**Definitions:**

4 Will-be-set-by-IN-TECH

original simulator and published modified versions, have similar problems with these second-order statistics. In (Xiao et al., 2006), Xiao and Zheng proposed a statistical SoS model for I2V channels which employs a zero-mean stochastic sinusoid as the specular LOS component, in contrast to previous Rician fading simulators that utilize a non-zero deterministic specular component. The statistical properties of the new simulators are confirmed by extensive simulation results, showing good agreement with theoretical analysis

Channel modeling in VANETs should be considered the both characteristics in I2V and IVC channels. Those I2V channel models may not fully reflect the mobility characteristics of VANET channels. Several works in the open literature have been studied in this area (Akki & Haber, 1989)-(Patel et al., 2003). The theoretical analysis of the IVC channels for urban and suburban land communication channels was first developed by Akki and Haber (Akki & Haber, 1989; Akki, 1994), and was extended by Vatalaro and Forcella in (Vatalaro & Forcella, 1997) to account for scattering in three dimensions (3-D), and by Linnartz and Fiesta in (Linnartz & Fiesta, 1996) to include LOS scenarios. Some channel measurement results for narrowband IVC communications have been presented in (Kovacs et al., 2002; Maurer et al., 2002; Cheng et al., 2007). R.Wang and D.Cox (Wang & Cox, 2002) introduced the discrete line spectrum method to simulate the IVC channels. Whereas the accuracy of this method was assured only for short-duration waveforms, Moreover, the numerical integrations required in determining the discrete set of frequencies and corresponding Doppler spectrum make the implementation complex and not easily reconfigurable for different Doppler frequencies or the Doppler frequency ratio. So it is not always suitable for real time hardware channel emulation or software simulation. A method based on inverse fast Fourier transform (IFFT) was presented by D.J.Young and N.C.Beaulieu (Young & Beaulieu, 2000). This method was more accurate and efficient than the method of discrete line spectrum, but the IFFT-based method requires a complex elliptic integration. The authors in (Patel et al., 2003) proposed a "double-ring" scattering model to simulate the IVC scattering environment and developed modifications of two SoS models (statistical and deterministic SoS models) often used to simulate I2V channels in (Patel et al., 2005). More recently, Wang and Liu (Wang et al., 2009) presented a scattering Rician IVC fading model with a LOS component by SoS method, which is based on the Rayleigh model proposed in (Patel et al., 2005). A new statistical SoS in (Zajic & Stuber, 2006) model is proposed for Rayleigh IVC fading channel to directly generate multiple uncorrelated complex envelope, which shows faster convergence than the model in (Patel et al., 2005) and adequate statistics with small simulation trials.

The statistical properties of Xiao and Zheng's simulators in (Xiao et al., 2006) are confirmed by extensive simulation results, showing good agreement with theoretical analysis in all cases and is a typical model with high quality for I2V channels. But with the development of mobile ad hoc networks, VANET channel modeling often involves the IVC channels, which is generally considered as the common case of the I2V channels. Therefore, in this chapter, we mainly focus on the modeling for IVC channels in VANETs. This motivates us to extend the new statistical SoS model in (Zajic & Stuber, 2006) by employ a LOS component to characterize the IVC channels of VANETs. Furthermore, the deterministic SoS model, proposed in (Patel et al., 2005), are employed to simulate Rayleigh IVC channel for its reduced-complexity and theoretical and simulation results verified the usefulness of the model. Seeking to find a more suitable Rician simulation model for VANET channels, we also introduce a LOS

component to extend the deterministic SoS model for comparison.

in all cases.


The angle between **v***x* and LOS component is 0◦ and the direction of **v***y* is perpendicular to the LOS component. From (Gregory, 2003), the Doppler frequency caused by LOS component in the IVC environment is | *fx*| = (|**v**2| cos *θ*<sup>2</sup> − |**v**1| cos *θ*1)/*λ*. The LOS component is given by

$$L = \sqrt{K} \exp[j\{2\pi(|\mathbf{v}\_2|\sin\theta\_2 - |\mathbf{v}\_1|\sin\theta\_1)\}t + \phi\_0] \tag{2}$$

where *K* is the ratio of the specular power to the scattering power, and the initial phase *φ*<sup>0</sup> is uniformly distributed over [−*π*,*π*).

With reference to (1) and (2), the received complex envelope of the IVC fading channel with a LOS component can be expressed as

$$Z(t) = \frac{Y(t) + \sqrt{K} \exp(j2\pi f\_0 t + \phi\_0)}{\sqrt{1+K}}\tag{3}$$

where *f*<sup>0</sup> = (|**v**2| cos *θ*<sup>2</sup> − |**v**1| cos *θ*1)/*λ*.

Assuming omnidirectional antennas and isotropic scattering conditions around the transmitter and the receiver, the statistical properties of the reference model are given as follows.

**5.1 Statistical SoS model**

*Yck*(*t*) = <sup>2</sup>

*Ysk*(*t*) = <sup>2</sup>

distributed on the interval [−*π*,*π*).

channels can be obtained as follows:

signal *Zk*(*t*) are given by

<sup>√</sup>*N*0*<sup>M</sup>*

<sup>√</sup>*N*0*<sup>M</sup>*

*N*<sup>0</sup> ∑ *n*=1

> *N*<sup>0</sup> ∑ *n*=1

where

follows:

Recently, A new model is proposed in (Zajic & Stuber, 2006) to directly generate multiple uncorrelated complex envelope, which has resolved the difficulty in producing time averaged auto- and cross-correlation functions that match the reference model (Akki & Haber, 1989).

<sup>163</sup> Sum-of-Sinusoids-Based Fading Channel Models

*f*1, *f*2, *αnk*, *βmk* and *φnmk* are the maximum radian Doppler frequencies, the random angle of departure, the random angle of arrival, and the random phase, respectively. It is assumed that *P* independent complex faded envelopes are required (*k* = 0, ..., *P* − 1) each consisting of *NM* sinusoidal components. The angles of departures and the angles of arrivals are chosen as

> + *θ* − *π* 4*N*<sup>0</sup>

*PM* <sup>+</sup> *<sup>ψ</sup>* <sup>−</sup> *<sup>π</sup>*

2*πk*

where *n* = 1, ..., *N*0, *m* = 1, ..*M*, *k* = 0, ..., *P* − 1. The angles of departures and the angles of arrivals in the *kth* complex faded envelope are obtained by rotating the angles of departures and the angles of arrivals in the (*<sup>k</sup>* <sup>−</sup> <sup>1</sup>)*th* complex envelope by 2*π*/(4*PN*0) and 2*π*/(2*PM*), respectively. The parameters *φnmk*, *θ* and *ψ* are independent random variables uniformly

With reference to (2),(11),(12),(13), the received complex envelope of the IVC fading

*Zk*(*t*) = *Yk*(*t*) + <sup>√</sup>*<sup>K</sup>* exp(*j*2*<sup>π</sup> <sup>f</sup>*0*<sup>t</sup>* <sup>+</sup> *<sup>φ</sup>*0)

The time-average auto-correlation and cross-correlation function of the in-phase and quadrature components, and the auto-correlation functions of the complex envelope of fading

*<sup>R</sup>*<sup>ˆ</sup> *ZckZck* (*τ*) = *<sup>R</sup>*<sup>ˆ</sup> *ZskZsk* (*τ*) = <sup>2</sup>*J*0(2*<sup>π</sup> <sup>f</sup>*1*τ*)*J*0(2*<sup>π</sup> <sup>f</sup>*2*τ*) + *<sup>K</sup>* cos (2*<sup>π</sup> <sup>f</sup>*0*τ*)

*<sup>R</sup>*<sup>ˆ</sup> *ZckZsk* (*τ*) = <sup>−</sup>*R*<sup>ˆ</sup> *ZskZck* (*τ*) = *<sup>K</sup>* sin(2*<sup>π</sup> <sup>f</sup>*0*τ*)

*<sup>R</sup>*<sup>ˆ</sup> *ZkZk* (*τ*) = <sup>2</sup>*J*0(2*<sup>π</sup> <sup>f</sup>*1*τ*)*J*0(2*<sup>π</sup> <sup>f</sup>*2*τ*) + *<sup>K</sup>* exp (*j*2*<sup>π</sup> <sup>f</sup>*0*τ*)

*Yk*(*t*) = *Yck*(*t*) + *jYsk*(*t*) (11)

cos [2*π f*2*t* cos (*βmk*)] cos [(2*π f*1*t* cos (*αnk*) + *φnmk*] (12)

sin [2*π f*2*t* cos (*βmk*)] sin [(2*π f*1*t* sin (*αnk*) + *φnmk*] (13)

*<sup>M</sup>* ) (15)

<sup>√</sup><sup>1</sup> <sup>+</sup> *<sup>K</sup>* (16)

<sup>2</sup>(<sup>1</sup> <sup>+</sup> *<sup>K</sup>*) (17)

<sup>2</sup>(<sup>1</sup> <sup>+</sup> *<sup>K</sup>*) (18)

<sup>1</sup> <sup>+</sup> *<sup>K</sup>* (19)

(14)

The *kth* complex faded envelope is given by (Zajic & Stuber, 2006)

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

*M* ∑ *m*=1

> *M* ∑ *m*=1

> > *<sup>α</sup>nk* <sup>=</sup> <sup>2</sup>*π<sup>n</sup>* 4*N*<sup>0</sup> + 2*πk* 4*PN*<sup>0</sup>

*<sup>β</sup>mk* <sup>=</sup> <sup>1</sup> 2 ( 2*πm <sup>M</sup>* <sup>+</sup>

The auto-correlation and cross-correlation functions of the in-phase and quadrature components, and the auto-correlation functions of the complex envelope of fading signal *Z*(*t*) are given by

$$R\_{Z\_i Z\_i}(\tau) = R\_{Z\_i Z\_i}(\tau) = \frac{2l\_0(2\pi f\_1 \tau)l\_0(2\pi f\_2 \tau) + K \cos(2\pi f\_0 \tau)}{2(1+K)}\tag{4}$$

$$R\_{Z\_i Z\_i}(\tau) = -R\_{Z\_i Z\_i}(\tau) = \frac{K \sin(2\pi f\_0 \tau)}{2(1+K)}\tag{5}$$

$$R\_{ZZ}(\tau) = \frac{2l\_0(2\pi f\_1 \tau)l\_0(2\pi f\_2 \tau) + K \exp(j2\pi f\_0 \tau)}{(1+K)} = \frac{2l\_0(2\pi af\_2 \tau)l\_0(2\pi f\_2 \tau) + K \exp(j2\pi f\_0 \tau)}{(1+K)}$$

(6) where *<sup>J</sup>*0(·) is the zeroth-order Bessel function of the first kind, *<sup>a</sup>* <sup>=</sup> *<sup>f</sup>*1/ *<sup>f</sup>*<sup>2</sup> is the ratio of two maximum Doppler frequencies (or vehicles speeds), and here assuming *f*<sup>1</sup> ≤ *f*2. *a* = 0 means that the transmitter is stationary and then equation (6) gives the auto-correlation for I2V channels, which indicates that I2V channels are a special case of IVC channels in VANETs.

Time-averaging is often used in place of ensemble averaging in simulation practice. The auto-correlation function of the real part of *Z*(*t*) for one trial (sample of the process) then becomes

$$\begin{split} \hat{R}\_{Z\_{\iota}Z\_{\iota}}(\tau) &= \lim\_{T \to \infty} \frac{1}{T} \int\_{0}^{T} \mathbf{Z}\_{\iota}(t) \mathbf{Z}\_{\iota}(t + \tau) dt \\ &= \frac{1}{N} \sum\_{n=1}^{N} \cos(2\pi f\_{1} \cos a\_{n} + 2\pi f\_{2} \cos \beta\_{n}) \tau \end{split} \tag{7}$$

Furthermore, the time averaging changes from trial to trial due to the random angle. As pointed out in (Xiao et al., 2006), the variance of the time average *Var*[*R*(·)] = *<sup>E</sup>*[|*R*(·) <sup>−</sup> *<sup>R</sup>*ˆ(·)<sup>|</sup> 2] provides a measure of the closeness of the model in simulating the desired channel with a finite number of sinusoids. A lower variance indicates that a smaller number of simulation trials are needed to achieve the desired statistical properties, and, the convergence of the corresponding model is faster. The time-averaged variances of the aforementioned correlation statistics are derived as

$$Var[\mathcal{R}\_{Z\_i Z\_\epsilon}(\tau)] = Var[\mathcal{R}\_{Z\_i Z\_\epsilon}(\tau)] = \left[\frac{1 + j\_0(4\pi a f\_2 \tau) j\_0(4\pi f\_2 \tau) - 2j\_0^2(2\pi a f\_2 \tau) j\_0^2(2\pi f\_2 \tau)}{2N}\right] / (1 + K)^2 \tag{8}$$

$$1 - J\_0(4\pi a f\_2 \tau) J\_0(4\pi f\_2 \tau) \tag{8}$$

$$\operatorname{Var}[\mathcal{R}\_{\mathbb{Z}\_t\mathbb{Z}\_t}(\tau)] = \operatorname{Var}[\mathcal{R}\_{\mathbb{Z}\_t\mathbb{Z}\_t}(\tau)] = [\frac{1 - f\_0(4\pi a f\_2 \tau)f\_0(4\pi f\_2 \tau)}{2N}]/(1+K)^2\tag{9}$$

$$\operatorname{Var}[\mathcal{R}\_{ZZ}(\tau)] = \operatorname{Var}[\mathcal{R}\_{ZZ}(\tau)] = [\frac{4 - 4f\_0^2 (2\pi af\_2 \tau) f\_0^2 (2\pi f\_2 \tau)}{N}] / (1 + \mathcal{K})^2 \tag{10}$$

In next sections, we use these statistics to compare the performance of the proposed models with this reference model.
