A Novel Segment-Based Model for Non-Stationary Vehicle-to-Vehicle Channels With Velocity Variations

By considering the velocity variations of both terminals and moving scatterers, a general segment-based model for non-stationary vehicle-to-vehicle (V2V) channels is proposed in this paper. The time evolving channel parameters, i.e., Doppler frequencies, angles of arrival and departure, path delays, and path powers, are analyzed and simplified by the Taylor series expansions. The proposed model can be applied for realistic V2V communication environments and explicitly reveal the impact of velocity variations on the channels. In addition, the theoretical statistical properties, i.e., the temporal correlation function (TCF), Doppler power spectrum density (DPSD), level-crossing rate (LCR), and average fading duration (AFD) are also analyzed and derived. Simulation results under four typical V2V scenarios show that the theoretical results agree well with the simulated and measured ones, which verifies the effectiveness of both the proposed model and the derivations.

SECTION I. Introduction

Vehicle-to-everything (V2X) communication can reduce the economic losses and increase the effectiveness of traffic by passing the information from a vehicle to any entity, i.e., vehicle-to-vehicle (V2V), vehicle-to-pedestrian (V2P), and vehicle-to-infrastructure (V2I). The V2V communication is also an application of the fifth generation (5G) communication systems in the intelligent transportation [1]–​[4]. However, the efficiency and reliability of V2V communication systems are affected by the scattering environment such as crowded space, density of vehicles, time-variant velocities, movements of scatterers, and etc. Moreover, the mobilities of both terminals and scatterers lead to the channel characteristic variations [5], [6], which make V2V channels different with the traditional fixed-to-mobile (F2M) channels [7]–​[9].

In the last decade, the geometry-based stochastic models (GBSMs) for V2V channels with the wide-sense stationary (WSS) assumption have been widely accepted [10]–​[12]. The corresponding statistical properties, i.e., the temporal correlation function (TCF), Doppler power spectrum density (DPSD), level-crossing rate (LCR) and average fading duration (AFD), can also be addressed in [11], [12]. However, measurement campaigns have demonstrated that the WSS assumption is only valid within a short time or distance interval [8], [13], i.e., 4.5 m for the light of sight (LoS) scenarios and 2.7 m for non-light-of-sight (NLoS) scenarios. Recently, several non-stationary V2V GBSMs have been presented in [14]–​[23]. Among them, the models in [14] and [15] assumed that the mobile transmitter (MT) and mobile receiver (MR) moved with a constant speed and direction, and the scatterers surrounding the terminals were fixed. The authors in [16] and [17] took the variant velocities of terminals into account but with fixed scatterers. The authors in [18]–​[22] considered the movements of both terminals and scatterers, but only the constant velocity is involved. Actually, the speeds and directions of terminals should be time-varying due to the traffic conditions and the scatterers, e.g. other vehicles and pedestrian, are usually in motion. In addition, it’s worth mentioning that it becomes very difficult to calculate channel parameters real time when complex mobility models are considered. To make a trade-off, a simplified V2V GBSM with velocity variations was studied in [23], but it was only suitable for the short time interval or linear trajectories. Moreover, only the TCF was analyzed and other important statistical properties were lacked. This paper aims to fill these gaps. The major contributions and novelties of this paper are summarized as follows:

1. Based on the idea of general non-stationary V2V GBSM, this paper proposes a segment-based channel model with velocities variations of both terminals and scatterers. The new model is suitable for the long time interval with arbitrary trajectory.

2. The time evolving algorithms of channel parameters, i.e., Doppler frequencies, angles of arrival and departure, path delays and path powers, are derived and simplified with the help of Taylor series expansions.

3. The theoretical expressions of TCF, DPSD, LCR, and AFD of proposed model under VM scattering scenarios are analyzed and verified by simulation and measurement results. These theoretical results can be used to observe the impact of velocity variations, i.e., the accelerations of the speeds and direction, on the channel characteristics.

The reminder of this paper is organized as follows. Section II gives a general GBSM for V2V channels with LoS and NLoS paths. In section III, a segment-based V2V GBSM with velocity variations is proposed. The corresponding channel parameters based on time-variant velocities are analyzed in section IV. Statistical properties of the proposed model are derived in section V. The simulations and validations are performed in section VI. Finally, some conclusions are given in section VII.

SECTION II. General GBSM for V2V Channels

A typical V2V communication scenario with arbitrary trajectories of the MT and MR is shown in Fig. 1. There are two coordinate systems denoted as the MT and MR coordinate systems with their origins at the central of MT and MR, respectively. The time-variant velocities of MT or MR is denoted by ${{\mathbf {v}}^{i}}(t)$ , where $i\in \{\text {MT,MR}\}$ is used to represent the MT or MR for brief. The V2V propagation channel includes several propagation paths, i.e., light-of-sight (LoS), single-bounce, double-bounce, and multiple-bounce. Moreover, the single- and double- bounce can be viewed as two special cases of non-light-of-sight (NLoS) path. Under general NLoS scenarios, the first and last clusters are denoted by $\mathbf {S}_{n}^{\text {MT}}$ and $\mathbf {S}_{n}^{\text {MR}}$ with time-variant velocities ${{\mathbf {v}}^{\mathbf {S}_{n}^{i}}}(t)$ , $i\in \{\text {MT,MR}\}$ , respectively. Based on the twin-cluster approach [24], [25], the rest clusters between $\mathbf {S}_{n}^{\text {MT}}$ and $\mathbf {S}_{n}^{\text {MR}}$ can be abstracted by a virtual link and described by the equivalent path delay and power. The detailed parameters used in this paper are listed in Table 1.

TABLE 1 Details of Parameter Definitions

FIGURE 1.

A typical V2V communication scenario with velocity variations.

Under the general scattering environment, the wideband complex channel impulse response (CIR) between the MT and MR can be expressed as (1) [2], where $N$ , $M$ are the numbers of multiple paths and sub-paths within each path, ${P^{\text {LoS}}}(t)$ , ${P^{\text {NLoS}}}(t)$ denote the path power of LoS and NLoS paths, respectively, and ${\tau ^{\text {LoS}}}(t)$ , $\tau _{n}^{\text {NLoS}}(t)$ denote the time delays of LoS and NLoS paths, respectively. In (1), shown at the bottom of this page,

${h^{\text {LoS}}}(t,\tau)$ , $h_{n,m}^{\text {NLoS}}(t,\tau)$ represent the complex channel gains of LoS and NLoS paths, respectively, and they can be further expressed as $$\begin{align*} {h^{\text {LoS}}}(t,\tau)=&{{\text {e}}^{\text {j}\cdot \text {(2}\pi \cdot \int _{0}^{t}{}{f^{\text {LoS}}}{(}t'\text {)d}t'+{\theta ^{\text {LoS}}}{)}}} \tag{2}\\ h_{n,m}^{\text {NLoS}}(t,\tau)=&{{\text {e}}^{\text {j}\left({2\pi \int _{0}^{t}{f_{n,m}^{\text {NLoS}}(t')\text {d}t}'+\theta _{n,m}^{\text {NLoS}}}\right)}} \tag{3}\end{align*}$$ where ${f^{\text {LoS}}}(t), f_{n,m}^{\text {NLoS}}(t)$ and ${\theta ^{\text {LoS}}}, \theta _{n,m}^{\text {NLoS}}$ are the time-variant Doppler frequencies and random phases, respectively. Taking the random phase $\theta _{n,m}^{\text {NLoS}}$ as an example, it includes two parts, i.e., the initial phase ${\theta _{n,m}^{\text {NLoS}}}'$ and the accumulated phase $\int _{-\infty }^{0}{f_{n,m}^{\text {NLoS}}(t')\text {d}t}'$ , and can be equivalently generated by the independent random variables uniformly distributed over $[-\pi,\pi$ ). Moreover, the Doppler frequency of LoS path can be defined as [17] $$\begin{equation*} {f^{\text {LoS}}}(t)=\frac {{{\mathbf {v}}^{\text {MT}}}(t)\cdot {{{\tilde {\mathbf {r}}}}^{\text {LoS,MT}}}(t)+{{\mathbf {v}}^{\text {MR}}}(t)\cdot {{{\tilde {\mathbf {r}}}}^{\text {LoS,MR}}}(t)}{\lambda } \tag{4}\end{equation*}$$ where $\lambda ={\text {c}}/{{f_{0}}}$ ; denotes the wavelength, ${f_{0}}$ and c represent the carrier frequency and speed of light, respectively, ${{\tilde {\mathbf {r}}}^{\text {LoS,}i}}(t)={{\left [{ \cos (\alpha _{}^{\text {LoS,}i}(t)),\sin (\alpha _{}^{\text {LoS,}i}(t)) }\right]}^{\text {T}}}$ , $i\in \{\text {MT,MR}\}$ is the unit direction vector with ${\alpha ^{\text {LoS,}i}}(t)$ denoting the time-variant AoA or AoD. In the presence of moving scatterers, the Doppler frequency of NLoS paths is defined as [23] $$\begin{align*}&\hspace {-0.4pc}f_{n,m}^{\text {NLoS}}(t)= \frac {{{\mathbf {v}}^{\text {MT,}\mathbf {S}_{n,m}^{\text {MT}}}}(t)\cdot {{{\tilde {\mathbf {r}}}}^{\text {MT,}\mathbf {S}_{n,m}^{\text {MT}}}}(t)}{\lambda } \\&\qquad\qquad\qquad\qquad\qquad\displaystyle {+\,\frac {{{\mathbf {v}}^{\mathbf {S}_{n,m}^{\text {MR}},\text {MR}}}(t)\cdot {{{\tilde {\mathbf {r}}}}^{\mathbf {S}_{n,m}^{\text {MR}},\text {MR}}}(t)}{\lambda }} \tag{5}\end{align*}$$ where the movements of clusters are took into account by using the relative velocity, i.e., $$\begin{align*} {{\mathbf {v}}^{i,\mathbf {S}_{n,m}^{i}}}(t)=&{v^{i,\mathbf {S}_{n,m}^{i}}}(t)\cdot {{\text {e}}^{\text {j}\cdot \alpha _{v}^{i,\mathbf {S}_{n,m}^{i}}(t)}} \\=&{{\mathbf {v}}^{i}}(t)-{{\mathbf {v}}^{\mathbf {S}_{n,m}^{i}}}(t) \tag{6}\end{align*}$$ ${{\tilde {\mathbf {r}}}^{i,\mathbf {S}_{n,m}^{i}}}(t)={{\left [{ \cos (\alpha _{n,m}^{i}(t)),\sin (\alpha _{n,m}^{i}(t)) }\right]}^{\text {T}}}$ and ${{\tilde {\mathbf {r}}}^{\mathbf {S}_{n,m}^{i},i}}(t)={{\left [{ \cos (\alpha _{n,m}^{i}(t)),\sin (\alpha _{n,m}^{i}(t)) }\right]}^{\text {T}}}$ denote the unit direction vectors from $\mathbf {S}_{n,m}^{i}$ to $i$ and from $i$ to $\mathbf {S}_{n,m}^{i}$ , respectively. It should be mentioned that (6) differs from the existing result in [16], where the authors did not take the movements of clusters into account.

SECTION III. Segment-Based V2V GBSM With Velocity Variations

The general GBSM of (1) is suitable for the V2V channels with velocity or trajectory variations, but it’s not practical to run the model since the time-variant channel parameters need to be calculated real time. Besides, it’s a principle model and cannot explicitly reveal the impact of velocity variations on the channel characteristics. In order to make the model more practical and tractable, we upgraded (1) into a segment-based V2V GBSM which can be used to observe the effects of the movements of terminals and scatterers and get the closed-form solution of system performance.

Firstly, the general CIR is divided into several segments in time domain, and the $k$ th segment of CIR can be expressed as

$$\begin{equation*} {h_{k}}(t,\tau)\triangleq h(t,\tau)\cdot \prod \nolimits _{{T_{0}}}{(t-k{T_{0}})} \tag{7}\end{equation*}$$ where $\prod \nolimits _{{T_{0}}}{(t)}$ denotes the windowing function given by $$\begin{equation*} \prod \nolimits _{T_{0}} {(t)} \stackrel { \Delta } = \begin{cases} 1,& 0 \le t \le {T_{0}}\\ 0,&\text {otherwise}. \end{cases} \tag{8}\end{equation*}$$ The time interval of each segment is related with the velocity variations. During each segment, the speed and movement direction can be approximated by two linear functions, i.e., $$\begin{align*} {v^{i,\mathbf {S}_{n,m}^{i}}}(t)=&{v^{i,\mathbf {S}_{n,m}^{i}}}({t_{0}})+{a^{i,\mathbf {S}_{n,m}^{i}}}\cdot (t-{t_{0}}) \tag{9}\\ \alpha _{v}^{i,\mathbf {S}_{n,m}^{i}}(t)=&\alpha _{v}^{i,\mathbf {S}_{n,m}^{i}}({t_{0}})+{b^{i,\mathbf {S}_{n,m}^{i}}}\cdot (t-{t_{0}}) \tag{10}\end{align*}$$ where ${v^{i,\mathbf {S}_{n,m}^{i}}}({t_{0}})$ and $\alpha _{v}^{i,\mathbf {S}_{n,m}^{i}}({t_{0}})$ denote the initial relative speed and movement direction at $t={t_{0}}$ , ${a^{i,\mathbf {S}_{n,m}^{i}}}$ and ${b^{i,\mathbf {S}_{n,m}^{i}}}$ represent the corresponding accelerations of speed and direction, respectively. Consequently, $h(t,\tau)$ of our proposed model can be rewritten as (11), shown at the bottom of the previous page,

where $K$ denotes the total number of segments in the simulation time interval.

Secondly, the Doppler frequency of NLoS paths can be rewritten as

$$\begin{align*} f_{n,m}^{\text {NLoS}}(t)=&\frac {{v^{\text {MT,}{\mathbf {S}}_{n}^{\text {MT}}}}(t)}{\lambda }\cos (\alpha _{n,m}^{\text {MT}}(t)-\alpha _{v}^{\text {MT,}{\mathbf {S}}_{n}^{\text {MT}}}(t)) \\&+\,\frac {{v^{\text {MR,}{\mathbf {S}}_{n}^{\text {MR}}}}(t)}{\lambda }\cos (\alpha _{n,m}^{\text {MR}}(t)-\alpha _{v}^{\text {MR,}{\mathbf {S}}_{n}^{\text {MR}}}(t)) \\=&f_{n,m}^{\text {MT,}{\mathbf {S}}_{n}^{\text {MT}}}(t)+f_{n,m}^{\text {MR,}{\mathbf {S}}_{n}^{\text {MR}}}(t). \tag{12}\end{align*}$$ We can expand $\cos (\alpha _{n,m,v}^{i}(t))$ by the Taylor series expansion at $t={t_{0}}$ as (13), shown at the top of this page,

where $\alpha _{n,m,v}^{i}(t)=\alpha _{n,m}^{i}(t)-\alpha _{v}^{i,\mathbf {S}_{n,m}^{i}}(t)$ is defined for brief. Using the first two items of the Taylor series in (13), $\cos (\alpha _{n,m,v}^{i}(t))$ can be approximated as $$\begin{equation*} \cos (\alpha _{n,m,v}^{i}(t))=\cos (\alpha _{n,m,v}^{i}({t_{0}}))+{c^{i}}({t_{0}})\cdot (t-{t_{0}}) \tag{14}\end{equation*}$$ where $$\begin{align*} {c^{i}}({t_{0}})\!=\!-\frac {{v^{i,\mathbf {S}_{n}^{i}}}({t_{0}})}{d_{n}^{i}({t_{0}})}\cdot {\sin ^{2}}(\alpha _{n,m,v}^{i}({t_{0}}))+b_{}^{i,\mathbf {S}_{n}^{i}}\cdot \sin (\alpha _{n,m,v}^{i}({t_{0}})). \\ \tag{15}\end{align*}$$ Substituting (14) into (5), the Doppler frequency at the MT or MR can be expressed by (16), shown at the top of this page.

Note that (16) explicitly reveals the impact of velocity variations on Doppler frequency. Moreover, the result derived in [23] can be viewed as the special case of $f_{n,m}^{i,\mathbf {S}_{n}^{i}}(t)$ at $t=0$ . Substituting (12) and (16) into (3), $h_{n,m}^{\text {NLoS}}(t)$ of the proposed model can be simplified as $$\begin{align*}&\hspace {-0.6pc}h_{n,m}^{\text {NLoS}}(t)=\frac {1}{\sqrt {M}}\sum \limits _{m=1}^{M}{{e^{\text {j}\cdot 2\pi (A_{n,m}^{\text {NLoS}}({t_{0}})\cdot {t^{3}}+B_{n,m}^{\text {NLoS}}({t_{0}})\cdot {t^{2}})}}} \\&\qquad\qquad\qquad\qquad\qquad\displaystyle {\cdot \, {e^{\text {j}\cdot ({2}\pi \cdot {(}C_{n,m}^{\text {NLoS}}({t_{0}})\cdot t+D_{n,m}^{\text {NLoS}}({t_{0}}))+{\theta _{n,m}})}}} \tag{17}\end{align*}$$ where $$\begin{align*}&\hspace {-1.2pc}A_{n,m}^{\text {NLoS}}({t_{0}}) \\[-2pt]=&\frac {{a^{\text {MT,}\mathbf {S}_{n}^{\text {MT}}}}({t_{0}}){c^{\text {MT}}}({t_{0}})}{3\lambda }+\frac {{a^{\text {MR,}\mathbf {S}_{n}^{\text {MR}}}}({t_{0}}){c^{\text {MR}}}({t_{0}})}{3\lambda } \tag{18a}\\[-2pt]&\hspace {-1.2pc}B_{n,m}^{\text {NLoS}}({t_{0}}) \\[-2pt]=&\frac {{a^{\text {MT,}\mathbf {S}_{n}^{\text {MT}}}}({t_{0}})\cdot \cos (\alpha _{n,m,v}^{\text {MT}}({t_{0}})-{c^{\text {MT}}}({t_{0}})\cdot {t_{0}})}{2\lambda } \\[-2pt]&+\,\frac {{a^{\text {MR,}\mathbf {S}_{n}^{\text {MR}}}}({t_{0}})\cdot \cos (\alpha _{n,m,v}^{\text {MR}}({t_{0}})-{c^{\text {MR}}}({t_{0}})\cdot {t_{0}})}{2\lambda } \\[-2pt]&+\,\frac {{v^{\text {MT,}\mathbf {S}_{n}^{\text {MT}},a}}({t_{0}})\cdot {c^{\text {MT}}}({t_{0}})}{2\lambda }+\frac {{v^{\text {MR,}\mathbf {S}_{n}^{\text {MR}},a}}({t_{0}})\cdot {c^{\text {MR}}}({t_{0}})}{2\lambda } \\[-2pt]\tag{18b}\\[-2pt]&\hspace {-1.2pc}C_{n,m}^{\text {NLoS}}({t_{0}}) \\[-2pt]=&\frac {{v^{\text {MT,}\mathbf {S}_{n}^{\text {MT}},a}}({t_{0}})\cdot \cos (\alpha _{n,m,v}^{\text {MT}}({t_{0}})-{c^{\text {MT}}}({t_{0}})\cdot {t_{0}})}{\lambda } \\[-2pt]&+\,\frac {{v^{\text {MR,}\mathbf {S}_{n}^{\text {MR}},a}}({t_{0}})\cdot \cos (\alpha _{n,m,v}^{\text {MR}}({t_{0}})-{c^{\text {MR}}}({t_{0}})\cdot {t_{0}})}{\lambda } \tag{18c}\\[-2pt]&\hspace {-1.2pc}D_{n,m}^{\text {NLoS}}({t_{0}}) \\[-2pt]=&-\left({\frac {{a^{\text {MT,}\mathbf {S}_{n}^{\text {MT}}}}({t_{0}})\cdot {c^{\text {MT}}}({t_{0}})}{3\lambda }\!+\!\frac {{a^{\text {MR,}\mathbf {S}_{n}^{\text {MR}}}}({t_{0}})\cdot {c^{\text {MR}}}({t_{0}})}{3\lambda }}\right)\!\cdot \! {t_{0}}^{3} \\[-2pt]&-\,\big(\frac {{a^{\text {MT,}\mathbf {S}_{n}^{\text {MT}}}}({t_{0}})\cdot (\cos (\alpha _{n,m,v}^{\text {MT}}({t_{0}}))-{c^{\text {MT}}}({t_{0}})\cdot {t_{0}})}{2\lambda } \\[-2pt]&+\,\frac {{a^{\text {MR,}\mathbf {S}_{n}^{\text {MR}}}}({t_{0}})\cdot \cos (\alpha _{n,m,v}^{\text {MR}}({t_{0}})-{c^{\text {MR}}}({t_{0}})\cdot {t_{0}})}{2\lambda } \\[-2pt]&+\,\frac {{v^{\text {MT,}\mathbf {S}_{n}^{\text {MT}},a}}({t_{0}})\cdot {c^{\text {MT}}}({t_{0}})}{2\lambda }+\frac {{v^{\text {MR,}\mathbf {S}_{n}^{\text {MR}},a}}({t_{0}})\cdot {c^{\text {MR}}}({t_{0}})}{2\lambda }\big)\cdot {t_{0}}^{2} \\[-2pt]&-\,\big(\frac {{v^{\text {MT,}\mathbf {S}_{n}^{\text {MT}},a}}({t_{0}})\cdot (\cos (\alpha _{n,m,v}^{\text {MT}}({t_{0}}))-{c^{\text {MT}}}({t_{0}})\cdot {t_{0}})}{\lambda } \\[-2pt]&+\,\frac {{v^{\text {MR,}\mathbf {S}_{n}^{\text {MR}},a}}({t_{0}})\cdot (\cos \alpha _{n,m,v}^{\text {MR}}({t_{0}})-{c^{\text {MR}}}({t_{0}})\cdot {t_{0}})}{\lambda }\big)\cdot {t_{0}} \\{}\tag{18d}\end{align*}$$ where ${v^{i,\mathbf {S}_{n}^{i},a}}(t)={v^{i,\mathbf {S}_{n}^{i}}}(t)-{a^{i,\mathbf {S}_{n}^{i}}}(t)\cdot t$ is defined for brief. In the similar way, the Doppler frequency of LoS path can also be approximated by the Taylor series expansion as $$\begin{align*}&\hspace {-1pc}{f^{\text {LoS,}i}}(t) \\\approx&\frac {{a^{i}}({t_{0}})\cdot {c^{\text {LoS,}i}}({t_{0}})}{\lambda }{t^{2}}\!+\!\big(\frac {({v^{i}}({t_{0}})-{a^{i}}({t_{0}})\cdot {t_{0}})\cdot {c^{\text {LoS,}i}}({t_{0}})}{\lambda } \\&+\,\frac {({a^{i}}({t_{0}})\cdot (\cos ({\alpha ^{\text {LoS,}i}}({t_{0}})-\alpha _{v}^{i}({t_{0}}))-{c^{\text {LoS,}i}}({t_{0}})\cdot {t_{0}}))}{\lambda }\big)\cdot t \\&+\,\!\frac {(\!{v^{i}}({t_{0}})\!-\!{a^{i}}({t_{0}})\!\cdot \! {t_{0}})\!\cdot \! (\!\cos ({\alpha ^{\text {LoS,}i}}({t_{0}})\!-\!\alpha _{v}^{i}({t_{0}}))\!-\!{c^{\text {LoS,}i}}({t_{0}})\!\cdot \! {t_{0}})}{\lambda } \\ \tag{19}\end{align*}$$ where $$\begin{align*}&\hspace {-0.6pc}{c^{\text {LoS,}i}}({t_{0}})=-\frac {{v^{\text {MT,MR}}}({t_{0}})}{{d^{\text {LoS}}}({t_{0}})}\cdot {\sin ^{2}}(\alpha _{}^{\text {LoS,}i}({t_{0}})-\alpha _{v}^{i}({t_{0}})) \\&\qquad\qquad\qquad\qquad\qquad\displaystyle {+\,{b^{\text {MT,MR}}}\cdot \sin (\alpha _{}^{\text {LoS,}i}({t_{0}})-\alpha _{v}^{i}({t_{0}}))} \tag{20}\end{align*}$$ and then ${h^{\text {LoS}}}(t)$ can be obtained as $$\begin{align*}&\hspace {-0.6pc}{h^{\text {LoS}}}(t,\tau)={{\text {e}}^{\text {j}\cdot {2}\pi \cdot ({A^{\text {LoS}}}({t_{0}})\cdot {t^{3}}+{B^{\text {LoS}}}({t_{0}})\cdot {t^{2}}{)}}} \\&\qquad\qquad\qquad\qquad\qquad\displaystyle {\cdot \, {{\text {e}}^{\text {j}\cdot \text {(2}\pi \cdot ({C^{\text {LoS}}}({t_{0}})\cdot t+{D^{\text {LoS}}}({t_{0}}))+{\theta ^{\text {LoS}}}{)}}}} \tag{21}\end{align*}$$ where $$\begin{align*}&\hspace {-1.2pc}{A^{\text {LoS}}}({t_{0}}) \\=&\frac {{a^{\text {MT}}}({t_{0}})\cdot {c^{\text {LoS,MT}}}({t_{0}})}{3\lambda }+\frac {{a^{\text {MR}}}({t_{0}})\cdot {c^{\text {LoS,MR}}}({t_{0}})}{3\lambda } \tag{22a}\\&\hspace {-1.2pc}{B^{\text {LoS}}}({t_{0}}) \\=&\frac {{a^{\text {MT}}}({t_{0}})\cdot (\cos (\alpha _{v}^{\text {LoS,MT}}({t_{0}}))-{c^{\text {LoS,MT}}}({t_{0}})\cdot {t_{0}})}{2\lambda } \\&+\,\frac {{a^{\text {MR}}}({t_{0}})\cdot (\cos (\alpha _{v}^{\text {LoS,MR}}({t_{0}}))-{c^{\text {LoS,MR}}}({t_{0}})\cdot {t_{0}})}{2\lambda } \\&+\,\frac {{v^{\text {MT},a}}({t_{0}})\!\cdot \! {c^{\text {LoS,MT}}}({t_{0}})}{2\lambda }\!+\!\frac {{v^{\text {MR},a}}({t_{0}})\!\cdot \! {c^{\text {LoS,MR}}}({t_{0}})}{2\lambda } \tag{22b}\\&\hspace {-1.2pc}{C^{\text {LoS}}}({t_{0}}) \\=&\frac {{v^{\text {MT},a}}({t_{0}})\cdot (\cos (\alpha _{v}^{\text {LoS,MT}}({t_{0}}))-{c^{\text {LoS,MT}}}({t_{0}})\cdot {t_{0}})}{\lambda } \\&+\,\frac {{v^{\text {MR},a}}({t_{0}})\!\cdot \! (\cos (\alpha _{v}^{\text {LoS,MR}}({t_{0}}))\!-\!{c^{\text {LoS,MR}}}({t_{0}})\!\cdot \! {t_{0}})}{\lambda } \tag{22c}\\&\hspace {-1.2pc}{D^{\text {LoS}}}({t_{0}}) \\=&-\left({\frac {{a^{\text {MT}}}({t_{0}})\cdot {c^{\text {LoS,MT}}}({t_{0}})}{3\lambda }+\frac {{a^{\text {MR}}}({t_{0}})\cdot {c^{\text {LoS,MR}}}({t_{0}})}{3\lambda }}\right)\cdot {t_{0}}^{3} \\&-\,\big(\frac {{a^{\text {MT}}}({t_{0}})\cdot (\cos (\alpha _{v}^{\text {LoS,MT}}({t_{0}}))-{c^{\text {LoS,MT}}}({t_{0}})\cdot {t_{0}})}{2\lambda } \\&+\,\frac {{a^{\text {MR}}}({t_{0}})\cdot (\cos (\alpha _{v}^{\text {LoS,MR}}({t_{0}}))-{c^{\text {LoS,MR}}}({t_{0}})\cdot {t_{0}})}{2\lambda } \\&+\,\frac {{v^{\text {MT},a}}({t_{0}})\cdot {c^{\text {LoS,MT}}}({t_{0}})}{2\lambda }+\frac {{v^{\text {MR},a}}({t_{0}})\cdot {c^{\text {LoS,MR}}}({t_{0}})}{2\lambda }\big)\cdot {t_{0}}^{2} \\&-\,\big(\frac {{v^{\text {MT},a}}({t_{0}})\cdot (\cos (\alpha _{v}^{\text {LoS,MT}}({t_{0}}))-{c^{\text {LoS,MT}}}({t_{0}})\cdot {t_{0}})}{\lambda } \\&+\,\frac {{v^{\text {MR},a}}({t_{0}})\!\cdot \! (\cos (\alpha _{v}^{\text {LoS,MR}}({t_{0}}))\!-\!{c^{\text {LoS,MR}}}({t_{0}})\!\cdot \! {t_{0}})}{\lambda }\big)\!\cdot \! {t_{0}} \\{}\tag{22d}\end{align*}$$ where ${v^{i,a}}(t)={v^{i}}(t)-{a^{i}}(t)\cdot t$ and $\alpha _{v}^{\text {LoS,}i}(t)={\alpha ^{\text {LoS,}i}}(t)-\alpha _{v}^{i}(t)$ are defined for brief. Finally, our proposed V2V channel model can be obtained by substituting (17) and (21) into (11). It’s worth mentioning that the proposed model can be easily extended to multiple-input multiple-output (MIMO) channels, while this paper only focuses on single-input single-output (SISO) channels and their statistical properties.

SECTION IV. Time Evolving of Channel Parameters

A. Time-Variant Distances

Since the clusters and terminals are moving, the distance between the MT and $\mathbf {S}_{n}^{\text {MT}}$ or between the MR and $\mathbf {S}_{n}^{\text {MR}}$ , denoted by $d_{n}^{i}(t), {}i\in \{\text {MT,MR}\}$ , depends on the relative movement between terminals and clusters. Suppose that the clusters $\mathbf {S}_{n}^{\text {MT}}$ or $\mathbf {S}_{n}^{\text {MR}}$ is the coordinate origins when we consider the relative geometry relationship between the MT and $\mathbf {S}_{n}^{\text {MT}}$ or between the $\mathbf {S}_{n}^{\text {MR}}$ and MR. The relative coordinate location of the MT or MR is $[d_{n}^{i}({t_{0}})\cdot \cos (\bar {\alpha }_{n}^{i}({t_{0}})) - \Delta L_{x}^{i}(t),d_{n}^{i}({t_{0}}) \cdot \sin (\bar {\alpha }_{n}^{i}({t_{0}})) - \Delta L_{y}^{i}(t)]$ and the Euclid distance between the MT and $\mathbf {S}_{n}^{\text {MT}}$ or the MR and $\mathbf {S}_{n}^{\text {MR}}$ can be expressed as (23), shown at the top of this page,

where $\bar {\alpha }_{n}^{i}({t_{0}})$ denotes the initial mean value of AoD or AoA at $t={t_{0}}$ , $d_{n}^{i}({t_{0}})$ denotes the initial distance between the MT and $\mathbf {S}_{n}^{\text {MT}}$ ((or between the MR and $\mathbf {S}_{n}^{\text {MR}}$ ) at $t={t_{0}}$ , $\Delta L_{x}^{i}(t)$ and $\Delta L_{y}^{i}(t)$ mean the distances of travel on $x$ axis and $y$ axis, respectively, and they can be further defined as $$\begin{align*} \Delta L_{x}^{i}(t)=&\int _{{t_{0}}}^{t}{}({v^{i,\mathbf {S}_{n}^{i}}}({t_{0}})+{a^{i,\mathbf {S}_{n}^{i}}}(t'-{t_{0}})) \\&\cdot \, \cos (\alpha _{v}^{i,\mathbf {S}_{n}^{i}}({t_{0}})+{b^{i,\mathbf {S}_{n}^{i}}}(t'-{t_{0}})){\text {d}}t' \tag{24}\\ \Delta L_{y}^{i}(t)=&\int _{{t_{0}}}^{t}{}({v^{i,\mathbf {S}_{n}^{i}}}({t_{0}})+{a^{i,\mathbf {S}_{n}^{i}}}(t'-{t_{0}})) \\&\cdot \, \sin (\alpha _{v}^{i,\mathbf {S}_{n}^{i}}({t_{0}})+{b^{i,\mathbf {S}_{n}^{i}}}(t'-{t_{0}})){\text {d}}t' \tag{25}\end{align*}$$ where the relative movement reduces to a constant direction when ${b^{i,\mathbf {S}_{n}^{i}}}=0$ . In order to simply the calculation, we expand $d_{n}^{i}(t)$ by the Taylor series expansion at $t={t_{0}}$ and take the first two terms as $$\begin{align*} d_{n}^{i}(t)=d_{n}^{i}({t_{0}})-{v^{i,\mathbf {S}_{n}^{i}}}({t_{0}})\cdot \cos (\bar {\alpha }_{n}^{i,\mathbf {S}_{n}^{i}}({t_{0}})-\alpha _{v}^{i,\mathbf {S}_{n}^{i}}({t_{0}}))\!\cdot \! (t-{t_{0}}).\!\!\!\! \\ \tag{26}\end{align*}$$ Moreover, the Euclid distance between the MT and MR is defined as (27), shown at the top of this page,

where ${d^{\text {LoS}}}({t_{0}})$ and ${\alpha ^{\text {LoS,MT}}}({t_{0}})$ denote the initial value of distance and AoD in LoS path at $t={t_{0}}$ , respectively, $\Delta {L_{x}}^{\text {MT,MR}}(t)$ and $\Delta {L_{y}}^{\text {MT,MR}}(t)$ represent the distances of travel between the MT and MR on $x$ axis and $y$ axis, $$\begin{align*} \Delta L_{x}^{\text {MT,MR}}(t)=&\int _{{t_{0}}}^{t}{}({v^{\text {MT,MR}}}({t_{0}})+{a^{\text {MT,MR}}}(t'-{t_{0}})) \\&\cdot \, \cos (\alpha _{v}^{\text {MT,MR}}({t_{0}})+{b^{\text {MT,MR}}}(t'\!\!-{t_{0}})){\text {d}}t' \qquad ~\tag{28}\\ \Delta L_{y}^{\text {MT,MR}}(t)=&\int _{{t_{0}}}^{t}{}({v^{\text {MT,MR}}}({t_{0}})+{a^{\text {MT,MR}}}(t'-{t_{0}})) \\&\cdot \, \sin (\alpha _{v}^{\text {MT,MR}}({t_{0}})+{b^{\text {MT,MR}}}(t'\!-{t_{0}})){\text {d}}t'\qquad \tag{29}\end{align*}$$ where ${v^{\text {MT,MR}}}({t_{0}})$ and $\alpha _{v}^{\text {MT,MR}}({t_{0}})$ denote the initial speed and moving direction of the relative velocity vector between the MT and MR as ${{\mathbf {v}}^{\text {MT,MR}}}(t)={{\mathbf {v}}^{\text {MT}}}(t)-{{\mathbf {v}}^{\text {MR}}}(t)={v^{\text {MT,MR}}}(t)\cdot {{\text {e}}^{\text {j}\cdot \alpha _{v}^{\text {MT,MR}}(t)}}$ . Similarly, ${d^{\text {LoS}}}(t)$ can be approximated by the Taylor series expansion at $t={t_{0}}$ as (30), shown at the top of this page.

B. Time-Variant AOAS and AODS

With the movements of terminals and clusters, the AoAs and AoDs are also time-variant. For the LoS path, the angles can be tracked by the geometrical relationships directly. However, the angles of NLoS path are random and difficult to track. In this paper, we model the AoAs and AoDs of NLoS path as

$$\begin{equation*} \alpha _{n,m}^{i}(t)=\bar {\alpha }_{n}^{i}(t)+\Delta {\alpha ^{i}} \tag{31}\end{equation*}$$ where $\bar {\alpha }_{n}^{i}(t)$ and $\Delta {\alpha ^{i}}$ denote the mean angle and random angle offset at the MT or MR. Measurement results [5] have revealed that the von Mises distribution is flexible and can be used to approximate many distributions, e.g., uniform and Gaussian. Besides, the VM distribution admits closed-form solutions for many useful statistical properties. The VM probability distribution function (PDF) is given by $$\begin{equation*} {P_{\Delta \alpha _{}^{i}}}(\alpha)=\frac {\exp ({\kappa ^{i}}\cos ({\alpha ^{i}}-{{{\bar {\alpha }}}^{i}}))}{2\pi {{\mathrm {I}}_{0}}({\kappa ^{i}})} \tag{32}\end{equation*}$$ where ${{\text {I}}_{0}}(\cdot)$ is the zeroth-order modified Bessel function of the first kind, ${{\bar {\alpha }}^{i}}$ denotes the mean angle, and ${\kappa ^{i}}$ means the factor related with the concentration of distribution. When ${\kappa ^{i}}=0$ , the VM PDF reduces to the classic uniform distribution.

In this paper, the VM distribution is adopted to generate initial angle offsets, while ${\kappa ^{i}}$ can be obtained by measurement results. Since the distribution is approximately unchanged during the simulation time, we only need to track the mean AoAs and AoDs. According to the geometrical relationship, we can obtain

$$\begin{equation*} \bar {\alpha }_{n}^{i}(t)=\begin{cases} \arccos \left({\dfrac {d_{n}^{i}({t_{0}})\cdot \cos (\bar {\alpha }_{n}^{i}({t_{0}}))-\Delta {L_{x}}^{i}(t)}{d_{n}^{i}(t)}}\right), \\ \quad ~ d_{n}^{i}({t_{0}})\cdot \cos (\bar {\alpha }_{n}^{i}({t_{0}}))-\Delta {L_{x}}^{i}(t)>0 \\ \pi -\arccos \left({\dfrac {d_{n}^{i}({t_{0}})\cdot \cos (\bar {\alpha }_{n}^{i}({t_{0}}))-\Delta {L_{x}}^{i}(t)}{d_{n}^{i}(t)}}\right), \\ \quad d_{n}^{i}({t_{0}})\cdot \cos (\bar {\alpha }_{n}^{i}({t_{0}}))-\Delta {L_{x}}^{i}(t) < 0 \\ \end{cases}~~ \tag{33}\end{equation*}$$ where $\bar {\alpha }_{n}^{i}({t_{0}})$ denotes the initial value at $t={t_{0}}$ . In order to make the calculation more tractable, we expand $\bar {\alpha }_{n}^{i}(t)$ by the Taylor series expansion at $t={t_{0}}$ and take the first two terms as $$\begin{equation*} \bar {\alpha }_{n}^{i}(t)\approx \bar {\alpha }_{n}^{i}({t_{0}})+g_{0}^{i}({t_{0}})\cdot (t-{t_{0}}) \tag{34}\end{equation*}$$ where $$\begin{equation*} g_{0}^{i}({t_{0}})=\begin{cases} -\dfrac {v_{0}^{i}({t_{0}})}{{d^{i,\mathbf {S}_{n}^{i}}}({t_{0}})}\cdot \sin (\bar {\alpha }_{n}^{i}({t_{0}})+\alpha _{v}^{i}({t_{0}})), \\ \quad {d^{i,\mathbf {S}_{n}^{i}}}({t_{0}})\cdot \cos (\bar {\alpha }_{n}^{i}({t_{0}}))-\Delta {L_{x}}^{i}({t_{0}})>0 \\ \dfrac {v_{0}^{i}({t_{0}})}{{d^{i,\mathbf {S}_{n}^{i}}}({t_{0}})}\cdot \sin (\bar {\alpha }_{n}^{i}({t_{0}})+\alpha _{v}^{i}({t_{0}})), \\ \quad {d^{i,\mathbf {S}_{n}^{i}}}({t_{0}})\cdot \cos (\bar {\alpha }_{n}^{i}({t_{0}}))-\Delta {L_{x}}^{i}({t_{0}}) < 0. \end{cases}~~ \tag{35}\end{equation*}$$

For the LoS path, the AoAs and AoDs can be tracked by

$$\begin{equation*} {\alpha ^{\text {LoS,}i}}(t)\approx {\alpha ^{\text {LoS,}i}}({t_{0}})+g_{0}^{\text {LoS,}i}({t_{0}})\cdot (t-{t_{0}}) \tag{36}\end{equation*}$$ where $$\begin{align*} g_{0}^{\text {LoS,}i}({t_{0}})\!=\!\begin{cases}\! -\dfrac {{v^{\text {MT,MR}}}({t_{0}})}{{d^{\text {LoS}}}({t_{0}})}\cdot \sin ({\alpha ^{\text {LoS,}i}}({t_{0}})+\alpha _{v}^{\text {MT,MR}}({t_{0}})), \\ \quad {d^{\text {LoS}}}({t_{0}})\cdot \cos ({\alpha ^{\text {LoS,}i}}({t_{0}}))-\Delta L_{x}^{\text {MT,MR}}({t_{0}})\!>\!0 \\ \dfrac {{v^{\text {MT,MR}}}({t_{0}})}{{d^{\text {LoS}}}({t_{0}})}\cdot \sin ({\alpha ^{\text {LoS,}i}}({t_{0}})+\alpha _{v}^{\text {MT,MR}}({t_{0}})), \\ \quad {d^{\text {LoS}}}({t_{0}})\cdot \cos ({\alpha ^{\text {LoS,}i}}({t_{0}}))-\Delta L_{x}^{\text {MT,MR}}({t_{0}})\! < \!0 \end{cases}\!\!\!\!\!\!\!\!\!\!\! \\ \tag{37}\end{align*}$$ and ${\alpha ^{\text {LoS,}i}}({t_{0}})$ denotes the initial value at $t={t_{0}}$ .

C. Time-Variant Path Delays and Powers

For the $n$ th NLoS path at time instant $t$ , the total delay includes the first bounce $\tau _{n}^{\text {MT}}(t)$ , virtual link ${{\tilde {\tau }}_{n}}(t)$ , and the last bounce $\tau _{n}^{\text {MR}}(t)$ . Thus, it can be calculated by (38), shown at the top of this page,

where c denotes the speed of light, ${{\tilde {\tau }}_{n}}(t)$ denotes the equivalent delay of virtual link and it can be updated by a first-order filtering method as ${{\tilde {\tau }}_{n}}(t-\Delta t)\cdot {{\text {e}}^{-\frac {\Delta t}{{\tau _{\text {dec}}}}}}+\left({1-{{\text {e}}^{-\frac {\Delta t}{{\tau _{\text {dec}}}}}}}\right)\cdot X$ [25], where $X\!~\sim ~\!U\left [{ {{d^{\text {LoS}}}(t)}/{\text {c}}\;,{\tau _{\text {max}}} }\right]$ , ${\tau _{\text {max}}}$ is the maximum delay, ${\tau _{\text {dec}}}$ denotes the decorrelation speed of time-variant delays, which is environment dependent and can be obtained by measurement campaigns. In addition, the time delay of LoS path can be approximately calculated by (33) as $$\begin{align*}&\hspace {-0.8pc}\tau _{n}^{\text {LoS}}(t) \\\approx&\frac {{d^{{\!\text {LoS}}}}({t_{0}}\!)\!-\!{v^{\text {MT,MR}}}({t_{0}}\!)\!\cdot \! \cos ({\alpha ^{{\!\text {LoS,MT}}}}({t_{0}}\!)\!-\!\alpha _{v}^{\text {MT,MR}}({t_{0}}\!))\!\cdot \! (t\!-\!{t_{0}}\!)}{\text {c}}. \\ \tag{39}\end{align*}$$ The powers of LoS and NLoS paths can be obtained by using the single slope exponential power delay model as $$\begin{align*} {P^{\text {LoS}}}(t)=&{{\text {e}}^{-\tau _{}^{\text {LoS}}(t)\cdot \frac {{\gamma _{\tau }}-1}{{\gamma _{\tau }}\cdot {\sigma _{\tau }}}}}\cdot {10^{-\frac {{\xi _{n}}}{10}}}. \tag{40}\\ {P_{n}^{\text {NLoS}}}'(t)=&{{\text {e}}^{-\tau _{n}^{\text {NLoS}}(t)\cdot \frac {{\gamma _{\tau }}-1}{{\gamma _{\tau }}\cdot {\sigma _{\tau }}}}}\cdot {10^{-\frac {{\xi _{n}}}{10}}} \tag{41}\end{align*}$$ where ${\xi _{n}}$ , ${\gamma _{\tau }}$ , and ${\sigma _{\tau }}$ denote the shadow term, delay distribution, and delay spread, respectively. Note that this model was adopted to calculate the powers of LoS and NLoS paths under V2V scenarios in [26], where the velocities of both terminals and scatterers are time-invariant. In our model, these environment dependent factors can be viewed unchanged within each segment, but for different segments we need different factors according to measurement results.

SECTION V. Statistical Properties of Proposed Model

A. Time-Variant TCF

The normalized TCF for non-stationary V2V channels should be time-variant and can be defined as [19]

$$\begin{equation*} \rho (t,\Delta t)=\frac {\text {E}\left [{ h(t){h^{*}}(t+\Delta t) }\right]}{\sqrt {\text {E}\left [{ {\left |{ h(t) }\right |^{2}} }\right]\cdot \text {E}\left [{ {{\left |{ {h^{*}}(t+\Delta t) }\right |}^{2}} }\right]}} \tag{43}\end{equation*}$$ where $\text {E}\left [{ \cdot }\right]$ represents the expectation function, ${()^{*}}$ is complex conjugate, $\Delta t$ is the time lag. Moreover, since the LoS and NLoS paths are usually uncorrelated, the total TCF can be rewritten as $$\begin{equation*} \rho (t,\Delta t)={\rho ^{\text {LoS}}}(t,\Delta t)+\rho _{n}^{\text {NLoS}}(t,\Delta t) \tag{44}\end{equation*}$$ where ${\rho ^{\text {LoS}}}(t,\Delta t)$ and $\rho _{n}^{\text {NLoS}}(t,\Delta t)$ denote the TCFs of LoS and NLoS paths, respectively. Substituting (21) into (43), the close-form TCF of LoS path can be obtained as (45), shown at the top of this page.

For the NLoS path, the TCF can be expressed as (46), shown at the top of this page,

where ${p_{\alpha _{n,m}^{\text {MT}},\alpha _{n,m}^{\text {MR}}}}(\alpha _{n}^{\text {MT}}(t),\alpha _{n}^{\text {MR}}(t))$ denotes the time-variant joint distribution of random angles, i.e., AoD $\alpha _{n}^{\text {MT}}(t)$ and AoA $\alpha _{n}^{\text {MR}}(t)$ . It should be mentioned that the TCFs in (46) is general and also suitable for different propagation conditions, i.e., single-bounce, and multiple-bounce paths. For general multi-bounce NLoS scenarios, the AoA and AoD are usually assumed independent. For the VM distribution, ${\kappa ^{i}}$ is almost constant during short analytical time intervals (several tens of milliseconds), angle offsets can be approximately time-invariant, i.e., $\alpha _{n}^{i}(t+\Delta t)-\bar {\alpha }_{n}^{i}(t+\Delta t)\approx \alpha _{n}^{i}(t)-\bar {\alpha }_{n}^{i}(t)$ . Holding this condition, it yields (47), shown at the top of this page.

B. Time-Variant DPSD

The time-variant DPSD $S(f,t)$ of the proposed model is defined by the Fourier transform of $\rho (t,\Delta t)$ with the respect to the time interval $\Delta t$ , which can be expressed as

$$\begin{equation*} S(f,t)=\int _{-\infty }^{\infty }{}\rho (t,\Delta t)\cdot {{\text {e}}^{-\text {j}\cdot {2}\pi \cdot f\cdot \Delta t}}\text {d}\Delta t. \tag{48}\end{equation*}$$ Note that the DPSD is also a function of time.

C. Time-Variant LCR and AFD

The time-variant LCR describes the number of times when the envelope crosses a certain threshold and can be calculated as [26]

$$\begin{align*} N(r,t)=&\frac {2r}{{{\pi ^{3/2}}}}\sqrt {\frac {B(t)(K(t) + 1)}{b_{0}}} {\text {e}^{ - K(t) - (K(t) + 1){r^{2}}}} \\&\cdot \, \!\int _{0}^{\pi /2} {\cosh (2\sqrt {K(t)(K(t) + 1)} r\cos \theta)} \\&\cdot \, \!\left [{\! {{{\text {e}}^{ \!- {(\nu (t)\sin \theta)^{2}}}} \!+\! \!\sqrt \pi \nu (t)\sin \theta {\text {erf}}(\nu (t)\!\sin \theta)} \!}\right]\!{\text {d}}\theta \qquad ~~ \tag{49}\end{align*}$$ where $K(t)={{P^{\text {LoS}}}(t)}/{{P^{\text {NLoS}}}(t)}$ ;, $\cosh (\cdot)$ is the hyperbolic cosine function, and $\text {erf(}\cdot {)}$ represents the error function. The parameters $\nu (t)$ and $B(t)$ can be respectively expressed as $$\begin{align*} \nu (t)=&\sqrt {{\frac{{K \cdot {b_{1}}{(t)^{2}}} }{ {b_{0} \cdot {b_{2}}(t) - {b_{1}}{(t)^{2}}}}}} \tag{50}\\ B(t)=&{b_{2}}(t){b_{0}}-{b}_{1}^{2}(t) \tag{51}\end{align*}$$ where $$\begin{equation*} {b_{m}}(t)={{\left.{ \frac {{{\text {d}}^{m}}\rho \left ({t,\Delta t }\right)}{{{\text {j}}^{m}}{\text {d}}\Delta {t^{m}}} }\right |}_{\Delta t=0}}. \tag{52}\end{equation*}$$ The time-variant AFD denotes the time duration of envelope staying below a certain threshold $r$ and can be proved as [27] $$\begin{equation*} L(r,t)=\frac {1-Q(\sqrt {2K(t)},\sqrt {2(K(t)+1){r^{2}}})}{N(r,t)} \tag{53}\end{equation*}$$ where $Q(\cdot, \cdot)$ represents the Marcum Q function.

SECTION VI. Simulation Results

In this section, the proposed model is simulated under four typical V2V scenarios and verified by comparing with the theoretical and measured results. In the simulation, we consider four trajectories of the MT and MR, i.e., the same direction, the opposite direction, single corner, and double corners, as shown in Fig. 2. The AoAs and AoDs are assumed to follow the VM distribution with $\kappa =1$ and the carrier frequency is 2.48 GHz. The moving scatterers are assumed uniformly distributed around two terminals and the random parameters are as follows, ${d_{n}^{i}}\sim \text {U(}5,30\text {) m}$ , $\left \|{ {{\mathbf {v}}^{\mathbf {S}_{n}^{i}}}(t) }\right \|\sim \text {U(}0,1{) }{\text {m}}/{\text {s}}$ ;, and $\alpha _{v}^{\mathbf {S}_{n}^{i}}(t)\sim \text {U(}-\pi,\pi {)}$ . The MT and MR are moving with different trajectories under four scenarios and the detailed velocity parameters can be found in Table 2.

TABLE 2 Velocity Parameters of the MT and MR

FIGURE 2.

Four different trajectories of the MT and MR.

In order to observe the impact of velocity variations on the Doppler frequencies under different scenarios. We take the LoS path as an example and calculate the theoretical Doppler frequencies by using (4). Meanwhile, the Doppler frequencies of proposed model and the model in [23] are also calculated and compared in Fig. 3. It is clearly shown that the simulated Doppler frequencies of our proposed model fit well with the theoretical ones, while the ones of [23] only does well during the initial time period. Actually, the model in [23] approximated the whole trajectory as one linear segment and it can be viewed as a special case of our method with ${t_{0}}=0$ . Moreover, it can be observed that the Doppler frequency changes with the time-variant mobility. The parameter of $\alpha _{v}^{i}$ affects the amplitude of Doppler frequency, while $b_{0}^{i}$ affects the variation trend of Doppler frequency.

FIGURE 3.

Theoretical and simulated Doppler frequencies.

By using (43)–​(47), the absolute values of TCFs at three time instants, i.e., $t=0\text {s, }2\text {s}$ and 5s are calculated and shown in Fig. 4. The simulated results of proposed model and the model in [23] are also given in Fig. 4, which are obtained by generating 1000000 channel fading samples and calculating the correlation between different time instants. The TCFs of our proposed model fit well with the simulated ones while the results of [23] only fit well with the simulated ones at $t=0 \text {s}$ . The reason is that our proposed model is expressed by the summation of several segments for the trajectory. It’s more accurate for arbitrary trajectories. However, the model in [23] approximated the trajectory by a linear model, which was only suitable for short time interval simulation. For validation purpose, the measured TCFs under Scenario II [7] are given in Fig. 4 and they also have a good agreement with our simulated results.

FIGURE 4.

Absolute values of theoretical, simulated, and measured TCFs at different time instants.

The theoretical and simulated DPSDs at three time instants, i.e., $t=0\text {s, 2s}$ and 5s are calculated and compared in Fig. 5. It is shown that the simulated results match well with the theoretical ones. The DPSD is composed of the Doppler frequencies of different rays. In Fig. 3, the Doppler frequency of our proposed model is more accurate than the one in [23], which indicates the DPSD of our proposed model should be better than the counterpart in [23]. The Doppler frequency shifting over time can be clearly observed and the maximum of Doppler shift in the scenario II is larger than the one in scenario I, which indicates the initial value of movement direction affects the Doppler shift. In addition, from the expression of (16) and (19) the other velocity parameters, i.e., $a_{0}^{i}$ , $b_{0}^{i}$ , and the initial values of speed would affect the Doppler shift, i.e., the maximum of Doppler shift, and the fading rate.

FIGURE 5.

Theoretical and simulated DPSDs at different time instants.

The theoretical and simulated LCRs and AFDs at three time instants, i.e., $t=0\text {s, 2s}$ and 5s are calculated and compared in Fig. 6 and Fig. 7, respectively. In Fig. 6, the simulated LCRs are calculated by counting the number of times the envelope crosses different threshold levels and being divided by the observation period. In Fig. 7, the simulated results are obtained by keeping track of all fading durations and being divided by the total number of fading. The good agreements between the theoretical and simulated results also show the correctness and generality of our proposed model.

FIGURE 6.

Theoretical and simulated LCRs at different time instants.

FIGURE 7.

Theoretical and simulated AFDs at different time instants.

SECTION VII. Conclusion

In this paper, we have proposed a new segment-based V2V GBSM, which allows velocity variations of terminals as well as moving scatterers. The channel parameters have been simplified by using Taylor series expansions, which could explicitly reveal the impact of velocity variations on the channel characteristics. Based on the proposed model, the Doppler frequency and the theoretical expressions of TCF, DPSD, LCR, and AFD have also been analyzed and derived. The good agreements between the theoretical, simulated, and measured statistical properties verify the correctness and generality of our proposed model. The proposed model can be used to effectively reproduce the realistic V2V channels for evaluating and optimizing the V2V communication systems.