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Extended Sum-of-Sinusoids-Based Simulation for Rician Fading Channels in Vehicular Ad Hoc Networks

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Abstract

In this paper, we propose an extended reference model and two novel Sum-of-Sinusoids (SoS) models (statistical and deterministic simulation models) propagation models considering the Rician K-factor and vehicle speed ratio in Vehicular Ad Hoc Networks (VANETs). Our models consider comprehensive scene of wave propagation in VANETs, including infrastructure-to-vehicle (I2V) channels with a LOS or NLOS environment, inter-vehicle communication (IVC) channels with a LOS or NLOS environment. The analysis of the statistical properties of the proposed models show that the statistics of the new models match those of the reference model at a large range of normalized time delays. The proposed models show improved approximations to the desired auto-correlation and faster convergence with the increase of Rician K-factor and vehicle speed ratio.

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Acknowledgments

This work were supported by the National Natural Science Foundation of China (No. 60762005), the Natural Science Foundation of Jiangxi Province for Youth (No. 2010GQS0153 and No. 2009GQS0070) and the Graduate Student Innovation Foundation of Jiangxi Province (No. YC10A032).

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Authors and Affiliations

  1. School of Information Engineering, Nanchang University, Nanchang, 330031, China

    Yuhao Wang & Xing Xing

  2. Department of Electrical and Computer Engineering, University of Calgary, Calgary, AB, T2N 1N4, Canada

    Siyue Chen

Authors
  1. Yuhao Wang
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  2. Xing Xing
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  3. Siyue Chen
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Corresponding author

Correspondence to Yuhao Wang.

Appendix I

Appendix I

1.1 Proof of Correlation Auto-Function of the In-Phase Component of the Statistical SoS Model

Proof

we first prove the (19)

$${R_{Z_{ck}Z_{ck}}(\tau)}=E[Z_{ck}(t)Z_{ck}(t+\tau)]=\frac{1}{1+K}\left\{\frac{4}{N_{0}M}E\left[\sum_{n,m=1}^{N_{0},M}{\hbox{cos}{(2\pi f_{1}t\hbox{cos}{\alpha_{nk}}+\phi_{nmk})}}{\hbox{cos}{(2\pi f_{2}t\hbox{cos}{\beta_{mk}})}}\, \cdot\,\sum_{p,q=1}^{N_{0},M}{\hbox{cos}(2\pi f_{1}(t+\tau)\hbox{cos}{\alpha_{pk}}+\phi_{pqk})}\hbox{cos}{(2\pi f_{2} (t+\tau) \hbox{cos}{\beta_{qk}})}\right]+K E\left[{\hbox{cos}{(2\pi f_{0}t+\phi_{0})}}\,\cdot\,\hbox{cos}{(2\pi f_{0}(t+\tau)+\phi_{0})}\right]+2{\sqrt{\frac{K}{N_{0}M}}} E\left[\sum_{n,m=1}^{N_{0},M}\hbox{cos}{(2\pi f_{1} t\hbox{cos}{\alpha_{nk}}+\phi_{nmk})}\hbox{cos}{(2\pi f_{2} t \hbox{cos}{\beta_{mk}})}\,\cdot\,\hbox{cos}{(2\pi f_{0}(t+\tau)+\phi_{0})}\right]+E\left[\hbox{cos}{(2\pi f_{0}t+\phi_{0})}\,\cdot\,2{\sqrt{\frac{K}{N_{0}M}}}\sum_{n,m=1}^{N_{0},M}\hbox{cos}{(2\pi f_{1}(t+\tau)\hbox{cos}{\alpha_{nk}}+\phi_{nmk})}\hbox{cos}{(2\pi f_{2}(t+\tau)\hbox{cos}{\beta_{mk}})}\right]\right\}=\frac{1}{1+K}\left\{\frac{1}{N_{0}M}E\left[\sum_{n,m=1}^{N_{0},M}\hbox{cos}{(2\pi f_{1}\tau\hbox{cos}{\alpha_{nk}})}\hbox{cos}{(2\pi f_{2}\tau\hbox{cos}{\beta_{mk}})}\right]\right\}+\frac{K}{2(1+K)}\hbox{cos}{(2\pi f_{0}\tau)}=\frac{1}{1+K}\left\{\frac{1}{N_{0}}\sum_{n=1}^{N_{0}}\frac{1}{2\pi}\int\limits_{-\pi}^{\pi}\hbox{cos}{\left[2\pi f_{1}\tau\hbox{cos}{\left(\frac{2\pi n}{4N_{0}}+\frac{2\pi k}{4PN_{0}}+\frac{\theta-\pi}{4N_{0}}\right)}\right]}d\theta\,\cdot\,\frac{1}{M}\sum_{m=1}^{M}\frac{1}{2\pi}\int\limits_{-\pi}^{\pi}\hbox{cos}{\left[2\pi f_{2}\tau\hbox{cos}{\left(\frac{2\pi m}{2M}+\frac{2\pi k}{2PM}+\frac{\psi-\pi}{2M}\right)}\right]}d\psi\right\}+\frac{K}{2(1+K)}\hbox{cos}{(2\pi f_{0}\tau)}=\frac{1}{1+K}\left\{\frac{1}{N_{0}}\sum_{n=1}^{N_{0}}\frac{1}{2\pi}\int\limits_{\frac{2\pi (n-1)}{4N_{0}}+\frac{2\pi k}{4PN_{0}}}^{\frac{2\pi n}{4N_{0}}+\frac{2\pi k}{4PN_{0}}}\hbox{cos}{(2\pi f_{1}\tau\hbox{cos}{\gamma_{n}})}\cdot4 N_{0}d\gamma_{n}\,\cdot\, \frac{1}{M}\sum_{n=1}^{M}\frac{1}{2\pi}\int\limits_{\frac{2\pi (m-1)}{2M}+\frac{2\pi k}{2PM}}^{\frac{2\pi M}{2M}+\frac{2\pi k}{2PM}}\hbox{cos}{(2\pi f_{2}\tau\hbox{cos}{\gamma_{m}})}\,\cdot\,2 Md\gamma_{m}\right\}+\frac{K}{2(1+K)}\hbox{cos}{(2\pi f_{0}\tau)}=\frac{1}{1+K}\left\{\frac{1}{N_{0}}\cdot\frac{1}{2\pi}\int\limits_{\frac{2\pi k}{4PN_{0}}}^{\frac{\pi}{2}+\frac{2\pi k}{4PN_{0}}}\hbox{cos}{(2\pi f_{1}\tau\hbox{cos}{\gamma_{n}})}\,\cdot\,4 N_{0}d\gamma_{n}\,\cdot\,\frac{1}{M}\cdot\frac{1}{2\pi}\int\limits_{\frac{2\pi k}{2PM}}^{\pi+\frac{2\pi k}{2PM}}\hbox{cos}{(2\pi f_{2}\tau\hbox{cos}{\gamma_{m}})}\,\cdot\,2 Md\gamma_{m}\right\}+\frac{K}{2(1+K)}\hbox{cos}{(2\pi f_{0}\tau)}=\frac{1}{1+K}\left[\frac{2}{\pi}\int\limits_{0}^{\frac{\pi}{2}}\hbox{cos}{(2\pi f_{1}\tau\hbox{cos}{\gamma_{1}})}d\gamma_{1}\cdot\frac{1}{\pi}\int\limits_{0}^{\pi}\hbox{cos}{(2\pi f_{2}\tau\hbox{cos} {\gamma_{2}})}d\gamma_{2}\right]\frac{K}{2(1+K)}\hbox{cos}{(2\pi f_{0}\tau)}=\frac{2J_{0}(2\pi f_{1}\tau)J_{0}(2\pi f_{2}\tau)+K\hbox{cos}{(2\pi f_{0}\tau)}}{2(1+K)}$$

This completes the proof of (27). Similarly, one can prove the (28) and (29), details are omitted for brevity.

1.2 Proof of Variance of Auto-Correlation of the Complex Envelope

Proof

We start with the first equality of (22) and derive

$$ {Var\{\hat{R}_{Z_{ck}Z_{ck}}(\tau)\}}=E\left[\left|\hat{R}_{Z_{ck}Z_{ck}}(\tau)-\frac{2J_{0}(2\pi f_{1}\tau)J_{0}(2\pi f_{2}\tau)+K\hbox{cos}{(2\pi f_{0}\tau)}}{2(1+K)}\right|^{2}\right]=E\left[|\hat{R}_{Z_{ck}Z_{ck}}(\tau)|^{2}\right]-\frac{J_{0}^{2}(2\pi f_{1}\tau)J_{0}^{2}(2\pi f_{2}\tau)}{(1+K)^{2}}-\left[\frac{K\hbox{cos}{2\pi f_{0}\tau}}{2(1+K)}\right]^{2}=E\left\{\frac{1}{(1+K)^{2}}\cdot\frac{1}{N_{0}^{2}M^{2}}\left[\sum_{n,m=1}^{N_{0},M}\hbox{cos}{(2\pi f_{1}\tau\hbox{cos}{\alpha_{nk}})}\hbox{cos}{(2\pi f_{2}\tau\hbox{cos}{\beta_{mk}})}\,\cdot\,\sum_{p,q=1}^{N_{0},M}\hbox{cos}{(2\pi f_{1}\tau\hbox{cos}{\alpha_{pk}})}\hbox{cos}{(2\pi f_{2}\tau\hbox{cos}{\beta_{qk}})}\right]\right\}-\frac{J_{0}^{2}(2\pi f_{1}\tau)J_{0}^{2}(2\pi f_{2}\tau)}{(1+K)^{2}}=\frac{1}{(1+K)^{2}}\cdot\frac{1}{N_{0}^{2}M^{2}}\left\{E\left[\sum_{n,m=1}^{N_{0},M}\hbox{cos}^{2}{(2\pi f_{1}\tau\hbox{cos}{\alpha_{nk}})}\hbox{cos}^{2}{(2\pi f_{2}\tau\hbox{cos}{\beta_{mk}})}\right]\,\cdot\,\sum_{n,m=1}^{N_{0},M}\sum_{p,q=1}^{N_{0},M}E\left[\hbox{cos}{(2\pi f_{1}\tau\hbox{cos}{\alpha_{nk}})}\hbox{cos}{(2\pi f_{2}\tau\hbox{cos}{\beta_{mk}})}\right] E\left[\hbox{cos}{(2\pi f_{1}\tau\hbox{cos}{\alpha_{pk}})}\quad (n\neq p \;\;\text{or}\;\;m\neq q)\,\cdot\,\hbox{cos}{(2\pi f_{2}\tau\hbox{cos}{\beta_{qk}})}\right]\right\}-\frac{J_{0}^{2}(2\pi f_{1}\tau)J_{0}^{2}(2\pi f_{2}\tau)}{(1+K)^{2}}=\frac{1}{(1+K)^{2}}\,\cdot\,\frac{1}{N_{0}^{2}M^{2}}\left\{N_{0}M\cdot\frac{1+J_{0}(4\pi f_{1}\tau)J_{0}(4\pi f_{2}\tau)}{4}+N_{0}^{2}M^{2}\left[J_{0}^{2}(2\pi f_{1}\tau)J_{0}^{2}(2\pi f_{2}\tau)-f_{c}(2\pi f_{1}\tau,2\pi f_{2}\tau)\right]\right\}-\frac{J_{0}^{2}(2\pi f_{1}\tau)J_{0}^{2}(2\pi f_{2}\tau)}{(1+K)^{2}}=\left[\frac{1+J_{0}(4\pi f_{1}\tau)J_{0}(4\pi f_{2}\tau)}{4N_{0}M}-f_{c}(2\pi f_{1}\tau, 2\pi f_{2}\tau)\right]/(1+K)^{2}$$

Similarly, we can validate the second equality of (30) and (31). Thus, we have

$$ {Var\{\hat{R}_{Z_{k}Z_{k}}(\tau)\}}=E\left[\left|\hat{R}_{Z_{k}Z_{k}}(\tau)-\frac{2J_{0}(2\pi f_{1}\tau)J_{0}(2\pi f_{2}\tau)+K\exp{(j2\pi f_{0}\tau)}}{1+K}\right|^{2}\right]=E\left[\left|2\hat{R}_{Z_{ck}Z_{ck}}(\tau)+j2\hat{R}_{Z_{ck}Z_{sk}}(\tau)-\frac{2J_{0}(2\pi f_{1}\tau)J_{0}(2\pi f_{2}\tau)}{(1+K)}-\frac{K\exp{j2\pi f_{0}\tau}}{(1+K)}\right|\right]^{2}=[\frac{1+J_{0}(4\pi f_{1}\tau)J_{0}(4\pi f_{2}\tau)}{N_{0}M}-4f_{c}(2\pi f_{1}\tau, 2\pi f_{2}\tau)]/(1+K)^{2}$$

This completes the proof.

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Wang, Y., Xing, X. & Chen, S. Extended Sum-of-Sinusoids-Based Simulation for Rician Fading Channels in Vehicular Ad Hoc Networks. Int J Wireless Inf Networks 19, 147–157 (2012). https://doi.org/10.1007/s10776-011-0167-8

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