A Three-Dimensional Fully Polarized Millimeter-Wave Hybrid Propagation Channel Model for Urban Microcellular Environments

Abstract

Millimeter-wave channel modeling is the basis of fifth-generation (5G) communication network design and applications. In urban microcellular environments, the roughness of wall surfaces can be comparable to the wavelengths of millimeter waves, resulting in walls that cannot be considered as smooth surfaces. Therefore, channel modeling methods based on only traditional three-dimensional ray tracing (RT) or the three-dimensional parabolic equation (PE) result in the limited computational accuracy of millimeter-wave channel models for urban environments. Based on the scattering theory of a rough surface and the typical scattering characteristics of a millimeter wave, the end field of the three-dimensional vector PE is regarded as the initial field of three-dimensional RT. Moreover, the number of scattered rays and scattering angles are introduced. Finally, a three-dimensional fully polarized millimeter-wave hybrid propagation channel model (3DFPHPCM) is proposed. The proposed model improves the computational accuracy of a single deterministic model. Millimeter-wave channel measurements in non-line-of-sight (NLOS) environments were carried out to verify and optimize the proposed 3DFPHPCM. The results show that the root mean square error (RMSE) and mean absolute error (MAE) of the proposed 3DFPHPCM are both minimized when compared to three-dimensional RT or the three-dimensional PE, which indicates that the proposed 3DFPHPCM has higher computational accuracy. Moreover, its runtime is the shortest among the methods. The results presented herein provide technical support for the layout of base stations.

1. Introduction

The millimeter-wave band generally covers 30 to 300 GHz. Since the introduction of fourth-generation (4G) mobile communication, frequencies below 30 GHz have been occupied by military and civilian applications, leaving few frequency resources available. To meet the heightened demand for mobile users and data services driven by the rapid development of mobile Internet and the Internet of Things, the millimeter-wave band has been applied in 5G mobile communications [1,2,3]. Compared to ultrashort waves and microwaves, millimeter waves [4,5] offer faster data transmission speeds, larger transmission bandwidths, and delays on the order of a millisecond, all of which have become advantages and technical bases for 5G millimeter waves. However, the characteristics of these high-frequency millimeter waves also introduce significant challenges for 5G technologies.

Wireless communication signals, especially mobile signals, are essential for cities to maintain normal and efficient operations. Urban microcellular environments are the basic application scenarios for transmitting and receiving wireless communication signals. In the millimeter-wave band, the distribution density and roughness of buildings in urban microcellular environments have a significant impact on the distribution of wireless communication signals in cities. The propagation characteristics and models of wireless channels are prerequisites for technology evaluation, equipment development, and the system design and network deployment of mobile communication. The propagation mechanism of the millimeter-wave band differs from that of the lower frequency band. As the millimeter-wave wavelength becomes shorter, the diffraction ability of the radio wave is reduced, and it is vulnerable to extreme weather conditions such as interference from the atmosphere, rainfall, snowfall, and fog, as well as other obstacles. However, the scattering effect is more pronounced in the millimeter wave band (as seen in [6]). Therefore, conducting research on channel modeling methods based on scattering effects plays an important role in the development of 5G technology. In general, there exist two methods for channel modeling. One is the statistical model [7,8,9] based on experimental data, which offers the advantages of easy and fast calculation but is hindered by poor prediction accuracy. Experiments must be conducted for various scenarios, and a large amount of experimental data must be acquired at an early stage. The other method is the deterministic model based on theory, which offers the advantages of high accuracy across a wide variety of application scenarios but is hindered by long calculation times and poor efficiency. Considering the very complex and diverse application scenarios of 5G, it takes considerable human and material resources to carry out complete statistical modeling. Therefore, conducting research on deterministic radio wave propagation methods in the millimeter wave is the key to performing multi-scene 5G channel modeling.

At present, the main deterministic models are the PE and RT [10,11,12]. RT can clearly portray the propagation trajectory of electric waves and provide a powerful and intuitive tool for the study of propagation characteristics. The PE can portray the spatial distribution of the field of electromagnetic wave propagation, which facilitates the reasonable planning of system design and network layout according to the field distribution.

Both methods offer distinct advantages in the study of electric wave propagation and have received considerable attention. There have been a number of studies on the use of the three-dimensional PE [13,14,15,16] to calculate the propagation characteristics of electromagnetic waves through buildings. The characteristics of the field distribution through a single building were analyzed by Levy using a scalar three-dimensional PE model in 1992 [17]. In 1996, Zaporozhets and Levy introduced the finite difference (FD) method to the solution of the three-dimensional PE. However, it is more difficult to apply this method in real-world scenarios with multiple buildings [18]. In 1999, Zelley et al. proposed the topographic boundary conditions of the three-dimensional PE and calculated the PE using the FD method [19]. In 1999, Zaporozhets proposed a strict three-dimensional vector PE model based on the FD method to solve the problems in the calculation of bistatic RCS for electric large-scale targets and urban wave propagation [20]. In 2003, Janaswamy elaborated the split-step Fourier transform (SSFT)-based algorithm for solving simple urban radio wave propagation characteristics using the three-dimensional PE in a single lossy medium for the first time [21]. Huibin et al. used the scalar three-dimensional PE method to calculate the wave propagation characteristics of single and multirow buildings based on the SSFT on flat terrain in 2006 [22]. In 2017, El Ahdab et al. considered the bidirectional propagation of the three-dimensional vector PE on flat terrain with peaks and irregular terrain comprising hills [23]. In 2018, Rasool et al. applied a 3DPE method to calculate the propagation factor over the flat terrain [24]. In 2019, Rasool et al. used a 3DPE method to calculate the field strength due to the forest on a lossy ground [25]. In 2022, Wang Zhiyi et al. proposed a scheme of ground clutter simulation based on the 3DPE model in simple urban environments [14]. However, only simple urban environments and terrain are computed using three-dimensional PE.

A number of studies have calculated the scattering effects of buildings. In 1999, Degli-Esposti proposed an effective roughness model to show that scattering effects are important for both received power and channel dispersion in various situations [26]. In 2001, Degli-Esposti [27] proposed a method based on the Lambertian model to simulate the scattering of buildings in urban wireless propagation environments. In 2007, Degli-Esposti et al. [28] conducted experiments which were compared with RT, specifically, a fusion of the Lambertian or directive models. The results showed that the simulations agree well with the measurements when a suitable directive model is used. In 2011, Mani et al. [29] examined the prediction accuracy of an advanced deterministic model for the depolarization of the channel and frequency selectivity in an indoor wireless propagation channel. Their results showed that RT, which considers the directive pattern model, significantly improved the prediction accuracy of path loss and cross-polarization discrimination, as well as the prediction of delay spread. In 2012, Mani et al. [30] compared the RT method including the directive scattering pattern model with experimental data and showed that upon including diffuse scattering, it was able to reproduce the angular spread of dense multipaths with an error of approximately 4° to 20°. Later, Vitucci et al. [31] added a Lambertian model to the RT method, which was compared with the post-processed data, and their results showed that diffuse scattering is a key component of the received power. In 2014, Mani et al. [32] developed a polarized diffuse scattering model based on the effective roughness model. In 2015, Pascual-Garcia et al. [33] proposed that choosing the appropriate parameter values for the effective roughness model could improve the accuracy of evaluating the power delay distribution at 60 GHz. In 2017, Navarro et al. [34] added a Lambertian model to the RT method to calculate the total electric field for horizontal and vertical polarizations. In 2021, Yaman et al. [35] accounted for the directive diffuse scattering model. In 2022, Andani et al. [36] proposed an empirical diffuse scattering in RT. In 2023, Chizhik et al. [37] set diffuse scattering to a value. Therefore, the scattering effect is crucial. However, the traditional RT emits a lot of rays. When the surrounding environment is more complex, the traced rays will frequently be reflected, diffracted, or scattered in RT, which results in considerable time being consumed in evaluating their traces and calculating the electric field. As a result, the calculation efficiency is very low.

The single deterministic model is inadequate in modeling radio wave propagation in the millimeter wave. Therefore, it is necessary to study the scattering effect and the fusion of RT and the PE. A method is proposed in this study—namely, the proposed 3DFPHPCM—based on three-dimensional RT and a three-dimensional PE. The method takes the scattering effect into account and introduces the end field of the three-dimensional vector PE to the initial field of three-dimensional RT that can reduce the number of transmitting rays and improve the accuracy of the single deterministic model.

The structure of the remainder of this paper is as follows. In Section 2, the relevant propagation modeling theory is introduced. In Section 3, a hybrid propagation channel model based on three-dimensional RT and the three-dimensional vector PE is presented. In Section 4, the millimeter-wave channel measurement at 39 GHz in urban microcellular environments is described. In Section 5, the hybrid propagation channel model is analyzed. Finally, the conclusions are presented in Section 6.

2. Relevant Propagation Modeling Theory

2.1. The Theory of the Three-Dimensional Vector PE

The three-dimensional PE is derived from the electromagnetic wave equation. The scalar three-dimensional PE can only express the propagation or scattering characteristics of a component of the electromagnetic wave in three-dimensional space and can be solved according to the characteristics of a component of the electromagnetic field. Therefore, the scalar three-dimensional PE cannot resolve phenomena such as the depolarization of electromagnetic waves as caused by impedance boundaries.

The three-dimensional vector PE differs from the three-dimensional scalar PE. The three-dimensional vector PE is an equation set consisting of three-dimensional scalar PEs that reflect the electromagnetic field components. Therefore, the three-dimensional vector PE can resolve phenomena such as the depolarization of electromagnetic waves as caused by the impedance boundaries.

The three-dimensional vector PE is derived from the vector wave equation. Considering only forward propagation, the expressions for the electric and magnetic vectors are shown below.

$$\overrightarrow{E}=(E_x,E_y,E_z)\begin{array}{ccc}& & \overrightarrow{H}=(H_x,H_y,H_z)\end{array}$$

(1)

In the uniform passive region, E and H satisfy Maxwell’s equations.

$$\{\begin{array}{c}\nabla \cdot D=0\\ \nabla \cdot B=0\end{array}\begin{array}{ccc}& & \{\begin{array}{c}\nabla \times H=-i\omega D\\ \nabla \times E=i\omega B\end{array}\end{array}$$

(2)

where the free-space wavenumber is $k_0=\omega \sqrt{{\mu }_0{\epsilon }_0}$ and ${\mu }_0$ and $\omega$ are the magnetic permeability in a vacuum and the angular frequency of the electromagnetic wave, respectively. Wave impedance in a vacuum is ${\eta }_0=\sqrt{{\mu }_0/{\epsilon }_0}\approx 120\pi$. The refractive index of the medium is $n=\sqrt{{\epsilon }_r}$. The dielectric constant of the medium is $\epsilon ={\epsilon }_r{\epsilon }_0$. The magnetic permeability of the medium is $\mu \approx {\mu }_0$, $D=\epsilon E$, and $B=\mu H$.

When $\overrightarrow{F}=0$ and $\overrightarrow{A}\ne 0$, an arbitrary transverse magnetic (TM) field is represented by ${\Psi }^e$. Let $\overrightarrow{A}=\widehat{z}{\Psi }^e$ denote the TM field for z. Then, the relationship between ${\Psi }^e$ and each component of the electromagnetic field is

$$\{\begin{array}{c}{E}_z^e={\displaystyle \frac{i{\eta }_0}{n^2{k}_0}}\left({\displaystyle \frac{{\partial }^2}{\partial z^2}}+k_0^2\right){\Psi }^e\\ E_y^e={\displaystyle \frac{i{\eta }_0}{n^2{k}_0}}{\displaystyle \frac{{\partial }^2{\Psi }^e}{\partial y\partial z}}\\ E_x^e={\displaystyle \frac{i{\eta }_0}{n^2{k}_0}}{\displaystyle \frac{{\partial }^2{\Psi }^e}{\partial x\partial z}}\end{array}\begin{array}{ccc}& & \{\begin{array}{c}{H}_z^e=0\\ H_y^e=-{\displaystyle \frac{\partial {\Psi }^e}{\partial x}}\\ H_x^e={\displaystyle \frac{\partial {\Psi }^e}{\partial y}}\end{array}\end{array}$$

(3)

When $\overrightarrow{F}\ne 0$ and $\overrightarrow{A}=0$, an arbitrary electric magnetic field is represented by ${\Psi }^e$. Let $\overrightarrow{F}=\widehat{z}{\Psi }^m$ denote the TE field for z. Then, the relationship between ${\Psi }^e$ and each component of the electromagnetic field is

$$\{\begin{array}{c}{E}_z^m=0\\ E_y^m={\displaystyle \frac{\partial {\Psi }^m}{\partial x}}\\ E_x^e=-{\displaystyle \frac{\partial {\Psi }^m}{\partial y}}\end{array}\begin{array}{ccc}& & \{\begin{array}{c}{H}_z^m={\displaystyle \frac{i}{{\eta }_0{k}_0}}\left({\displaystyle \frac{{\partial }^2}{\partial z^2}}+k_0^2\right){\Psi }^m\\ H_y^m={\displaystyle \frac{i}{{\eta }_0{k}_0}}{\displaystyle \frac{{\partial }^2{\Psi }^m}{\partial y\partial z}}\\ H_x^m={\displaystyle \frac{{\partial }^2{\Psi }^m}{\partial x\partial z}}\end{array}\end{array}$$

(4)

In the homogeneous passive region, any scalar electromagnetic field component can be set as $\Psi ={\Psi }^e+{\Psi }^m$, which satisfies the homogeneous scalar wave equation ${\nabla }^2\Psi +k_0^2{n}^2\Psi =0$ in the passive region. ${\nabla }^2={\partial }^2/\partial x^2+{\partial }^2/\partial y^2+{\partial }^2/\partial z^2$ is the Laplace operator. Replacing $\Psi$ with $e^{i{k}_{0}x}\Psi (x,y,z)$ and substituting $e^{i{k}_{0}x}\Psi (x,y,z)$ into ${\nabla }^2\Psi +k_0^2{n}^2\Psi =0$ yields

$${\displaystyle \frac{{\partial }^2\Psi }{\partial x^2}}+2i{k}_0{\displaystyle \frac{\partial \Psi }{\partial x}}+{\nabla }_t^2\Psi +k_0^2(n^2-1)\Psi =0$$

(5)

where ${\nabla }_t^2={\partial }^2/\partial y^2+{\partial }^2/\partial z^2$ denotes the transverse Laplace operator.

Equation (5) can be decomposed as

$$[\partial /\partial x+i{k}_0(1-Q)][\partial /\partial x+i{k}_0(1+Q)]\Psi =0$$

(6)

where $Q=\sqrt{{\nabla }_t^2/k_0^2+n^2}$ is referred to as the pseudodifferential operator. The approximation method of Feit–Fleck is $Q=\sqrt{1+A+B}\approx \sqrt{1+A}+\sqrt{1+B}-1$.

The forward and backward propagation (or scattering) of the wave is represented in Equation (6), respectively. If only the forward propagation is considered and the Feit–Fleck method is used for the approximate representation of the pseudodifferential operator $Q$, make $A={\nabla }_t^2/k_0^2$ and $B=n^2-1$; then,

$$Q=\sqrt{k_0^2+{\nabla }_t^2+k_0^2(n^2-1)}/k_0\approx \sqrt{{\nabla }_t^2+k_0^2}/k_0+n-1$$

(7)

The forward three-dimensional wide-angle PE is then obtained as

$${\displaystyle \frac{\partial \Psi (x,y,z)}{\partial x}}=i{k}_0\left(\sqrt{k_0^2+{\nabla }_t^2}/k_0+n-2\right)\Psi (x,y,z)$$

(8)

That is,

$$\{\begin{array}{c}{\displaystyle \frac{\partial {\Psi }_x(x,y,z)}{\partial x}}=i{k}_0\left(\sqrt{k_0^2+{\nabla }_t^2}/k_0+n-2\right){\Psi }_x(x,y,z)\\ {\displaystyle \frac{\partial {\Psi }_y(x,y,z)}{\partial x}}=i{k}_0\left(\sqrt{k_0^2+{\nabla }_t^2}/k_0+n-2\right){\Psi }_y(x,y,z)\\ {\displaystyle \frac{\partial {\Psi }_z(x,y,z)}{\partial x}}=i{k}_0\left(\sqrt{k_0^2+{\nabla }_t^2}/k_0+n-2\right){\Psi }_z(x,y,z)\end{array}$$

(9)

In summary, the three-dimensional vector PE of the magnetic field (or electric field) is essentially a system of equations consisting of three-dimensional scalar PEs that can reflect three field components; thus, the three-dimensional vector and scalar PEs are in identical forms in the equations.

Equation (8) is a first-order partial differential equation about $x$. In the Cartesian coordinate system, we let the initial value of the electromagnetic field be $\Psi (x_0,y,z)$ and solve the partial differential equation on this basis. Therefore, the forward propagation equation can be solved as follows.

$$\Psi (x,y,z)=\exp [i{k}_0\Delta x(\sqrt{k_0^2+{\nabla }_t^2}/k_0+n-2)]\Psi (x_0,y,z)$$

(10)

where $\Delta x=x-x_0$ is the step size.

According to Equation (10), the essence of solving the three-dimensional PE is that the wavefronts in the $(y,z)$ plane are stepped forward along the $x$-axis and the subsequent wavefront is determined by the previous wavefront.

Separating the constant term and the refractive index term in Equation (10), we obtain

$$\Psi (x,y,z)=\exp [i{k}_0\Delta x(n-2)]\exp [i\Delta x\sqrt{k_0^2+{\nabla }_t^2}]\Psi (x_0,y,z)$$

(11)

In Equation (11), the refractive index term is first considered a constant, and a two-dimensional Fourier transform is applied to the remainder of $\Psi (x,y,z)$ in the $(y,z)$ plane, where

$$\mathcal{F}[\exp (i\Delta x\sqrt{k_0^2+{\nabla }_t^2})\Psi (x_0,y,z)]=\exp (i\Delta x\sqrt{k_0^2-k_y^2-k_z^2})\mathcal{F}[\Psi (x_0,y,z)]$$

(12)

Therefore, the field in the next step can be found according to Equation (11) as

$$\Psi (x+\Delta x,y,z)=\exp [i{k}_0\Delta x(n-2)]{\mathcal{F}}^{-1}\{\exp (i{k}_x\Delta x)\mathcal{F}[\Psi (x_0,k_y,k_z)]\}$$

(13)

where $k_x=\sqrt{k_0^2-k_y^2-k_z^2}$ denotes the component of $k_0$ in the $x$ direction.

2.2. The Theory of Three-Dimensional RT

When the wavelength of electromagnetic waves is very short and the medium changes slowly in space, the propagation and scattering characteristics of electromagnetic waves are localized. That is, the field in the vicinity of the observation point need not be solved according to the distribution over the entire initial surface but only by the field over a certain portion of that surface. This allows us to assume that the wavefront of an electromagnetic wave is viewed locally as the same as the wavefront of a plane wave. Therefore, in the high-frequency condition, it can be assumed that

$$\overrightarrow{E}={\overrightarrow{E}}_0{e}^{-j{k}_d\phi }\begin{array}{cc}& \end{array}\overrightarrow{H}={\overrightarrow{H}}_0{e}^{-j{k}_d\phi }$$

(14)

where $\phi =\phi (x,y,z)$ is a function of spatial position, and it is also an eigenfunction that determines the equiphase plane. ${\overrightarrow{E}}_0$ and ${\overrightarrow{H}}_0$ are also functions of spatial position.

Substituting Equation (14) into the Maxwell equation for the passive region and using the geometrical-optical approximation of the high-frequency field, $\lambda \to 0$, $k_d\to \infty$, $|\nabla \epsilon |\lambda /\epsilon \ll 1$, and $|\nabla \mu |\lambda /\mu \ll 1$.

We obtained the field equations based on the geometrical optics approximation.

$$\nabla \phi \times {\overrightarrow{H}}_0+c\epsilon {\overrightarrow{E}}_0=0$$

(15)

$$\nabla \phi \times {\overrightarrow{E}}_0-c\mu {\overrightarrow{H}}_0=0$$

(16)

$${\overrightarrow{E}}_0\cdot \nabla \phi =0$$

(17)

$${\overrightarrow{H}}_0\cdot \nabla \phi =0$$

(18)

In urban microcellular environments, $\mu$ is mostly constant and $\mu ={\mu }_0$. Therefore, Equations (15) and (16) can be written as

$${\overrightarrow{E}}_0=Z_0{\overrightarrow{H}}_0\times \nabla \phi /n$$

(19)

$$Z_0{\overrightarrow{H}}_0=\nabla \phi /n\times {\overrightarrow{E}}_0$$

(20)

where $n=\sqrt{\epsilon /{\epsilon }_0}$ is the refractive index of the medium and $Z_0=\sqrt{\mu /\epsilon }$ is the characteristic impedance of the medium. Equations (19) and (20) show that ${\overrightarrow{E}}_0$, ${\overrightarrow{H}}_0$, and $\nabla \phi$ are mutually orthogonal vectors; thus, the geometric optical field is locally a plane wave.

Bringing Equation (16) into Equation (15) yields

$$\nabla \phi \times (\nabla \phi \times {\overrightarrow{E}}_0)+n^2{\overrightarrow{E}}_0=0$$

(21)

and

$$\nabla \phi \times (\nabla \phi \times {\overrightarrow{E}}_0)=({\overrightarrow{E}}_0\cdot \nabla \phi )-{(\nabla \phi )}^2{\overrightarrow{E}}_0$$

(22)

Therefore,

$$[n^2-{(\nabla \phi )}^2]{\overrightarrow{E}}_0=0$$

(23)

As E₀ has a non-zero solution, there must exist

$${(\nabla \phi )}^2=\nabla \phi \cdot \nabla \phi =n^2$$

(24)

where $\phi$ is referred to as the eikonal, which includes the variation in the electrical parameters of the medium. Equation (24) is referred to as the eikonal equation. This equation leads to the eikonal or phase function $\phi (x,y,z)$, which is a fundamental equation in geometric optics. Based on Fermat’s principle, the ray between two points is the curve that the optical path ${\displaystyle {\int }_{c}nds}$ takes as the extreme value. In a homogeneous medium, the optical path is proportional to the geometric distance. The straight line between two points is the shortest; thus, the optical path consists of a straight line.

3. A Hybrid Propagation Channel Model

3.1. The Proposed 3DFPHPCM

The proposed 3DFPHPCM is used to model the environment first, and the three-dimensional PE derived from the waveguide equation is used to combine the absorption boundary and impedance boundary conditions of the actual rough ground with the initial field. Moreover, the step-by-step Fourier transform algorithm is adopted to obtain the magnetic field $H_y,E_x,E_y,E_z,H_x,H_z$ as described in Section 2.1. As there is almost no restriction on the applicable conditions of three-dimensional RT, the initial field $E_{total},H_{total}$ of three-dimensional RT can be obtained directly.

According to the relationships among $\overrightarrow{k}$, $\overrightarrow{E}$, and $\overrightarrow{H}$ in Equation (25), the propagation direction of the ray in the three-dimensional RT can be obtained.

$$\overrightarrow{H}=\widehat{k}\times \overrightarrow{E}/\eta$$

(25)

Thereafter, the propagation mechanism of the rays and the building is determined (i.e., direct, reflected, and diffracted rays). Finally, the scattering effect is considered in the three-dimensional RT, several scattered rays are taken, and the direction of scattered rays is determined via sampling to enable us to propose the presented 3DFPHPCM. The steps used in the proposed 3DFPHPCM to calculate the different polarizations (HH, VV, VH, and HV) are given below.

• Preparatory work: The first step involves reading the file of the environment data, including the vertex coordinate, the sides, and the outer normal vector of the triangular surface. The second step involves initializing the information on various electromagnetic parameters, including the height of the transmitting and receiving antennas, operating frequency, transmitting power, wavelength, dielectric constant, and dielectric conductivity;
• Computation of the initial field information for three-dimensional RT: Firstly, the environment is modeled. Secondly, the three-dimensional PE derived from the waveguide equation is used to combine the absorbing boundary and impedance boundary conditions of the actual rough ground with the initial field, and the step-by-step Fourier transform algorithm is adopted to obtain the field component information at the boundary of the region. Finally, the field component information at the boundary of the region is used as the initial field information of the three-dimensional RT;
• Determination of whether the ray is a direct ray: We determine whether the connecting line of the receiving antenna and transmitting antenna contain an intersection point in the triangular surface. If there is no intersection, there is no obstruction between the transmitting antenna and the receiving antenna. Thereafter, we record the relevant information;
• Determination of single scattered rays: We determine which rays are single scattered rays from the transmitting to the receiving antenna;
• Determination of double scattered rays: We determine which rays are double scattered rays from the transmitting to the receiving antenna;
• Calculation: We calculate the path loss for different polarizations (HH, VV, VH, and HV) at each receiving point based on the ray path.

The pseudocode for the proposed 3DFPHPCM is shown in Algorithm 1.

Algorithm 1. Calculate the total Path Loss L

1: procedure total Path Loss L

2:   Hy                     ▷ Calculate magnetic field Hy based on three-dimensional PE

3:   Ex, Ey, Ez, Hx, Hz        ▷ Calculate Ex, Ey, Ez, Hx, Hz based on Maxwell’s equations

4:   k                       ▷ Calculate wave vector k at each grid point

5:   cc                       ▷ Calculate the total numbers of rays cc

6:    for (each ray) do

7:     if the ray is a direct ray

8:       record the ray

9:     elseif the ray is single scattered ray

10:      record the ray

11:    elseif the ray is double scattered ray

12:      record the ray

13:    end if

14:  end for

15:  L ← Calculate the total Path Loss L

16:  return L

17: end procedure

Based on the above steps, the detailed pseudocode of the serial algorithm for three-dimensional RT is developed as shown in Algorithm 2.

The implementation of the proposed 3DFPHPCM involves the sampling of scattered rays, as shown in Section 3.2.

Algorithm 2. Calculate the total electric field E for different types of rays

1: procedure total electric field E

2:    for (each ray) do

3:     if the ray intersects with triangular surface

4:      m1 ← Connecting the scattered ray m1 between the single scattered point and the receiving point

5:       if the scattered ray m1 has no intersection with the remaining triangular surface and is within the preset angle

6:         E1 ← Calculate the electric field E1 of a single scattered ray

7:       else

8:        m2 ← Sampling of scattered ray m2 based on a single scattered point within a preset angular range

9:        if scattered ray m2 intersects with the remaining triangular surface

10:         m3 ← Connecting the scattered ray m3 between the double scattered point and the receiving point

11:          if the scattered ray m3 has no intersection with the remaining triangular surface

12:            if the scattered ray m3 is within the preset angle

13:              E2 ← Calculate the electric field E2 of a double scattered ray

14:            end if

15:          end if

16:        end if

17:      end if

18:     end if

19:    end for

20:   E ← Calculate the total electric field

21:  return E

22: end procedure

3.2. The Theory of the Ray-Splitting Technique

The ray-splitting method is used to study double scattering. Sampling is performed around the direction of specular reflection to simulate the spatial distributions of the scattered ray energy. The number of samples is positively correlated with the accuracy of the calculation. Therefore, the greater the number of scattered rays, the more accurate and time-consuming the calculation is. It is therefore necessary to consider the tradeoff between computational accuracy and time. An optimal solution is to reduce the number of scattered rays as much as possible while maintaining a certain degree of computational accuracy. As the spatial distribution of the energy of the scattered ray is not uniform, we can improve the sampling efficiency by appropriately increasing the sampling density with a higher scattered energy density.

When sampling the scattered rays, the local coordinate system of the surface is established. In Figure 1, the intersection position of the incident rays and the triangular surface is set as the origin $O^{\prime }$ of the coordinate system, and the unit normal vector of the triangular surface is set as $\widehat{n}$. The direction of specular reflection is the $z^{\prime }$-axis, and there is no fixed rule for the selection of the $x^{\prime }$-axis and the $y^{\prime }$-axis directions, though they should satisfy the right-hand rule. In this study, the cross product of the unit normal vector of the triangular surface and the specular reflection ray is set as the $y^{\prime }$-axis, and the $z^{\prime }$-axis is finally determined according to the right-hand rule such that ${\widehat{y}}^{\prime }$ is $\widehat{n}\times {\widehat{z}}^{\prime }$ and ${\widehat{x}}^{\prime }={\widehat{y}}^{\prime }\times {\widehat{z}}^{\prime }$.

The scattering angle $\theta$ and the scattering azimuth angle $\phi$ can thus be obtained as

$$\begin{array}{l}\theta ={\cos }^{-1}[1-{\zeta }_1\cdot (1-\cos {\theta }_1)]\\ \phi =2\pi {\zeta }_2\end{array},$$

(26)

where $\theta$ and $\phi$ are both random numbers between 0 and 1.

4. Millimeter-Wave Channel Measurement at 39 GHz

The channel measurements in NLOS environments were conducted in the urban microcellular environments of Qingdao, China, as shown in Figure 2.

An omnidirectional antenna was used at the transmitter (Tx), with an operating frequency of 26.5–40 GHz. Moreover, the receiver (Rx) adopted a horn antenna (as shown in Figure 3) and its operating frequency was also 26.3–40 GHz. The MIMO channel measurement system is shown in Figure 4. The relevant measurement parameters are shown in

Table 1. The measurement parameters in the urban microcellular environment.

Physical Parameters	Value
Frequency (GHz)	39
Bandwidth (GHz)	1
Transmitting power (dBm)	0
The height of Rx/Tx antenna (m)	1.5/9
Transmitting antenna/gain (dBi)	Omnidirectional/6.19
Receiving antenna/gain (dBi)	Horn/26.92
The stepping of horn antenna	5°
Polarization method of transmitting antenna	V
Polarization method of receiving antenna	V

.

The specific measurement process is shown in Figure 5. During measurement, Tx was fixed, and its coordinates were (2.9, 54.7, 9). A total of 10 points were selected for Rx, which was moved in increments of five meters along the gray line, namely, y = 62.9, z = 1.5, x = (10, 15, 20, 25, 30, 35, 40, 45, 50, and 55). Moreover, Rx was moved in the horizontal and vertical directions using a rotary table at each receiving point. In Figure 5, the 3D label indicates experimental data where the azimuth and elevation angles were measured, and the 2D label indicates experimental data where only the elevation angle was 0° and azimuth angles were measured at 5° intervals. The solid red lines represent the x-axis (along the direction of the building length) and the y-axis (along the direction of the building width), and the z-axis is measured along the direction of the building height. Moreover, the green lines represent the lawn. The power transmitted by the Tx is 0 dBm, the measurement bandwidth is 1 GHz, and both Tx and Rx are vertically polarized.

The 3D measurements were performed for azimuth angles at 5° intervals and elevation angles at −10°, 0°, 10°, 20°, and 30° at the receiving end when the receiving antenna was very close to the transmitting antenna. The farther the receiving antenna is from the transmitting antenna, the smaller the effect of the elevation angle on the path loss. Therefore, these data were collected for azimuth angles of every 5° and elevation angles of −10°, 0°, and 10° at the Rx in NLOS environments. The complex relative permittivity of the wall is ${\epsilon }_r=3+0.005i$, which is taken from the reference literature.

5. Analysis of the Hybrid Propagation Channel Model

The 3DFPHPCM proposed in this study was established based on the three-dimensional vector PE and the three-dimensional RT. Only the line-of-sight, single scattering, and double scattering are considered in the theoretical study. When combined with the experimental measurements in urban microcellular environments, the variation in the path loss at each receiving point with the horizontal distance between the transmitting and receiving antennas was investigated for different numbers of scattered rays and different scattering angles.

5.1. Effect of the Number of Scattered Rays on the Proposed 3DFPHPCM

The path loss variation with the number of scattered rays at each reception point was first studied for a fixed scattering angle in this study. The trend of the path loss at each reception point with the horizontal distance between the transmitting and receiving antennas for a scattering angle of 5° and different numbers of scattered rays is shown in Figure 6. ‘P_mea’ indicates the measured value, ‘SP’ indicates the sum-of-individual-ray-powers (SP) method, and ‘N’ indicates the number of scattered rays. The curves are more consistent with the experimental data when the number of ray bars is 9 and 29. The RMSE, MAE, and runtime for different numbers of scattered rays at a scattering angle of 5° are shown in

Table 2. RMSE, MAE, and runtime for different numbers of scattered rays at a scattering angle of 5°.

Number of Scattered Rays	RMSE (SP)	MAE (SP)	Running Time (s)
9	4.1778	3.4927	94.56
19	14.8409	9.5007	148.482
29	5.6016	4.9940	199.29
39	9.8880	8.3241	252.172
49	16.2592	11.4766	305.003
59	20.9067	18.4632	359.282
69	10.3565	8.0150	408.97
79	11.3172	8.9271	462.32
89	10.2659	8.2918	515.726

. As the number of scattered rays increases, the runtime of the procedure increases accordingly, as shown in Table 2. The two sets of data with the smallest RMSE and MAE can be found in Table 2. When the number of scattered rays is 29, the RMSE is 5.6016, the MAE is 4.9940, and the runtime is 94.56 s. When the number of scattered rays is nine, the RMSE is 4.1778, the MAE is 3.4927, and the runtime is 199.29 s. Therefore, the theoretical calculation and the experimental data are in good agreement, and the runtime is short when the number of scattered rays is nine. Moreover, the number of scattered rays affects the runtime. The conformity of the theoretical calculations and experimental data is examined in the following section when the number of scattered rays is approximately nine.

As shown in Figure 7, the consistency between the theoretical calculations and experimental data was examined when the scattering angle was 5° and the number of scattered rays was approximately nine in this study. The theoretical calculation and experimental data are more consistent when the number of scattered rays is 9 and 15, as shown in the figure. The RMSE, MAE, and runtime when the number of scattered rays is nine at a scattering angle of 5° are shown in

Table 3. RMSE, MAE, and runtime for approximately 9 scattered rays at a scattering angle of 5°.

Number of Scattered Rays	RMSE (SP)	MAE (SP)	Running Time (s)
2	9.1440	8.3327	53.631
3	7.0643	5.4402	60.079
4	8.7474	6.7801	66.464
5	7.6453	6.6384	70.802
6	15.2676	12.0192	76.33
7	7.8952	6.7775	82.401
8	12.1419	8.3684	88.081
9	4.1778	3.4927	94.56
10	11.1638	8.4646	99.757
11	13.3374	10.6575	104.055
12	11.0998	8.8231	109.868
13	11.1954	9.4365	115.048
14	9.4557	7.7238	119.369
15	5.9410	4.6944	124.746
16	11.1536	9.5969	131.245
17	16.3286	9.5217	137.002
18	10.4477	8.6636	142.715

. As above, the runtime increases as the number of scattered rays increases; however, the increase is not obvious. When the number of scattered rays is 15, the RMSE is 5.941 and the MAE is 4.6944, both of which are greater than the RMSE and the MAE when the number of scattered ray bars is nine. The RMSE and MAE of the other number of scattered rays are also larger than those when the number of scattered rays is nine. The above findings indicate that the theoretical calculation and experimental data are in good agreement when the number of scattered rays is nine; therefore, the number of scattered rays was set to nine in the subsequent study.

5.2. Effect of Scattering Angles on the Proposed 3DFPHPCM

Second, the trend of path loss at each reception point was investigated for a fixed number of scattered rays when the scattering angle was different. ‘A’ is denoted as the scattering angle in Figure 8. The variation in path loss with distance at different scattering angles is not obvious. Therefore, it is necessary to study the RMSE and MAE for different numbers of rays.

Table 4. RMSE, MAE, and runtime at different scattering angles when the number of scattered rays is 9.

Scattering Angle (°)	RMSE (SP)	MAE (SP)	Running Time (s)
3	54.4215	39.3436	93.022
4	46.0455	28.7651	93.917
5	4.1778	3.4927	94.56
6	32.7552	25.6358	95.009
7	41.8929	40.2752	95.218
8	43.2574	41.9560	95.657
9	51.0809	50.0220	95.46
10	51.3264	50.1893	96.974
11	50.5802	48.8019	97.507
12	54.1937	52.8255	98.980

shows the RMSE, MAE, and runtime for different scattering angles when the number of scattered rays is nine. There is little difference in the runtime at different scattering angles, indicating that the scattering angle has almost no effect on the runtime. Only when the scattering angle is 5° is the RMSE 4.1778 and the MAE 3.4927, which is the only angle among all angles where both RMSE and MAE are less than 10. For the other scattering angles, the RMSE and MAE are greater than 10, which is much larger than the result when the scattering angle is 5°. The above results indicate that, when the scattering angle increases, not only will the number of scattered rays increase but also the received electric field. Therefore, the difference between the experimental measurement and the theoretical calculation is significant. As a consequence, it is necessary to study the conformity of theoretical calculations and experimental data at a smaller range of angles, for example, approximately 5°.

Thereafter, the variation in the path loss with distance was investigated near the scattering angle of 5° when the number of scattered rays was nine. The trend of the path loss with distance between the transmitting and receiving antennas (at each reception point) is shown in Figure 9 when the scattering angles are between 4.1° and 5.9°. The experimental data and the theoretical calculations are in good agreement when the scattering angle is 5°. The RMSE, MAE, and runtime when the number of scattered rays is nine are shown in

Table 5. RMSE, MAE, and runtime at a scattering angle of 5° when the number of scattered rays is 9.

Scattering Angle (°)	RMSE (SP)	MAE (SP)	Running Time (s)
4.1	44.5926	29.6713	93.851
4.2	34.1570	20.7556	93.437
4.3	30.9607	17.1313	94.504
4.4	10.2343	9.4571	93.897
4.5	12.2252	9.3473	92.677
4.6	11.1887	10.4644	93.037
4.7	11.7482	9.6383	93.562
4.8	8.9051	7.5535	93.147
4.9	8.8014	7.4915	93.977
5	4.1778	3.4927	94.56
5.1	5.2127	4.3813	93.166
5.2	19.8508	13.0281	94.447
5.3	17.5563	14.1417	93.418
5.4	9.0696	7.0469	94.029
5.5	22.2665	17.9165	94.067
5.6	12.2798	10.3050	93.704
5.7	18.7043	15.2454	92.994
5.8	20.2656	18.2173	94.551
5.9	32.5403	29.0101	93.771

when the scattering angle is approximately 5°. The difference in runtimes at different scattering angles is minimal and nearly the same as in Table 5, indicating that the scattering angle exhibits no effect on the runtime. The RMSE and MAE are less than 10 for scattering angles of 4.8°, 4.9°, 5°, and 5.1°. The RMSE and MAE are 8.9051 and 7.5535 for scattering angles of 4.8°, 8.8014 and 7.4915 for scattering angles of 4.9°, 5.2127 and 4.3813 for scattering angles of 5.1°, and 4.1778 and 3.4927 for scattering angles of 5°, respectively, which are less than the RMSE and MAE at 4.8° and 4.9°. Therefore, the scattering angle can be determined as 5°. In conclusion, the experimental data and the theoretical calculations are in good agreement when the number of scattered rays is nine and the scattering angle is 5° for urban microcellular environments. The scattering angle has no effect on the runtime whereas the number of scattered rays affects the runtime.

5.3. Comparison and Analysis at Different Polarizations

Two sets of data with both RMSE and MAE values less than 10 were selected to analyze the trend of path loss with the horizontal distance between the transmitting and receiving antennas for different polarizations, i.e., when the number of scattered rays is nine and the scattering angles are 4.8°, 4.9°, 5.1°, and 5°, as shown in Figure 10 and Figure 11. ‘VH’ indicates that the transmitting antenna is vertically polarized and the receiving antenna is horizontally polarized. ‘P_mea_VV’ indicates that the VV polarization was selected in the measurement. Overall, the path loss increases and the received electric field decreases with increasing distance in different polarizations. The path loss in the ‘HH’ and ‘HV’ polarizations is less than in the ‘VV’ and ‘VH’ polarizations. Although the scattering angles are very close, the path loss and the electric field obtained at each receiving point are different when the scattering angles are 4.8°, 4.9°, 5.1°, and 5° and the number of scattered rays is nine. Moreover, there is a large difference between these values. The above results indicate that the scattering angle exhibits a great influence on the received electric field at each receiving point.

5.4. Comparison and Analysis of the Proposed 3DFPHPCM with a Single Deterministic Model

In this study, the proposed 3DFPHPCM is compared and analyzed with a single deterministic model (conventional three-dimensional RT and the three-dimensional PE). ‘SP_N9_A5’ denotes the proposed 3DFPHPCM. It can be seen in Figure 12 that the proposed 3DFPHPCM is close to the experimental data, while the three-dimensional PE and three-dimensional RT deviate from the experimental data significantly. As shown in

Table 6. RMSE and MAE when using the different methods.

Different Method	RMSE (SP)	MAE (SP)	Running Time (s)
Three-dimensional PE	12.0166	10.7764	5.4892 × 103
Three-dimensional RT	8.5142	7.3333	302.344
SP_N9_A5	4.1778	3.4927	94.56

, the RMSE and MAE of the proposed 3DFPHPCM are 4.1778 and 3.4927, respectively; in comparison, those of the three-dimensional PE are 12.0166 and 10.7764, respectively, and those of three-dimensional RT are 8.5142 and 7.333, respectively, which are much greater values than the RMSE and MAE of the proposed 3DFPHPCM. Moreover, its runtime is the shortest among the methods. These results demonstrate that the proposed 3DFPHPCM has higher accuracy.

6. Conclusions

In this study, in order to examine the proposed 3DFPHPCM, the three-dimensional vector PE was used to provide the initial field strength for three-dimensional RT, allowing the aforementioned 3DFPHPCM to be developed—taking into account that the initial ray produces different polarization effects on the wall and the ground—in order to analyze the effects of different scattering angles and the different number of scattered rays on prediction accuracy in the millimeter wave. In addition, measurements at 39 GHz were carried out in urban microcellular environments. The results showed that, when the number of scattered rays increases, the runtime increases accordingly. Meanwhile, the change in the scattering angle has no effect on the runtime. When the scattering angle was 5° and the number of scattered rays was nine, the RMSE and MAE between the theoretical calculation and experimental data were 4.1778 and 3.4927, respectively, and the runtime was 94.56 s, all of which reached the minimum values, as confirmed through multiple tests. Compared with traditional RT and the three-dimensional PE, the proposed 3DFPHPCM had the smallest RMSE and MAE, higher computational accuracy, and the shortest runtime. The results presented herein provide a theoretical basis for the optimization of base station locations. Our proposed model is only suitable for urban microcellular environments, and we will conduct experiments for other scenarios in the future to optimize the proposed model.