p^{\prime}$ ($q=q^{\prime}$), the STFCF is
reduced to the receive (transmit) space cross-correlation function (CCF).
Similarly, by setting $q=q^{\prime}$, $p=p^{\prime}$, and $\Delta t=0$, the
STFCF is reduced to the time-variant frequency correlation function (FCF).

##  IV Results and Analysis

In the simulations, we assume that the generation rate
$\lambda_{\mathrm{G}}=80/$m [31, Table I] and the recombination rate
$\lambda_{\mathrm{R}}=4/$m, so that the mean number of clusters is fixed as
$20$ [12]. The percentage of moving clusters is $P_{F}=0.3$ [31]. The
parameter estimation of the proposed 5G small-scale fading channel model is
based on the minimum mean square error (MMSE) criterion, i.e.,
$\epsilon=|\hat{\mathsf{f}}-\mathsf{f}(\mathsf{P})|^{2}$, where
$\hat{\mathsf{f}}$ is the measured statistical property such as space CCF and
complementary cumulative distribution function (CCDF) of coherence bandwidth,
$\mathsf{f}$ is the statistical property of the channel model, and
$\mathsf{P}$ is the parameter set of the channel model. Note that the
parameter set $\mathsf{P}$ for a specific statistical property is not
necessary to include all parameters of the channel model, since only a small
number of parameters will significantly affect this statistical property.
Besides the parameter set $\mathsf{P}$, the rest parameters are generated
randomly according to the WINNER II channel model [12]. Based on the MMSE
criterion, an exhaustive search of the parameter set $\mathsf{P}$ will be
performed using an optimization procedure until $\epsilon$ reaches a certain
threshold and then the parameter set will be obtained. Details of the
parameter estimation procedure are introduced in Appendix C. The cluster
parameters finally obtained for simulations are listed in Table III. Their
distributions can be referred to Table II.

TABLE III: Simulation parameters for different simplified 5G channel models. 3D wideband massive MIMO | 3D wideband HST conventional MIMO
---|---
Parameters | Mean | Standard deviation | Parameters | Mean | Standard deviation
$M_{n}$ | 20 | 0 | $M_{n}$ | 20 | 0
$\tau_{n}$ | 930 ns | 930 ns | $\tau_{n}$ | 930 ns | 930 ns
$\tau_{m_{n}}$ | 0 ns | 0 ns | $\tau_{m_{n}}$ | 0 ns | 0 ns
$\left(D^{R}_{n},D^{T}_{n}\right)$ |  $\left(25,30\right)$ m |  $\left(15,10\right)$ m | $\left(D^{R}_{n},D^{T}_{n}\right)$ |  $\left(25,30\right)$ m |  $\left(15,10\right)$ m
$\phi_{n}^{A}$ | 0.78 rad | 1.15 rad | $\phi_{n}^{A}$ | 0.78 rad | 0.90 rad
$\phi_{n}^{E}$ | 0.78 rad | 0.18 rad | $\phi_{n}^{E}$ | 0.78 rad | 0.18 rad
$\varphi_{n}^{A}$ | 1.05 rad | 0.54 rad | $\varphi_{n}^{A}$ | 1.05 rad | 0.54 rad
$\varphi_{n}^{E}$ | 0.78 rad | 0.11 rad | $\varphi_{n}^{E}$ | 0.78 rad | 0.11 rad
$D^{a}_{c}$ | 30 m | 0 m | $D^{a}_{c}$ | 50 m | 0 m
$D^{s}_{c}$ | 100 m | 0 m | $D^{s}_{c}$ | 100 m | 0 m
$\varsigma$ | 30 s | 0 s | $\varsigma$ | 7 s | 0 s
$\left(\psi^{R}_{A},\psi^{R}_{E}\right)$ |  $\left(\frac{\pi}{4},\frac{\pi}{4}\right)$ rad | $(0,0)$ |  $\left(\psi^{R}_{A},\psi^{R}_{E}\right)$ rad | $\left(\frac{\pi}{4},\frac{\pi}{4}\right)$ | $(0,0)$
$\left(\psi^{T}_{A},\psi^{T}_{E}\right)$ |  $\left(\frac{\pi}{3},\frac{\pi}{4}\right)$ rad | $(0,0)$ |  $\left(\psi^{T}_{A},\psi^{T}_{E}\right)$ rad | $\left(\frac{\pi}{3},\frac{\pi}{4}\right)$ | $(0,0)$
2D wideband V2V conventional MIMO | 3D mmWave conventional MIMO
Parameters | Mean | Standard deviation | Parameters | Mean | Standard deviation
$M_{n}$ | 20 | 0 | $M_{n}$ | 15 | 3.87
$\tau_{n}$ | 930 ns | 930 ns | $\tau_{n}$ | 305 ns | 305 ns
$\tau_{m_{n}}$ | 0 ns | 0 ns | $\tau_{m_{n}}$ | 3 ns | 3 ns
$\left(D^{R}_{n},D^{T}_{n}\right)$ |  $\left(25,30\right)$ m |  $\left(15,10\right)$ m | $\left(D^{R}_{n},D^{T}_{n}\right)$ |  $\left(5,5\right)$ m |  $\left(3,3\right)$ m
$\phi_{n}^{A}$ | 0.78 rad | 0.91 rad | $\phi_{n}^{A}$ | 0.78 rad | 0.91 rad
$\phi_{n}^{E}$ | 0 rad | 0 rad | $\phi_{n}^{E}$ | 0.78 rad | 0.18 rad
$\varphi_{n}^{A}$ | 1.04 rad | 0.53 rad | $\varphi_{n}^{A}$ | 1.04 rad | 0.53 rad
$\varphi_{n}^{E}$ | 0 rad | 0 rad | $\varphi_{n}^{E}$ | 0.78 rad | 0.11 rad
$D^{a}_{c}$ | 30 m | 0 m | $D^{a}_{c}$ | 30 m | 0 m
$D^{s}_{c}$ | 10 m | 0 m | $D^{s}_{c}$ | 100 m | 0 m
$\varsigma$ | 5 s | 0 s | $\varsigma$ | 7 s | 0 s
$\left(\psi^{R}_{A},\psi^{R}_{E}\right)$ |  $\left(\frac{\pi}{4},\frac{\pi}{4}\right)$ rad | $(0,0)$ | $\left(\psi^{R}_{A},\psi^{R}_{E}\right)$ |  $\left(\frac{\pi}{4},\frac{\pi}{4}\right)$ rad | $(0,0)$
$\left(\psi^{T}_{A},\psi^{T}_{E}\right)$ |  $\left(\frac{\pi}{3},\frac{\pi}{4}\right)$ rad | $(0,0)$ | $\left(\psi^{T}_{A},\psi^{T}_{E}\right)$ |  $\left(\frac{\pi}{3},\frac{\pi}{4}\right)$ rad | $(0,0)$

To validate the correctness of the proposed general 5G GBSM, the analytical
ACFs of the simulation model and corresponding simulation results of the
wideband conventional MIMO channel model for both $\mathrm{Cluster}_{1}$ and
$\mathrm{Cluster}_{2}$ are compared in Fig. 4. Half-wavelength linear arrays
are assumed at both the transmitter and receiver. It should be noted that
these ACFs are normalized with respect to $\mathrm{Cluster}_{1}$. It can be
seen from Fig. 4 that the simulated ACFs align well with the analytical ACFs
of the simulation model for both $\mathrm{Cluster}_{1}$ and
$\mathrm{Cluster}_{2}$.

Figure 4: Comparison between the normalized analytical and simulated ACFs of
$\mathrm{Cluster}_{1}$ and $\mathrm{Cluster}_{2}$ of the wideband conventional
MIMO channel model ($f_{c}=2$ GHz, $\|\mathbf{D}\|=200$ m, $M_{R}=M_{T}=2$,
$\Delta v^{R}=\Delta v^{T}=0$ m/s, $|\mathbf{v}^{T}|=0$,$|\mathbf{v}^{R}|=5$
m/s, $D_{c}^{a}=50$ m, $D_{c}^{s}=100$ m, $M_{1}=M_{2}=81$, $\varsigma=7$ s,
NLOS).

The receiver normalized space CCF of the 3D wideband massive MIMO channel
model (simulation) in Table III, the measured averaged space CCF in the NLOS
scenario in [24, Fig. 10], and the space CCF of the WINNER II channel model
are compared in Fig. 5. The horizontal axis of Fig. 5 has been normalized with
respect to half wavelength. The measurement in [24] was performed in an
outdoor environment with a 128-element polarized virtual linear array covering
both LOS and NLOS scenarios. Each polarized antenna pair was also separated by
half wavelength. When the antenna index difference is larger than $2$, the
correlation coefficients drop to a relatively low level and different sub-
channels can be considered as uncorrelated. In this case, the fitting is not
important anymore. It is clear that the space CCF of the proposed massive MIMO
channel model aligns well with the measured data when the antenna index
difference is less than $3$. However, the WINNER II channel model
overestimates antenna correlations because cluster evolution was not
considered on the array axis.

Figure 5: The receiver normalized space CCFs of the 3D wideband massive MIMO
channel model (simulations), measurement in [24], and WINNER II channel model
($f_{c}=2.6$ GHz [24], $\|\mathbf{D}\|=200$ m, $M_{R}=M_{T}=32$, $\Delta
v^{R}=\Delta v^{T}=0$ m/s [24], $|\mathbf{v}^{T}|=|\mathbf{v}^{R}|=0$ [24],
$D_{c}^{a}=30$ m, $D_{c}^{s}=100$ m, $\kappa=-8$ dB, polarized antennas,
NLOS).

The CCDFs of the stationary intervals of the 3D wideband HST conventional MIMO
channel model (with velocity 60 m/s) in Table III, the measurement in [44,
Fig. 4(a)] with an omnidirectional antenna, and the WINNER II channel model
are shown in Fig. 6. The median of the stationary interval is approximately
$40$ ms. Clearly, the proposed HST channel model can fit the measurement
result very well. The WINNER II channel model has a much higher stationary
interval due to the fact that it does not have cluster power evolution in the
time domain.

Figure 6: The CCDFs of the stationary intervals for the 3D wideband HST
conventional MIMO channel model (simulation), measurement in [44], and WINNER
II channel model ($f_{c}=932$ MHz [44], $\|\mathbf{D}\|=200$ m,
$M_{R}=M_{T}=2$, $\Delta v^{R}=\Delta v^{T}=0.5$ m/s, $\|\mathbf{v}^{R}\|=60$
m/s [44], $\|\mathbf{v}^{T}\|=0$ m/s [44], $D_{c}^{a}=50$ m, $D_{c}^{s}=100$
m, $\varsigma=7$ s, NLOS).

The $90$% coherence bandwidth measures the bandwidth in which the channel can
be regarded as flat. The cumulative distribution function (CDF) of $90$%
coherence bandwidth of the 2D wideband V2V conventional MIMO channel model in
Table III is illustrated in Fig. 7, which is also validated by that of the
measured suburban V2V channel in [63, Fig. 5(a)]. The measurement in [63] was
performed in a suburban scenario with both transmitter and receiver moving in
the same direction. The $90$% coherence bandwidth of the WINNER II channel
model has smaller spread because its PDP is not time variant.

Figure 7: The CDFs of $90$% coherence bandwidths of the 2D wideband V2V
conventional MIMO channel model, measurement (suburband scenario) in [63], and
WINNER II channel model ($f_{c}=5.9$GHz [63], $\|\mathbf{D}\|=400$m, $\Delta
v^{R}=\Delta v^{T}=0.5$m/s, $\|\mathbf{v}^{T}\|=\|\mathbf{v}^{R}\|=25$m/s,
$D_{c}^{a}=30$m, $D_{c}^{s}=10$m, $\varsigma=5$s, NLOS).

A snapshot of the simulated normalized angular power spectrum (APS) at the
receiver with a half-wavelength linear array of the 2D mmWave massive MIMO
channel model is illustrated in Fig. 8. The APS is estimated with the smooth
multiple signal classification (MUSIC) algorithm [62][65] using a sliding
window of $3$ consecutive antennas. With these sliding windows, the smooth
MUSIC algorithm is able to resolve more AoAs than the conventional MUSIC
algorithm. It can be observed that the number of clusters is small because of
the high carrier frequency of mmWave. Meanwhile, appearance and disappearance
of clusters on the array axis can also be seen due to the massive MIMO antenna
array.

Figure 8: A snapshot of the simulated normalized APS at the receiver of the 2D
mmWave massive MIMO channel model ($f_{c}=58$ GHz, $M_{T}=2$, $M_{R}=32$,
$\|\mathbf{D}\|=6$ m, $\Delta v^{R}=\Delta v^{T}=0$ m/s,
$|\mathbf{v}^{T}|=|\mathbf{v}^{R}|=0$, $D_{c}^{a}=30$ m, $D_{c}^{s}=100$ m,
NLOS).

Fig. 9 presents the CCDFs of the root mean square (RMS) delay spreads of the
3D mmWave conventional MIMO channel model in Table III, the measurements in
[64, Fig. 5], and the WINNER II channel model. The measurements in [64] were
performed in indoor scenarios with omnidirectional antennas in azimuth.
Scenario 1 and Scenario 2 are the Room H scenario and Room F scenario in [64,
Fig. 5], respectively. It is clear that measurement results for different
scenarios in [64, Fig. 5] can be fitted properly by the proposed 3D mmWave
channel model. The WINNER II channel model does not support mmWave small-scale
channel characteristics well as it overestimates the RMS delay spread and has
smaller variations. The RMS delay spread of the mmWave channel model is
between $20$ ns and $50$ ns.

Figure 9: The CCDFs of RMS delay spreads of the 3D mmWave conventional MIMO
channel model, measurement in [64], and WINNER II channel model ($f_{c}=58$
GHz [64], $\|\mathbf{D}\|=6$ m, $\Delta v^{R}=\Delta v^{T}=0$ m/s,
$\|\mathbf{v}^{T}\|=\|\mathbf{v}^{R}\|=0$ m/s, $D_{c}^{a}=30$ m,
$D_{c}^{s}=100$ m, $\varsigma=7$ s, NLOS).

##  V Conclusions

In this paper, a unified framework for 5G wireless small scale fading channel
models has been proposed, based on the WINNER II and SV channel models. The
proposed general 3D non-stationary 5G channel model can model massive MIMO,
V2V, HST, and mmWave communication scenarios, as well as considering time
evolution feature of channels and arbitrary antenna array layouts. The
modeling of time evolution of channels includes cluster evolution on the time
axis, geometrical relationship updates, and evolution of delays and powers of
rays. It has been shown that the simulated statistical properties of the
proposed general 3D non-stationary 5G GBSM can fit well the corresponding
measurements. The proposed general 5G GBSM can be reduced to various
simplified channel models by properly setting certain parameters. For future
work, applications of the proposed channel model to 5G system simulators and
parameter estimations of the proposed 5G channel model from more channel
measurements need to be considered. Also, the general 5G GBSM can be further
extended by considering cooperative MIMO channel model with multi-link
correlations, HST channel model with time-variant Rician factor, HST channel
model in tunnel scenarios, time-variant antenna pattern, and the Rician factor
evolution in both time and array domains in massive MIMO scenarios.

##  Appendix A Calculation of Antenna Patterns in (9)

In a 3D space, the geometry of antenna pattern depends on the orientation of
antennas. The LCS is obtained by a sequence of rotations from the GCS [14].
First, a rotation of $\alpha^{T}$ about $x_{\mathrm{G}}$ axis is operated.
Second, a rotation of $\beta^{T}$ about the new $y_{\mathrm{G}}$ axis is
operated. Third, a rotation of $\gamma^{T}$ about the new $z_{\mathrm{G}}$
axis is operated. These three operations can be expressed as (38) [14].

$\displaystyle\mathbf{R}=\begin{bmatrix}\cos\gamma^{T}&-\sin\gamma^{T}&0\\\
\sin\gamma^{T}&\cos\gamma^{T}&0\\\
0&0&1\end{bmatrix}\begin{bmatrix}\cos\beta^{T}&0&\sin\beta^{T}\\\ 0&1&0\\\
-\sin\beta^{T}&0&\cos\beta^{T}\end{bmatrix}\begin{bmatrix}1&0&0\\\
0&\cos\alpha^{T}&-\sin\alpha^{T}\\\
0&\sin\alpha^{T}&\cos\alpha^{T}\end{bmatrix}.$ (38)

Let $\mathbf{a}$ and $\mathbf{b}$ be positions vectors in the GCS, and let
$\left(\tilde{x},\tilde{y},\tilde{z}\right)^{\mathrm{T}}$ be the coordinates
of $\mathbf{a}-\mathbf{b}$ in the LCS. Then,
$\left(\tilde{x},\tilde{y},\tilde{z}\right)^{\mathrm{T}}=\mathbf{R}(\mathbf{a}-\mathbf{b})$.
Let $w(x,y)$ be the four-quadrant inverse tangent function [66] of $x$ and
$y$, and let $\tilde{\theta}=w(\tilde{y},\tilde{x})$ and
$\tilde{\phi}=w(\tilde{z},\sqrt{\tilde{x}^{2}+\tilde{y}^{2}})$. Then, the
antenna patterns in (9) can be computed as
$F_{H}(\mathbf{a},\mathbf{b})=G(\tilde{\theta},\tilde{\phi})\cos\tilde{\theta}$
and
$F_{V}(\mathbf{a},\mathbf{b})=G(\tilde{\theta},\tilde{\phi})\sin\tilde{\theta}$.
In this paper, dipole antennas are assumed at the transmitter side. In this
case [67],
$G(\tilde{\theta},\tilde{\phi})=\sqrt{1.64}\frac{\cos\left(\frac{\pi}{2}\cos\tilde{\phi}\right)}{\sin\tilde{\phi}}.$
At the receiver side, calculation follows a similar procedure. Omnidirectional
antennas are assumed at the receiver side. In this case,
$G(\tilde{\theta},\tilde{\phi})=1$. Both antenna patterns can be replaced by
the actual antennas used. The rotation angles $\alpha^{T}$, $\alpha^{R}$,
$\beta^{T}$, $\beta^{R}$, $\gamma^{T}$, and $\gamma^{R}$ are all set as
$\frac{\pi}{15}$ for simplicity, which can be modified according to realistic
settings.

##  Appendix B Calculation of (27)

Let us consider the time interval between $t$ and $t+\Delta t$, where $\Delta
t$ is small. Then, the ray mean power difference
$\Delta\tilde{P}_{n,m_{n}}(t)$ between these two time instants is computed as

$\displaystyle\Delta\tilde{P}_{n,m_{n}}(t)=\tilde{P}_{n,m_{n}}(t+\Delta
t)-\tilde{P}_{n,m_{n}}(t).$ (39)

With the assumption of the inverse power law, the ray mean power is inversely
proportional to the $\eta$th ($\eta>1$) power of travel distance, i.e.,
$\tilde{P}_{n,m_{n}}(t)=\frac{\mathcal{C}}{\left[\tau_{n}(t)+\tau_{n,m_{n}}\right]^{\eta}c^{\eta}}$,
where $\mathcal{C}$ is a constant. The derivative of $\tilde{P}_{n,m_{n}}(t)$
with respect to $t$ is obtained by

$\displaystyle\frac{\Delta\tilde{P}_{n,m_{n}}(t)}{\Delta t}$
$\displaystyle=\left(\frac{\mathcal{C}}{\left[\tau_{n}(t)+\tau_{n,m_{n}}\right]^{\eta}c^{\eta}}\right)^{\prime}$
$\displaystyle=-\eta\frac{\mathcal{C}}{\left[\tau_{n}(t)+\tau_{n,m_{n}}\right]^{\eta+1}c^{\eta}}\left(\tau_{n}(t)\right)^{\prime}$
$\displaystyle=-\eta\frac{\mathcal{C}}{\left[\tau_{n}(t)+\tau_{n,m_{n}}\right]^{\eta+1}c^{\eta}}\frac{\left[\tau_{n}(t+\Delta
t)-\tau_{n}(t)\right]}{\Delta t}.$ (40)

Thus, the ray mean power evolution in terms of time can be derived as

$\displaystyle\frac{\Delta\tilde{P}_{n,m_{n}}(t)}{\tilde{P}_{n,m_{n}}(t)}=\frac{-\eta}{\tau_{n}(t)+\tau_{n,m_{n}}}\frac{\left[\tau_{n}(t+\Delta
t)-\tau_{n}(t)\right]}{\Delta t}\Delta t.$ (41)

It follows that

$\displaystyle\tilde{P}_{n,m_{n}}(t+\Delta
t)=\tilde{P}_{n,m_{n}}(t)\frac{(\eta+1)\tau_{n}(t)-\eta\tau_{n}(t+\Delta
t)+\tau_{n,m_{n}}}{\tau_{n}(t)+\tau_{n,m_{n}}}.$ (42)

For simplicity, in this paper we use $\eta=2$ following the inverse square
law. Then, (27) is obtained.

##  Appendix C Parameter Estimation Procedure of the Proposed Model

A typical parameter estimation method can be found in [68]. This paper follows
a similar procedure as in [68], i.e., estimating parameters minimizing the
difference in statistical properties between the model and measurement. The
parameter estimation procedure in this paper directly estimates parameters to
fit the statistic properties of the channel. The parameters of the proposed
model can be divided into three categories, i.e., parameters determined via
reasonable assumptions, parameters determined via parameter estimation
methods, and parameters randomly generated. The categories of parameters are
listed in Table IV. Let $\mathsf{P}$ be the set of parameters to be estimated,
i.e.,
$\mathsf{P}=\left\\{\mathrm{std}\left[\phi_{n}^{A}\right],\mathrm{std}\left[\phi_{n}^{E}\right],\mathrm{std}\left[\varphi_{n}^{A}\right],\mathrm{std}\left[\varphi_{n}^{E}\right],D^{a}_{c},D^{s}_{c},\varsigma\right\\}$
and ${\mathsf{f}}(\mathsf{P})$ be a statistical property with channel impulse
responses generated using $\mathsf{P}$. The estimated parameters
$\hat{\mathsf{P}}$ can be obtained via optimization methods, e.g., exhaustive
search, by minimizing
$\hat{\mathsf{P}}=\arg\min_{\mathsf{P}}|\hat{{\mathsf{f}}}-\mathsf{f}(\mathsf{P})|^{2}$
where $\hat{\mathsf{f}}$ is the measured statistical property. This can be
summarized into a number of steps.

  1. Step 1:

Define target threshold $\epsilon_{\mathrm{T}}$ and initialize parameters
$\hat{\mathsf{P}}$.

  2. Step 2:

Generate channel coefficients with parameters $\hat{\mathsf{P}}$.

  3. Step 3:

Calculate the statistical property $\mathsf{f}(\hat{\mathsf{P}})$ using the
generated channel coefficients.

  4. Step 4:

Compute the error between the model and measurement
$\epsilon=|\hat{\mathsf{f}}-\mathsf{f}(\hat{\mathsf{P}})|^{2}$.

  5. Step 5:

If $\epsilon\leqslant\epsilon_{\mathrm{T}}$, output $\hat{\mathsf{P}}$;
Otherwise, generate a new set of $\hat{\mathsf{P}}$ and go to Step 2.

TABLE IV: Details of parameter estimation. Parameters | Mean | Standard deviation
---|---|---
$M_{n}$ | Determined via assumption | 0
$\tau_{n}$ | Randomly generated via WINNER II | Randomly generated via WINNER II
$\tau_{m_{n}}$, $D_{n}^{T}$, $D_{n}^{R}$ | Determined via assumption | Determined via assumption
$\phi_{n}^{A}$, $\phi_{n}^{E}$, $\varphi_{n}^{A}$, $\varphi_{n}^{E}$ | Randomly generated via WINNER II | Determined via parameter estimation
$D^{a}_{c}$ | Determined via parameter estimation | 0 m
$D^{s}_{c}$ | Determined via parameter estimation | 0 m
$\varsigma$ | Determined via parameter estimations | 0 s

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|  Shangbin Wu received his B.S. degree incommunication engineering from South
China Normal University, Guangzhou, China, in 2009, M.Sc. degree in wireless
communications with distinction from University of Southampton, Southampton,
UK, in 2010, and Ph.D. degree in electrical engineering from Heriot-Watt
University, Edinburgh, UK in 2015. From 2010 to 2011, he worked as a LTE R&D
engineer responsible for LTE standardization and system level simulation in
New Postcom Equipment Ltd., Guangzhou, China. From October 2011 to August
2012, he was with Nokia Siemens Network, where he worked as a LTE algorithm
specialist, mainly focusing on LTE radio resource management algorithm design
and system level simulations. He has been with Samsung R&D Institute UK as a
5G researcher since November 2015.
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|  Cheng-Xiang Wang (S’01-M’05-SM’08-F’17) received the BSc and MEng degrees
in Communication and Information Systems from Shandong University, China, in
1997 and 2000, respectively, and the PhD degree in Wireless Communications
from Aalborg University, Denmark, in 2004. He was a Research Fellow with the
University of Agder, Grimstad, Norway, from 2001 to 2005, a Visiting
Researcher with Siemens AG Mobile Phones, Munich, Germany, in 2004, and a
Research Assistant with the Hamburg University of Technology, Hamburg,
Germany, from 2000 to 2001. He has been with Heriot-Watt University,
Edinburgh, U.K., since 2005, where he was promoted to a Professor in 2011. He
is also an Honorary Fellow of the University of Edinburgh, U.K., and a
Chair/Guest Professor of Shandong University and Southeast University, China.
He has authored 2 books, one book chapter, and over 300 papers in refereed
journals and conference proceedings. His current research interests include
wireless channel modeling and (B)5G wireless communication networks, including
green communications, cognitive radio networks, high mobility communication
networks, massive MIMO, millimetre wave communications, and visible light
communications. Prof. Wang is a Fellow of the IET and HEA, and a member of the
EPSRC Peer Review College. He served or is currently serving as an Editor for
nine international journals, including the IEEE TRANSACTIONS ON VEHICULAR
TECHNOLOGY since 2011, the IEEE TRANSACTIONS ON COMMUNICATIONS since 2015, and
the IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS from 2007 to 2009. He was the
leading Guest Editor of the IEEE JOURNAL ON SELECTED AREAS IN COMMUNICATIONS,
Special Issue on Vehicular Communications and Networks. He is also a Guest
Editor of the IEEE JOURNAL ON SELECTED AREAS IN COMMUNICATIONS, Special Issue
on Spectrum and Energy Efficient Design of Wireless Communication Networks and
Special Issue on Airborne Communication Networks, and the IEEE TRANSACTIONS ON
BIG DATA, Special Issue on Wireless Big Data. He served or is serving as a TPC
Member, TPC Chair, and General Chair of over 80 international conferences. He
received nine Best Paper Awards from the IEEE Globecom 2010, the IEEE ICCT
2011, ITST 2012, the IEEE VTC 2013, IWCMC 2015, IWCMC 2016, the IEEE/CIC ICCC
2016, and the WPMC 2016. He is recognized as Web of Science 2017 Highly Cited
Researcher.
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|  el-Hadi M. Aggoune (M’83-SM’93) received the M.S. and Ph.D. degrees in
electrical engineering from the University of Washington (UW), Seattle, WA,
USA. He taught graduate and undergraduate courses in electrical engineering at
many universities in the United States and abroad. He served at many academic
ranks including Endowed Chair Professor. He is listed as Inventor in two
patents assigned to the Boeing Company, USA and the Sensor Networks and
Cellular Systems (SNCS) Research Center, University of Tabuk, Saudi Arabia.
His research is referred to in many patents, including patents assigned to
ABB, Switzerland and EPRI, USA. He authored many papers in IEEE and other
journals and conferences. Dr. Aggoune is serving on many technical committees
for conferences worldwide as well as reviewer for many journals. One of his
Labs won the Boeing Supplier Excellence Award. Dr. Aggoune won the IEEE
Professor of the Year Award, UW Branch. He is a Professional Engineer
registered in the State of Washington. He is currently serving as Professor
and the Director of SNCS Research Center, University of Tabuk. His research
interests include wireless sensor networks, energy systems, and scientific
visualization.
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|  Mohammed M. Alwakeel (SM’14) was born in Tabuk, Saudi Arabia. He received
his B.S. and M. S Degrees from King Saud University, Riyadh, Saudi Arabia, and
his Ph.D. Degree in Electrical Engineering from Florida Atlantic University,
Boca Raton, Florida. He served as Communications Network Manager at the Saudi
National Information Center in Riyadh. He served as faculty member at King
Abdulaziz University and then Associate Professor and Dean of the Computers
and Information Technology College at the University of Tabuk, Tabuk, Saudi
Arabia. After that, he was a full professor at Computers and Information
Technology College, and the Vice Rector for Development and Quality at the
University of Tabuk. Currently, he is a member of Alshura Council (The
Consultative Council of the Kingdom of Saudi Arabia). His current research
interests include teletraffic analysis, mobile satellite communications, and
sensor networks and cellular systems.
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|  Xiaohu You (F’12) received his Ph.D. Degrees from Southeast University,
Nanjing, China, in Electrical Engineering in 1988. Since 1990, he has been
working with National Mobile Communications Research Laboratory at Southeast
University, where he is currently a professor and the director. He has
contributed over 150 IEEE journal papers in the areas of signal processing and
wireless communications. From 1998 to 2016, he was the Principal Expert of the
3G, 4G and 5G Research Projects under China National 863 High-Tech Program,
respectively. Professor You served as the general chairs of IEEE WCNC 2013 and
IEEE VTC 2016. He was the recipient of the China National 1st Class Invention
Prize in 2011, and was selected as IEEE Fellow in same year.
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