# On the Statistical Properties of Nakagami-Hoyt Vehicle-to-Vehicle Fading Channel under Nonisotropic Scattering

Authors: Muhammad Imran Akram; Asrar U. H. Sheikh
Journal: International Journal of Antennas and Propagation
 (2012)

---

## Abstract

This paper presents the statistical properties of the vehicle-to-vehicle Nakagami-Hoyt (Nakagami-q) channel model under non-isotropic condition. The spatial time correlation function (STCF), the power spectral density (PSD), squared time autocorrelation function (SQCF), level crossing rate (LCR), and the average duration of Fade (ADF) of the Nakagami-Hoyt channel have been derived under the assumption that both the transmitter and receiver are nonstationary having nonomnidirectional antennas. A simulator that uses the inverse-fast-fourier-transform- (IFFT-) based computation method is designed for this model. The simulator and analytical results are compared.


---

## Body

## 1. Introduction

With passage of time, applications of wireless communication, their usefulness, and reliability are increasing. The recent applications where wireless communications is extensively used include wireless local area network (WLAN), multimedia messaging cellular telephone systems, satellite systems, femtocells, Bluetooth, and Zigbee devices. The wireless devices are also extensively used in in-car security systems equipment, home television systems private mobile systems, and so forth. In the conventional wireless communication systems, all the mobile stations communicate with each other via fixed base stations which are normally placed at elevated locations. Since mobile station is likely to be surrounded by objects having different shapes and orientations, the direct propagation path may not always present, and communication results from scattering that occurs near the mobile station. The channel between the transmitter and receiver is usually multipath fading channel; the signal fading occurs due to terminal mobility.An in-depth knowledge leads to an accurate model of mobile propagation channel which is essential for a simulator design that provides dependable performance results. Over the past many years, several mobile channel models have been proposed for links between fixed base station and mobile station. These include short-term fading models like the well-known Rayleigh, Rice [1], Hoyt [2], Nakagami-m [3], and Weibull [4]. For long-term fading model, lognormal distribution has been used [5, 6]. Several composite fading models combining the effects of short- and long-term fading (Nakagami-lognormal [3], Suzuki [7], and Rice-lognormal [8]) have also been proposed.Over the past decade, the research has been focused on vehicle-to-vehicle (V2V) or mobile-to-mobile (M2M) communication systems where no base station is present and both the transmitterTx and the receiver Rx are in motion. The V2V communication finds applications in mobile ad-hoc wireless networks, intelligent highway systems, emergency, military, and security vehicles. The antennas are mounted on the top or on the side of vehicles, which move with different velocities and resulting in time varying channels. The buildings and other obstacles surrounding the terminals act as scatterers thereby generating multipaths. Depending upon the vehicular positions, the line of sight (LOS) may or may not be present.During the past decade, a large number of research projects have been done on V2V communications [9–12]. Reference [13] presents a survey of the vehicular channel characterization under different environments (highway, rural, urban, and suburban). It shows the path loss exponent, Root Mean Square (RMS) delay spread, and mean Doppler spread under these conditions. The statistical model for vehicle-to-vehicle communication was first proposed by Akki and Haber [14], and its statistical properties were reported in [15]. Based on the work of [15], many V2V simulators were designed and implemented. Reference [16] presented a discrete line spectrum-based approach to simulate the channel. The work reported in [17] is based on sum of sinusoids (SOS) approach for simulator design. In [18], the simulation of multiple input multiple output (MIMO) V2V is presented. The simulator proposed in [19] is based on Kullback-Leibler divergence which is compared with IFFT based approach of simulator design. Reference [20] uses Gaussian quadrature rule for simulator design. Reference [21] proposes an efficient SOS-based approach for V2V simulator design. All the simulator design approaches mentioned above were designed for V2V Rayleigh fading channels.A small number of works have used non-Rayleigh fading channel models. The second order statistics of Nakagami-Hoyt channel have been derived in [22]. Reference [23] derives the statistical properties of double Nakagami-Hoyt channel.In many real world scenarios, nonisotropic scattering is often experienced by both the mobile transmitter and receiver. It has been shown in [24–26] that in dense urban environments, non-isotropic scattering around the mobile station exists. Reference [27] derives the second order statistics of V2V Ricean fading channel under non-isotropic conditions and compares the theoretical results with the measured data. Reference [28] derives the autocorrelation function of Rice process under non-isotropic condition. Reference [29] presented V2V model for Rayleigh fading under non-isotropic condition. Many nonuniform distributions have been discussed for angle of arrival (AOA) and angle of departure (AOD). These include Gaussian, Laplacian, quadratic, and Von Misses distributions. Von Misses distribution (assumed in this paper), a generic case described in [24], covers the other distributions (Gaussian, Laplacian, cosine, and uniform distributions) as its special cases.In this paper, a novel V2V Nakagami-Hoyt channel model under non-isotropic scattering condition is proposed. The existing channel models [2, 15, 30] are treated as its special cases. Analytical expressions for second order statistical properties including STCF, PSD, SQCF, LCR, and ADF of the envelop of the proposed model have also been derived. An IFFT based simulator has been developed to validate the first and second order statistical parameters of the proposed model. Mean Square Error (MSE) of autocorrelation function is also plotted to show the simulator accuracy. To the best of authors' knowledge, no work has been reported on the statistical parameters of the proposed model and its simulation.The remainder of this paper is organized as follows.  Section2 presents details of the Nakagami-Hoyt V2V channel model and its first and second order statistics. Section 3 describes the simulation method, results of the simulator, and their comparison with the analytical results. Finally, Section 4 concludes the paper.


## 2. The Nakagami-Hoyt Channel Model

The Nakagami-Hoyt (also known asq) distribution is the distribution of the modulus of a complex Gaussian random variable whose components are uncorrelated with zero mean and unequal variances.In this section, we briefly describe the proposed channel model along with its usefulness and derive the first and second order statistics of Hoyt fading channel under the assumptions that the channel is narrow band, and the receiver and transmitter are moving with velocitiesV1 and V2, respectively and the non-isotropic scattering (i.e, AOA and AOD have nonuniform distributions).A Hoyt process,R(t), is obtained by complex Gaussian random process as
(1)μ(t)=μ1(t)+jμ2(t),R(t)=|μ(t)|,
where j=-1, μ1(t) and μ2(t) are complex Gaussian random processes with zero means and variances σ12 and σ22, respectively. |·| indicates the L2 norm.The parametersq and a are defined as
(2)q=σ1σ2,a=V2V1.Reference [14] proposed a statistical model for the mobile-to-mobile Rayleigh fading channel. This model is modified for the Nakagami-Hoyt frequency flat fading channel. The baseband equivalent channel impulse response is given as
(3)μ1(t)=∑n=1N‍rncos((w1n+w2n)t+ϕn).
Also
(4)μ2(t)=1q∑n=1N‍rnsin((w1n+w2n)t+ϕn),
where N is the number of propagation paths; rn and ϕn are the uniformly distributed amplitude and phase of nth path, respectively. win is given by
(5)win=2πfdicos(αin),i=1,2,
where fd1 and fd2 are the maximum Doppler frequency due to the motion of receiver and transmitter, respectively. α1n and α2n are the AOA and AOD of the nth path with respect to the velocity vector of receiver and transmitter, respectively.Assuming thatα1n, α2n, and ϕn are independent for all n with α1n and α2n are non-uniformly distributed having Von Mises PDF described in [24]. The PDF of the Von Mises distribution is given by
(6)pα(α)=exp[κcos(α-ν)]2πI0(κ),κ>0,
where I0(·) is the zero-order modified Bessel function of the first kind, ν is the mean direction of the AOD or AOA, and κ is the concentration parameter which controls the width of the scatterers. Figure 1 illustrates the PDF pα(α) with different values of κ for ν=0. If κ=0, then the von Mises PDF reduces to a uniform distribution (isotropic scattering).Figure 1
Von Mises PDF showing nonisotropic scattering.


### 2.1. Usefulness of the Proposed Channel Model

The measurements made in the rural environment demonstrated that the channel is more accurately modeled only when the variance of the in-phase and the quadrature components are not identical [22]. This observation was further supported in [31] where the model matches the measured data for the cases of unequal variances. For V2V communication, reference [32] observed that when the distance between the vehicles exceeds 70 m, the Nakagami m-factor is observed to be less than unity, which corresponds to the case of unequal variances of the component of complex Gaussian process. Further as found from the V2V measurements [33, 34], the m value of each tap of the channel model described is found to be less than unity (0.75–0.89) which from [22] corresponds to the value of q (0.5–0.707).The channel models described previously do not consider the following two scenarios simultaneously. First, when the antennas are not omnidirectional, and scatterers around the receiver and transmitter are not uniformly distributed. This is normally the case in V2V communications when transmit and receive antennas are present inside the vehicles. Second, when the fading is severe and the channel gains are no longer Rayleigh ieq<1. The proposed model is a generalized model. IT is applicable to these cases for V2V when a≠0 and base to vehicle when a=0 and covers the previously existing models as its special cases.


### 2.2. First Order Statistics

The probability density function of the envelopeR(t) is given by [2]
(7)pR(x)=xσ1σ2exp[-x24(1σ12+1σ22)]×I0[x24(1σ12-1σ22)],x≥0,
where I0(·) denotes the zeroth-order modified Bessel function of the first kind. The pdf of the corresponding phase process ν(t)=arctan[μ2(t)/μ1(t)] is given by
(8)pν(θ)=σ1σ22π(σ22cos2θ+σ12sin2θ),0≤θ<2π.Since the pdfs are independent of time so they will remain independent for V2V Nakagami-Hoyt channels. The meanmμ and RMS Rrms values can be easily obtained as
(9)mμ=E[μ(t)]=E[μ1(t)]+jE[μ2(t)]=0Rrms=E[μ(t)2]=E[μ1(t)2]+E[μ2(t)2]=σ12+σ22=σ21+q2.


### 2.3. Second Order Statistics

In this section, The STCF, PSD, SQCF, LCR and ADF of the Nakagami-Hoyt V2V fading process are derived. These quantities are useful in estimation of burst error, mobile velocity and Markov modeling of fading channels [35–37].


#### 2.3.1. Spatial Time Correlation Function

For derivation of spatial time correlation function ofNakagami-Hoyt V2V channel, [14] is used as a reference, which describes the case for Rayleigh distribution. The results are obtained for more general case where σ1≠σ2.The spatial time correlation function of the envelope is given by [38]
(10)R(x1,x2,t1,t2)=12〈μx2(t2)μx1*(t1)〉,
where 〈·〉 is the statistical average μx1(t1) and μx2(t2) are the complex envelop received at positions x1 and x2 at time t1 and t2, respectively as given in [14]. It is shown in Appendix A that the general form of the spatial time correlation function is given by (11)R(Δx,Δt)=∑n=01‍(1+qcos(nπ))24q2σ12∏i=12‍I0(κi2-K2Mi2+j2κiKMicosνicos(nπ))I0(κi),where
(12)Mi(Δx,Δt)=ViΔt+(2-i)ΔxK=2π/λ, Δt=t2-t1 and Δx=x2-x1.R
(
Δ
x
,
Δ
t
) is the correlation functions of two signal envelopes obtained at two locations Δx apart and at two time instant Δt apart.The time correlation function is obtained by settingΔx = 0 in (11) (13)Rμ(Δt)=R(0,Δt)=∑n=01‍(1+qcos(nπ))24q2σ12∏i=12‍I0(κi2-K2(ViΔt)2+j2κiKViΔtcosνicos(nπ))I0(κi).For a more specific case (isotropic scattering), substituting κi=0 and νi=0 in (11) and simplifying, (14)R(Δx,Δt)=1+q22q2σ12J0(KV2Δt)J0(KV1Δt+KΔx)J0(·) is the zero-order Bessel function. It can be shown that for q=1, the space time correlation function for vehicle to vehicle Rayleigh fading channel is obtained.
(15)R(Δx,Δt)=σ12J0(KV2Δt)J0(KV1Δt+KΔx)
which matches with the result of [14]. Further by setting V2=0 (i.e., transmitter stationary) in (15), we get Tx (stationary) and Rx (mobile) Rayleigh channel
(16)R(Δx,Δt)=σ12J0(KV1Δt+KΔx).
The time correlation is obtained by setting Δx = 0 in (15) as
(17)Rμ(Δt)=R(0,Δt)=1+q22q2σ12J0(KV2Δt)J0(KV1Δt).
The spatial correlation function is obtained by setting Δt=0 in (15) as
(18)Rμ(Δx)=R(Δx,0)=1+q22q2σ12J0(KΔx).
Similarly, for Rayleigh fading, the time and spatial correlation functions are obtained as
(19)Rμ(Δt)=σ12J0(KV2Δt)J0(KV1Δt),Rμ(Δx)=σ12J0(KΔx).


#### 2.3.2. Power Spectral Density

The power spectral densityS(f) of the wide sense stationary (WSS) process is obtained by taking the Fourier transform of the time autocorrelation function as
(20)S(f)=∫-∞∞‍Rμ(Δt)e-j2πfΔtdΔt.
For non-isotropic scattering (κi≠0, νi≠0), the integral is evaluated using the Fourier transform of (13).For isotropic scattering, the power spectral density is obtained by taking the Fourier transform of (17) as
(21)S(f)=1+q22q2σ12∫-∞∞‍J0(KV2Δt)J0(KV1Δt)e-j2πfΔtdΔt.
The integral can be evaluated using [39]. From [14], the reduced form is
(22)S(f)=1+q22q2π2fm1aσ12×K[(1+a)2a1-(f(1+a)fm1)2],
where K(·) is the elliptical integral function of first kind; fm1, fm2 are the maximum Doppler shifts due to the motion of the receiver and transmitter, respectively with fmi=Vi/λ. Therefore, fm2=afm1.Now, for the casea=0, we have V2=0. Hence, the PSD is obtained as
(23)S(f)=1+q22q2σ12∫-∞∞‍J0(KV1Δt)e-j2πfΔtdΔt
which is evaluated in [39] as
(24)S(f)=1+q22q2σ12πfm12-f2
which is the expression for PSD of base to mobile Hoyt channel. Again the Rayleigh base to mobile PSD expression is obtained by setting q=1 in (23) as
(25)S(f)=σ12πfm12-f2.


#### 2.3.3. Squared Time Autocorrelation Function

The squared time autocorrelation function is used in computation of carrier to noise ratio (CNR). For the proposed channel, it is defined as(26)Rμ2(Δt)=E[|μ(t)|2|μ(t-Δt)|2].
In Appendix C, it is derived and the final form obtained is
(27)Rμ2(Δt)=σ14(1+q2)2q4+2[R112(Δt)+R222(Δt)+2R122(Δt)],
where R11 and R22 are the autocorrelation of in-phase and quadrature components, respectively, where R12 is cross-correlation between them.


#### 2.3.4. Level Crossing Rate and Average Duration of Fade

The level crossing rate of the processR(t) is obtained by solving the following integral:
(28)NR(r)=∫0∞‍z˙pRR˙(r,z˙)dz˙,
where pRR˙ is the joint PDF of R(t) and its time derivative R˙(r). From [22], LCR for stationary to mobile Hoyt channel is given by
(29)NR(r)=r(2π)3/2σ1σ2×∫02π‍e[-(r2/2σ12σ22)(σ22cos2(θ)+σ12sin2(θ))]×β1cos2(θ)+β2sin2(θ)dθ.
The expression will remain the same for V2V Nakagami-Hoyt channel. The only thing that will differ here will be the values of β1 and β2. These can be found in Appendix B using the relationship βi=-Rμ¨(0). Also, substituting β2=β1/q2 and ρ=R/Rrms, the LCR is obtained as
(30)NR(ρ)=ρβ1(q2+1)(2π)3/2qσ1×∫02π‍e[-(ρ2(q2+1)/2q2)(cos2(θ)+q2sin2(θ))]×q2cos2(θ)+sin2(θ)dθ.
For isotropic scattering, from Appendix B, substituting the values of β1=(2πσ1fm1)2(1+a2), the expression becomes
(31)NR(ρ)=(1+a2)(q2+1)fm1ρ2qπ×∫02π‍e[-(ρ2(q2+1)/2q2)(cos2(θ)+q2sin2(θ))]×q2cos2(θ)+sin2(θ)dθ.
It is easy to show that by substituting a=0 and q=1, the above equation is reduced to the expression for base to mobile Rayleigh LCR given by [40].The average duration of fade of a signal is defined as average duration of time for which the signalr spends below a specified threshold R0. It is given by [41]
(32)τ-=P(r<R0)NR(r),
where P(r<R0) is the cumulative density function obtained by
(33)P(r<R0)=∫0R0‍pR(x)dx.
This is obtained by integrating (7). Hence substituting (31) and (33) in (32), ADF can be directly obtained.


## 2.1. Usefulness of the Proposed Channel Model

The measurements made in the rural environment demonstrated that the channel is more accurately modeled only when the variance of the in-phase and the quadrature components are not identical [22]. This observation was further supported in [31] where the model matches the measured data for the cases of unequal variances. For V2V communication, reference [32] observed that when the distance between the vehicles exceeds 70 m, the Nakagami m-factor is observed to be less than unity, which corresponds to the case of unequal variances of the component of complex Gaussian process. Further as found from the V2V measurements [33, 34], the m value of each tap of the channel model described is found to be less than unity (0.75–0.89) which from [22] corresponds to the value of q (0.5–0.707).The channel models described previously do not consider the following two scenarios simultaneously. First, when the antennas are not omnidirectional, and scatterers around the receiver and transmitter are not uniformly distributed. This is normally the case in V2V communications when transmit and receive antennas are present inside the vehicles. Second, when the fading is severe and the channel gains are no longer Rayleigh ieq<1. The proposed model is a generalized model. IT is applicable to these cases for V2V when a≠0 and base to vehicle when a=0 and covers the previously existing models as its special cases.


## 2.2. First Order Statistics

The probability density function of the envelopeR(t) is given by [2]
(7)pR(x)=xσ1σ2exp[-x24(1σ12+1σ22)]×I0[x24(1σ12-1σ22)],x≥0,
where I0(·) denotes the zeroth-order modified Bessel function of the first kind. The pdf of the corresponding phase process ν(t)=arctan[μ2(t)/μ1(t)] is given by
(8)pν(θ)=σ1σ22π(σ22cos2θ+σ12sin2θ),0≤θ<2π.Since the pdfs are independent of time so they will remain independent for V2V Nakagami-Hoyt channels. The meanmμ and RMS Rrms values can be easily obtained as
(9)mμ=E[μ(t)]=E[μ1(t)]+jE[μ2(t)]=0Rrms=E[μ(t)2]=E[μ1(t)2]+E[μ2(t)2]=σ12+σ22=σ21+q2.


## 2.3. Second Order Statistics

In this section, The STCF, PSD, SQCF, LCR and ADF of the Nakagami-Hoyt V2V fading process are derived. These quantities are useful in estimation of burst error, mobile velocity and Markov modeling of fading channels [35–37].


### 2.3.1. Spatial Time Correlation Function

For derivation of spatial time correlation function ofNakagami-Hoyt V2V channel, [14] is used as a reference, which describes the case for Rayleigh distribution. The results are obtained for more general case where σ1≠σ2.The spatial time correlation function of the envelope is given by [38]
(10)R(x1,x2,t1,t2)=12〈μx2(t2)μx1*(t1)〉,
where 〈·〉 is the statistical average μx1(t1) and μx2(t2) are the complex envelop received at positions x1 and x2 at time t1 and t2, respectively as given in [14]. It is shown in Appendix A that the general form of the spatial time correlation function is given by (11)R(Δx,Δt)=∑n=01‍(1+qcos(nπ))24q2σ12∏i=12‍I0(κi2-K2Mi2+j2κiKMicosνicos(nπ))I0(κi),where
(12)Mi(Δx,Δt)=ViΔt+(2-i)ΔxK=2π/λ, Δt=t2-t1 and Δx=x2-x1.R
(
Δ
x
,
Δ
t
) is the correlation functions of two signal envelopes obtained at two locations Δx apart and at two time instant Δt apart.The time correlation function is obtained by settingΔx = 0 in (11) (13)Rμ(Δt)=R(0,Δt)=∑n=01‍(1+qcos(nπ))24q2σ12∏i=12‍I0(κi2-K2(ViΔt)2+j2κiKViΔtcosνicos(nπ))I0(κi).For a more specific case (isotropic scattering), substituting κi=0 and νi=0 in (11) and simplifying, (14)R(Δx,Δt)=1+q22q2σ12J0(KV2Δt)J0(KV1Δt+KΔx)J0(·) is the zero-order Bessel function. It can be shown that for q=1, the space time correlation function for vehicle to vehicle Rayleigh fading channel is obtained.
(15)R(Δx,Δt)=σ12J0(KV2Δt)J0(KV1Δt+KΔx)
which matches with the result of [14]. Further by setting V2=0 (i.e., transmitter stationary) in (15), we get Tx (stationary) and Rx (mobile) Rayleigh channel
(16)R(Δx,Δt)=σ12J0(KV1Δt+KΔx).
The time correlation is obtained by setting Δx = 0 in (15) as
(17)Rμ(Δt)=R(0,Δt)=1+q22q2σ12J0(KV2Δt)J0(KV1Δt).
The spatial correlation function is obtained by setting Δt=0 in (15) as
(18)Rμ(Δx)=R(Δx,0)=1+q22q2σ12J0(KΔx).
Similarly, for Rayleigh fading, the time and spatial correlation functions are obtained as
(19)Rμ(Δt)=σ12J0(KV2Δt)J0(KV1Δt),Rμ(Δx)=σ12J0(KΔx).


### 2.3.2. Power Spectral Density

The power spectral densityS(f) of the wide sense stationary (WSS) process is obtained by taking the Fourier transform of the time autocorrelation function as
(20)S(f)=∫-∞∞‍Rμ(Δt)e-j2πfΔtdΔt.
For non-isotropic scattering (κi≠0, νi≠0), the integral is evaluated using the Fourier transform of (13).For isotropic scattering, the power spectral density is obtained by taking the Fourier transform of (17) as
(21)S(f)=1+q22q2σ12∫-∞∞‍J0(KV2Δt)J0(KV1Δt)e-j2πfΔtdΔt.
The integral can be evaluated using [39]. From [14], the reduced form is
(22)S(f)=1+q22q2π2fm1aσ12×K[(1+a)2a1-(f(1+a)fm1)2],
where K(·) is the elliptical integral function of first kind; fm1, fm2 are the maximum Doppler shifts due to the motion of the receiver and transmitter, respectively with fmi=Vi/λ. Therefore, fm2=afm1.Now, for the casea=0, we have V2=0. Hence, the PSD is obtained as
(23)S(f)=1+q22q2σ12∫-∞∞‍J0(KV1Δt)e-j2πfΔtdΔt
which is evaluated in [39] as
(24)S(f)=1+q22q2σ12πfm12-f2
which is the expression for PSD of base to mobile Hoyt channel. Again the Rayleigh base to mobile PSD expression is obtained by setting q=1 in (23) as
(25)S(f)=σ12πfm12-f2.


### 2.3.3. Squared Time Autocorrelation Function

The squared time autocorrelation function is used in computation of carrier to noise ratio (CNR). For the proposed channel, it is defined as(26)Rμ2(Δt)=E[|μ(t)|2|μ(t-Δt)|2].
In Appendix C, it is derived and the final form obtained is
(27)Rμ2(Δt)=σ14(1+q2)2q4+2[R112(Δt)+R222(Δt)+2R122(Δt)],
where R11 and R22 are the autocorrelation of in-phase and quadrature components, respectively, where R12 is cross-correlation between them.


### 2.3.4. Level Crossing Rate and Average Duration of Fade

The level crossing rate of the processR(t) is obtained by solving the following integral:
(28)NR(r)=∫0∞‍z˙pRR˙(r,z˙)dz˙,
where pRR˙ is the joint PDF of R(t) and its time derivative R˙(r). From [22], LCR for stationary to mobile Hoyt channel is given by
(29)NR(r)=r(2π)3/2σ1σ2×∫02π‍e[-(r2/2σ12σ22)(σ22cos2(θ)+σ12sin2(θ))]×β1cos2(θ)+β2sin2(θ)dθ.
The expression will remain the same for V2V Nakagami-Hoyt channel. The only thing that will differ here will be the values of β1 and β2. These can be found in Appendix B using the relationship βi=-Rμ¨(0). Also, substituting β2=β1/q2 and ρ=R/Rrms, the LCR is obtained as
(30)NR(ρ)=ρβ1(q2+1)(2π)3/2qσ1×∫02π‍e[-(ρ2(q2+1)/2q2)(cos2(θ)+q2sin2(θ))]×q2cos2(θ)+sin2(θ)dθ.
For isotropic scattering, from Appendix B, substituting the values of β1=(2πσ1fm1)2(1+a2), the expression becomes
(31)NR(ρ)=(1+a2)(q2+1)fm1ρ2qπ×∫02π‍e[-(ρ2(q2+1)/2q2)(cos2(θ)+q2sin2(θ))]×q2cos2(θ)+sin2(θ)dθ.
It is easy to show that by substituting a=0 and q=1, the above equation is reduced to the expression for base to mobile Rayleigh LCR given by [40].The average duration of fade of a signal is defined as average duration of time for which the signalr spends below a specified threshold R0. It is given by [41]
(32)τ-=P(r<R0)NR(r),
where P(r<R0) is the cumulative density function obtained by
(33)P(r<R0)=∫0R0‍pR(x)dx.
This is obtained by integrating (7). Hence substituting (31) and (33) in (32), ADF can be directly obtained.


## 2.3.1. Spatial Time Correlation Function

For derivation of spatial time correlation function ofNakagami-Hoyt V2V channel, [14] is used as a reference, which describes the case for Rayleigh distribution. The results are obtained for more general case where σ1≠σ2.The spatial time correlation function of the envelope is given by [38]
(10)R(x1,x2,t1,t2)=12〈μx2(t2)μx1*(t1)〉,
where 〈·〉 is the statistical average μx1(t1) and μx2(t2) are the complex envelop received at positions x1 and x2 at time t1 and t2, respectively as given in [14]. It is shown in Appendix A that the general form of the spatial time correlation function is given by (11)R(Δx,Δt)=∑n=01‍(1+qcos(nπ))24q2σ12∏i=12‍I0(κi2-K2Mi2+j2κiKMicosνicos(nπ))I0(κi),where
(12)Mi(Δx,Δt)=ViΔt+(2-i)ΔxK=2π/λ, Δt=t2-t1 and Δx=x2-x1.R
(
Δ
x
,
Δ
t
) is the correlation functions of two signal envelopes obtained at two locations Δx apart and at two time instant Δt apart.The time correlation function is obtained by settingΔx = 0 in (11) (13)Rμ(Δt)=R(0,Δt)=∑n=01‍(1+qcos(nπ))24q2σ12∏i=12‍I0(κi2-K2(ViΔt)2+j2κiKViΔtcosνicos(nπ))I0(κi).For a more specific case (isotropic scattering), substituting κi=0 and νi=0 in (11) and simplifying, (14)R(Δx,Δt)=1+q22q2σ12J0(KV2Δt)J0(KV1Δt+KΔx)J0(·) is the zero-order Bessel function. It can be shown that for q=1, the space time correlation function for vehicle to vehicle Rayleigh fading channel is obtained.
(15)R(Δx,Δt)=σ12J0(KV2Δt)J0(KV1Δt+KΔx)
which matches with the result of [14]. Further by setting V2=0 (i.e., transmitter stationary) in (15), we get Tx (stationary) and Rx (mobile) Rayleigh channel
(16)R(Δx,Δt)=σ12J0(KV1Δt+KΔx).
The time correlation is obtained by setting Δx = 0 in (15) as
(17)Rμ(Δt)=R(0,Δt)=1+q22q2σ12J0(KV2Δt)J0(KV1Δt).
The spatial correlation function is obtained by setting Δt=0 in (15) as
(18)Rμ(Δx)=R(Δx,0)=1+q22q2σ12J0(KΔx).
Similarly, for Rayleigh fading, the time and spatial correlation functions are obtained as
(19)Rμ(Δt)=σ12J0(KV2Δt)J0(KV1Δt),Rμ(Δx)=σ12J0(KΔx).


## 2.3.2. Power Spectral Density

The power spectral densityS(f) of the wide sense stationary (WSS) process is obtained by taking the Fourier transform of the time autocorrelation function as
(20)S(f)=∫-∞∞‍Rμ(Δt)e-j2πfΔtdΔt.
For non-isotropic scattering (κi≠0, νi≠0), the integral is evaluated using the Fourier transform of (13).For isotropic scattering, the power spectral density is obtained by taking the Fourier transform of (17) as
(21)S(f)=1+q22q2σ12∫-∞∞‍J0(KV2Δt)J0(KV1Δt)e-j2πfΔtdΔt.
The integral can be evaluated using [39]. From [14], the reduced form is
(22)S(f)=1+q22q2π2fm1aσ12×K[(1+a)2a1-(f(1+a)fm1)2],
where K(·) is the elliptical integral function of first kind; fm1, fm2 are the maximum Doppler shifts due to the motion of the receiver and transmitter, respectively with fmi=Vi/λ. Therefore, fm2=afm1.Now, for the casea=0, we have V2=0. Hence, the PSD is obtained as
(23)S(f)=1+q22q2σ12∫-∞∞‍J0(KV1Δt)e-j2πfΔtdΔt
which is evaluated in [39] as
(24)S(f)=1+q22q2σ12πfm12-f2
which is the expression for PSD of base to mobile Hoyt channel. Again the Rayleigh base to mobile PSD expression is obtained by setting q=1 in (23) as
(25)S(f)=σ12πfm12-f2.


## 2.3.3. Squared Time Autocorrelation Function

The squared time autocorrelation function is used in computation of carrier to noise ratio (CNR). For the proposed channel, it is defined as(26)Rμ2(Δt)=E[|μ(t)|2|μ(t-Δt)|2].
In Appendix C, it is derived and the final form obtained is
(27)Rμ2(Δt)=σ14(1+q2)2q4+2[R112(Δt)+R222(Δt)+2R122(Δt)],
where R11 and R22 are the autocorrelation of in-phase and quadrature components, respectively, where R12 is cross-correlation between them.


## 2.3.4. Level Crossing Rate and Average Duration of Fade

The level crossing rate of the processR(t) is obtained by solving the following integral:
(28)NR(r)=∫0∞‍z˙pRR˙(r,z˙)dz˙,
where pRR˙ is the joint PDF of R(t) and its time derivative R˙(r). From [22], LCR for stationary to mobile Hoyt channel is given by
(29)NR(r)=r(2π)3/2σ1σ2×∫02π‍e[-(r2/2σ12σ22)(σ22cos2(θ)+σ12sin2(θ))]×β1cos2(θ)+β2sin2(θ)dθ.
The expression will remain the same for V2V Nakagami-Hoyt channel. The only thing that will differ here will be the values of β1 and β2. These can be found in Appendix B using the relationship βi=-Rμ¨(0). Also, substituting β2=β1/q2 and ρ=R/Rrms, the LCR is obtained as
(30)NR(ρ)=ρβ1(q2+1)(2π)3/2qσ1×∫02π‍e[-(ρ2(q2+1)/2q2)(cos2(θ)+q2sin2(θ))]×q2cos2(θ)+sin2(θ)dθ.
For isotropic scattering, from Appendix B, substituting the values of β1=(2πσ1fm1)2(1+a2), the expression becomes
(31)NR(ρ)=(1+a2)(q2+1)fm1ρ2qπ×∫02π‍e[-(ρ2(q2+1)/2q2)(cos2(θ)+q2sin2(θ))]×q2cos2(θ)+sin2(θ)dθ.
It is easy to show that by substituting a=0 and q=1, the above equation is reduced to the expression for base to mobile Rayleigh LCR given by [40].The average duration of fade of a signal is defined as average duration of time for which the signalr spends below a specified threshold R0. It is given by [41]
(32)τ-=P(r<R0)NR(r),
where P(r<R0) is the cumulative density function obtained by
(33)P(r<R0)=∫0R0‍pR(x)dx.
This is obtained by integrating (7). Hence substituting (31) and (33) in (32), ADF can be directly obtained.


## 3. Simulation and Results

The simulator described in this paper uses Smith spectrum method used in [42]. This method is IFFT based and was slightly modified to generate Hoyt Fading signal envelope. This method requires frequency domain generation and processing of random signal followed by inverse Fourier transform to obtain a time domain sequence with the desired properties. The block diagram of the proposed simulator is shown in Figure 2. To implement the simulator, the following steps are performed.(1)
Input number of frequency samples(N) and time samples (M).(2)
Specify maximum doppler frequency of the receiverfm1 in Hz.(3)
Specify the value of parametera such that maximum Doppler frequency of transmitter fm2=afm1.(4)
Specify the value ofq.(5)
Generate twoN/2 samples Gaussian quadrature components of zero mean and unity variance. Generate the remaining N/2 components by conjugating them. This forms the negative frequency components.(6)
GenerateN points spectrum S(f).(7)
The frequency spacing between the adjacent spectral lines is given byΔf=2fm1(1+a)/N.(8)
The time resolution is given by1/Δf(M-1).(9)
Multiply the in-phase and quadrature components byS(f) and perform the IFFT of the resultant individual. Normalize both the resulting in-phase and quadrature to make their variance unity.(10)
Quadrature component will yieldμ2 while in-phase component after multiplying with q will yield μ1.(11)
The root of the sum of squared envelop of both will generate random variable Nakagami Hoyt distribution for the given value ofq.(12)
The phase distribution is obtained by using the phase random variabletan-1(μ1/μ2).Figure 2
Block diagram of IFFT based simulator.The simulation was run with the following parameters, carrier frequencyf=900MHz, velocity of receiver V1=72km/hr which means fm1=60Hz, three different values of q=1,0.5,0.3 and three different values of a=1,0.5,0. The simulator sample output for q=0.5, a=0.5 is shown in Figure 3. The amplitude and phase pdf plots are shown in Figures 4 and 5, respectively. The corresponding theoretical output of (7) and (8), respectively was also plotted for comparison.Figure 3
Output of the hoyt simulator.Figure 4
Hoyt amplitude PDF plotq=0.5, a=0.5, k=3.Figure 5
Hoyt phase PDF plotq=0.5,a=0.5, k=3.The power spectral density plots forκ=(0,1,2,3) are shown in Figure 6. The plots for other values of q are the scaled version of this and are not shown here. It has been evident from the plot that S(f) has peaks at f=±(fm1-fm2). It has been shown that plot is symmetric for κ=0 indicating isotropic scattering whereas κ≠0 results in asymmetric PSD (non-isotropic scattering).Figure 6
PSD plot forq=0.5.The autocorrelation plots forq=0.5 and a=0.5 with 4 different values of κ=(0,1,2,3) are shown in Figure 7. Plot for κ=0 indicates V2V isotropic scattering. The normalized autocorrelation functions plots have been shown in Figure 8. The plots for a=1,0.5,0, q=0.5, and κ=3 are compared with the theoretical expression given in (17). Since the real and imaginary components are Gaussian, it can be found from the plots that the normalized autocorrelation functions are still Bessel but with different shape than the one shown in [15]. Also the plots are function of κ and ν.Figure 7
Time autocorrelation function of real part of envelope.Figure 8
Time Autocorrelation function of Real part of Envelope forq=0.5, k=3.The normalized squared autocorrelation plots of real part forq=0.5, κ=3, a=0.5 is shown in Figure 9. The plot is compared with the theoretical derived expression.Figure 9
Squared Autocorrelation function of Real part forq=0.5, k=3.The LCR and ADF of envelop forq=(1,0.5,0.3) and κ=3 are plotted in Figures 10, 11, 12, 13, 14, and 15 for three different values of a=(1,0.5,0). The curves are matched with their theoretical expressions given by (31) and (32). q=1 shows the Rayleigh envelop whereas a=0 indicates base to mobile communication plots for LCR and ADF.Figure 10
Level crossing rates forq=1, k=3.Figure 11
Level crossing rates forq=0.5, k=3.Figure 12
Level crossing rates forq=0.3, k=3.Figure 13
Average duration of fade forq=1, k=3.Figure 14
Average duration of fade forq=0.5, k=3.Figure 15
Average duration of fade forq=0.3, k=3.The mean square error (MSE) of the time autocorrelation function is given by(34)MSE=E[(RμΔt-Rμ^Δt)2],
where RμΔt and Rμ^Δt are the theoretical and estimated autocorrelation functions, respectively. Figure 16 shows the mean square error of time autocorrelation function as a function of number of frequency sample points N. The figure is obtained for q=0.5, a=0.5, κ=3, and varying N in power of 2 in the range 2048–32768. It is evident from the curve that the MSE reduces when the number of sample points is increased. Hence, more accurate simulator is obtained at the cost of increasing the complexity of simulator.Figure 16
Mean square error of autocorrelation functionq=0.5, a=0.5, k=3.


## 4. Conclusion

The second order statistical properties for vehicle to vehicle Nakagami-Hoyt channels, under the non-isotropic scattering conditions at both the transmitter and receiver, have been developed. These include expressions for space time correlation function, power spectral density, squared time autocorrelation, level crossing rates, and average duration of fade. The Nakagami-Hoyt V2V simulator has also been developed to verify the above mentioned theoretical expressions. It has been found that the theoretical results match closely with the simulated data verifying the validity of the model.



---

*Source: 179378-2012-11-07.xml*