2. Estimation of the Complex CIRs in WSSUS MIMO V2V Channels
As shown in
Figure 1, we consider a geometry-based stochastic WSSUS channel model for MIMO V2V point-to-point communications, where the uniform linear array (ULA) at the transmitter and receiver are respectively composed of
and
antenna elements. The proposed channel model can be extended to end-to-end channel models as we properly adjust the geometric parameters of the channel model. In the proposed channel model, the radio propagation environment is characterized by scattering with LoS and NLoS components between the MT and MR. For the NLoS components, the waves undergo multiple rays within the cluster, wherein every ray is approximated to experience the same propagation distance from the center of the corresponding array [
23]. Notice that the MT and MR move with time-varying speeds and arbitrary trajectories in practical V2V communication scenarios [
24]; however, if we consider the time-varying moving directions and speeds of the transmitter and receiver in the proposed estimation algorithm for V2V channel modeling, the derivations will be very complicated, and therefore will be our future work. Instead, we assume that the MT and MR move at constant speeds
and
, respectively; while their moving directions indicated by the
x-axis are denoted by
and
, respectively. This assumption has been widely proposed in the existing literature [
25,
26]. In fact, we only depict the
ℓ-th cluster in such channel model, while other clusters can be depicted in similar solutions, and therefore we omit them here for brevity.
In this part, we suppose that the propagation characteristics satisfy the WSSUS assumption, which indicates that the channel statistical properties in time domain remain invariant over a relatively short period of time, and that the clusters with different propagation delays are uncorrelated. For such channel model, the statistical proposed properties are characterized by a matrix
of size
, where
represents the complex CIR between the
p-th (
) antenna of the MT array and
q-th (
) antenna of the MR array. Let us suppose that the LoS and NLoS propagation components are independent to each other, we have that [
27]
where
for the LoS rays is deterministic, which can be written by [
28]
where
denotes the Rician factor,
is the carrier frequency,
c is the speed of light, and
is the carrier wavelength. The
is the distance of the waves from the
p-th transmit antenna directly propagate to the
q-th receive antenna, i.e., LoS components. Based on the geometry-based relations in
Figure 1, we have
where
and
where
. The
accounts for the distance from the center of the MT antenna array to that of the MR array. The
and
are the adjacent antenna element spacings at the MT and MR, respectively;
and
are the orientations of the transmit and receive antenna arrays indicated by the
x-axis, respectively. Furthermore, in (1),
for NLoS rays is random, which statistical properties are introduced by the initial random phase
. It is assumed that the phase
satisfies the uniform distribution in the interval from
to
. Assume that there are
N sub-paths within the
ℓ-th cluster. Then, we can obtain
where
and
are, respectively, the AoD and AoA related to the
n-th ray within the
ℓ-th cluster. The
and
are, respectively, the distances from the
p-th transmit antenna and
q-th receive antenna to the
ℓ-th cluster. In the following, we assume that every ray within the
ℓ-th cluster is approximately at the same angle and same distance from the center of the corresponding array [
23], e.g.,
,
,
,
. Then, we can obtain
where
and
denote the distances from the centers of the MT and MR antenna arrays to the cluster, respectively, which are expressed as
Subsequently, we define the moving time period of the MT and MR is from 0 to
T. It can be observed that the complex CIR
in the WSSUS V2V channel model is related to the moving time
t of the MT and MR, we define
as the signal transmitted by the
p-th antenna of the MT array. By taking the convolution operation between the complex CIR
and signal
in terms of the moving time
t, the received signal of the
q-th antenna element for different time
t can be written by
where ∗ stands for the convolution operation and
is the complex noise of the
q-th receive antenna. By substituting (1) into (10), the received signal
can be further written by
where
. Next, we convert the continuous received signal
in
to
K discrete sequence of samples [
18], where the spacing between two different samples is denoted by
. At the MR, the signal of the
k-th (
) sample of the
q-th receive antenna can be written by
where
stands for the complex noise of the
k-th sample of the
q-th antenna of the MR array. Assume that the
follow the white Gaussian distribution [
29], i.e.,
, i.e.,
. Define
as the received signal vector of the
q-th antenna of the MR array, which is written by
Based on the aforementioned derivations, the received signal vector of the WSSUS MIMO V2V communication system, denoted by
, is written by
It is obvious that the received signal vector
in (14) is in complex domain, which brings us relatively high computational complexities on the estimates of the angular parameters at the MT in the WSSUS V2V channel model, especially for complicated communication systems. To solve this challenge, it is required to convert the signal
from the complex domain to the real-value domain. Let us define a vector
as [
30]
where
and
, which are both composed of
real elements, stand for the real and imaginary parts of the complex received signal vector
, respectively. Define
and
as the
k-th sample of the discrete sequences
and
, respectively, which are written by
where
and
are, respectively, the real and imaginary parts of the
k-th sample of the
q-th receive antenna of the complex white Gaussian noise, i.e.,
and
. Based on these derivations, we suppose that
and
, with the vectors
and
accounting for the mean elements of the
discrete sequences of the real received signal vector
, respectively,
is an identity matrix of the size
. The mean values of the
k-th sample of the vectors
and
are denoted by
and
, respectively. We have that
It is assumed that the
elements in the received signal vector
are independent to each other; hence, the probability density function (PDF) of the AoD
for the normal distribution can be written by [
31]
where
denotes the
dimensional covariance matrix
of the normal distributed real vector
, i.e.,
. The
stands for the expectation of the
, which means
. By substituting the aforementioned expressions into (20), we can further obtain
By taking the natural Logarithm of (21), we have
It is worth mentioning that the AoA in WSSUS V2V channel models can be obtained by using the conventional solutions, such as compressive sensing-based [
32] and deep learning-based methods [
33]. In light of this, we will estimate the AoD in the following parts. Affiliate that, we are able to obtain the estimated expressions of the complex CIR to estimate the physical properties of the WSSUS channel models. To achieve this challenge, we should solve the maximum likelihood estimation (MLE) problem with respect to the optimization AoD
, which can be expressed as
Accordingly, the AoD
in WSSUS channel models for V2V communications can be derived by adopting the Newton-Raphson algorithm [
34]. In light of this, the distances from the
p-th transmit antenna and
q-th receive antenna can be respectively estimated as
Subsequently, the propagation distances of the waves from the
p-th transmit antenna and
q-th receive antenna to the cluster are respectively estimated as
By substituting the estimated parameters in (26) and (27) into (5), the complex CIR of the WSSUS MIMO V2V channel model for the NLoS propagation components are estimated as
Thus far, the complex CIR of WSSUS MIMO V2V channel models has been derived and investigated. Their investigations of V2V channel propagation characteristics will be discussed later in
Section 4.
3. Estimation of the Complex CIRs in Non-WSSUS MIMO V2V Channels
It is important to mention that when the MT and MR move from one position to another position, as shown in
Figure 2, it is required to derive the real-time model parameters, such as the AoD/AoA and path lengths, to characterize the time and frequency non-WSSUS nature of the MIMO channel models for V2V communications. Furthermore, we should estimate the real-time complex CIRs to investigate the non-WSSUS V2V channel propagation properties [
35]. It deserves however to mention that the estimation of the time-varying complex CIRs seems to be impossible, which is mainly on account of the high computational complexity of such process. To solve this issue, we define
as a time interval during which the mobile terminals are stationary, which correspond to the time instant we begin to observe the channel. In such stage, we propose a computational method to estimate the angular parameters at the MT. Afterwards, for the state when the channel is not stationary, we estimate the real-time angular parameters at the MT and MR according to the estimated initial AoD and the model parameters of the motion of the transceivers. Finally, when we substitute the estimates of the time-varying expressions of the AoD and AoA into the complex CIR, the statistical propagation properties of the non-WSSUS channel models for V2V communications can be characterized.
It deserve to mention that the non-WSSUS channel models for V2V communications are represented by the matrix
of size
, where
accounts for the complex CIR of the propagation links between the
p-th transmit antenna and
q-th receive antenna, which is written by [
35,
36]
where
and
denote the real-time propagation delays of the waves that emerge from the center position of the MT array to that of the MR array via the LoS and NLoS components, respectively. We define
as the complex CIR between the
p-th MT antenna and the
q-th MR antenna at the time instant
. It can be observed that the complex CIR
in the non-WSSUS V2V channel model depends on the moving time
t and propagation delay
, we define
as the signal transmitted by the
p-th antenna of the MT array, which is obviously different from the previous definition of the transmitted
in the WSSUS V2V channel model in
Section 2. By taking the convolution operation between the complex CIR
and signal
in terms of the delay
, the received signal of the
q-th antenna element for different delay
is written by
where
accounts for the complex noise of the
q-th receive antenna. We define
as the complex noise vector, which elements are supposed to be independent to each other. The complex received signal vector, denoted by
, can be expressed as
where
stands for the
matrix of the complex fading envelope for the LoS rays during the initialization, where the real and imaginary parts of the complex fading envelope
are denoted as
and
, respectively, which can be expressed as
where
represents the propagation distance of the waves from the
p-th transmit antenna to the
q-th receive antenna via the LoS component. In the following, let us convert the vector
in (31) from the complex-value domain to the real domain for the sake of high-efficient investigation of the non-WSSUS MIMO V2V channel characteristics, which is similar to the previous discussions. To achieve this goal, we define
as [
30]
where
and
account for the real and imaginary parts of the complex matrix
, respectively. Define
as
Furthermore, in (31),
is a
matrix of the complex fading envelope for the NLoS rays, where the real and imaginary parts of the complex fading envelope
are denoted by
and
, respectively, which can be expressed as
Similar as before, we define
as the complex matrix for the NLoS components in real domain, which can be expressed as [
30]
where
and
account for the real and imaginary parts of the complex matrix
, respectively. Define
as
In addition, we define
as
Based on the aforementioned derivations, the received signal vector
can be rewritten in real-value domain, i.e.,
. According to (31), we have that
At the MR, we define
as the
q-th signal vector in real domain, which can be written by
where
We convert the continuous received signal
and
to
K discrete sequences of samples. In this case, the discrete received signal of the
q-th receive antenna element in real-value domain, that is
, can be further expressed as
Suppose that the received signal vectors
and
satisfy the white Gaussian distribution, i.e.,
and
, where
and
stand for the
K mean elements of the vectors
and
, respectively. Let us define
and
as the mean values of the
k-th samples of the
and
, respectively, which can be expressed as
where
and
account for the real and imaginary parts of the complex signal transmitted by the
p-th antenna of the MT array for the LoS components, respectively. The
and
are the real and imaginary parts of the complex signal transmitted by the
p-th transmit antenna for NLoS rays, respectively. In light of this, the received signal vector of the non-WSSUS MIMO V2V communication system at the time instant
, denoted by
, can be estimated as
Therefore, the PDF of the AoD
in the non-WSSUS channel models for V2V communications for the normal distribution is written by
Similar to the estimates of the AoD in the WSSUS channel models for V2V communications in
Section 2, the
in the non-WSSUS channel models can be estimated as
Accordingly, the initial propagation distances from the centers of the transmit and receive antenna arrays to the cluster, they are
and
, respectively, can be estimated as
Subsequently, the real-time propagation distances from the center positions of the transmit and receive antenna arrays to the cluster, they are
and
, respectively, can be estimated as
Next, the real-time propagation distances from the
p-th transmit antenna and
q-th receive antenna to the cluster, they are
and
, respectively, can be estimated as
where
and
stand for the estimated angular parameters at the MT and MR, respectively, which can be written by
According to (51)–(58), the complex CIR of the non-WSSUS MIMO V2V channel model for the NLoS components can be estimated as
Thus far, the complex CIR of non-WSSUS MIMO V2V channel models has been derived and investigated. Their investigations of V2V channel propagation characteristics will be discussed later in
Section 4.
5. Numerical Results and Discussions
In this section, we first describe the AoD estimations in the WSSUS and non-WSSUS MIMO V2V channel models. In the proposed estimation algorithms, when the V2V channel is WSSUS, we should record the second sample from the NLoS propagation rays while the first sample is in correspondence with the LoS propagation rays. In this way, we can estimate the AoD in the WSSUS channel model. However, when the channel is non-stationary, the signal received at the MR consist of the components with different propagation delays. In contrast, in the estimation process of the initial stage, we assume that the MR receives the signal propagated from the MT for a while. In such stage, we need to record the second sample corresponding to the NLoS propagation rays, as well as the first sample corresponds to the LoS propagation rays. In this way, we can estimate the initial AoD in the non-WSSUS channel model.
To validate the efficiency of the proposed algorithms for estimating the statistical properties of the WSSUS and non-WSSUS MIMO channel models for V2V communications, we summarize some basic parameters as follows:
GHz,
,
,
, and
. In the following, we use the mean squared error (MSE) to measure the coarse AoD estimate. Define the MSE as [
34]
where
accounts for the estimation of the AoD
of the
u-th Monte Carlo trial.
Figure 3 plots the MSEs of the angular parameters at the MT for total number of trials
and different sample numbers
K in the WSSUS V2V channel model. It can be seen that the impacts of the total number of trials on the estimate performances of the angular parameters are insignificant. Furthermore, the MSEs drop as the number of the discrete sequence rises, which shows that the proposed computational solutions are excellent for estimating the V2V channel propagation characteristics, especially as the value of the
K is large. The above results are consistent with the conclusions in [
38], and therefore further validates the accuracy of the derivations and conclusions. In addition, we can observe that when the distance
rises from 50 m to 200 m, the MSEs for estimating the AoD increase slowly [
39]. In
Figure 4, we illustrate the MSEs for estimating the complex CIR of the WSSUS V2V channel models. It is observed that the estimation errors for estimating the complex CIR
for NLoS propagation components drop gradually as the number of samples
K increases, which are in consistent with the simulation results in [
40,
41], thereby demonstrating that the estimation performance, such as the estimating precision, etc., can be nicely used for WSSUS V2V channel modeling. Furthermore, we notice that when the distance
rises from 50 m to 200 m, the MSEs of the complex CIR
of the WSSUS MIMO V2V channel model rise slowly, which are in agreements with the behaviors in
Figure 3.
Figure 5 illustrates the MSEs for estimating the AoD in the non-WSSUS MIMO V2V channel models. It is obvious that when the moving directions of the transmitter and receiver are set as
and
, respectively, which means the transceivers move in opposite directions, the MSEs rise slowly as the moving time
t increases from 1 s to 5 s. In
Figure 6, we illustrate the MSEs for estimating the complex CIR
of the non-WSSUS channel models for MIMO V2V communications. It is observed that when the moving time of the MT/MR rises from 1 s to 5 s, the MSEs of the complex CIR of the non-WSSUS MIMO V2V channel model increase slowly, which are in consist with the conclusions in
Figure 5.
It deserves to mention that Equation (
39) for
Figure 7 is a Bessel function, which fluctuates as we increase the antenna spacing; however, we notice that the general trend is to decrease with the increasing of the antenna spacing. The aforementioned simulation results fit the measurements in [
42] very well, which further shows the accuracy of the analysis and derivations of the spatial CCFs of the WSSUS MIMO V2V channel model. For V2V communications, the propagation characteristics are obviously influenced by the moving properties of the transmitter and receiver. In light of this, in
Figure 7, we plot the estimated ST CCFs of the WSSUS channel models for MIMO V2V communications with respect to the moving directions/velocities of the MT and MR. It is obvious that the ST CCFs are independent of the moving directions
and
. Furthermore, when the moving velocities
and
increase from 5 m/s to 20 m/s, the ST CCFs decrease gradually.
By using (63),
Figure 8 shows the estimated ST CCFs of the non-WSSUS channel models for V2V communications in terms of the moving time/directions/velocities of the transmitter and receiver. It is observed that when the moving time
t is fixed, the spatial correlation for the case of the transmitter and receiver moving towards each other (i.e.,
and
) is larger than that for the case of the transmitter and receiver moving away from each other (i.e.,
and
). The difference between the aforementioned two cases becomes more obvious as the
t rises from 1 s to 5 s. Furthermore, we notice that when the moving directions of the transmitter and receiver are set as
and
, respectively, the ST CCFs rise as the
t increases from 1 s to 3 s. However, when the moving directions of the transmitter and receiver are set as
and
, respectively, the ST CCFs decrease slowly as the
t increases from 1 s to 3 s.
Figure 9 illustrates the estimated ST CCFs of the WSSUS and non-WSSUS MIMO V2V channel models for different Rician factors
. It can be seen that when the Rician factor
increases from 0.01 to 5, the proportion of the rays with NLoS interactions in the MIMO V2V channel models drops slowly, which results in the rising of the values of the spatial correlations.
By using (65),
Figure 10 depicts the estimated temporal ACFs of the non-WSSUS channel models for V2V communications. It can be seen that when the
t is fixed, the temporal correlations for the case of
and
are relatively larger than those for the case of
and
. The difference between these two cases becomes more obvious as the
t increases from 1 s to 5 s. Furthermore, notice that when the transmitter and receiver move mutually, the temporal correlations rise as the
t increases. However, when the MT and MR move away from mutually, the temporal correlations drop as the
t rises.
Figure 11 illustrates the estimated temporal ACFs of the non-WSSUS channel models for V2V communications for different distances
. It is obviously observed that there are different properties in the temporal correlations at different
t. Notice that similar conclusions are in
Figure 8. It also can be seen that the temporal correlations drop slowly as the time difference
rises. They are consistent with the results in [
17], thereby validating the aforementioned simulations and analysis. Furthermore, we notice that by increasing the
from 50 m to 200 m, the propagation delay of the waves emerging from the transmitter to the receiver rises gradually, which results in the increasing of the correlations in temporal domain.