A Non-WSSUS Channel Simulator for V2X Communication Systems

: This paper presents a simulator of non-wide sense stationary uncorrelated scattering (non-WSSUS) multipath fading channels for the performance analysis of vehicle-to-everything (V2X) communication systems. The proposed simulator is constructed with the combination of the Monte Carlo and sum-of-cisoids (SOC) principles, and it is suitable for multicarrier transmission schemes such as those deﬁned for dedicated short-range communications (DSRC) and cellular-based V2X (C-V2X) communications. The channel simulator provides an accurate and ﬂexible solution to reproduce the time and frequency (TF) correlation properties of non-WSSUS vehicular channels under arbitrary isotropic and non-isotropic scattering conditions. Furthermore, the proposed simulator allows velocity variations and non-linear trajectories of the mobile stations (MSs). To demonstrate the practical value of the presented simulator, we evaluate the bit error rate (BER) performance of two channel estimation techniques that are considered for IEEE 802.11p transceivers, namely the least squares (LS) estimator and the spectral temporal averaging (STA) technique. The BER performance of both channel estimators was analyzed by considering three propagating scenarios for road safety applications. Our results show that the non-stationary characteristics of the vehicular multipath fading channel have nearly no effects on the LS estimator’s BER performance. In contrast, the performance of the STA estimator is signiﬁcantly affected by the channel’s non-stationary characteristics. A variation of the original STA technique that applies only a temporal averaging is introduced in this work to improve the system’s BER in non-WSSUS channels. and C.A.G.; software, J.J.J.-R. and C.A.G.-V.; validation, C.A.G., J.M.L.-R., D.U.C.-D., and R.V.; formal analysis, J.J.J.-R., C.A.G.-V., C.A.G., J.M.L.-R., D.U.C.-D., and R.V.; investigation, J.J.J.-R. and C.A.G.-V.; resources, J.J.J.-R., C.A.G., J.M.L.-R., D.U.C.-D., and R.V.; writing, original draft preparation, J.J.J.-R., C.A.G.-V., and C.A.G.; writing, review and editing, J.M.L.-R., D.U.C.-D., and R.V.; supervision, C.A.G.; project administration, J.J.J.-R., C.A.G., J.M.L.-R., and D.U.C.-D.; funding acquisition, J.J.J.-R., C.A.G., J.M.L.-R., D.U.C.-D., and R.V.


Introduction
The worldwide growth of the vehicle fleet has caused high levels of pollution due to inefficient traffic control and increased concerns about the number of deaths and injuries from vehicular accidents. Intelligent transportation systems (ITSs) are a group of technologies that address these problems by managing vehicle operations and assisting drivers with safety and contextual information [1,2]. ITSs require a wireless communication platform to keep a vehicle connected with other vehicles and with other elements on the road [3][4][5][6][7][8][9][10]. The technical requirements of this vehicular communication

Motivation and Related Work
The work presented in this paper is motivated by the need for a flexible vehicular channel simulator that enables the emulation of propagation scenarios similar to those found in road safety applications [1,2]. These scenarios are characterized, on the one hand, by a small coverage area, according to the distance covered by vehicles during the transmission of warning messages, and on the other hand, by the vehicles' high mobility, which may include speed variations and changes in the vehicles' trajectories [3][4][5]. The relevance of a simulation framework with these capabilities lies in the fact that most of the existing vehicular channel simulators do not incorporate the aforementioned properties. The early work in modeling and simulation of vehicular channels assumed: (i) the uniform motion of the vehicles, (ii) the transmission of narrowband signals, and (iii) the fulfillment of the wide-sense stationary (WSS) condition, e.g., [33][34][35][36]. Subsequent work was focused on extending these initial efforts with respect to small-scale wideband channels by assuming WSS statistical properties in the time and frequency domains, i.e., with respect to locally WSSUS vehicular channels [37][38][39]. Nonetheless, empirical evidence showed that the vehicular radio channels do not fulfill the WSSUS condition [23][24][25], and this issue recently motivated a further wave of research initiatives aiming at the formulation of new statistical reference models for the simulation of locally non-WSSUS wideband channels. Contributions in this regard include the work presented in [40][41][42][43][44].
Most of the papers dealing with the modeling of locally non-stationary vehicular channels, either narrowband or wideband, assume uniform motion of the MSs, that is vehicles moving on a linear trajectory and with a constant speed. To the best of the authors' knowledge, Iqbal and Abhayapala were the first to formulate a non-stationary mobile radio channel model that considers the effects of non-uniform motion in an F2M wireless link, where the MS is moving with a constant acceleration on a linear trajectory [45]. This F2M channel model was extended by Pätzold and Borhani in [46] with respect to non-linear trajectories. The proposals in [46] were applied by Dahech et al. [47] to characterize non-stationary V2X channels allowing for speed and trajectory variations of the vehicles [47]. Based on the concepts discussed in [45][46][47], novel simulation models for non-stationary vehicular channels under velocity variations of the MSs were proposed independently in [48,49]. In the aforementioned papers, the spotlight was on narrowband channels. In [31], Gutiérrez et al. proposed a statistical model of locally non-stationary wideband V2X channels that incorporates the effects of speed and trajectory variations. The scope of the latter paper was limited to the formulation and statistical analysis of the channel model. As an extension, in this paper, we provide a methodology for the simulation of the channel model in [31]. The simulation framework presented here was conceived of as a tool for the performance analysis of vehicular communication systems based on multicarrier modulation techniques, such as the DSRC and C-V2X systems reviewed next.

Multicarrier Data Frames of the DSRC and LTE-V2X Standards
In this section, we present a brief overview of the IEEE 802.11p and LTE-V2X specifications for the physical layer of vehicular communication systems. In the case of the LTE-V2X, we review the characteristics of the PC5 interface, which is considered for V2V, V2I, and V2P applications. Specifically, the parameters of the multicarrier schemes for both sets of specifications, as well as their frame structures are presented for a typical 10 MHz bandwidth. Then, we describe a mathematical reference model of the received multicarrier data frame that is valid for both sets of specifications.

General Overview of IEEE 802.11p and LTE-V2X Specifications
The IEEE 802.11p standard was developed as an amendment to the specifications for WLANs to address wireless communication in vehicular environments [4,5]. This standard presents the specifications for DSRC systems in the 5.9 GHz band. The IEEE 802.11p transceivers use OFDM as a modulation technique, in which a block of data is transmitted through equidistant subcarriers. The data transmission can be performed over three different bandwidths, which can be 5, 10, or 20 MHz.
The preferred bandwidth is 10 MHz since it is more robust against the time dispersion of the transmitted signal [4]. The IEEE 802.11p standard considers a total of 64 subcarriers, among which 48 are used for data transmission and four are used as pilot subcarriers. The remaining 12 subcarriers do not transmit information to avoid interference between adjacent bands. The parameters of an IEEE 802.11p OFDM symbol for a 10 MHz bandwidth are shown in Table 1 [11]. Furthermore, the frame structure considers short training symbols (STSs) with a duration of 1.6 µs and long training symbols (LTSs) of 6.4 µs. In addition, the specifications consider a guard interval (GI) of 1.6 µs to reduce intersymbol interference. Figure 1 shows the frame structure for a typical 10 MHz bandwidth, as specified in the IEEE 802.11p. First, ten STSs (t 1 , . . . , t 10 ) are considered as a preamble for signal detection, automatic gain control, and diversity selection (t 1 , . . . , t 7 ), as well as for coarse frequency offset estimation and timing synchronization (t 8 , t 9 , and t 10 ). Next, there are two LTSs (T 1 and T 2 ) that follow two GIs, which are placed after the tasks in the preamble. The LTSs T 1 and T 2 are employed for channel estimation and fine frequency offset estimation. Then, the frame considers a header that consists of a GI and a signal block where the bit rate and frame length are defined. Finally, the frame ends with the the data field, which includes one GI of 1.6 µs for each information block. The IEEE 802.11 standard does not restrict the frame length, but it recommends a size between one and 4096 bytes according to the application requirements [11].  Figure 1. IEEE 802.11p frame structure with a 10 MHz bandwidth [11].
With respect to LTE-V2X, 3GPP included a set of specifications to the LTE standard for applications in vehicular environments [14]. Two operation modes are defined in such 3GPP specifications. The first mode allows vehicles to communicate with the cellular network for cloud-based applications, whereas the second mode enables V2X short-range communications. This latter mode is based on the PC5 interface that is considered for direct D2D communications [14]. Such a mode is referred to as sidelink transmission [9]. The PC5 interface operates in the 5.9 GHz band using SC-FDMA as the transmission scheme. The specifications of the PC5 interface consider resource blocks (RBs) that consist of 12 subcarriers. Depending on the bandwidth, the transmission scheme is scalable by carrier aggregation, which adds new resource blocks to the symbol. For a typical 10 MHz bandwidth, the specifications define 50 RBs, which occupy a total of 600 data subcarriers [14]. The main parameters of LTE-V2X employing the PC5 interface for a 10 MHz bandwidth are shown in Table 1 [14]. The subcarrier spacing is set to 15 kHz (i.e., 180 kHz per RB). Additionally, the frame duration is fixed to 1 ms, which allows the transmission of 14 symbols including GIs. The duration of each symbol and GI is 66.31 µs and 4.69 µs, respectively. The frame structure is shown in Figure 2, where the symbols are enumerated from 0 to 13. The LTE-V2X frame includes nine data symbols, four demodulation reference signal (DMRS) symbols and one empty symbol for T X − R X switch and timing adjustment [9,14]. The last symbol allows tracking the channel for the next transmission. These parameters are likely to be employed by emerging C-V2X technologies since 5G specifications are also considering the use of the PC5 interface for direct D2D applications in vehicular environments [15].

Mathematical Model of the Received Signal in Multicarrier Systems
The channel simulator proposed in this paper is based on a signal model that is suitable for multicarrier communications that transmit the information employing data frames. This signal model is valid for the data frames specified by the IEEE 802.11p and 3GPP LTE-V2X standards. We consider N p subcarriers for data transmission. Therefore, we model the transmitted symbols in the complex baseband equivalent as column vectors of N p elements where each element corresponds to the data in the n-th subcarrier. Then, considering a frame composed of K multicarrier (OFDM) symbols, the transmitted signal is described by an N p × K matrix X ∈ C N p ×K .
Considering the transmission of the data frame, the received signal can be expressed as: where Y, H, and N ∈ C N p ×K represent respectively the received symbols matrix, the channel transfer function matrix, and the additive white Gaussian noise (AWGN) matrix. Furthermore, the operator • represents the Hadamard product.

Reference Channel Model
In this section, we describe the reference channel model that assumes non-linear trajectories and velocity variations of the MSs. We introduce first the reference propagation environment at the beginning of the transmission. Then, we present the channel's transfer function and describe its parameters under the one-ring scattering assumption.

Description of the Reference Propagation Environment
The proposed channel simulator in this work is based on the reference model for vehicular communications presented in [31]. Such a model considers velocity variations and non-linear trajectories of the MSs. This model is particularly well suited for analyzing the channel's non-stationarities arising at a small-scale level due to the time-varying nature of the propagation delays [42] and the frequency dependence of the Doppler shift [50]. In this model, the transmitted signal reaches the receiving antenna by single interactions with L fixed (static) IOs randomly located in the propagation environment. As a result of such interactions, L replicas of the transmitted signal are observed at the receiving antenna. Figure 3 shows the reference propagation environment when the MSs begin the transmission at t = 0. The black dots represent the IOs, whereas the black triangle and the black square denote the transmitter (T X ) and receiver (R X ) stations, respectively. The initial positions of T X and R X are represented by O O O y O O O , respectively, and they are separated by a distance D. Vectors v q and a q for q ∈ {T, R} represent the velocity and constant acceleration components, respectively, of T X and R X . The velocity vectors define the initial speeds and directions of motion of the MSs. The initial speeds of T X and R X are, respectively, ν T and ν R . The angles γ T and γ R correspond to the directions of the velocity components of T X and R X , respectively. On the other hand, the acceleration components define the magnitudes and directions for the MS changes of motion over time. The acceleration magnitudes of T X and R X are a T and a R , respectively. The directions of the acceleration vectors are ρ T and ρ R for T X and R X , respectively. Furthermore, p T denotes the position of the -th IO with respect to the reference point O O O, and p R describes the R X position seen from the -th IO. The -th IO is separated by a distance d T from T X and a distance d R from R X . Finally, φ T represents the angle of departure (AOD) and φ R the angle of arrival (AOA), both of them from the interaction with the -th IO. The vectors introduced in this section, as well as the constants that complete the description of the propagation environment are summarized in Table 2, where the notation x = M x ∠Θ x indicates a vector x with magnitude M x and direction Θ x .

Description Parameter
Velocity vector of

Mathematical Model of the Channel Transfer Function
Following [31], the transfer function for non-WSSUS channels with Rayleigh fadings is represented as follows: where j 2 = −1, m corresponds to the frequency subcarrier index, k is the data symbol index, ∆ f denotes the separation between subcarriers, L represents the total number of IOs around R X , g and τ [k] are the amplitude and the time-varying propagation delay of the -th replica of the transmitted signal, and θ stands for the phase shift caused by the interaction of the transmitted signal with the -th IO. The carrier frequency is given by f c = c/λ, with c denoting the speed of light and λ the wavelength of the transmitted signal. Furthermore, Π T 0 [k] is defined as: where T s is the duration of a single symbol. The function Π T 0 [k] denotes a rectangular windowing function introduced to constrain H[m, k] within a time interval T 0 where the plane wave approach is valid. The gains g are assumed positive random variables, whereas the phase shifts θ are independent and identically distributed (i.i.d.) random variables with a circular uniform distribution in [−π, π). The time-varying propagation delays τ [k] in (2) are defined by: where f S andḟ A [k] characterize the Doppler frequency shifts caused by the initial speed and the acceleration of the MSs, respectively, and they are given by: with f q max = ν q /λ andḟ q max [k] = (kT s a q )/λ for q ∈ {T, R}, which correspond to the maximum Doppler frequency shifts due to the initial speed and the acceleration of T X and R X , respectively. The parameters ν q and a q for q ∈ {T, R} represent the initial speeds and accelerations of the MSs. Variables γ q and ρ q for q ∈ {T, R} are deterministic quantities that characterize the initial direction of motion of the MSs and the direction of their acceleration components, respectively.
The AODs and AOAs are circular symmetric random variables having distributions p T φ (φ) and p R φ (φ), respectively. We consider that the function p R φ (φ) follows a von Mises distribution given by [51]: where µ R is the mean of the distribution, κ is known as the concentration parameter, and I 0 (κ) is the modified zeroth order Bessel function of the first kind [51]. When κ increases its value, the values of φ R accumulate around its mean value. A particular case of this function is when κ = 0. In this case, the values φ R follow a uniform distribution over [−π, π). The von Mises distribution in (7) allows modeling arbitrary propagation scenarios with isotropic scattering (κ = 0) and non-isotropic scattering (κ > 0). Considering that only one side of the radio link is affected by local interactions with IOs and that such interactions occur in the receiver side, the AODs φ T can be defined as [31]: where d T and d R depend on φ R and φ R as follows: when the IOs are assumed to be located on a ring of radius d centered on O O O .

The Proposed Vehicular Channel Simulator
For the simulation of the reference channel model given by the transfer function H[m, k] in (2), we follow the principle of deterministic channel modeling [52]. According to this principle, the simulation of a non-realizable stochastic reference channel model can be accomplished by generating a single realization or sample function in each simulation run. Therefore, a realization of the reference channel can be computed by following the transfer function [53,54]: where g = 1 L for all is an observation of the amplitude, θ ∈ [−π, π) is a single realization of the phase shift that follows a uniform circular distribution, and the time-varying propagation delays τ [k] are given by: with: where φ T , φ R ∈ [−π, π) are realizations of the AOD and AOA, respectively, that follow the distributions of the circular random variables φ T and φ R , as described in Section 4.2. The rest of the parameters L, f c , D, d, ν q , a q , γ q , and ρ q for q ∈ {T, R}, as well as µ R and κ for the distribution of φ R are deterministic quantities introduced by the user in order to generate arbitrary propagation environments. This simulation model can be classified as an SOC model of Class XII according to the classification described in [55]. The described channel simulator is suitable for the performance analysis of V2X communication systems based on multicarrier transmission schemes, such as specified in the current V2X communication standards. For this purpose, we consider that the system uses N p subcarriers to transmit K symbols. Then, the simulation model in (11) can be used as H m,k = H[m − N p /2, k − 1] for m = 1, . . . , N p and k = 1, . . . , K to generate the channel transfer function matrix. This expression allows us to start the transmission at time t = 0 and generate a spectrum centered at the carrier frequency f c . A realization of the channel model requires us to specify the bandwidth B and the carrier frequency f c , as well as the spacing in time and frequency according to the system specifications (e.g., IEEE 802.11p or LTE-V2X) for a total of K symbols and N p subcarriers. Furthermore, the propagation scenario should be initialized by specifying the number of IOs, the scattering parameters, and the components for the dynamics of the MSs. Besides, for the plane wave propagation approach to be valid in the simulation channel model, the far-field condition must be met. Therefore, we consider defining arbitrary distance values such that D d. Table 3 shows the pseudocode for the function GenerateChannel, which produces a single realization of the channel based on the proposed simulation model. This function takes as input the motion parameters for the MSs (ν T , ν R , a T , a R , γ T , γ R , ρ T , ρ R ), the number of IOs L, the scattering parameters (µ R and κ), the distance between MSs D, the radius d of the ring of IOs, and the number of total symbols K. We consider that the system parameters are defined inside the function following the desired V2X specifications. First, we set all the system parameters according to the V2X standard for a bandwidth B. Then, we generate the corresponding random variables and compute the Doppler shifts following the simulation model with the user inputs for an arbitrary propagation scenario. Finally, a realization of the channel transfer function is computed and returned to simulate data transmission. Table 3. Pseudocode of the proposed channel simulator.
Set system parameters according to the standard: Bandwidth B Carrier Frequency f c Total number of subcarriers N p Symbol duration T s Subcarriers spacing ∆ f Compute gains g for the L IOs Generate random variables for the L IOs: Phase shift θ AOAs φ R with von Mises distribution AODs φ T following (8) Compute maximum Doppler frequencies due to initial speed f T max and f R max Compute Doppler frequency shift f S due to initial speed Compute time-varying maximum Doppler frequencies due to acceleratioṅ The simulation of a V2X communication system requires several blocks in addition to the channel simulator. These blocks realize different communication tasks such as modulation, equalization, and demodulation. Figure 4 shows a block diagram for the simulation of a V2X communication system, which consists of three main blocks: the transmitter, the wireless propagation channel, and the receiver. First, in the transmitter side, the training and data generator are used to obtain random information frames to be transmitted. We consider that K symbols are generated with N p subcarriers. Then, the information symbols are modulated, i.e., the information is encoded from the message source in a way that is suitable for transmission. Next, the modulated signal is sent through the multiplicative non-WSSUS V2X channel that is also corrupted by AWGN. The V2X channel in this block diagram is obtained using the GenerateChannel function with an arbitrary propagation scenario. In the receiver side, we consider channel estimation and equalization blocks to mitigate the impact of the wireless channel. Finally, the signal is demodulated to decode the message information.  Table 4 shows the pseudocode to simulate a V2X communication system to evaluate its performance in terms of the BER. The implementation should consider a set of specifications (e.g., IEEE 802.11p or LTE-V2X) to set the frequency and time parameters. The first step is to set the amount of data symbols F on each frame. Next, the total number of symbols is specified according to the standard. For the IEEE 802.11p, two training symbols are added at the beginning of each frame to obtain the initial channel estimation [11]. Therefore, each frame contains K = F + 2 OFDM symbols. On the other hand, the number of symbols is fixed in the LTE-V2X specifications. There are 14 symbols in total, from which nine are used for data [14]. The next step is to establish the desired range of the energy per bit-to-noise spectral density ratio (Eb/No) to simulate data transmission over different noise levels. The parameters of the multicarrier scheme (number of data and total subcarriers, number of pilots) are then configured according to the specifications for the desired bandwidth. For instance, we can follow the parameters in Table 1 for a 10 MHz bandwidth. Moreover, the specifications for DSRC and C-V2X communications differ in the type of pilots. In the case of IEEE 802.11p, it defines four subcarriers as pilots, whereas the LTE-V2X specifications consider four complete symbols. After defining those values, the parameters for the propagation scenario are established, and a matrix that represents the transmitted symbols (including data and pilots) is generated randomly for each noise level. Since the simulator is based on the Monte Carlo principle, n realizations of the transmission process are necessary to obtain an average performance metric. For each iteration, a channel transfer function is calculated by calling the function GenerateChannel, which depends on the deterministic parameters described above. Subsequently, the data transmission is simulated following Equation (1). This simulator can be employed to evaluate different schemes, such as channel estimation. Therefore, we perform such transmission tasks and assess their performance over the generated channel. Finally, the BER is averaged for all iterations. Number of IOs L MSs initial speeds, acceleration components, and direction parameters ν T , ν R , a T , a R , γ T , γ R , ρ T , and ρ R Scattering parameters µ R , κ Initial distance D (between T X and R X ) Ring radius d (between R X and the -th IO) Generate training and data symbols randomly Modulate training and data symbols Organize training and data symbols in matrix X Initialize accumulated number of errors

Simulation Examples
This section presents the performance analysis of a V2X communication system under different propagation scenarios employing binary phase-shift keying (BPSK) modulation. This framework can be adapted to any multicarrier system and modulation scheme, such as the ones specified by the IEEE 802.11p and 3GPP LTE-V2X standards. In particular, we used the proposed simulator to evaluate the BER performance of a DSRC system based on the IEEE 802.11p standard in scenarios where road safety applications might be employed. We evaluated the performance of two channel estimation techniques that are considered for IEEE 802.11p transceivers, namely the LS estimator and the STA technique, which are described in the following subsections.

Channel Estimation Techniques for the IEEE 802.11p Standard
Channel estimation and equalization techniques are widely used in wireless communications to recover the signal that propagated through the channel. In the IEEE 802.11p standard, channel estimation is performed using the LTSs T 1 and T 2 that are inserted in the preamble of each frame. For this standard, the signal model shown in (1)  , where x e k ∈ C N p ×1 is the k-th column of the matrix X e for k = 1, 2.

Least Squares Estimator
The LS channel estimation over the k-th OFDM symbol and the m-th subcarrier is given by: for m = 1, 2, . . . , N p and k = 1, 2, . . . , F, where H d LS represents the estimated channel matrix. From (15), we can see that the LS estimator is used when we assume a time-invariant channel for the transmission of the whole frame, so that the estimated channel H d LS m,k is the same for all k, i.e., for all OFDM information data symbols. After the channel estimation stage, the received OFDM symbols pass to an equalization process. Considering zero-forcing equalization, the equalized signal is given by: for m = 1, 2, . . . , N p and k = 1, 2, . . . , F, whereỸ d LS is the equalized data matrix.

Spectral Temporal Averaging Technique
The STA technique presented in [32] takes as the initial point the LS estimator in (15). This estimation scheme is intended for time-varying channels, as it tracks the channel variations according to: for m = 1, 2, . . . , N p and k = 2, 3, . . . , F where H d ACT denotes an estimate of the channel matrix. The estimates H d ACT m,k given in (18) are affected by AWGN. For this reason, an averaging operation is computed for each subcarrier as follows: for m = 1, 2, . . . , N p and k = 2, 3, . . . , F, whereH d is a matrix that contains all the averages in the frequency domain and β is an integer variable that controls the number of subcarriers in the averaging process. In general, when we have a frequency selective channel, the value of β must be small. On the other hand, for non-selective channels β must be higher [32]. Then, the estimated channel is given by: for m = 1, 2, . . . , N p and k = 2, 3, . . . , F, where α is an updating and weighting factor restricted to α > 1 and (1 − 1/α) > 0. A large value of α is used in slow-fading channels, while a small value is preferred for fast-fading channels [32]. Finally, the channel estimation shown in Equation (20) is used in (17) to equalize the next OFDM data symbol and repeat the whole process until the end of the frame.

Simulation Setup
For the simulation experiments, we considered the geometrical one-ring scattering model with a ring of IOs surrounding the R X under isotropic (κ = 0) and non-isotropic (κ > 0) scattering conditions. The propagation scenarios were defined in such a way that they resemble situations where road safety applications might have a fundamental role. The evaluated scenarios are: (i) an intersection where two vehicles are approaching each other; (ii) an overtaking maneuver where a vehicle changes its trajectory to pass the one in the front; and (iii) a highway situation with two vehicles moving on opposite lanes. We assumed a set of general parameters to compare the performance in the different propagation scenarios. The number of IOs surrounding the receiver was L = 20, the initial distance between T X and R X : D = 500 m, and the distance of the ring of IOs surrounding R X : d = 30 m. For the cases that include acceleration components (a T and a R ), we considered a magnitude of 20 m/s 2 . We acknowledge that these acceleration terms are extremely high for an urban environment, but they help us to illustrate that despite their excessive magnitudes, the performance differences from a constant velocity scenario are almost imperceptible by the small transmission times. The particular parameters that define the motion of the MSs for each scenario are described below.

•
Scenario I: Intersection. The MSs move in such a way that they approach each other with angles γ T = 60 • and γ R = 120 • , and both T X and R X acceleration components point in the same direction of their corresponding initial speeds (i.e., γ T = ρ T and γ R = ρ R ). The motion of the MSs resembles a safety scenario where vehicles circulate on two streets that intersect at one point. This scenario is depicted in Figure 5. • Scenario II: Overtaking maneuver. A vehicle accelerates and changes its direction of motion aiming to overtake the one in front of it. For this scenario, we assume that both vehicles start moving in the same direction γ T = γ R = 0 • , but the transmitting station has an acceleration component with angle γ R = 15 • , which modifies its trajectory over time. In the case of the receiving vehicle, this one has no acceleration component, i.e., a R = 0. Figure 6 illustrates this scenario where the two vehicles start on the same lane. • Scenario III: Opposite lanes. Two vehicles are approaching each other on opposite parallel lanes of a highway environment. This scenario is shown in Figure 7, where we consider the angles γ T = ρ T = −10 • and γ R = ρ R = 170 • to describe the motion of the vehicles in opposite directions over parallel lanes.

Results of Changes of the Initial Speed
First, we evaluated the BER performance for a transmission of F = 64 data OFDM symbols in each frame at different relative speeds between the MSs by varying both the initial speed and acceleration components. For this simulation, a propagation environment with isotropic scattering was considered (κ = 0), that is a scenario that resembles an urban environment with IOs that uniformly surround R X . The BER performance curves for Scenario I are shown in Figure 8, for both the LS estimator and the STA technique. In the case of the LS estimator, the performance decreased when the initial speeds of the MSs were increased from ν T = ν R = 20 km/h (relative speed of 40 km/h) to ν T = ν R = 100 km/h (relative speed of 200 km/h) regardless of the propagation scenario. This condition occurred since the Doppler frequency shifts increases as the relative speed also increases. Therefore, the channel presents more variations, and the LS estimation becomes less accurate.
In fact, the BER performance of the LS estimator is independent of the channel non-stationarities and MSs acceleration components since the channel estimation is carried out with the two initial training symbols on each frame. Consequently, the time variations in a data frame caused by the non-stationarities and acceleration terms could be negligible for the LS estimator. This result is illustrated in Figure 8, which is consistent with [56] for WSSUS channels and with [57] for non-WSSUS channels with constant speed. Concerning the STA technique, the BER performance was poor compared to the LS estimator. Such performance was almost equal to the case where there is no channel estimation at all since the statistics of the non-WSSUS process change as the data transmission proceeds, and the non-stationarities directly impact the weighted average that was carried out in the frequency domain [57,58]. In this particular case, the channel effect on the system performance due to the acceleration of the MSs was minimal and masked by its inherent non-stationarities [59]. In addition to the results presented in Figure 8, Table 5 shows the Eb/No levels that attained 10% and 1% BER for the low mobility (20 km/h and 40 km/h) and high mobility (80 km/h and 100 km/h) conditions in the three propagation scenarios described above. The legend NA denotes that the BER level was not achieved by using the corresponding estimation scheme. In this table, we can observe that the performance of the LS estimator in low mobility conditions was similar for the three scenarios, reaching 10% BER with comparable Eb/No levels. In such conditions, an Eb/No regime greater than 5.5 dB guarantees a BER lower than 10% for all the scenarios. On the other hand, for high mobility conditions, we observed that Eb/No levels greater than 10 dB were required to attain a 10% BER. As observed in Figure 8, the BER performance of this estimator was greater than 0.1% even at high Eb/No regimes for initial speeds of 100 km/h. Furthermore, the results showed that the performance of the LS estimation in Scenario I was the least affected by the high mobility conditions, reaching BER levels lower than 10% with an Eb/No greater than 16.51 dB when the initial speeds were 100 km/h. For the STA technique, its BER performance always remained greater than 10% in these experiments regardless of the propagation scenario although using a large value of α.

Results of Changes of the Frame Length
For these simulation experiments, the BER performance was evaluated in low mobility conditions transmitting different numbers of OFDM data symbols by considering initial speeds of ν T = ν R = 40 km/h, acceleration magnitudes of a T = a R = 20 m/s 2 , and a scenario with isotropic scattering (κ = 0). Figure 9 shows that the BER performance of the LS estimation in Scenario I decreased as the size of the frame increased. This drop in performance is due to the channel variations between the transmitted OFDM data symbols, which causes the initial LS estimation to gradually lose its validity until it becomes practically obsolete for very large frames. In this case, the worst performance for the LS estimator occurred after the frame length exceeded 128 OFDM data symbols. Regarding the STA technique, the results showed that as the updating factor increased (e.g., from α = 2 to α = 20), the performance of the STA technique improved, as shown in Figure 9. In such a case, as the parameter α in (20) increased, the estimated channel tends to be similar to the LS estimation. Furthermore, the performance of the STA technique was benefited in low mobility scenarios with small frame lengths, as observed for F = 32. Finally, by increasing the frame length and/or decreasing the updating factor α, the performance of the STA technique approached the case where there was no channel estimation because the weighted average in (20) became imprecise in non-WSSUS channels, whose statistics changed before the transmission of a data frame ended [58].  Table 6 presents the required Eb/No levels to achieve 10% and 1% BER by employing the LS and STA techniques in low mobility conditions, assuming frame lengths ranging from 16 to 128 OFDM symbols in the three propagation scenarios described previously. This table shows that as the frame length increased, the performance of both the LS estimator and STA technique with α = 20 decreased. As observed in Table 5 for changes in the initial speed, the BER performance of the STA technique with α = 2 always remained greater than 10%. For frames of 64 OFDM symbols or less, the performance of the LS and STA (with α = 20) techniques was similar in the three considered scenarios approaching the 10% BER with comparable Eb/No levels. In particular, in the case of short frames with 32 symbols or less, an Eb/No level greater than 4.61 dB and 7.15 dB for the LS estimator and STA technique (with α = 20), respectively, guaranteed a BER lower than 10% for all the propagation scenarios. Furthermore, for these short frame lengths, the performance of the LS estimator and STA technique with α = 20 reached 1% for Eb/No regimes greater than 17.06 dB and 39.99 dB, respectively. Finally, Eb/No levels greater than 5.41 dB and 10.7 dB reached BER performances lower than 10% employing the LS estimator for frame lengths of 64 and 128, respectively. For the STA technique, considering frames with more than 64 OFDM symbols, the BER performance always remained greater than 1% when using α = 20. As observed, the STA estimator performed well assuming short data frames and low mobility conditions. However, this technique demanded larger Eb/No values than the LS estimator. Table 6. Required Eb/No levels to attain 10% and 1% BER over Scenarios I, II, and III for different frame lengths by considering initial speeds of ν T = ν R = 40 km/h and acceleration components of a T = a R = 20 m/s 2 (NA-not achieved).

Results over Non-Isotropic Scattering
The following experiment was carried out with the objective of analyzing scenarios where the IOs that surrounded the R X were distributed in a particular region. This condition emulates a scenario where there is a large amount of free space and the IOs are concentrated in one given direction, e.g., rural environments. In this simulation, high mobility conditions with initial speeds of ν T = ν R = 100 km/h, acceleration magnitudes of a T = a R = 20 m/s 2 , and a fixed frame length of F = 32 were considered to transmit data over highly non-isotropic scattering scenarios with κ = 20 and three different means µ R = 0, π/2, and 3π/4. We compared the results with the isotropic scenario using κ = 0. Figure 10 shows the BER performance curves for this experiment according to Scenario I (see Figure 5). The plot shows that the performance of the LS estimator improved remarkably in non-isotropic conditions, i.e., when the IOs were concentrated in a region. In this scenario, the maximum propagation delay decreases as the IOs concentrate in a region that is closer to the transmitting vehicle, so the frequency selectivity of the channel is reduced [42]. Furthermore, the performance of the LS estimator depends on the mean µ R of the AOAs and also on the directions of motion (γ T and γ R ) of the MSs. For example, taking as a reference the propagation scenario shown in Figure 5, assuming κ = 20, µ R = 3π/4, γ T = ρ T = 60 • , and γ R = ρ R = 120 • , the IOs were concentrated in a region that was closer to T X , so that the maximum propagation delay was smaller, which benefited the performance of the LS estimator. Similarly, by considering µ R = π/2 or µ R = 0, which meant that the IOs were further away from T X , the maximum propagation delay was larger resulting a more frequency-selective channel, and as a consequence, a poorer performance of the LS estimator. Finally, the LS estimator performance was almost insignificantly affected by the magnitude and direction of the MSs acceleration (i.e., a T , a R , ρ T , and ρ R ) since the estimation was carried out by the two initial OFDM training symbols on the frame. These results are consistent with [57] for non-WSSUS channels, where there are no acceleration components. In the case of the STA technique, the main drop in its performance was caused by the channel non-stationarities, regardless of the initial speed or acceleration components of the MSs. This was due to the dynamic channel estimation in (18), which becomes obsolete rapidly even between two consecutive OFDM symbols. Therefore, although considering the data transmission over highly non-isotropic scattering scenarios, the performance of the STA technique with small values of α always approaches the case where there is no channel estimation, as observed in Figure 10.  Table 7 shows the required Eb/No levels to attain 10%, 1%, and 0.1% BER over the described propagation scenarios for isotropic and non-isotropic scattering conditions. We considered initial speeds of ν T = ν R = 100 km/h and acceleration components of a T = a R = 20 m/s 2 , with a constant frame length of F = 32 OFDM symbols. The results showed that the BER performance of both LS and STA (with α = 20) techniques depended on the placement of the IOs. This behavior occurred since the propagation delays depends on the AOAs, which are determined by the location of the IOs that surround the receiving vehicle. The BER performance decreases when the maximum propagation delay increases, which occurrs when the IOs were located further from the transmitting vehicle. Therefore, the initial positions and motion profiles of the MSs also impacted the BER performance in non-isotropic conditions. The BER performance of both estimation techniques in isotropic conditions was always lower than for the non-isotropic scenarios. Such performance was always greater than 1% in the considered simulation conditions. In the case of the LS estimator, its performance attained a 10% BER with similar Eb/No levels. However, depending on the mean AOA µ R , the Eb/No requirement could increase significantly. For instance, in Scenario II, a 10% BER performance was yielded with 4.43 and 8.14 dB, when the mean AOAs were 3π/4 and 0, respectively. With respect to the STA scheme with α = 20, its performance was also affected by the mean AOA that defined the region where the IOs concentrated. This technique required higher Eb/No levels than observed for the LS estimator. In Scenario III, a 10% BER could be reached employing 7.03 and 19.03 dB, when the IOs were located around 0 and 3π/4, respectively, which was a significant increase in the Eb/No requirement. In addition, the performance of the LS estimator could reach a 0.1% BER for high Eb/No regimes in some of the non-isotropic cases. Table 7. Required Eb/No levels to attain 10%, 1%, and 0.1% BER over Scenarios I, II, and III for isotropic and non-isotropic scattering conditions by considering initial speeds of ν T = ν R = 100 km/h and acceleration components of a T = a R = 20 m/s 2 , with a constant frame length of F = 32 (NA-not achieved).

Scenario I Scenario II Scenario III
Mean AOA 0 0 Since the reference model in (2) was developed for non-WSSUS channels, its second order statistics depend on both time and frequency, as demonstrated in [42]. Consequently, the averages made with the STA technique in the frequency domain are not entirely accurate in this type of channel, as can be seen in the simulation experiments presented earlier in this work. Therefore, we propose a modification to the STA technique that consists of eliminating the frequency averaging process for each subcarrier shown in (19), so that the new channel estimation is given by: where H d ACT m,k is the dynamic channel estimation shown in (18). The BER performance curves corresponding to Scenario I by using the proposed estimation in (21) are shown in Figure 11. We assumed isotropic scattering (κ = 0) in a low mobility scenario at initial speeds of ν T = ν R = 40 km/h, acceleration magnitudes of a T = a R = 20 m/s 2 , and frame lengths of F = 32 and 64, and 128 OFDM symbols. We can observe that the performance of the STA technique obtained with (21) improved significantly in comparison to the case where the frequency averaging process was included. Even if the MSs were accelerating, the frame length was increased or a small value of α was used for the estimation process, and the BER performance was better than when the frequency averaging was considered. The improvement compared to the performance of the LS estimator was due to the dynamic update of the channel estimation, which is performed for each symbol, instead of maintaining the initial estimate for all the data frame.
In addition to the BER curves in Figure 11, Table 8 shows the required Eb/No levels to attain 10%, 1%, and 0.1% BER in low mobility conditions over the three described propagation scenarios by considering frame lengths ranging from 16 to 128 OFDM symbols. From Table 8, we can observe that the performance of the modified STA technique decreased as the frame length increased. However, significant performance improvement was observed compared to that obtained by employing the original STA technique (cf. Table 6). Such improvement allowed yielding BER levels of 1% when the Eb/No regimes were greater than 19.05 dB for a frame length of 64, and greater than 22.46 dB for a frame length of 128. For these frame lengths, the original STA technique did not attain such BER performance. Furthermore, this modification also allowed reaching 0.1% BER levels for high Eb/No regimes, which is an improvement with respect to the performance of the LS estimator in the same simulation conditions. This improvement is because the modified STA technique performes a dynamical estimation that trackes the channel variations from symbol to symbol. This action is not performed by the LS technique, whose initial estimate is used to equalize all the symbols in the frame. Although requiring higher Eb/No levels, the proposed modification enabled attaining BER performances that were not reachable when employing the LS estimator. In addition, the modified STA scheme overcame the limitations of its original version in non-WSSUS channels by removing the spectral averaging operations.

Conclusions
This article presents a simulator that allows reproducing non-WSSUS multipath fading channels under arbitrary isotropic or non-isotropic scattering conditions. Furthermore, the proposed simulator allows velocity variations and non-linear trajectories of the MSs. This simulator is based on the combination of the SOC method and the Monte Carlo principle, and it can be adapted to the multicarrier transmission parameters of the schemes specified in the IEEE 802.11p and LTE-V2X standards. Therefore, the channel simulator can be used to evaluate the performance of V2X communication systems based on those standards for different scenarios of interest. In this paper, the proposed channel simulator was applied to the BER performance analysis of two channel estimation techniques for DSRC transceivers based on the IEEE 802.11p standard, namely the LS estimator and the STA estimator. Our proposal was evaluated in three propagating scenarios for road safety applications. The simulation results showed that the performance of the LS estimator largely depended on the motion profiles of the MSs, the frame length, and the scattering scenario. This is due to the underlying channel estimation procedure since the estimate is computed using only the two initial training symbols and is maintained constant until the end of the frame. Therefore, the estimation can lose its validity over time with the channel variations. An important observation of the LS estimator is that neither the non-stationarities of the channel, nor the acceleration of the MSs have a significant impact on its BER performance. Regarding the STA technique, the non-stationarities of the channel affect its performance, in addition to the losses caused by high mobility conditions, frame length, or scattering scenario. Therefore, in this work, we propose a variant of the STA technique as an alternative method for the estimation of non-WSSUS channels in DSRC systems. The modified STA scheme improves its estimation performance remarkably for Eb/No levels greater than 15 dB, while for smaller values, its performance worsens since the dynamic updates are affected by the high level of noise. This variant of the STA technique is an option that must be analyzed in greater detail for its possible implementation in DSRC systems. In addition, new estimation techniques are necessary to deal with the inherent non-stationarities that are found in vehicular channels. In this sense, the simulation framework presented here can be used as a reference for the evaluation of new transmission schemes for a V2X communication system under arbitrary scattering and mobility conditions.

Conflicts of Interest:
The authors declare no conflict of interest.

Abbreviations
The following abbreviations are used in this manuscript: 3GPP