I. INTRODUCTION
A. Backgrounds
AS the next generation of information and communication technology advances, the development of sixthgeneration (6G) networks is progressively emerging as a significant opportunity for national competition and social advancement [1], [2]. Simultaneously, a new phase of global scientific and technological revolution and industrial transformation is accelerating, with the widespread commercialization of new-generation information and communication technologies poised to generate vast amounts of transmitted data. This surge in global mobile data transmission poses considerable challenges. The expansion of higher frequencies is an inevitable trend, leading to the need for increased frequency bandwidth. This makes the communication and sensing to be integrated and symbiotic, evolving into the technical direction of integrated sensing and communication (ISAC), one of the leading trends of 6G technology [3]. Massive multiple-input multiple-output (MIMO) technology is expected to find extensive application in multidimensional and multiscale communication scenarios, leveraging its abundant frequency bandwidth resources, exceptionally high data transmission rates, and ease of integration with large-scale antenna arrays. Ultimately, this technology is poised to become a pivotal supporting element for future high-speed wireless communication systems [4], [5].
It is worth mentioning that channel models are of the significant importance to the design, optimization, and performance evaluation of air-to-ground (A2G) wireless communication systems. In light of this, it is critical to propose a physical dynamic channel model that can reflect faithfully the impacts of the physical characteristics of RIS array on the electromagnetic waves in a wireless environment between the Unmanned aerial vehicle (UAV) transmitter and ground receiver [6], [7]. To develop a more practical communication model for high-speed UAV wireless propagation systems, the modeling process must account for the highly dynamic, non-stationary motion trajectories of UAVs [8]. Due to the considerable mobility of UAVs, crucial parameters such as received signal power, time delay, and angle exhibit timevarying characteristics. This rapid time-varying behavior of parameters results in non-stationarity in the channel across both time and frequency domains [9]. The non-stationarity in the UAV communication channel across time and frequency domains implies rapid changes in the channel’s statistical properties. Consequently, channel statistics inherently vary over time in systems employing communication aided by advanced technologies. UAVs, with their multifunctionality, high mobility, ease of deployment, and cost-effectiveness, can function as auxiliary communication platforms in airspace. As 6G wireless networks advance, the importance of UAVto- vehicle communication is becoming increasingly prominent [10], [13].
B. Prior Works
Currently, numerous studies have explored channel modeling for UAV wireless communication systems, categorizing models into deterministic and statistical ones. Deterministic channel modeling typically involves extensive measurements or the use of ray-tracing to obtain corresponding channel parameters and is primarily suitable for specific scenarios [11], [12]. In [14], deterministic models established through raytracing can effectively analyze both narrowband and wideband channels, enabling the computation of specific channel impulse responses (CIRs) at the cost of computational complexity. In contrast, physical models have been widely used to describe IRS-assisted communication scenarios due to their higher accuracy and generality. In [15], a 3D non-stationary MIMO statistical communication model was introduced, examining the effects of UAV transmitter motion and altitude properties in 3D space on communication statistics. [16] presented a flexible 3D reference model with two cylinders and adjustable height for MIMO air-to-air (A2A) communications in UAV channels, covering a wide range of UAV-to-UAV communication scenarios. [17] introduced a 3D MIMO channel model for A2G communications in UAV environments, offering a new algorithm for angular estimation to evaluate future UAVMIMO communication systems performance. [18] proposed a 3D non-stationary physical model for A2A UAV communication scenarios, designed to adapt to various simulation environments with its random movement capability. In [19], UAV transmitters and ground vehicle receivers were strategically positioned at the focal points of a boundary ellipsoid, with numerical findings emphasizing the significant impact of receive antenna angles on the theoretical propagation characteristics of the A2G communication model.
However, obstructions such as trees or tall buildings can easily block signals in urban environments during UAVto- vehicle communications [20]. Intelligent reflecting surface (IRS) stands out in addressing these challenges due to its unique qualities of low cost, low energy consumption, programmability, and ease of deployment [21]–[24]. The authors in [25] demonstrate the significant potential of IRS by addressing its distance dependence and emphasizing the critical importance of phase. Furthermore, smart IRS implementations can achieve higher power gains through optimal phase control. The study in [26] focuses on solving the mmWave channel estimation issue in communication systems assisted by IRS. In [27], the author investigates the use of IRS technology to optimize channel capacity in line-ofsight (LoS)-free indoor mmWave scenarios, concluding that increasing the number of elements enhances channel capacity through IRS reflection element optimization. Additionally, a proposed MIMO wideband non-stationary channel model with IRS assistance is presented in [28]. Thanks to the tunability of its parameters, the communication model proved applicable in describing a wide array of wireless high-speed propagation scenarios. The results illustrated that low correlation in IRSassisted communication systems can benefit both temporal and spatial domains. In [29], the authors devised a novel two-stage channel estimation method for generic IRS-assisted multiuser mmWave communication systems. Additionally, in [30], the authors examined a hybrid far- and near-field communication model in vehicular scenarios, supported by IRS, which accounted for both types of propagation conditions. Meanwhile, in [31], a narrowband model for IRS-assisted communication systems was proposed, addressing the complexities of complex CIR and Doppler shifts. Subsequent research in [32] delved into exploring the channel’s statistical characteristics and investigating various phase shift methods.
Existing literatures have demonstrated that the adoption of double-IRS technology in wireless communication systems significantly enhance the communication system performance, which mainly lead to an increased achievable rate compared to single-RIS assisted systems. In this context, various studies have been done on double-RIS assisted wireless communication systems in recent years [33]. To investigate the enhancement of channel performance through double-IRS, the authors in [34] introduced a physical double-IRS MIMO communication model where the transmitter, receiver, and cluster were all in motion states. It was further demonstrated that the double-IRS channel model exhibits temporal nonstationary features. A stochastic channel model for a cooperative MIMO system with a 3D-based double-IRS was employed in [35], aiming to describe A2G communications in UAV scenarios. [36] proposes a double-IRS based hybrid beamforming architecture for multiuser multiple-input-singleoutput (MU-MISO) mmWave communications. Simulations showed that this architecture outperforms current single-IRS mmWave communication systems in terms of communication quality. The authors in [37] explored a collaborative passive beamforming approach in a multi-user communication system with double-IRS assistance, illustrating the enhanced effectiveness of the double-IRS system over the traditional single-IRS system through the utilization of multiplicative beamforming gains. In [40], the authors compared the bit error rate and sum rate of the proposed double-IRS assisted communication system with amplifiers to that of the traditional IRS system. Numerous simulation findings have demonstrated the superiority of the considered IRS technology for wireless communication systems compared to those without the IRS. In [41], the authors discussed the design of a multi-user mmWave MIMO system with double-IRS assistance, considering multi-antenna users and inter-RIS signal reflection. The superiority of the double-IRS approach in improving system performance was validated by the simulation results. Despite extensive research on double-IRS assisted communication systems, research on double-IRS assisted UAV-to-vehicle scenarios remains limited. Further research is needed to fully explore the potential of double-IRS assisted UAV-to-vehicle communication systems. It is worth mentioning that the channel models mentioned above cannot accurately describe the channel characteristics of double-IRS assisted A2G wireless propagation systems. These drive us to propose a general stochastic physical model that can be used to describe communication scenarios in A2G communication systems with the assistance of double-IRS technology.
C. Main Contributions
Motivated by the above the current research gaps, this paper innovatively propose a non-stationary physical dynamic model for MIMO wireless channels in A2G communication scenarios, where the double-IRSs technology is the first time to be considered to assist the signal transmission in the proposed communication scenarios. The considered communication model employs a physical framework to delineate frequency-selective fading in channels, offering insights for future communication system design with the assistance of double-IRSs. The primary contributions and innovations of this article are summarized as follows:
Firstly, we presents a novel 3D dynamic physical communication model with the assistance of double-IRS technology, which aims at boosting the propagation performance of A2G wireless communication systems. Here, we are the first time to consider the impacts of the UAV physical statistics, such as its motion trajectories and motion speeds, on the channel propagation features in not only time domain, but also in space and frequency domains. This provided communication model can be used to accommodate a variety of other scenarios by setting different values of the model parameters.
Secondly, we based on the proposed communication model, are the first time to derive the expressions of the correlation statistics of the A2G channels with the assistance of the double-IRS technology. To be specific, they are the space correlation features, time correlation features, as well as frequency correlation features. Simulation results reveal that the physical statistics of the arrays for double-IRS have vital influences on the channel statistical properties, offering insights for designing double-IRS-assisted UAV-to-vehicle wireless communication systems.
Finally, the impact of engineering implementation of double-IRS suited on the surface of the buildings in the real world was discussed, such as the heights of the IRSs. Simulation results showed that non-ideal double-IRS can lead to a certain degree of performance loss.
The subsequent sections of this paper are structured as follows: Section II presents the system model for physical A2G communication with double-IRSs. It discusses the derivation and analysis of the complex CIRs of the considered communication channel. The statistical characteristics of the physical communication channel are meticulously examined in Section III. Section IV elaborates on the correlation features of the path links across space, time, and frequency domains. Section V offers numerical results and corresponding analyses. Finally, Section VI encapsulates the conclusions drawn from this study.
Notation: Matrices, vectors, and scalars are represented by bold uppercase, bold lowercase, and regular lowercase letters, respectively. For instance, a matrix is denoted by [TeX:] $$\mathbf{Z},$$ a vector by [TeX:] $$\mathbf{z},$$ and a scalar by z. Symbols such as [TeX:] $$\mathbf{E}[\cdot],(\cdot)^*,[\cdot]^T \text { and }\|\cdot\|$$ signify the expectation, complex conjugate, transpose, and Frobenius norm operations, respectively. The imaginary unit is denoted by [TeX:] $$j=\sqrt{-1}.$$
II. SYSTEM MODEL
For A2G wireless communication scenarios, the LoS propagation paths between the transmitter in the air and the receiver on the ground are not always blocked, which are mainly due to the dynamic features of the transmitter and those of the receiver. As a matter of fact, it experiences blockage occasionally with the motion of the transmitter and receiver. Existing research works have shown that when the LoS links between the transmitter in the air and the receiver on the ground are heavily blocked by objects such as buildings, double-IRS, which are suited on the buildings of two different buildings, can be employed to provide a strong LoS link from the transmitter in the air to the RIS and from the RIS to the receiver on the ground, thus boosting the communication performance. Referring to Fig. 1, our analysis revolves around a physical A2G communication model featuring omnidirectional uniform linear arrays (ULAs) denoted as P and Q on the UAV and vehicle sides, respectively [38], [39]. The global Cartesian coordinate system has its origin at the midpoint of the UAV antenna array in the azimuth plane. Specifically, the positive x-axis is defined as the line connecting this midpoint with the vehicle side’s antenna array midpoint, while the y-axis follows the right-hand rule. Here, we establish the horizontal plane as the plane containing the x-axis and y-axis, while the z-axis extends vertically upwards from the origin. The UAV and vehicle move at speeds denoted by [TeX:] $$v_T \text{ and } v_R,$$ respectively. Importantly, the motion of the UAV and vehicle does not alter the coordinate system. For the considered double-IRSs, the [TeX:] $$\mathrm{IRS}_1$$ is placed on the surface of one building in the yzplane, which the IRS2 is placed on the surface of another building in the xz-plane. With the help of such arrangement, it effectively ensure that the information in the 3D of the Cartesian coordinate system is fully accessible. The initial altitudes of UAV is [TeX:] $$H_0.$$ Table I lists some definitions of the key parameters of the communication model for quick reference.
Proposed propagation model for IRS-empowered UAV-to-vehicle wireless communication scenarios.
EXPLICATIONS OF CRUCIAL MODEL PARAMETERS IN RIS-ENABLED UAV-TO-VEHICLE CHANNELS.
In the considered physical A2G model, we denote the vector distances from the midpoint of the UAV antenna arrays to the [TeX:] $$p \text {th }(p=1,2, \cdots, P)$$ transmitting antenna and from the vehicle antenna arrays’ midpoint to the [TeX:] $$q \text {th }(q=1,2, \cdots, Q)$$ receiving antenna as
It is evident from the physical communication model that vehicle has no motion feature parameters in the vertical component. In light of this, the vector distances from the origin to the midpoint of the antenna arrays at the UAV and vehicle sides, denoted by [TeX:] $$\mathbf{A}_T(t) \text { and } \mathbf{A}_R(t),$$ respectively, can be written by
Subsequently, the path length from the origin of the global coordinate system to the centers of [TeX:] $$\mathrm{IRS}_1 \text { and } \mathrm{IRS}_2$$ can be expressed as [TeX:] $$\mathbf{d}_{\mathrm{IRS}_1}=\left[0, y_{\mathrm{IRS}_1}, z_{\mathrm{IRS}_1}\right]^{\mathrm{T}}$$ and [TeX:] $$\mathbf{d}_{\mathrm{IRS}_2}=\left[x_{\mathrm{IRS}_2}, D_0, z_{\mathrm{IRS}_2}\right]^{\mathrm{T}}$$, respectively, where [TeX:] $$D_0$$ represents the perpendicular distance from [TeX:] $$\mathrm{IRS}_2$$ to the xz-plane. In light of this, the distance from the center of [TeX:] $$\mathrm{IRS}_1$$ to the center of [TeX:] $$\mathrm{IRS}_2$$ can be described as [TeX:] $$\xi_{\mathrm{IRS}_2}^{\mathrm{IRS}_1}=\left\|\mathbf{d}_{\mathrm{IRS}_1}-\mathbf{d}_{\mathrm{IRS}_2}\right\|.$$ For the [TeX:] $$\left(m_1, n_1\right) \text {th }$$ element in [TeX:] $$\mathrm{IRS}_1$$, i.e., [TeX:] $$m_1=1,2, \cdots, M_1$$ and [TeX:] $$n_1=1,2, \cdots, N_1$$, as well as the [TeX:] $$\left(m_2, n_2\right) \text {th }$$ element in [TeX:] $$\mathrm{IRS}_1$$, i.e., [TeX:] $$m_2=1,2, \cdots, M_2$$ and [TeX:] $$n_2=1,2, \cdots, N_2$$, their distances with the origin that can be represented as
We consider a cluster between the UAV and the MR to simulate the scattering environment of the communication model. The distance between the cluster and the origin is represented by the distance vector [TeX:] $$\mathbf{d}_{\mathrm{clu}}=\left[x_{\mathrm{clu}}, y_{\mathrm{clu}}, z_{\mathrm{clu}}\right]^{\mathrm{T}}.$$
III. COMPLEX CHANNEL IMPULSE RESPONSE
As shown in Fig. 2, we can obviously note that the radio signals transmitted by the UAV experience different kinds of the propagation links before arriving at the vehicle, where each kind consists of multi-path. Four different propagation paths are considered based on the geometry of the proposed model, including:
The wave travels along the path of the [TeX:] $$\mathrm{UAV} \rightarrow \mathrm{IRS}_1 \rightarrow$$ vehicle.
The wave travels along the path of the [TeX:] $$\mathrm{UAV} \rightarrow \mathrm{IRS}_2 \rightarrow$$ vehicle.
The wave travels along the path of the [TeX:] $$\mathrm{UAV} \rightarrow \mathrm{IRS}_1 \rightarrow \mathrm{IRS}_2 \rightarrow$$ vehicle.
The wave travels along the path of the [TeX:] $$\mathrm{UAV} \rightarrow \text{ cluster } \rightarrow$$ vehicle.
Propagation angles and distances are considered in the proposed MIMO UAV-to-vehicle communication model with the assistance of double-IRSs technology for different links. (a) [TeX:] $$\mathrm{UAV} \rightarrow \mathrm{IRS}_1 \rightarrow$$ vehicle; (b) [TeX:] $$\mathrm{UAV} \rightarrow \mathrm{IRS}_2 \rightarrow$$ vehicle; (c) [TeX:] $$\mathrm{UAV} \rightarrow \mathrm{IRS}_1 \rightarrow \mathrm{IRS}_2 \rightarrow$$ vehicle; (d) [TeX:] $$\mathrm{UAV} \rightarrow \text{ cluster } \rightarrow$$ vehicle.
It is generally assumed that these four links are independent, and we can calculate the complex gain of each path separately and add them together.
For the considered communication model, its physical features can be captured by a matrix [TeX:] $$\mathbf{H}(t, \tau)=\left[h_{p q}(t, \tau)\right]_{Q \times P}$$ of size [TeX:] $${Q \times P}.$$ The [TeX:] $$h_{p q}(t, \tau)$$ represents complex coefficient of the propagation path corresponding to the pth antenna at the UAV side and the qth antenna at the vehicle side, which can be represented as
In (7), [TeX:] $$\tau^{\operatorname{IRS}_1}(t)=\left(\xi_T^{\mathrm{IRS}_1}(t)+\xi_R^{\mathrm{IRS}_1}(t)\right) / c$$ denotes the transmission delay of the [TeX:] $$\mathrm{UAV} \rightarrow \mathrm{IRS}_1 \rightarrow$$ vehicle link, with c being the speed of radio signal. Here, [TeX:] $$\xi_T^{\mathrm{IRS}_1}(t)$$ and [TeX:] $$\xi_R^{\mathrm{IRS}_1}(t)$$ represent the path lengths from the midpoints of antenna arrays at the UAV and vehicle sides to that of the [TeX:] $$\mathrm{IRS}_1$$ array, respectively, defined as [TeX:] $$\xi_T^{\mathrm{IRS}}(t)=\left\|\mathbf{d}_{\mathrm{IRS}_1}-\mathbf{A}_T(t)\right\|$$ and [TeX:] $$\xi_R^{\mathrm{IRS}}(t)=\left\|\mathbf{d}_{\mathrm{IRS}_1}-\mathbf{A}_R(t)\right\|$$, correspondingly. Furthermore, [TeX:] $$\tau^{\mathrm{IRS}_2}(t)=\left(\xi_T^{\mathrm{IRS}_2}(t)+\xi_R^{\mathrm{IRS}_2}(t)\right) / c$$ is the transmission delay of the [TeX:] $$\mathrm{UAV} \rightarrow \mathrm{IRS}_2 \rightarrow$$ vehicle link, where [TeX:] $$\xi_T^{\mathrm{IRS}_2}(t)$$ and [TeX:] $$\xi_R^{\mathrm{IRS}_2}(t)$$ represent the path lengths from the midpoints of antenna arrays at the UAV and vehicle sides to that of the [TeX:] $$\mathrm{IRS}_2$$ array, defined as [TeX:] $$\xi_T^{\mathrm{IRS}_2}(t)=\left\|\mathbf{d}_{\mathrm{IRS}_2}-\mathbf{A}_T(t)\right\|$$ and [TeX:] $$\xi_R^{\mathrm{IRS}_2}(t)=\left\|\mathbf{d}_{\mathrm{IRS}_2}-\mathbf{A}_R(t)\right\|$$, correspondingly. Moreover, we have
where [TeX:] $$\xi_{T . \ell}^{\text {cluster }}(t)=\left\|\mathbf{d}_{\text {clu }}-\mathbf{A}_T(t)\right\|$$ and [TeX:] $$\xi_R^{\mathrm{clu}, \ell}(t)=\| \mathbf{d}_{\mathrm{clu}}-\mathbf{A}_R(t) \|$$ denote the transmission delays of the [TeX:] $$\mathrm{UAV} \rightarrow \mathrm{IRS}_1 \rightarrow \mathrm{IRS}_2 \rightarrow$$ vehicle link and [TeX:] $$\mathrm{UAV} \rightarrow \text{ cluster } \rightarrow$$ vehicle link, respectively.
[TeX:] $$\mathrm{UAV} \rightarrow \mathrm{IRS}_1 \rightarrow$$ vehicle link the channel coefficients [TeX:] $$h_{p q}^{\mathrm{IRS}_1}(t)$$ of the (p, q)th UAV-vehicle antenna pair are written by the following manner:
where represents the wavelength of the radio signals. It is worth mentioning that when the [TeX:] $$\mathrm{IRS}_1$$ follows a uniform configuration for the reflection phase, [TeX:] $$\varphi_{m_1, n_1}(t)$$ follows a uniform distribution. The real-time reflection amplitude and phase of the [TeX:] $$\left(m_1, n_1\right)$$ unit in [TeX:] $$\mathrm{IRS}_1$$ can be expressed as [TeX:] $$\chi_{m_1, n_1}(t) \text { and } \varphi_{m_1, n_1}(t),$$ respectively. Furthermore, in (10), [TeX:] $$\xi_p^{\left(m_1, n_1\right)}(t) \text { and } \xi_q^{\left(m_1, n_1\right)}(t)$$ are respectively the distances from the pth antenna at the UAV side and qth antenna vehicle side to the [TeX:] $$\left(m_1, n_1\right)$$th unit in the [TeX:] $$\mathrm{IRS}_1$$, respectively, which can be denoted as
In (10), [TeX:] $$\theta_T^{\left(m_1, n_1\right)}(t) \text { and } \gamma_T^{\left(m_1, n_1\right)}(t)$$ respectively account for the angle of departure of the [TeX:] $$\mathrm{UAV} \rightarrow \mathrm{IRS}_1 \rightarrow$$ vehicle link with respect to the [TeX:] $$\left(m_1, n_1\right)$$th unit in [TeX:] $$\mathrm{IRS}_1$$ in the azimuth and elevation dimensions at the UAV side, and can be identified as
Similarly, [TeX:] $$\theta_R^{\left(m_1, n_1\right)}(t) \text { and } \gamma_R^{\left(m_1, n_1\right)}(t)$$ are respectively the angle of arrival of the [TeX:] $$\mathrm{UAV} \rightarrow \mathrm{IRS}_1 \rightarrow$$ vehicle link with respect to the [TeX:] $$\left(m_1, n_1\right)$$th unit in [TeX:] $$\mathrm{IRS}_1$$ in the azimuth and elevation dimensions at the vehicle side, and can be identified as
For the [TeX:] $$\mathrm{UAV} \rightarrow \mathrm{IRS}_2 \rightarrow$$ vehicle link, the channel coefficients [TeX:] $$h_{p q}^{\mathrm{IRS}_2}(t)$$ of the (p, q)th antenna pair are written by the following manner:
where the real-time reflection amplitude and phase of the unit at the [TeX:] $$\left(m_2, n_2\right)$$th position in [TeX:] $$\mathrm{IRS}_2$$ can be expressed as [TeX:] $$\chi_{m_1, n_1}(t) \text { and } \varphi_{m_1, n_1}(t),$$ respectively. We can substitute [TeX:] $$[\cdot]_2$$ for all subscripts [TeX:] $$[\cdot]_1$$ as the model parameters for the [TeX:] $$\mathrm{UAV} \rightarrow \mathrm{IRS}_1 \rightarrow$$ vehicle link and the [TeX:] $$\mathrm{UAV} \rightarrow \mathrm{IRS}_2 \rightarrow$$ vehicle link are obviously similar. For the sake of brevity, we will not include the angles and distances related to the [TeX:] $$\mathrm{IRS}_2$$ propagation link.
For the [TeX:] $$\mathrm{UAV} \rightarrow \mathrm{IRS}_1 \rightarrow \mathrm{IRS}_2 \rightarrow$$ vehicle propagation link, the channel coefficients [TeX:] $$h_{p q}^{\mathrm{IRS}_{12}}(t)$$ of the (p, q)th antenna pair are expressed as follows
where [TeX:] $$\xi_{\left(m_2, n_2\right)}^{\left(m_1, n_1\right)}$$ denotes the distance between the [TeX:] $$\left(m_1, n_1\right)$$th unit in [TeX:] $$\mathrm{IRS}_1$$ and the [TeX:] $$\left(m_2, n_2\right)$$th unit in [TeX:] $$\mathrm{IRS}_2$$, which can be derived by [TeX:] $$\xi_{\left(m_2, n_2\right)}^{\left(m_1, n_1\right)}=\left\|\mathbf{d}_{m_1, n_1}-\mathbf{d}_{m_2, n_2}\right\| .$$
For the [TeX:] $$\mathrm{UAV} \rightarrow \text{ cluster } \rightarrow$$ vehicle link, the channel coefficients [TeX:] $$h_{p q}^{\mathrm{clu}}(t)$$ of the (p, q)th antenna pair can be expressed as follows
The real-time distances of the propagation links from the cluster centre to the midpoints of the ULA of the UAV and the ULA of the MR can be denoted as [TeX:] $$\xi_{T, \ell}^{\mathrm{clu}}(t) \text { and } \xi_{R, \ell}^{\mathrm{clu}}(t),$$ respectively. [TeX:] $$\left\{\varphi_{\ell}\right\}_{\ell=1,2, \cdots, L}$$ is the randomly selected uniformly distributed and mutually independent phase values, which can be denoted as [TeX:] $$\varphi_{\ell} \sim \mathrm{U}[-\pi, \pi] .$$ The cluster often contains a significant number of NLoS rays, which tends towards infinity, i.e., [TeX:] $$L \rightarrow \infty,$$ and is normally assumed to be represented by the variable L. Here, we can calculate the expression of the [TeX:] $$\xi_{T, \ell}^{\mathrm{clu}}(t) \text { and } \xi_{R, \ell}^{\mathrm{clu}}(t)$$ as follows:
Therefore, the real-time angle of departure of the NLoS propagation link are [TeX:] $$\theta_{T, \ell}^{\text {clu }}(t) \text { and } \gamma_{T, \ell}^{\text {clu }}(t),$$ respectively, and can be defined as
Furthermore, the real-time angle of arrival of the NLoS propagation link are denoted by [TeX:] $$\theta_{R, \ell}^{\text {clu }}(t) \text { and } \gamma_{R, \ell}^{\text {clu }}(t),$$ respectively, and can be described as
IV. COMPLEX CIRS OF THE PROPOSED PROPAGATION MODEL
In this section, we will discuss the channel propagation characteristics of double-IRSs assisted UAV-to-vehicle propagation environments, specifically investigating the correlation features of the considered path links in space, time, as well as frequency domains. Here, the spatial-temporal correlation function of a physical model assesses channel correlation across time and space. It derives normalized spatial-temporal cross-correlation functions from distinct complex channel coefficients [TeX:] $$h_{p q}(t, \tau) \text { and } h_{p^{\prime} q^{\prime}}(t, \tau),$$ where [TeX:] $$p^{\prime}=1,2, \cdots, P$$ and [TeX:] $$q^{\prime}=1,2, \cdots, Q.P$$ It is widely assumed that different propagation paths and different propagation delays are uncorrelated with each other. In light of this, we can easily obtain the following function:
where [TeX:] $$\Delta t$$ denotes the time difference; [TeX:] $$p, p^{\prime}, q, \text { and } q^{\prime}$$ denote the antenna indices. [TeX:] $$\Delta p=\left|p^{\prime}-p\right| \delta_T / \lambda$$ represents the normalized antenna spacing between the [TeX:] $$p \text {th and } p^{\prime} \text {th }$$ antennas on the UAV side, while [TeX:] $$\Delta q=\left|q^{\prime}-q\right| \delta_R / \lambda$$ denotes the normalized antenna spacing between the [TeX:] $$q \text {th and } q^{\prime} \text {th }$$ antennas on the vehicle side. Due to the independent nature of the time-varying complex CIRs, as evident in (7), the space-time correlation function can be written by
where [TeX:] $$\rho_{(p, q),\left(p^{\prime}, q^{\prime}\right)}^{\mathrm{IRS}_1}(t, \Delta p, \Delta q, \Delta t)$$ stands for the correlation features of [TeX:] $$\mathrm{UAV} \rightarrow \mathrm{IRS}_1 \rightarrow$$ vehicle propagation link in space domain, and it can be expressed as
Since the spatial correlation features of [TeX:] $$\mathrm{IRS}_2$$ propagation link in space domain is comparable to that of [TeX:] $$\mathrm{IRS}_2$$ link, we may derive it by substituting [TeX:] $$[\cdot]_2$$ subscripts for all [TeX:] $$[\cdot]_1$$ subscripts in (28). To keep the formulation brief, we omit this section.
Then, [TeX:] $$\rho_{(p, q),\left(p^{\prime}, q^{\prime}\right)}^{\mathrm{IRS}_{12}}(t, \Delta p, \Delta q, \Delta t)$$ denotes the correlation features of [TeX:] $$\mathrm{UAV} \rightarrow \mathrm{IRS}_1 \rightarrow \mathrm{IRS}_2 \rightarrow$$ vehicle propagation link in space domain, which can be expressed as
A common assumption is that the NLoS propagation links conforms to the complex Gaussian distribution, which implies that the number of NLoS rays within the cluster is assumed to tend to infinity, i.e., [TeX:] $$L \rightarrow \infty$$. This will lead to a phenomenon that the complex CIR of NLoS propagation links to satisfy the Central Limit Theorem. In light of this, the correlation features of the [TeX:] $$\mathrm{UAV} \rightarrow \text{ cluster } \rightarrow$$ vehicle in space domain in the considered communication model can be expressed as
where [TeX:] $$\theta_R^{\text {clu }}$$ and elevation angle [TeX:] $$\gamma_R^{\mathrm{clu}}$$ represents the joint distribution of the receiver azimuth angle [TeX:] $$\theta_R^{\text {clu }}$$ and elevation angle [TeX:] $$\gamma_R^{\mathrm{clu}}$$ of the probability density function (PDF). According to (28) and (32), We can observe that the spatial features changes in tandem with the azimuths of the UAV and MR, as well as the configuration of the IRS.
Subsequently, setting [TeX:] $$p=p^{\prime} \text { and } q=q^{\prime}$$ in the above spatial features, we can obtain [TeX:] $$\Delta p=\Delta q=0.$$ Then we fix the antenna spacings [TeX:] $$\delta_T=\delta_R=\lambda / 2,$$ the temporal features of the considered communication model can be calculated as follows:
It notable that the temporal aspects of this communication model directly tie to the motion time t of the UAV/vehicle, alongside the time difference [TeX:] $$\Delta t,$$ indicating that it is nonstationary across the time domain. Furthermore, it shows that the physical features of different kinds of propagation links in the considered communication model in the fixed scattering region are uncorrelated. This suggests that conventional physical model, which ignores the effects of non-stationarity, might underestimate the correlation of the channel elements
Here, the correlation features of the path links in the considered communication model can be expressed as
where the frequency separation is denoted by [TeX:] $$\Delta f.$$ Afterwards, using the Fourier transform, we can obtain the time-invariant transfer function, which can be written in the following manner:
Finally, we can obtain (34) by replacing it with (35), which can be written in the following manner:
Based on the derivation above, we conclude that the correlation features of the path links in this communication model are independent of a fixed frequency f, but instead rely on the frequency difference [TeX:] $$\Delta f$$ and the duration of motion t. This demonstrates non-stationarity in the time domain while maintaining stationarity in the frequency domain.
V. SIMULATION RESULTS AND DISCUSSIONS
In this section, we will delve into the simulation results to better understand the propagation characteristics of the communication model and juxtapose them with the anticipated outcomes.
A. Simulation Setup
Unless otherwise indicated, the parameters of the suggested communication model in our numerical simulation are as follows: For IRS propagation links, [TeX:] $$x_{\mathrm{IRS}_1}=80 \mathrm{~m}, y_{\mathrm{IRS}_1}=30 \mathrm{~m},$$ [TeX:] $$z_{\mathrm{IRS}_1}=20 \mathrm{~m}, \chi_{m_1, n_1}(t)=1, M_1=N_1=30, d_{M_1}=d_{N_1}=\lambda / 4,$$ whereas [TeX:] $$x_{\mathrm{IRS}_2}=20 \mathrm{~m}, y_{\mathrm{IRS}_2}=30 \mathrm{~m},$$ [TeX:] $$z_{\mathrm{IRS}_2}=40 \mathrm{~m}, \chi_{m_2, n_2}(t)=1, M_2=N_2=30, d_{M_2}=d_{N_2}=\lambda / 4$$; for UAV transmitter: [TeX:] $$v_T=10 \mathrm{~m} / \mathrm{s}, \eta_T=\pi / 2 \zeta_T=\pi / 6$$; for vehicle receiver: [TeX:] $$v_R=10 \mathrm{~m} / \mathrm{s}, \eta_T=\pi / 2$$; [TeX:] $$\phi_T=\phi_R=\pi / 3, \text { and } \psi_T=\psi_R=\pi / 4, \delta_T=\delta_R=\lambda / 2,$$ [TeX:] $$P=30, Q=100, t=2 \mathrm{~s}, H_0=40 \mathrm{~m}, D_0=50 \mathrm{~m}, \text { and } f_c=5 \mathrm{GHz}.$$
B. Performance Analysis
By utilizing (28)–(30), the correlation characteristics of path links within the considered communication model for various propagation links are depicted in Fig. 3. It is evident that the values of the correlation features of propagation links in space domain drop slowly as we increase the value settings of the antenna intervals. This observation can also be seen in the reference [30], corroborating the accuracy of our simulated results. It deserves to consider that not only the (28) and (29), but also the (30) are all Bessel functions, and thereby leading to a fluctuation observation with the rising value setting of the antenna spacing. Nonetheless, the general changing curves of the spatial correlation features are to drop when increasing the antenna spacing. Particularly, the fluctuation of the spatial features is notably reduced for the double-IRS link compared to other links, suggesting that the introduction of double-IRS can enhance the propagation characteristics of the communication model.
Spatial features of the considered communication model under different combinations of propagation links.
In Fig. 4, we examine the correlation features of propagation links in the spatial domain within the considered communication model for various UAV heights. It’s observed that as [TeX:] $$z_{\mathrm{IRS}_2}$$ increases from 20 m to 150 m, the spatial features exhibit a fluctuating downward trend, eventually stabilizing. These simulation results align with those found in [35], further highlighting the influence of IRS height on signal propagation. The fluctuations in spatial features depicted in the figure stem from variations in [TeX:] $$z_{\mathrm{IRS}_2}$$, impacting the distance between UAV/MR and IRS. Thus, it’s essential to account for the effect of IRS height on transmission characteristics in practical configurations.
Spatial features of the considered communication model under different heights of the [TeX:] $$\mathrm{IRS}_1 \text{ and } \mathrm{IRS}_2$$.
By using (31), Fig. 5 illustrates the correlation characteristics of the path links in the time domain within the considered model. The temporal features display a decreasing trend as [TeX:] $$\Delta t$$ increases, with the [TeX:] $$\mathrm{IRS}_{12}$$ link exhibiting the most pronounced decrease in temporal features. Consequently, it can be inferred that the introduction of IRS restricts the temporal features in NLoS links, thereby reducing the channel coherence time. This simulation result aligns with findings in [42], confirming the accuracy of the experimental results. Furthermore, it’s noteworthy that double-IRS significantly influences the temporal features.
Temporal features of the considered communication model under different combinations of propagation links when t = 2 s.
In Fig. 6, it is evident that the curve varies with the motion duration, showcasing the temporal correlations’ sensitivity to time, thereby indicating the non-smooth nature of the model in the time domain. Similar results are documented in [43], further enhancing the validity of our conclusions. Moreover, the temporal features exhibit decreasing trends as the time interval [TeX:] $$\Delta t$$ expands. This declining trend becomes more pronounced with increasing motion time t. Furthermore, the temporal features of the double-IRS links demonstrate a significantly slower decline compared to other links, a pattern consistent with findings in [34], which lends additional support to the model’s accuracy. It is evident that the presence of double-IRS has a notable impact on the temporal features of the communication model.
Temporal ACFs of the considered communication model in terms of the different time motion.
Fig. 7 illustrates the various trends of the temporal features corresponding to different heights of the UAV. The curve demonstrates a notable impact of the UAV’s altitude on the temporal correlations. Additionally, as [TeX:] $$H_0$$ increases from 10 m to 100 m, the temporal ACFs exhibit a decreasing trend, coinciding with the rise in UAV height. These observations align with findings from [42], corroborating the deductions regarding temporal correlations. This outcome arises from the variation in [TeX:] $$H_0$$, which affects the distance between the UAV and the IRS. Consequently, it is imperative to consider the UAV’s altitude when configuring practical setups.
Temporal features of the considered communication model under differen heights of UAV when t = 2 s.
By using (32), Fig. 8 illustrates the normalized correlation features of the path links within the examined communication model. It is evident that as we increase the value of the frequency interval [TeX:] $$\Delta f,$$ the normalized frequency features of the path links exhibit a fluctuating downward trend before eventually stabilizing. This observation is consistent with findings in [44], affirming the accuracy of the presented simulation results. The emergence of this phenomenon is attributed to the inclusion of IRS links in the communication model. Furthermore, it is observed that the normalized frequency features fluctuate slightly less for double-IRS links compared to single-IRS links, highlighting the impact of double-IRS configurations on the communication model.
Normalized frequency features of the considered communication model in terms of the different combinations of propagation links (P = 30, Q = 40, and t = 2 s).
Fig. 9 depicts the changing trend of the normalized frequency features with varying heights of [TeX:] $$z_{\mathrm{IRS}_2}.$$ The curves clearly illustrate the significant non-stationarity of the frequency features as [TeX:] $$z_{\mathrm{IRS}_2}$$ changes. This observation aligns with the findings in Fig. 3, further emphasizing the impact of [TeX:] $$\mathrm{IRS}_2$$ height on communication. The variation in [TeX:] $$z_{\mathrm{IRS}_2}$$ alters the distance between the UAV/vehicle and the IRS, thereby influencing the normalized frequency features. Consequently, when deploying the IRS, careful consideration of its height is imperative due to its substantial effect on communication performance.
Normalized frequency features of the considered communication model under different heights of the [TeX:] $$\mathrm{IRS}_1 \text { and } \mathrm{IRS}_2$$ when t = 2 s.
VI. CONCLUSIONS
In this paper, we have developed a comprehensive 3D physical communication model for ISAC in UAV-to-vehicle scenarios, leveraging double-IRS technology to enhance radio signal transmissions. The UAV operates in aerial space while the vehicle maneuvers at ground level. We have formulated the complete channel matrix of our communication model by deriving the complex CIR for four distinct propagation links, considering the effects of multipath propagation. Through our investigation of the communication model, we have analyzed the correlation features of propagation links across spatial, temporal, and frequency domains. Our simulation results underscore the significant impact of UAV altitude and IRS positional configuration on the propagation process, underscoring the necessity of accounting for these factors in practical IRS deployment scenarios. Furthermore, our findings highlight the superiority of the double-IRS configuration over the single- IRS counterpart in terms of channel characteristics, signaling the importance of integrating double-IRS setups into UAV-tovehicle communication models.
In the future, we plan to explore three potential directions: i) incorporating acceleration to study communication scenarios at varying speeds, thus improving the realism of the double- IRS-assisted communication model; ii) calculating the power gain of various propagation links by considering both smallscale multipath fading and large-scale path loss; iii) developing more efficient algorithms to enhance the double-IRS-assisted UAV-to-vehicle communication modeling approach.