**6.1 Effects of Rician factor in VANETs**

The results of Figs. 2-3 are obtained using *a* = 1, *f*<sup>1</sup> = *f*<sup>2</sup> = 50*Hz*, *N*<sup>0</sup> = *M* = *NC* = *P* = 8. For a fair comparison, we use *N* = 4*N*<sup>0</sup> × 2*M* = 512 sinusoids for simulation of the reference model.

<sup>0</sup> <sup>2</sup> <sup>4</sup> <sup>6</sup> <sup>8</sup> <sup>10</sup> −1

(a) a=0

Normalized Time Delay τ

−0.5

0

0.5

Re[Rzz(τ)]

more comprehensive than the existing ones.

**6.2 Effects of vehicle speed ratio in VANETs**

compared with the reference model.

1

<sup>0</sup> <sup>2</sup> <sup>4</sup> <sup>6</sup> <sup>8</sup> <sup>10</sup> −1

(b) a=0.2

Normalized Time Delay τ

Reference model Statistical model(1 simulation trial) Statistical model(10 simulation trials)

MEDS model

−0.5

Reference model Statistical model(1 simulation trial) Statistical model(10 simulation trials)

MEDS model

0

0.5

Re[Rzz(τ)]

<sup>169</sup> Sum-of-Sinusoids-Based Fading Channel Models

<sup>0</sup> <sup>2</sup> <sup>4</sup> <sup>6</sup> <sup>8</sup> <sup>10</sup> −1

(c) a=1

It should be noted that the plot of the auto-correlation function when *K* = 0 keeps agreement with (Patel et al., 2005) and the result of the variance of auto-correlation is almost similar to that in (Zajic & Stuber, 2006), which indicate that our models and performance analysis are

The simulation results presented in Figs. 4-5 are obtained using *K* = 0, *f*<sup>2</sup> = 50*Hz*, *f*<sup>1</sup> = 0, 10 and 50*Hz* when the corresponding value of speed ratio *a* equals to 0, 0.2 and 1. The imaginary part of the auto-correlation of the complex envelope for the proposed two SoS models is always equal to 0, which is in line with the ideal situation and shows better performance

Fig. 4 shows the real part of correlation properties of the above models with different *a* in VANETs. It is observed that the proposed models provide a better approximation to the desired auto-correlation when *a* increases. From Figs. 5a-5c, we found that the variances of the auto-correlation of our models tend to be lower with a larger value of *a*. So the proposed models perform better with a smaller relative speed difference. A physical interpretation

Fig. 4. Real part of the auto-correlation function of the complex envelope with different *a*

Normalized Time Delay τ

1

Reference model Statistical model(1 simulation trial) Statistical model(10 simulation trials)

with Rician K-factor and Vehicle Speed Ratio in Vehicular Ad Hoc Networks

MEDS model

−0.5

0

0.5

Re[Rzz(τ)]

1

Fig. 3. Variance of the auto-correlation and cross-correlation function with different *K*

Fig. 2 shows the correlation properties of the aforementioned models with different *K* factors in VANETs. For a large range of normalized time-delay (0 ≤ *f*1*TS* ≤ 4), the proposed simulation models keep good agreement with the reference model, without exhibiting any sort of periodicity as encountered in Wang and Cox's model (Wang & Cox, 2002). For the same time delay *τ*, the magnitude of the channel correlation tends to be larger. As the *K* factor increases, the proposed models get closer to the reference model.

Fig. 3 compares the variances of the auto- and cross-correlation functions for the proposed simulation models. As shown in Figs. 3a-3c, the variances of the auto-correlation functions decrease as *K* factor increases. It indicates that the simulation models perform better under a larger amount of LOS components. When *K* is larger, the LOS components become more dominant over the scattering components, which avoids the deviation caused by the finite scatters. We can also observe that the variances of the auto-correlation of our models are higher than the reference model. It is noted that the difference between the statistical model and the reference model becomes smaller when the number of the simulation trials is increased. Simulation results show that the statistical model achieves better convergence by averaging 10 simulation trials. Fig. 3d shows that the variances of the cross-correlation for the proposed models are lower than the reference model.

12 Will-be-set-by-IN-TECH

0.001 0.002 0.003 0.004 0.005 0.006 0.007 0.008 0.009 0.01

> 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 <sup>1</sup> x 10−3

Fig. 3. Variance of the auto-correlation and cross-correlation function with different *K*

Fig. 2 shows the correlation properties of the aforementioned models with different *K* factors in VANETs. For a large range of normalized time-delay (0 ≤ *f*1*TS* ≤ 4), the proposed simulation models keep good agreement with the reference model, without exhibiting any sort of periodicity as encountered in Wang and Cox's model (Wang & Cox, 2002). For the same time delay *τ*, the magnitude of the channel correlation tends to be larger. As the *K* factor

Fig. 3 compares the variances of the auto- and cross-correlation functions for the proposed simulation models. As shown in Figs. 3a-3c, the variances of the auto-correlation functions decrease as *K* factor increases. It indicates that the simulation models perform better under a larger amount of LOS components. When *K* is larger, the LOS components become more dominant over the scattering components, which avoids the deviation caused by the finite scatters. We can also observe that the variances of the auto-correlation of our models are higher than the reference model. It is noted that the difference between the statistical model and the reference model becomes smaller when the number of the simulation trials is increased. Simulation results show that the statistical model achieves better convergence by averaging 10 simulation trials. Fig. 3d shows that the variances of the cross-correlation for the

Var[RZcZs(τ)]

Var[RZcZc(τ)]

<sup>0</sup> <sup>2</sup> <sup>4</sup> <sup>6</sup> <sup>8</sup> <sup>10</sup> <sup>0</sup>

Reference model Statistical model(1 simulation trial) Statistical model(10 simulation trials)

MEDS model

Normalize Time Delay τ

Reference model(K=0) Reference model(K=1) Reference model(K=5) Statistical model MEDS model

(b) K=1

<sup>0</sup> <sup>2</sup> <sup>4</sup> <sup>6</sup> <sup>8</sup> <sup>10</sup> <sup>0</sup>

Normalize Time Delay τ

(d) Variance of the cross-correlation function

<sup>0</sup> <sup>2</sup> <sup>4</sup> <sup>6</sup> <sup>8</sup> <sup>10</sup> <sup>0</sup>

Reference model Statistical model(1 simulation trial) Statistical model(10 simulation trials)

Reference model Statistical model(1 simulation trial) Statistical model(10 simulation trials)

MEDS model

MEDS model

Normalize Time Delay τ

(a) K=0

<sup>0</sup> <sup>2</sup> <sup>4</sup> <sup>6</sup> <sup>8</sup> <sup>10</sup> <sup>0</sup>

Normalize Time Delay τ

increases, the proposed models get closer to the reference model.

proposed models are lower than the reference model.

(c) K=5

0.005 0.01 0.015 0.02 0.025 0.03 0.035

> 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 <sup>1</sup> x 10−3

Var[RZcZc(τ)]

Var[RZcZc(τ)]

Fig. 4. Real part of the auto-correlation function of the complex envelope with different *a*

It should be noted that the plot of the auto-correlation function when *K* = 0 keeps agreement with (Patel et al., 2005) and the result of the variance of auto-correlation is almost similar to that in (Zajic & Stuber, 2006), which indicate that our models and performance analysis are more comprehensive than the existing ones.
