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spectral density (PSD) of a Rayleigh mobile-to-mobile chan- nel has been derived. The behavior of the Doppler PSD has also been studied in [9] for the case of ...
SIMULATION OF MOBILE-TO-MOBILE RADIO FADING CHANNELS Nazih Hajri1 , Neji Youssef1 1

Matthias P¨atzold 2

and

Ecole Sup´erieure des Communications de Tunis, 2083 EL Ghazala, Ariana, Tunisia Faculty of Engineering and Science, Agder University College, Grimstad, Norway [email protected] , [email protected], [email protected]

2

ABSTRACT In this paper, we discuss the applicability of deterministic parameter computation methods to the design of simulation models for mobile-to-mobile multipath fading channels. It is shown that the Rice’s sum-of-sinusoids concept, employing the mean square error method (MSEM), the Lp -norm method (LPNM) and the method of exact Doppler spread (MEDS) can directly be applied to the design of so-called Rayleigh and “double-Rayleigh” mobileto-mobile radio fading channel models. The performance of the designed mobile-to-mobile channel simulator is analyzed with respect to the autocorrelation function (ACF) and level crossing rate (LCR) of fading processes, and also by considering the bit error probability (BEP) of noncoherent differential phase shift keying (DPSK) modulation schemes.

[5] the IFFT filtering technique has been applied to simulate single-input single-output (SISO) Rayleigh fading. In [6], the simulation of multiple-input multiple-output (MIMO) channels based on the concept of Rice’s sum-of-sinusoids has been studied. This paper provides further insight into the application of the concept of deterministic channel modelling to simulate SISO mobile-to-mobile fading channels. Specifically, we apply extended versions of known methods [3], which have originally been developed for the simulation of conventional multipath fading channels, to compute the parameters of sinusoids used for our case. A good agreement is found between the first and second order statistics of the resulting simulation model and those of the reference model. The organization of the paper is as follows. Section 2 presents an overview on mobile-to-mobile radio multipath fading channels. Section 3 contains the simulation results of these models. Finally, Section 4 concludes the paper.

1. INTRODUCTION Mobile-to-mobile communications have become a major topic of research during the last few years. Applications of this type of communications are best exemplified by intervehicle exchange of information for cooperative driving, road-safety and improved traffic management[1, 2]. In order to design related communication systems that ensure high-quality radio links, as it is required, e.g., in safety applications, the investigation of the characteristics of the mobile-to-mobile radio propagation channel is an important part that needs to be investigated thoroughly. Various studies have considered the subject of channel modelling with its two important aspects of large scale path loss and small scale fading. The statistics of multipath fading channels were first addressed in [4], where the Doppler power spectral density (PSD) of a Rayleigh mobile-to-mobile channel has been derived. The behavior of the Doppler PSD has also been studied in [9] for the case of three-dimensional scattering environments. Path loss measurements for intervehicle communications are reported in [1] and [2]. Besides the modelling of the mobile-to-mobile fading channel and the investigation of its statistical properties, the development of proper channel simulators also plays a major role in the design and performance evaluation of related communication systems. Despite its importance, only recently the topic of simulation of mobile-to-mobile radio channels has started to gain interest. For example, in

2. REVIEW OF MOBILE-TO-MOBILE RADIO FADING CHANNELS By assuming various propagation scenarios, several models have been proposed for the description of the statistics of multipath fading channels in mobile-to-mobile radio transmission links. Akki et al. [4] derived a model for which the complex envelope corresponding to the narrow band fading channel is given by µ(t) = =

µ1 (t) + jµ2 (t) lim

N →∞

N X

cn exp j[2π(fT,n + fR,n )t + θn ](1)

n=1

where N represents the number of distinct propagation paths between the transmitter and the receiver, cn is the amplitude of the signal received from the nth path, fT,n and fR,n are the Doppler frequencies caused by the motion of the transmitter and the receiver, respectively, and θn denotes the random phase uniformly distributed in the interval (0, 2π]. If N → ∞, then we can invoke the central limit theorem to show that µ(t) tends to a wide-sense stationary complex Gaussian process with zero-mean. This implies that the fading amplitude and the channel phase, at any given time instant, are described by the Rayleigh probability density function (PDF) and the uniform PDF, respectively. It has been shown in [4] that in case of isotropic

scattering and N → ∞, the ACF of the process µi (t) (i = 1, 2) is described by the so-called “double-Doppler” ACF Γµi µi (τ ) = σ02 J0 (2πfT,max τ )J0 (2πfR,max τ )

(2)

where σ02 denotes the variance of the real-valued Gaussian random process µi (t) (i = 1, 2), J0 (.) is the zeroth order Bessel function of the first kind, and fT,max and fR,max are the maximum Doppler frequencies caused by the motion of the transmitter and the receiver, respectively. The second model assumes the “double-ring” scattering environment shown in Figure 1 [5]. In this case, the complex envelope of the received fading signal is described by the complex Gaussian process given by µ(t) = lim

M →∞ N →∞

M,N X

cm,n exp j[2π(fT,m + fR,n )t + θm,n ]

where σ12 , σ22 are the mean powers of the two independent Rayleigh processes, and K0 (.) is the zeroth order modified Bessel function of the second kind. The ACF of the so-called “double-Rayleigh” process is also given by the “double-Doppler” ACF described by (2) [4]. The reference models described above are not realizable since the number of scatters are infinite. However, efficient simulation models can be obtained by applying the concept of deterministic channel modelling. This will be the subject of the next section. 3. SIMULATION OF MOBILE-TO-MOBILE RADIO FADING CHANNELS According to the concept of deterministic channel modelling, the reference models described in the previous section can basically be approximated by the following process

m,n=1

(3) where M and N stand for the number of the propagation paths due to the scatterers around the transmitter and the receiver, respectively. The ACF, Γµi µi (τ ), of the Gaussian processes, µi (t), corresponding to this model is also given by (2) [4] when isotropic scattering is assumed.

VR X

Tx

µ ˜(t) = µ ˜1 (t) + j µ ˜2 (t).

(6)

For the model described by (1), µ ˜i (t) is given by µ ˜i (t) =

Ni X

i i )t + θi,n ], + fR,n ci,n cos[2π(fT,n

i = 1, 2

n=1

(7) i i where the parameters ci,n , fT,n , fR,n and θi,n have to be determined such that the statistics of µ ˜i (t) is close to that of the reference model µi (t).

Rx

VTX

3.1. Rayleigh Fading Channel 3.1.1. Single-Sum-of-Sinusoids Model Figure 1. Double-ring scattering model. The third channel model is also obtained by assuming the double-ring scattering model where the distance between the transmitter and the receiver is considered to be large. In this case, the received signal can be described by the product of two complex Gaussian processes according to [7] µ(t) =

lim

M X

M →∞ N →∞

·

N X

Am exp j[2πfT,m t + θT,m ]

m=1

Bn exp j[2πfR,n t + θR,n ]

(4)

For the simulation of this fading channel model, we apply the MSEM [3] to compute the parameter sets {ci,n }, i i {fT,n }, and {fR,n }. These quantities are computed in such a way that the mean square error, EΓµi µi , given by Z τmax 1 ˜ µ µ (τ ))2 dτ (8) EΓµi µi = (Γµi µi (τ ) − Γ i i τmax 0 i becomes minimal [3], where τmax = 2[fT ,maxN+f . R,max ] Here, Γµi µi (τ ) (i = 1, 2) is the ACF of the reference ˜ µ µ (τ ) (i = 1, 2) is the model expressed by (2), while Γ i i ACF of the deterministic process µ ˜i (t) given by [3]

n=1

where Am , θT,m , Bn and θR,n are independently distributed random amplitudes and phases due to local scatterers around the transmitter and the receiver, respectively. This expression is more general than (3), because the signal amplitudes corresponding to the propagation paths are considered as independent random variables and not as a product of independent uniformly distributed random variables. The PDF of the fading amplitude, |µ(t)|, corresponds to that of the product of two independent Rayleigh processes. This PDF is known as the “double-Rayleigh” distribution [7] r r ), r ≥ 0 (5) pR (r) = 2 2 K0 ( σ1 σ2 σ1 σ2

˜ µ µ (τ ) = Γ i i

Ni 2 X ci,n i i cos[2π(fT,n + fR,n )τ ]. 2 n=1

(9)

By applying the MSEM, the discrete frequencies are given by i fT,n =

fR,max fT,max i (2n − 1), fR,n = (2n − 1) (10) 2Ni 2Ni

for i = 1, 2. The formula for the gains ci,n is ½ Z τmax 1 ci,n = 2σ0 J0 (2πfT,max τ ) τmax 0

(11) 1/2

i i ·J0 (2πfR,max τ ) cos[2π(fT,n + fR,n )τ ]dτ }

.

Figure 2 shows the ACF of the simulation model in comparison to that of the reference model for N1 = 25 and N2 = 26. We can see from this figure that the correspondence between the desired ACF and the simulation model’s one is good over the time interval [0, τmax = 0.04223 s]. 1.2 Reference Model Simulation "Single Sum" Model (MSEM) 1

ACF

0.8

0.6

˜ µ µ (τ ) can be exwhere Γµi µi (τ ) is given by (2) and Γ i i pressed as M X N X c2i,m,n i i cos[2π(fT,m + fR,n )τ ]. 2 m=1 n=1 p (13) In (13), the gains ci,n,m are given by ci,n,m = σ0 2/N M . The performance of the simulation model can be studied ˜ µ µ (τ ) and the from Figures 4 and 5, where the ACF Γ i i LCR of |˜ µ(t)| are compared with the corresponding ideal quantities.

˜ µ µ (τ ) = Γ i i

0.4

1

0.2

0.8

Reference Model Simulation "Double Sum" Model

0.6

−0.2 0

0.01

0.02

τ

0.03

0.04

0.05

ACF

0

0.06

0.4

0.2

˜ µ µ (τ ) of the simulation model in comFigure 2. ACF Γ i i parison to the ACF Γµi µi (τ ) of the reference model for N1 = 25, N2 = 26, fT,max = fR,max = 148 Hz, and σ02 = 1.

Besides the ACF, the LCR of the envelope of the simulated fading process |˜ µ(t)| is an important statistical quantity that must be verified. Figure 3 shows an excellent agree-

0

−0.2

−0.4 0

0.005

0.01

0.015

τ

0.02

0.025

0.03

0.035

0.04

Figure 4. ACF of the simulation model as compared with that of the reference model for N1 = 5, M1 = 6, N2 = 6, M2 = 6, fT,max = fR,max = 148 Hz, and σ02 = 1.

250

250

Simulation "Single Sum" Model (MSEM) Reference Model

Simulation " Double Sum" Model ( L2−Norm) Reference Model

200

200

LCR

150

LCR

150

100

100 50

50 0 0

0.5

1

1.5

r

2

2.5

3

0 0

Figure 3. LCR of the simulation model compared to that of the reference model for N1 = 400, N2 = 401, fT,max = fR,max = 148 Hz and σ02 = 1.

ment between the simulated LCR and the reference quantity reported in [3, eq. (3. 28)].

½ i µi

:=

1 τmax

Z 0

τmax

˜ µ µ (τ )|p dτ |Γµi µi (τ ) − Γ i i

1.5

r

2

2.5

3

Figure 5. Comparison of the LCR of the simulation model with that of the reference model for N1 = 7, M1 = 8, N2 = 8, M2 = 8, fT,max = fR,max = 148 Hz, and σ02 = 1.

¾1/p (12)

where Γµµ (Ts ) = 2Γµi µi (Ts ), Ts is the symbol duration, Eb is the average energy per bit, and N0 denotes the noise

In this case the reference model is given by (3). We use the LPNM to compute the parameters of the corresponding simulation model. This method is aiming to minimize the following error function [3] (p)

1

The study of the performance of the channel simulator with respect to the BEP is also of great importance. Here, we consider the data transmission of DPSK signals over a Rayleigh mobile-to-mobile fading channel using noncoherent demodulation. The BEP of noncoherent DPSK signaling is given by [8] Ã ! Γµµ (Ts ) 1 1− (14) Pe = 0 2 1+ N Eb

3.1.2. Double-Sum-of-Sinusoids model

EΓµ

0.5

0.5

Reference Model Simulation Model

0.45

0.4

0.35

0.3

pR(r)

power. The approximate BEP P˜e when using the deterministic channel simulator can be obtained by replacing ˜ µµ (Ts ) = 2Γ ˜ µ µ (Ts ) Γµµ (Ts ), in (14), by the ACF Γ i i ˜ µ µ (Ts ) is described by (13)) , i.e., (here, Γ i i à ! ˜ µµ (Ts ) Γ 1 ˜ 1− . (15) Pe ≈ 0 2 1+ N Eb

0.25

0.2

0.15

0.1

0.05

Figure 6 shows the comparison of the BEP of the Rayleigh “double-sum” simulation model with that of the corresponding theoretical quantity. The computation of the parameters of the channel simulator was performed by using the LPNM. As can be seen from this figure, the simulation quantity and the theoretical one are in good agreement.

0

0

0.5

1

1.5

2

2.5

r

3

3.5

4

4.5

Figure 7. Comparison of the PDF of the simulation model with that of the reference model for N1 = 7, M1 = 8, N2 = 8, M2 = 8, fT,max = fR,max = 148 Hz, and σ02 = 1.

0

10

Simulation Reference Simulation Simulation

−1

Model Model Model Model

( ( ( (

BEP

10

with respect to the ACF, LCR, and BEP. All the statistical properties of the simulation models have been found to match the required theoretical statistical properties sufficiently. Thus, these methods can be used to accurately simulate mobile-to-mobile fading channels.

Ts.(fT,max+fR,max)=0.01) Ts.(fT,max+fR,max)=0.01) T .(f +f )=0.06) s T,max R,max Ts.(fT,max+fR,max)=0.06)

−2

10

−3

10

5. REFERENCES

(N1,M1,N2,M2)=(5,6,6,6) −4

10

0

5

10

15

20

Eb/N0 [dB]

25

30

35

40

Figure 6. BEP P˜b for noncoherent DPSK systems (double-ring) by using the LPNM (σ02 = 1/2). 3.2. Double-Rayleigh Fading Channel Finally, we consider the simulation of the “double-Rayleigh” fading channel described by (4). For this case, we use the MEDS to compute the channel parameters [3]. This method identifies the Doppler coefficients Ai,m and Bi,n with the following quantities p p 1/2 1/2 Ai,m = σ0 2/Mi , Bi,n = σ0 2/Ni (16) and the Doppler frequencies as i fT,m = fT,max sin [

π 1 (m − )], 2Mi 2

(17)

1 π (n − )]. 2Ni 2

(18)

i fR,n = fR,max sin [

The PDF of the “double-Rayleigh” simulation model fits accurately the corresponding theoretical quantity as can be seen in Figure 7. The comparison of the ACFs (not shown here for reasons of brevity) was found to be satisfactory.

[1] T. Tank and J. P. M. G. Linnartz, “Vehicle-to-Vehicle Communications for AVCS Platooning,” IEEE Trans. Veh. Technol, vol. 46, no. 2, pp. 528–536, May 1997. [2] R. Verdone , “Multi-Hop R-ALOHA for InterVehicle Communications at Millimeter Waves,” IEEE Trans. Veh. Technol, vol. 46, no. 4, pp. 992–1005, Nov. 1997. [3] M. P¨atzold, Mobile Fading Channels, Wiley, 2002. [4] A. S. Akki, and F. Haber, “A Statistical Model of Mobile-to-Mobile Land Communication Channel,” IEEE Trans. Veh. Technol, vol. 35, no. 1, pp. 2–7, 1986. [5] C. S. Patel, G. L. St¨uber, and T. G. Pratt, “Simulation of Rayleigh Faded Mobile-to-Mobile Communication Channels,” VTC’03-Fall, vol. 1, pp. 163–167, Oct. 2003. [6] B. O. Hogstad, M. P¨atzold, N. Youssef, and D. Kim, ”A MIMO Mobile-to-Mobile Channel Simulator: Part II- The Simulation Model,” PIMRC’05, Berlin, Germany, Sept. 2005. ISBN 3-8007-2909. [7] I. Z. Kovacs, P. C. F. Eggers, K. Olesen, and L. G. Petersen, “Investigations of Outdoor-to-Indoor Mobileto-Mobile Radio Communication Channels,” VTC’02Fall, pp. 430–434, Sept. 2002.

4. CONCLUSION

[8] W. C. Jakes, Ed., Microwave Mobile Communications, Piscataway, N. J.: IEEE Press, 1994.

In this paper, we have focussed on the simulation of mobileto-mobile fading channels using Rice’s sum-of-sinusoids. The MSEM, LPNM, and the MEDS, originally developed for simulating conventional fading channels, have been applied to simulate various mobile-to-mobile fading models. The performance of the simulators has been investigated

[9] F. Vatalaro and A. Forcella, “Doppler Spectrum in Mobile-to-Mobile Communications in the Presence of Three-Dimentional Multipath Scattering,” IEEE Trans. Veh. Technol, vol. 46, no. 1, pp. 213 –219, Feb. 1997.

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