Geometry based Stochastic Channel Modeling
using Ambit Processes

Vehicle-to-vehicle (V2V) and vehicle-to-infrastructure (V2I) communication systems form crucial components of intelligent transportation systems [1, 2]. In order to support the development and deployment of V2V and V2I communication systems, channel models characterizing important features of vehicular wireless channels are required. In recent years, several channel models based on different modeling techniques have been proposed [3, 4, 5]. These channel models offer varying degrees of flexibility in terms of simplicity, accuracy, and scalability to enable modeling of various channel parameters.

In order to balance the trade-off between accuracy and complexity of the channel model, geometry based stochastic channel model (GBSCM)s have found wide acceptance for modeling vehicular channels [6, 5, 7]. In GBSCMs, scatterers are placed in the Euclidean plane based on probability density functions (PDF)s which are derived empirically from measurements, and the channel impulse response (CIR) is evaluated using simplified ray tracing techniques. Depending on the modeling aspects, GBSCMs further classified into regular-GBSCMs and irregular-GBSCMs. In regular-GBSCMs [8], scatterers are distributed across certain regular shapes including rings and ellipses, whereas in irregular-GBSCMs, scatters are distributed across arbitrary shapes. Despite the simplicity and analytical tractability of regular-GBSCMs, they do suffer from inaccuracies in modeling vehicular channels. In general, placement of scatterers on the regular shapes does not always ensure modeling of real-world scenarios [6]. On the other hand, owing to the more realistic distribution of scatterers, irregular-GBSCMs show a better agreement with the measurement data [9].

Despite the accurate modeling of vehicular channels, irregular-GBSCMs tend to suffer from high simulation complexity and reduced analytical tractability due to the involvement of a large number of scatterers used to characterize the multi-path environment. Due to the high computational complexity resulting from the involvement of a large number of scatterers, existing methods traditionally employ approximations to model scattering environments. One such approach [10, 8] relies on specific placement of a finite number of scatterers in the scattering region so that the approximated model captures channel characteristics with a fair amount of accuracy. However, this approach fails to achieve desirable statistical properties of the reference model [11]. The second approach [12] involves the introduction of discrete prolate spheroidal sequences to represent scattering environments thereby reducing the dimensionality of the problem. The approach in [12] is based on a key assumption that wireless channels are wide-sense stationary, and variations in direction of arrival or departure as well as multi-path power during the simulation interval are negligible. However, in realistic propagation scenarios, the vehicular wireless channel is, in general, non-stationary [6], and therefore, channel simulations for longer time duration mandate multi-path powers, path delays, and Doppler frequencies to be calculated in every channel instance using direct geometrical calculations. Moreover, the approach in [12] does not model the birth-death event of multi-path components.

To address the computational complexity, in this paper, we present an irregular GBSCM in which we model scatterer locations and their modification relative to a moving wireless node using a novel approach known as ambit process. We have two main objectives: (i) To develop a general analytical framework for irregular-GBSCM characterizing spatial consistency, non-stationarity, birth-death events of multi-path components, (ii) Simulate channel variations resulting from change in direction of multi-path components and their appearance/disappearance in a computationally efficient way.

The rest of the paper is organized as follows. Section II presents the system model as the basis for the proposed geometrical channel modeling and the fundamentals of ambit processes. Section III elaborates a novel 2D convolution based algorithm for simulating V2I channels. Section IV includes simulation results and Section V concludes the paper.

In this section, we present an algorithm for simulating time-varying CIR of non-stationary V2I channels. The computational complexity of the algorithm is significantly reduced through a 2D convolution operation. For this purpose, we transform the kernel function, $f$ in (6) in the form of a semi-stationary function [13]. Therefore, we decompose $f$ in (6) into the format given in (III).

The approximation, $d_{s}(t)\approx d(BS_{x},BS_{y};\int_{0}^{u}v(\zeta)\text{d}\zeta,y_{s})$ is expected to be sufficiently accurate since $d(BS_{x},BS_{y};\int_{0}^{u}v(\zeta)\text{d}\zeta,y_{s})\gg d(\int_{0}^{t}v(\zeta)\text{d}\zeta,y_{i};\int_{0}^{u}v(\zeta)\text{d}\zeta,y_{s})$ in general. For realization, we initially assume constant velocity $v_{0}$ for $v(t)$ and introduce time variability later. By treating the terms $\exp(kd(v_{0}t,y_{i};v_{0}u,y_{j}))$ and $\exp(kd(BS_{x},BS_{y};v_{0}u,y_{j}))$ as constants for a given time instance $t$, the kernel function in (III) is represented in the form,

where

The symbol $*$ denotes linear convolution operator. Now, we apply a convolution technique to evaluate the time-varying channel impulse response by extending the method introduced in [15], which was originally proposed for simulating one dimensional Lèvy processes. We first modify the expression in (II-B) by separating time and space integration. For this purpose, we assume the ambit set has the form, $A(t,y_{i})=(t,0)+A$, where $A=A(0,y_{i})$. Hence,

We assume that $\overline{f}(x,y_{i}-y_{j})=0$ for $x\leq 0$. Moreover, based on the independent scattering property of Lèvy bases [13], we write, $L(\text{d}u,\text{d}y)=\text{d}L(u,y_{i})$. Therefore, (III) becomes

We see that the resulting ambit process can be represented in terms of a $2$D grid with step sizes, $\Delta>0$ and $\Lambda>0$. Therefore, we have,

We note that $\overline{f}(x,y)=0$, if $(x,y)\notin A(0,y_{i})$. Similar to the approach in [15], we assume $\overline{f}$, $\widehat{f}$, and $\sigma$ are approximately constant and equal to their respective values at $((l-1)\Delta,(m-1)\Lambda)$. Then,

where $\nabla L_{l,m}=L(l\Delta,m\Lambda)-L((l-1)\Delta,(m-1)\Lambda)$, $l,m\in\mathbb{N}$ are the increments of the underlying Lèvy process. Now, we devise a simulation algorithm for V2I channel in Algorithm Algorithm based on the formulation in (III).

The algorithm has two phases; initialization and channel simulation. During the initialization phase, all the required channel parameters in the form of matrices $X_{mat}$ and $Y_{mat}$ are precomputed for the 2D convolution operation to be used in the channel simulation phase. For this purpose, we first sample time, space ($y-$ dimension), and delay dimensions with respective number of points $P$, $M$, and $D$. The parameters $P$, $M$, and $D$ are determined based on maximum propagation delay $\tau_{max}$, delay resolution $\delta\tau$, simulation time $T_{max}$, and radius of the scattering region $R$. Then, for each space sample index, we construct two matrices with columns corresponding to delay and rows corresponding to time. For the first matrix $X_{mat}$, we compute the function $\overline{f}$ for all the time and space sample points. The computed $\overline{f}$ is mapped into the locations of $X_{mat}$ specified by the time and delay sample points. For the second matrix $Y_{mat}$, a similar procedure is adopted with the function defined by the product of $\widehat{f}$, $\sigma$, and $\nabla$. In the simulation phase, we perform 2D convolution of row wise folded $X_{mat}$ and $Y_{mat}$ for each space sample points as shown in step 8 of Algorithm. The resulting 2D matrices for each space sample index are added.

The computational benefit is obtained owing to the fact that the proposed V2I channel simulator tracks modifications in the received signal power, propagation delay, and phase of each multi-path component through the convolution operation. Computationally efficient algorithms exist for implementing convolution of sequences. It should be noted that impulse response for a longer time duration $T_{max}$ can be efficiently computed by breaking the matrix $Y_{mat}$ into multiple small matrices and apply overlap-add or overlap-save method to compute their convolution.

The effect of time-varying velocity of MU on channel impulse response is introduced in steps 11-16 of Algorithm. The time varying velocity of MU results in sampling of the MU’s location at a non-uniform rate depending on the velocity of MU at a given instant of time. The impulse response of a non-stationary V2I channel $h_{N}(t,\tau)$ is simulated using a non-uniform sampling rate conversion technique given in steps 12-16 of the Algorithm. We assume that the actual space-sampling interval (along x-axis) for simulating V2I channel is sufficiently small so that adjacent samples of the channel coefficients are highly correlated. Therefore, we apply a linear interpolation technique to determine the channel coefficient at the required spatial point as given in step 15-16 of Algorithm. The non-stationary time-varying channel impulse response is stored in variable $h_{N}$.

To validate the proposed channel modeling framework, we carry out numerical simulations. We consider that a BS is arbitrarily located at $(-100,20)$ and the initial position of a MU is at $(0,0)$. The MU is assumed to be moving away from the BS with an initial velocity, $v_{0}=40$ km/hr as shown in Fig. 2. We assume that BS transmits signal at a frequency $f_{c}=2.6$ GHz. Moreover, $G_{s}=G_{L}=(\lambda/4\pi)^{2}$ and path loss exponent, $\gamma=1.7$ (chosen based on the average of path loss exponents reported in [6]). Numerical results corresponding to the temporal autocorrelation function (ACF) and Doppler power spectral density (PSD) are generated by averaging the output of 500 independent realization of the Poisson Lèvy field. Moreover, LoS component is not accounted in the simulations.

We simulate non-stationary V2I channels by accounting time-varying velocity for MU. We assume that the MU accelerates with a uniform rate of acceleration $a$. The simulation plots for normalized temporal ACF of the channel at different time instances are shown in Fig. 3. The plots show that as the velocity of the MU increases with time due to the acceleration, the temporal ACF decreases rapidly with the time difference. This implies that channel decorrelates quickly with time. The time-varying Doppler PSD corresponding to the received signal at MU is presented in Fig. 4. The shape of the Doppler PSD closely follows the measured Doppler PSD for typical V2I channels. The Doppler PSD evaluated at different time instances varies significantly due to the presence of non-stationary properties of the simulation environment. This is essentially due to the inclusion of time varying velocity for MU and the birth-death process on time-space axis. The plots in Fig. 4 indicate that maximum Doppler shift increases for higher rate of accelerations since more acceleration leads to higher velocities thereby increasing the Doppler shift.

Firstly, we compare the complexity of the proposed simulation algorithm and the direct approach for simulating time varying vehicular wireless channels. The direct approach (used in [12] for tracking channel variations outside the stationary interval) involves computation of channel parameters such as path loss and propagation delay based on the relative modification of scatterer position with respect to moving MU using geometrical calculations. We conduct simulations for varying $R_{s}$ and $N_{s}$, ranging from $R_{s}=100=N_{s}=100$ to $R_{s}=N_{s}=5000$ (which corresponds to a model for diffuse components) in a desktop computer configured with INTEL CORE i7^® microprocessor and $16$ GB RAM. The savings in simulation time as function of scatterer density and arrival rate is shown in Fig. 5(a) as an empirical cumulative distribution function (CDF) obtained from multiple realization of the scattering environment. The median of the simulation time for generating $4$s of channel using the direct approach is approximately $10$ times the simulation time for the proposed approach with $R_{s}=N_{s}=5000$. We also plot the error between the time varying received power (with unit transmit power) simulated using the direct approach and the proposed simulation algorithm. The error plot is depicted in Fig. 5(b) in terms of the ratio of received power simulated separately using the direct approach and the proposed simulation algorithm. The median of the error is found to be $-0.31$ dB with standard deviation equals to $0.12$ dB which can be attributed to the fact that the approximated path loss computed for the proposed simulation algorithm is less than or equal to that considered in the direct approach.

In this paper, we introduce a novel method which serves as a tool to characterize vehicular wireless channels based on the theory of ambit processes. In particular, we define suitable ambit sets and kernel functions to model scattering regions and channel properties, respectively. Key features of vehicular channels such as channel non-stationarity, multi-path death events, spatio-temporal channel correlation are studied through the proposed modeling framework. The flexibility and the tractability offered by the underlying theory of ambit processes provides a useful platform for studying important characteristics of vehicular communication channels thus paving the way for development of new channel models within the scope of next generation communication systems. Additionally, the proposed computationally efficient simulation algorithm models vehicular channels and yields several significant characteristics of vehicular communication channels. Future work involves validation of certain modeling aspects in the proposed framework with measurement data from practical scenarios.