Statistical Characteristics of 3D MIMO Channel Model for Vehicle-to-Vehicle Communications

Spatial and temporal characteristics of the propagation channel have a signi ﬁ cant in ﬂ uence on multiantenna method applicability for ﬁ fth-generation- (5G-) enabled Internet of Things (IoT). In this paper, the statistical characteristics of a novel three-dimensional (3D) geometric-based stochastic model for next-generation vehicle-to-vehicle (V2V) multiple-input multiple-output (MIMO) communications under the nonisotropic scattering environment are investigated. In both line-of-sight (LoS) and non-line-of-sight (NLoS) conditions, the proposed model investigates the spatial, frequency, and temporal domain statistical distribution of multipath received signals by using the time-variant transfer function for indoor environments. The probability density function (PDF) of separation distance between the transceiver antennas, angle-of-arrival (AoA), and angle-of-departure (AoD) in the azimuth and elevation planes is derived by using closed-form expressions. For the space, time, and frequency correlation function (STF-CF), a precise analytical expression is derived based on MIMO antenna system. We further determine the e ﬀ ects of several model parameters on the V2V channel performance, such as tunnel width, antenna array spacing, Ricean K -factor, and moving velocity. The statistical characteristics of the MIMO channel model are validated by simulation results, con ﬁ rming the ﬂ exibility and e ﬀ ectiveness of our proposed model in the tunnel scenario.


Introduction
Recently, the wireless communication technologies like multiple input multiple output (MIMO), channel coding, cooperative communication, and internet of vehicles (IoV) have played a crucial role towards the development of future communication networks [1][2][3][4][5][6][7][8][9][10]. MIMO channel, which uses hundreds or even more antennas, has received a lot of attention. It has been demonstrated that MIMO technique is capable to improve the energy and spectrum efficiency, especially in rich scattering environment. The rapid advancement of the 5G-enabled Internet-of-Things (IoT) communication systems for vehicle-to-vehicle (V2V) radio communications has acquired much more attentions [11][12][13][14]. Unlike traditional fixed to mobile cellular networks, the V2V channel uses low elevation for multiple antennas when both the transmitting and receiving antennas are conceived in motion. The radio propagation parameters between the transmitting (M T ) and receiving (M R ) antennas must be designed to facilitate the thorough investigation of V2V radio communications [15]. Therefore, the realistic propagation channel model is required which can provide a quick and easy algorithm to approximate the statistical characteristics of V2V radio channel [16,17].
To determine the optimized performance of the V2V radio communications in tunnel environments, precise channel models are under consideration. The V2V channel models can be classified as stochastic models and deterministic models (mostly based on the ray tracing (RT) method). Stochastic models are classified into two types: regular-shaped geometrical-based stochastic models (RS-GBSMs) [18][19][20][21][22] and non-regular-shaped geometricalbased stochastic models (NGBSMs). The latter, also named as parametric models, are established from channel measurements, whereas the former are constructed from the regular geometrical-shape of the scattering objects. In [18], it is proved that the line-of-sight (LoS) path between the transceiver antennas is likely to be interrupted by buildings and barriers. As a result, Ricean channels must be used to investigate V2V communication systems. In [19], a precise visual scattering model is proposed for the multibounced V2V propagation channels in crowded urban communication environment; however, they did not address the influence of moving car velocity on the V2V channel propagation characteristics. In addition, [20,21] introduce the 3D RS-GBSMs for microcell and macrocell communication systems, respectively.
However, the majority of the RSGBSMs models are based on narrowband channel assumptions, where the rays have almost the same propagation delay [22], which is not an accurate portrayal of V2V wireless communication system. In [23], the narrowband and wideband V2V channel characteristics based on propagation delays have been investigated through the real-time measurement campaign. The wide-sense stationary uncorrelated scattering (WSSUS) assumptions are used mostly in previous channel models; however, it is only applicable for short time intervals [24]. The WSSUS means actually firstand second-order stationarity in both time and frequency domains (wide-sense stationarity in time, uncorrelated scattering in delay) and holds in successive time and frequency interval. According to the measurement campaign in [25,26], the stationary interval, in which the WSSUS assumption is still valid for V2V environments, is substantially smaller than the observation interval. This implies that in V2V channels, the WSSUS assumptions are not being fulfilled in most of the cases. Therefore, the stochastic channel models must need to consider the channel's nonstationarity properties to compensate these shortcomings.
Similarly, in [27], the algebraic framework of the transceiver antennas in the Doppler spectrum based on moving velocity and arbitrary delays has been investigated and velocity vectors are considered to characterize the arbitrary moving direction in V2V radio communication environment. In [28], a novel 3D-GBSM model based on an ellipsoid Gaussian scattering distribution is proposed that can jointly characterize the elevation angles, azimuth angles, and distances from the transmitter (receiver) to the first (last)-bounce scatterers. The spatial consistency of the channel is supported by tracking the locations of the transmitter, receiver, and scatterers. Reference [29] proposes a statistical wideband V2V MIMO channel model based on the Unitary matrix transformation algorithm. The V2V MIMO channel model for other taps is estimated by examining the propagation delays of the first tap with a Unitary matrix transformation algorithm. In [30], a nonstationary stochastic model has been presented between the vehicle and terminal using 3D antenna arrays and random trajectories with fixed velocity consideration. In [31], the delay-based Doppler PDF for V2V radio channels is computed by using a prolate spheroidal system. In [32], the authors proposed a V2V scattering model with the assumption of randomly distributed scatterers across the semicircular shaped tunnel sidewalls. In the context of randomly moving clusters, [33] proposes a nonstationary GBSM channel model for future intelligent V2V communications. Because of the nonstationarity induced by moving clusters, the clusters' movements are described using a random walk procedure.
Also, [34] presents a regular-shaped 3D-GBSM model for V2V MIMO system for nonisotropic fading channel. However, the channel statistics of PDF, time delays, and tunnel width impact at different locations, which is critical for V2V MIMO systems, have not been investigated in the given model. Therefore, in this paper, the 3D MIMO V2V channel is modeled using the rectangular single-bounce scattering tunnel (RSBST) model, rather than the geometrical semicircular or circular tunnel propagation models. The inspiration came from the idea that both of the geometrical rectangular and semicircular tunnels are generally used in MIMO systems. As arched and rectangular tunnel environments have different propagation characteristics, therefore, we feel that our proposed model is simple and more suitable for indoor rectangular tunnel environment. To our knowledge, no investigation of radio wave propagation in rectangular tunnel from the perspective of scatterer distribution and movement of V2V MIMO systems along tunnel width has been performed.
In this paper, we investigate the new geometrical RSBST model and compute its geometrical statistics with other scattering models to examine the influence of size and crosssectional shape of tunnels on the V2V radio channel configurations. The key contribution of this study is to analyze the effect of different RSBST model parameters on V2V channel statistics and to obtain an optimized expression for scatterer distribution. The distribution of scatterers has been derived by considering the z-axis influence on the system performance and computed the 3D STF-CF of the V2V propagation channel in LoS and NLoS propagation environments. The impacts of numerous model parameters, such as tunnel diameter, antenna array spacing, Ricean K-factor, and moving velocity, on the V2V channel statistics are investigated as well.
The remainder of this paper is arranged in the following manner. Section 2 explains the proposed RSBST system model. In Section 3, different characteristics of the 3D-GBSM channel, such as TVTF, STF-CF, and power angular spread, are derived. In Section 4, the analytical results are presented and critically analyzed. Finally, Section 5 summarizes the paper.

Analysis of Geometric RSBST Model
The geometrical RSBST propagation model for the V2V MIMO system inside a rectangular-shaped tunnel is described in this section. Figure 1 represents a typical propagation scenario in an L-length tunnel. As illustrated in 2 Wireless Communications and Mobile Computing Figure 2, we assume that the scatterers are distributed randomly beside the tunnel sidewalls and ceiling. Due to cross-sectional size, shape, and geometry, the characteristics of a rectangular tunnel are substantially different from those of a circular or arch tunnel [35]. For example, in an arch tunnel, the tunnel width (W) is dependent on the tunnel radius due to semicircular geometry, whereas in a rectangular-shaped tunnel, the width (W) of the tunnel is independent from the height (H) of tunnel. As a result, when the scatterers are distributed on the tunnel ceiling and sidewalls, the z-axis affects the statistics of the V2V radio channel in a rectangular tunnel. Thus, we consider the impact of tunnels' z-axis when computing the STF-CF for nextgeneration V2V MIMO channel configurations in our proposed 3D RSBST model. In addition, Figure 3 shows the geometrical representation of RSBST V2V scattering model. As illustrated in Figure 3, we employ a Cartesian coordinate approach to investigate the distribution of random scatterers S pqr on the tunnel sidewalls and ceiling for ∀m, 1 ≤ p ≤ P, ∀q, 1 ≤ q ≤ Q, and ∀r, 1 ≤ r ≤ R. On the rectangle tunnel sidewalls and ceiling, we suppose that the infinite number of scattering components (i:e:, P, Q, R⟶∞) is distributed randomly [36].
However, the WSSUS assumptions can be still fulfilled for V2V radio channels over the small time periods [26]. Therefore, the V2V radio channel inside the tunnel is assumed to be WSSUS in our analysis. The locations of the receiver (MS R ) and transmitter (MS T ) are defined by cartesian coordinates ðx R X , y R X , z R X Þ and ðx T X , y T X , z T X Þ, when MS R and MS T are both moving in the opposite or same directions with velocity v. φ Rv and φ Tv determine the angle of motion for M R and M T antenna elements, respectively.
Additionally, the MS R and MS T are equipped with a multiple M R and M T antenna elements, and the antenna element spacing is represented by δ R and δ T , respectively. The path length between the mth transmitting antenna A m T ðm = 1, 2, ⋯:,M T Þ and the scatterer S pq is denoted by d m,pq T , whereas the distance between the nth receiving antenna A n R ðn = 1, 2, ⋯, N R Þ and the scatterer S pq is denoted by d pq,n T . Moreover, the d m,n TR specifies the distance between the mth element of transmitting array and the nth element of receiving array in terms of LoS path. In our work, we assume single-bounce scattering (SB) approach and further analyze the scattering effects due to randomly distributed scatterers between transceiver antennas, also known as effective scattering elements. The elevation angles and orientation of the M R (M T ) antenna array elements with respect to the xy plane can be represented by ϕ R ðϕ T Þ and γ R ðγ T Þ, respectively. The symbols β p,q,r T and α p,q T signify the elevation angle of departure (EAoD) and azimuth angle of departure (AAoD) of the signals that are impinging on the S pq scattering elements, whereas the angles β p,q,r R and α p,q R represent the elevation angle of arrival (EAoA) and azimuth angle of arrival (AAoA) of the signals which are scattered from S pq scattering elements, respectively. The EAoD, EAoA, AAoD, and AAoA in coordinate system ðx p , y q Þ of scattering elements are given as follows: where i = R or i = and, the index i relates to the receiver or transmitter. The functions f ðx p , y q Þ and gðx p , y q Þ are defined as follows:

Wireless Communications and Mobile Computing
STF-CF, and power angular spread will be calculated from the geometrical RSBST scattering model and provided as a sum of LoS and diffuse components.

TVTF. The TVTF is denoted as H nm ðt, f Þ, which is the sum of LoS and diffuse components as
where H LoS nm ðt, f Þ and H diffuse nm ðt, f Þ represent the TVTF of LoS and diffuse components, respectively. The H LoS nm ðt, f Þ of LoS components is written as follows: where In (8) where where the symbols f T m , f R m , k diffuse R , and τ p,q,r nm represent the maximum Doppler frequency of MS T and MS R , Ricean K -factor, and the delays of diffuse components, respectively. In addition, d m,pq T and d pq,n R denote the distances between scatterer S p,q,r to mth transmitting antenna array A m T ðm = 1, 2, ⋯:,M T Þ and from the scatterer S p,q,r to the nth receiving antenna array components A n R ðm = 1, 2, ⋯:,M R Þ, respectively.

STF-CF.
The STF-CF is the correlation of H nm ðt, f Þ and H n ′ m ′ ðt + τ, f + υÞ in (4), which can be calculated as where ð·Þ * represents the conjugate value operator and Eð·Þ denotes the expected value operator. ρ LoS nm,n ′ m ′ ðδ T , δ R , τ, vÞ and ρ diffuse nm,n ′ m ′ ðδ T , δ R , τ, vÞ represent the STF-CF of the LoS and the diffuse components, respectively. Thus, the STF-CF ρ LoS nm,n′m′ ðδ T , δ R , τ, vÞ can be calculated as follows: where Moreover, ρ diffuse nm,n ′ m ′ ðδ T , δ R , τ, vÞ can be estimated as Note that the propagation delays ðτ mm ′ p,q,r Þ, and the Doppler frequencies ð f p,q,r T Þ are based on x m , y n , and z r coordinates. Due to the infinite number of scattering elements in proposed V2V model, the random coordinates x m , y n , and z r can be replaced by independent variables x, y, and z. Furthermore, the x and y are assumed to be uniformly The expression z = W/2 · tan α can be used to calculate the distribution of variable z, where z is randomly distributed in the range of ½0 tan −1 ð2H/WÞ. By considering the transformation of given variables, the PDF of variable z can be derived as follows: The a z ðzÞ of variable z for different widths ðWÞ is shown in Figure 4. We can see that the a z ðzÞ of z in (23) drops down the uniform value within the range of z ∈ ½0, H, as the width (W) of tunnel increases.
Thus, by considering PDFs of randomly distributed variables x, y, and z, the joint PDF a xyz ðxyzÞ can be calculated as a xyz xyz ð Þ= a x x ð Þ · a y y ð Þ · a z z ð Þ, Similarly, by exploiting the joint PDF a xyz ðxyzÞ, the STF- (20) can be rewritten as It should be noted that the propagation angles β T ðx, y, zÞ , β R ðx, y, zÞ, α T ðx, y, zÞ, and α R ðx, y, zÞ in (27) and (28) are dependent on the x, y, and z coordinates of S p,q,r .

Power Angular Spread. The power angular spread (PAS)
can be configured by exploiting the Bartlett method [37]. For electromagnetic analysis, the Bartlett beamforming technique is used to estimate the angle of departure and angle of arrival in azimuth and elevation planes. The Bartlett  A denote the antenna array spacing along the y-axis, antenna array spacing along the x-axis, elevation angle along the positive y-axis, and azimuth angle along the positive x − axis. Furthermore, the signal vector of the arrays can be written as where nðtÞ denotes the noise power. Also, if K sources have the same time slot and frequency, the received signals can be written as follows: By assuming that the received signals are coming through the azimuth direction, so the output of the array by Bartlett can be calculated as follows: where σ 2 represents the noise power variance and ½· H specifies the transpose conjugate of the given vector. The significant solution of (29) is as follows: This means that the weighting vector is the same as the incident wave spatial characteristics. The final expression for estimating the PAS is given here only. Reference [37] contains further information about the Bartlett algorithm.
where R xx is the covariance matrix of received signals, and it may be calculated as where YðiÞ ∈ C M R ×M T denotes the received signals at the ith delay bin and vec½· is the linear transformation operator, which sequentially turns the matrix columns into a single column vector.
In V2V scenario, the geometric path length is usually time-variant due to the relative movement between the M S R and MS T antenna arrays. Compared to several existing 3D and 2D fixed-to-mobile (F2M) and V2V channel models, the proposed channel model can be utilized more efficiently in tunnel environment; for example, as the relative time is equal to zero, the proposed model tends to be wide-sensestationary (WSS) channel, which is similar to the F2M channel [38]. However, the proposed model represents nonstationary V2V channels where the relative time is not equal to zero, as described in [39]. Moreover, the proposed 3D model can reflects a wide range of communication scenarios by modifying the geometrical model parameters, as indicated in Sections 2 and 3. For example, the proposed channel model can represent the 3D semiellipsoid channel when the relative time is set to zero, as demonstrated in [40].  Figure 5. The tunnel is divided into two sections, the first of which is a 20 m long tunnel platform or entrance with a 6.5 m high    The following parameters are considered to produce numerical results: ϕ T = ϕ R = 60°, γ T = γ R = 45°, φ T = φ R = 30°, ðx T = 0, y T = 0, z T = 3:1 mÞ, ðx R = 0, 2,⋯,200 m, y R = 3:5 m, z R = 2:7 mÞ, and f R m ðHzÞ = ½120 240 360. The MS T is fixed to the tunnel sidewalls at the entrance, whereas the MS R moves along the tunnel axis. The tunnel walls are reinforced with concrete material. The propagation angles β T ðx, y, zÞ, β R ðx , y, zÞ, α T ðx, y, zÞ, and α R ðx, y, zÞ are estimated according to (27) and (28). The first 50 m distance in tunnel is conceived as LoS path, where the rest of 150 m is considered as NLoS path.   11 Wireless Communications and Mobile Computing the influence of path loss and signal-to-noise ratio so that channel correlation can be compared in a fair way. After fixing the SNR, channel matrix can express the multipath effects in a proper way. In our simulations, the bandwidth is 160 MHz, which means the sampling interval of two adjacent multipath components can be as low as 6.25 ns (1/ 160 MHz). Because the coherent bandwidth of the channel is lower than the long-term-evolution for metro (LTE-M) signal bandwidth, we used wideband channel assumptions for the calibrations. In this paper, we have considered the reflections from the tunnel sidewalls and ceiling only for simulation results.

Angular Distribution and Propagation
Delays. In Figure 6, we have shown power angular distribution by considering the elevation angle of departure (EAoD), azimuth angle of departure (AAoD), elevation angle of arrival (EAoA), and azimuth angle of arrival (AAoA) at 100 m separation distances between the transceiver antenna arrays (x R − x T = 100 m). Here, the number of scatterers is conceived as 2000 [42]. We can see that EAoA and AAoA both have negative magnitude which is because of z r < z i and y q < y i , respectively. The power and angular spread is configured by using the Bartlett method [37]. From Figure 6, we can observe that the propagative angles reproduce two main beams at −58°of elevation and azimuth of −83°and −46°for EoA and AoA (see Figure 6(a)) and at 84°elevation and azimuths of 88°and 133°for EAoD and AAoD (see Figure 6(b)), respectively. Moreover, the propagation delays ðτÞ are shown in Figure 7 by using RSBST approach at 5.6 GHz operating frequency. The results represent the cumulative distribution function (CDF) of propagation delays in the presence of different tunnel widths (W = 5 m, 7 m, and 9 m). It can be seen that propagation delays are increased by increasing the tunnel width. A modal theory can explain this pretty well [43]. Multiple propagation modes are triggered at the shorter MS T to MS R distances. At a greater distance, particularly in the case of NLoS propagation, high-order model attenuation becomes critical, and only the low-order primary model is retained. As a result, the reflected environment's delay spread is relatively steady. The tunnel has a smaller path loss exponent in the given frequency band, but it still displays the waveguide effect. Furthermore, the signal's shorter wavelength ðλÞ increases the specular reflections, which boosts the waveguide effect in the tunnel for higher frequency transmissions. When the multipath components are obstructed by the tunnel sidewalls and ceiling, the waveguide effect is considerably decreased. Also, due to the enormous 1500 m radius of curvature ðRÞ, there are only a few path loss exponents in the NLoS region. Figure 8, the theoretical vs. the simulation analysis of joint PDF of proposed RSBST model for different tunnel widths (W) is shown by using (24). Here, antenna array spacing is set as δ T = δ R 0:5 λ. It can be shown that joint PDF grows as the separation distance between the MS T and the MS R reduces and decreases as the width (W) of the tunnel increases. Further-more, the simulated results are in good agreement with the theoretical values of a xyz ðxyzÞ, verifying the validity of the proposed model. In a multipath channel, the initial geometric path length has a significant impact on the PDF of the transceiver separation distance. It is worth noting that the MS R and MS T are installed in two semiellipsoid scenarios in [44], and the PDF statistics are compatible with our proposed 3D model. Due to the symmetry of the geometric radio channel model, each joint PDF a xyz ðxyzÞ curve in Figure 8 is symmetric, which is also compatible with [44]. Thus, the proposed 3D MIMO model can be used to characterize the real tunnel environment. Figure 9 presents the STT-CF for different K-factor values, when f R m = 360 Hz, coherence time τ = 1 ns, and separating distance between transceiver arrays is x R − x T = 200 m. As shown in Figure 9, the STF-CF decreases as the tunnel width ðWÞ or MS T and MS R antenna array spacing ðδ T , δ R Þ increases, and the STF-CF increases as the K-factor increases. Meanwhile, the STF-CF simulation values are in good agreement with the theoretical values, indicating that the proposed model is suitable. The LoS components have a negative impact on MIMO channel performance. Figure 10 represents the STF-CF for different normalized time lag τ · f R m and Doppler shift ð f R m = ½120 Hz ð23 km/hÞ , 240 Hz ð46 km/hÞ, and 360 Hz ð69 km/hÞÞ for MS R when K = 3, and distance between MS T and MS R is set to x R − x T = 200 m. The STF-CF of the simulation results fits very well with the theoretical results, as illustrated in Figure 10. It is worth noting that when temporal separation increases, the STF-CF decreases. In our proposed model, we have considered the LoS and diffused components which provide different STF-CF values instead of 1 under different antenna spacing and normalized time lag, as given in [45]. When the Doppler shift f R m raises, the STF-CF reduces rapidly. The moving velocity of vehicles affects the V2V channel statistics of the MIMO system, as shown in [46], and this behavior can also be shown in Figure 10.

Conclusions
In this paper, we have presented a novel 3D scattering model for V2V MIMO radio communications in geometrical rectangle tunnel environment. Moreover, the statistical characteristics and analytical results of the proposed model allow one to understand the real tunnel propagation environment for 5G-enabled IoT communications, as it assumes the randomly distributed scatterers along the tunnel sidewalls and ceiling. A generic analytical expression for the STF-CF has been obtained by considering the nonisotropic scattering conditions in both LoS and NLoS propagation environments. Near-and far-field effects are taken into account in this RSBST model to characterize the spherical wavefronts, resulting in AAoA, EAoA, EAoD, and AAoD statistics and PDF and STF-CF responses on the MIMO antenna arrays. It is proved that the nonstationarity, including moving direction φ v and propagation delays, affects the MIMO channel performance. From the analytical results, we can deduce that the STT-CF is seriously affected by propagation 12 Wireless Communications and Mobile Computing delay, K-factor, movement velocity of receiving antenna array, and antenna array spacing. The proposed model results are aligned very well with previous propagation models, indicating that the proposed massive MIMO channel model can be generalized.

Data Availability
The data that support the research findings are available on request. The data are not publicly available due to the privacy of research participants.