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AN EFFICIENT DETERMINISTIC SIMULATION MODEL FOR LAND MOBILE SATELLITE CHANNELS Yingchun Li, Matthias P¨atzold, Ulrich Killat, Frank Laue The authors are with the Department of Digital Communication Systems, Technical University of Hamburg-Harburg, Germany new analytical model that is presented in Fig. 1. From Fig. 1, we observe that two independent coloured zero mean real Gaussian noise processes 1(t) and 2(t) are used to produce a complex Gaussian noise process

Abstract — In this paper, a new stochastic model for land mobile satellite channels is presented. From the analytical reference model, an efficient deterministic computer simulation model is derived by replacing all coloured Gaussian noise processes by finite sums of weighted sinusoids with equally distributed phases. A numerical optimization procedure has been applied in order to adapt the statistics of our proposed analytical model to an equivalent real-world land mobile satellite channel for light and heavy shadowing environments. Various theoretical and simulation results show that both models, analytical and simulation, have an excellent performance.

(t) = 1 (t) + j2 (t)

(1)

with cross-correlated inphase and quadrature components 1 (t) and 2(t). The behaviour of the cross-correlation between 1 (t) and 2 (t) can be controlled by a phase coefficient  ( 2 (0; )). A direct line of sight (LOS ) component m = m1 + jm2 = ej (2)

is also taken into consideration in that model. Thereby  and  denote the amplitude and phase of the LOS component, respectively. Note that we only consider the case where m is independent of time, i.e., the Doppler frequency of the LOS component is assumed to be equal to zero. From the non-zero mean complex Gaussian noise process  (t) = (t) + m a further stochastic process  (t) can be obtained by taking the absolute value

I. INTRODUCTION The digital transmission of signals over mobile channels is disturbed by random variations of the envelope and phase of the received signal. In particular for non frequency-selective channels these random variations can be modelled simply by multiplying the transmitted signal with a proper stochastic process. The discovery and description of appropriate stochastic model processes and its adaptation to real-world channels by tuning the model parameters is for a long time a subject of research. In this paper, we will introduce an analytical model for the stochastic process with the characteristic that the underlying inphase and quadrature Gaussian components are cross-correlated. Such a cross-correlation is equivalent to an asymmetrical Doppler power spectral density and is therefore in better agreement with realistic propagation situations. The investigation of the statistical properties, such as the probability density function (PDF ), the level-crossing rate (LCR), and the average duration of fades (ADF ), are also topics of the paper. Furthermore, for our proposed analytical model an efficient simulation model will be presented. Although the simulation model is completely deterministic, its statistical properties are in an excellent conformity with the analytical reference model. Finally, all the parameters of the analytical model and the therefrom derived simulation model are optimized in such a way that the corresponding PDF , LCR, and ADF coincide with measurement results of an equivalent real-world land mobile satellite channel for light and heavy shadowing environments.

(t) = j (t)j =

p

(1 (t) + m1 )2 + (2 (t) + m2 )2

:

(3)

The above stochastic process  (t) with underlying cross-correlated quadrature components is proposed as an appropriate stochastic model for modelling the statistics of large classes of land mobile satellite channels. µ1(t)

ν1(t)

µρ (t) cos α m1= ρ cos θρ ν2(t)

Hilbert Transformer

sin α

Fig. 1

ξ (t)

µ2(t)

m2= ρ sin θρ

Analytical model for the stochastic process

 (t)

A. The Doppler Power Spectral Density Function A widely accepted Doppler power spectral density ( PSD) function for mobile fading channel models is the Jakes PSD [1], which has a symmetrical shape due to the assumption that the incoming directions of the received multipath waves are uniformly distributed in the interval [0; 2). If some of the multipath waves are blocked by obstacles or absorbed by the electromagnetic properties of the physical environment, then the resulting Doppler PSD of the complex Gaussian noise process (t), S (f ), becomes asymmetrical.

II. THE ANALYTICAL MODEL AND ITS STATISTICAL PROPERTIES In this section, we investigate the statistical properties of a 1

Next, we will show how to get an asymmetrical Doppler PSD function S (f ). Therefore, we use for the Doppler PSD of the processes i (t), Si i (f ) (i = 1; 2), the following shapes

(

Si i (f ) =

2fmax

2 p i

1 (f=fmax )2

? 0

; jf j < i fmax ; ; jf j  i fmax ;

B. Probability Density Function, Level-Crossing Rate, and Average Duration of Fades For the derivation of the PDF and the LCR of the stochastic process  (t), we applied the techniques described in [2] and [3]. For the PDF of  (t), p (z ), we found the following result

(4)

 2 + 2 ? 2z cos( ?  ) z  p (z) = 2 sin  exp ?
2 o sin2  o ? n cos  o exp [z2 sin 2 + 2 sin 2 ? 2z sin( +  )] d ; 2 o sin2 

where fmax is the maximum Doppler frequency, i that determines the mean power of the process i (t), 1 = 1, i.e., the PSD S1 1 (f ) is equal to the Jakes PSD, and 2 2 (0; 1], which means that the PSD S2 2 (f ) results in a cut-off Jakes PSD. From Fig. 1, the following expressions can be obtained

1 (t) = 1 (t) + 2 (t) ;

(5a)

2 (t) = [1 (t) + 2 (t)] cos  + [1 (t) ? 2 (t)] sin  ;

(5b)

where z

=

2[r1 1 ( ) + r2 2 ( )] + j 2[r1 1 ( ) ? r2 2 ( )] sin  :

=

2[1 + sgn(f ) sin ]
S1 1 (f ) + 2[1 ? sgn(f ) sin ]
S2 2 (f ) ;

12 2

 + 2 arcsin 2 2



;

(6)

p  Z  2 2 r r +  ? 2r cos( ?  )
exp ? N (r) = (2)3=2 o sin  ? 2 o sin2  p p 1 + cos  sin 2
fexp[?g2 (r;)] + g(r;)[1 + erf (g(r;))]g
o n cos  [r2 sin 2 + 2 sin 2 ? 2r sin( +  )] d ; (10) exp 2 2 o sin 

(7)

which has an asymmetrical shape for  2 (0; ). An example for the asymmetrical Doppler PSD function S (f ) is depicted in Fig. 2, where  = 15o , 12 = 0:04, 22 = 1, and 2 = 0:4 have been chosen.

where

g(r; ) = o f sin( ?  ) ?pcos [r cos 2 ?  cos( +  )]g ; o sin  2 (1 + cos 
sin 2) _

o = ?2(fmax )2



= ? o ? _ 2o = o ; 12 2

S µ µ (f)

-κ 2 ⋅f max f

κ 2 ⋅f max

0

(11a)

(11b)



2 1 + 2 [arcsin 2 ? sin(2 arcsin 2 )]  2

i h p _ o = ?2fmax 12 ? 22 (1 ? 1 ? 22 ) :

-f max

(9)

denotes the mean power of the processes i(t) (i = 1; 2). That PDF p (z ) contains the Rice, Rayleigh, and one-sided Gaussian density as special cases. For example, let  = =2 and  6= 0 ( = 0), then (8) reduces to the Rice (Rayleigh) density; and in the limit  ! 0 and  = 0, one yields the one-sided Gaussian density. The LCR of the process  (t), N (r), is a measure for the average number of crossings per second at which the envelope  (t) crosses a specified signal level r with positive slope. The final result for N (r) is given by

Taking the Fourier transform of the above equation and using the relation Si i (f ) = ?jsgn(f )Si i (f ), one obtains the Doppler PSD S (f ) as function of Si i (f ) in the following form

S (f )

(8)

 0, and o = rii (0) =

where the notations 1(t) and 2(t) denote the corresponding Hilbert transform of the processes 1(t) and 2 (t), respectively. The autocorrelation function 1 of the complex Gaussian noise process (t), r ( ), can be expressed by the autocorrelation functions rii ( ) and the cross-correlation functions ri i ( ) of the processes i(t) and i(t) as follows

r ( )



Z

z

2 is a constant

;

(11c) (11d)

We remark that o represents the curvature of the autocorrelation function r1 1 ( ) and r2 2 ( ) at  = 0, i.e. o = ri i (0); and _ o is the difference between the time derivative of the cross-correlation functions r2 2 ( ) and r1 1 ( ) at  = 0, i.e. _ o = r_2 2 (0) ? r_1 1 (0). The ADF of the process  (t), T? (r), is a measure for the expected value for the length of time intervals over which the signal envelope  (t) is below a specified level r. The ADF T? (r) is in general defined by [1]

f max

→

Fig. 2 An asymmetrical Doppler PSD S (f )

P (r) T? (r) = N?(r) ; 

1 In our paper, the autocorrelation function r ( ) is definded by r ( ) =   E f (t)(t +  )g, where E f
g denotes statistical averaging.

2

(12)

where P? (r) indicates the probability that the process  (t) is found below the level r. P? (r) can be derived from (8) according to

P? (r) =

Zr 0

p (z)dz

   (n ? 1 ) ; n = 1; 2;: : :; N ; fi;n = fmax sin 2N i 0 2 i

 2 + 2 ? 2z cos( ?  ) z  = exp ?
2 o sin2  0 2 o sin  ? n cos  o exp [z2 sin 2 + 2 sin 2 ? 2z sin( +  )] ddz : 2 2 o sin  Zr

z

Z

A detailed introduction into the theory of deterministic simulation systems can be found in [4] and [5]. In this paper, we applied the method of exact Doppler spread (cf. [4]) in order to compute the parameters fi;n ; ci;n, and i;n . According to that method, for the Jakes PSD and the cut-off Jakes PSD the discrete Doppler frequencies fi;n are defined by



Ni e, and the Doppler factors ci;n are given where Ni0 = d 2 arcsin i  by

(13)

r ci;n = i N20 : i

The performance of our proposed model will be demonstrated by adapting the statistical properties of the analytical model to measurement results. But before we discuss this topic in detail in Section IV, we will present in the next section an efficient deterministic simulation model.



A proper deterministic simulation model for the proposed analytical model can be derived from Fig. 1, if we replace the processes 1(t) and 2(t) by the following sums of Ni sinusoids

Ni X n=1

ci;n cos(2fi;n t + i;n ) ; i = 1; 2 ;

(14)

where ci;n , fi;n , and i;n are called Doppler factors, discrete Doppler frequencies, and Doppler phases, respectively. After carrying out some additional network transformations, the structure of the deterministic simulation model as shown in Fig. 3 results. It is important to note that no Hilbert transformer and no Filters for shaping the PSDs are required, and observe also the role of the phase coefficient .

2 ~_

~ = ? ~o ? ~o ; ~o = 1 2

cos(2π f 1,1 t + θ 1,1 )

cos(2 π f 1,N t + θ1,N ) 1

1

m 1 = ρ cos θ ρ

~ = o

~ µ1 (t)

c 2,1

?2

2

( N1 X n=1

cos(2 π f 2,1 t + θ 2,1 )

cos(2 π f 2,Ν t + θ2,Ν ) 2

~_ o = ?

c 2,Ν2

2

~ µρ (t)

~ ξ (t)

cos(2 π f 1,1 t + θ 1,1 − α )

1

c 1,Ν1

1

m 2 = ρ sin θ ρ ~ µ2 (t)

c 2,1 cos(2π f 2,1 t + θ 2,1 + α )

cos(2π f 2,Ν t + θ 2,Ν + α ) 2

( N1 X n=1

c21;n +

(c1;n f1;n

( N1 X n=1

)2

c f

+

2 1;n 1;n

N2 X n=1 N2 X n=1

+

)

c22;n ;

(17b)

) (c2;n f2;n

N2 X n=1

;

)2

(17c)

)

c f

2 2;n 2;n

:

(17d)

2 ~ max Figs. 4 and 5 show us the deviations of ~o and =f 2 , respectively. Thereby, from the ideal quantities o and =fmax we have selected the optimized parameters 1; 2; 2 for an equivalent land mobile satellite channel (see Section IV) for light and heavy shadowing as listed in Table 1. Analytical expressions for the PDF , LCR, and ADF of the deterministic simulation model can also be derived. The resulting expressions have exactly the same form as we have found for the analytical model, we only have to replace the quantities ; o ; o , and _ o ~ ~o ; by the corresponding quantities of the simulation model ; ~ , and ~_ , respectively. Further performance studies of the o o simulation model will be discussed in the next section.

c 1,1

cos(2 π f 1,Ν t + θ1,Ν − α )

(17a)

o

c 1,1

c 1,N1

(16)

The Doppler phases i;n are combined in a vector i = (i;1 ; i;2 ; : : :; i;Ni ) and can be identified with a permutation of the elements of the vector i = (2 N1i ; 2 N2i ;


; 2) for i = 1; 2. Thus, all parameters of the deterministic simulation model are determined, and we have fulfilled the preconditions for an investigation of the degradation effects caused by substituting the stochastic processes i(t) of the analytical model by the deterministic functions ~i(t) of the simulation model. Therefore, we compare the characteristic quantities of the analytical model ; o ; o , and _ o (see (11b), (9), (11c), and (11d), respectively ) with the corresponding characteristic ~ ~o ; ~o , and ~_ o , quantities of the simulation model, ; respectively. The lastnamed quantities can be expressed as function of fi;n; ci;n, and Ni as follows

III. THE DETERMINISTIC SIMULATION MODEL

~i (t) =

(15)

c 2,Ν2

2

Fig. 3 Structure of the deterministic simulation system 3

0.12 0.11

˜o ψo , ψ

0.09

E2 ( ) = f

Simulation Model Analytical Model

0.1

+

0.07

Heavy Shadowing

m=1

(M h X m=1

 i2 ) 2 r r r m m m  W2 (  ) P+ (  ) ? P+ (  ) ; 1

(18)

0.16

two appropriate weighting functions. Observe that we only consider N ( rm ) and P+ ( rm ) = 1 ? P? ( rm ) in the above error function and ignore T? ( rm ), that is because T? ( rm ) is completely defined by P? ( rm ) and N ( rm ) (see therefore (12) ). The minimization of (18) can be performed by applying any elaborated numerical optimization procedure. Our optimized parameters of the analytical model for the equivalent satellite channel are listed in Table 1. The results of the CDF; LCR, and ADF of the analytical model are shown in the Figs. 6-8, respectively. Simulation results are also given, where for light shadowing N1 = N2 = 15 sinusoids and for heavy shadowing N1 = N2 = 7 sinusoids have been selected. For both shadowing situations, the number of samples Ns of the simulated channel output sequence ~(kTa ) (k = 1; 2; : : :; Ns ) was Ns = 3  106 , whereby a sampling interval Ta of Ta = 3  10?4 has been chosen for the evaluation of the statistics.

0.14

Table 1: Optimized parameters of the analytical model

0.03 0.02 5

10

15

20

25

→

Ni

Fig. 4 Comparison of the quantities o (analytical model) and ~o (simulation model) as function of Ni

0.22 Simulation Model Analytical Model

0.2 0.18

~

1 (M h  r i2 ) 2 X r r m m m  W1 (  ) N (  ) ? N (  )

where denotes the parameter vector =(1 ; 2; 2; ; , ), M is the number of measurement values, and W1 (
); W2(
) are

0.06 0.04

β / f 2 max , β / f 2 max

max

Light Shadowing

0.08

0.05

1

Light Shadowing

0.12

Shadowing Light Heavy

Heavy Shadowing 0.1 0.08 0.06 5

10

15 Ni

20

1

2

2





0.1031 0.0893

0.9159 0.7468

0.2624 0.1651

0.3492 0.3988

1.057 0.2626



53:1o 30:3o

In the Figs. 6-8 results from the analytical model and the simulation model almost coincide and can hardly be distinguished in these figures. Fig. 6 shows a plot of the cumulative distribution function P+ ( r ) of the analytical and the simulation model. The measurement results are also shown. For heavy shadowing, the analytical model shows in comparison with the measured values little differences at low signal levels and some deviations at median signal levels but is in good agreement at high levels. While for the case of light shadowing, the results show reasonably good fits at low and high signal levels, but some deviations at median signal levels also exist. For both cases, an extremely good agreement between the analytical and the simulation model was found throughout the whole signal range. Generally, the results of both models, analytical and simulation, indicate a slightly higher fading behaviour. Fig. 7 shows a comparison of the normalized level-crossing rates between the analytical model, simulation model, and measurement results. For light and heavy shadowing environments, the normalized level-crossing rate of the analytical model and the simulation model is throughout the whole signal range in an extremely good agreement with the measurement data. A comparison of the normalized average duration of fades is shown in Fig. 8. Besides small deviations for light shadowing and little differences at low and high signal levels for heavy shadowing, the results of the analytical and simulation model are in fairly good agreement with the measurement data.

25

→

2 Fig. 5 Comparison of the characteristic quantities =fmax (ana2 ~ lytical model) and =fmax (simulation model)

IV. OPTIMIZATION AND SIMULATION RESULTS In this section, we present some applications of the analytical model investigated in this paper. The measurement results of cumulative distribution functions (CDF ) P+ (r), level-crossing rates N (r), and average duration of fades T? (r) are adopted from [6] as object functions for the optimization of the parameters of the analytical model. The measurement environment for this equivalent satellite channel was a rural area with light and heavy shadowing. We remark that the same measurement results have been considered in [7]. This enables a fair and direct performance comparison of the two different models. In order to find optimal values for the parameters of the analytical model (1; 2; 2; ; ; ), which determine the behaviour of the cumulative distribution function P+ (r) = 1 ? P? (r), the level-crossing rate N (r), and the average duration of fades T? (r), we consider the following error function 4

V. CONCLUSION 1

Cumulative Distribution

0.9

In our paper, we have investigated the statistics of a new stochastic process  (t), whose inphase and quadrature components are cross-correlated. The presented results show that the analytical model and the therefrom derived deterministic simulation model can provide a good approximation of the CDF , LCR, and ADF to measured values throughout the whole signal range. The proposed analytical model and the corresponding deterministic simulation model should thus be useful for modelling and simulating of large classes of realistic land mobile satellite channels. The multiplication of the proposed process  (t) with a lognormal process is straightforward. The resulting product process will be called a generalized Suzuki process. Such a generalized Suzuki process contains the classical Suzuki process [8], the modified Suzuki process [9], and the extended Suzuki processes of Type I [2] and Type II [3] as special cases, and offers therefore a greater flexibility.

Light Shadowing

0.8 0.7 0.6

Heavy Shadowing

0.5 0.4 0.3 Measurement [6] Analytical Model Simulation Model

0.2 0.1 0 -30

-25

-20

-15

-10 r/ρ (dB)

-5

0

5

Fig. 6 Cumulative distribution function P+ (r=) for light and heavy shadowing

Normalized Level-Crossing Rate

REFERENCES 10

10

Measurement [6] Analytical Model Simulation Model

0

[1] W.C. Jakes, Ed., Microwave Mobile Communications. New York: IEEE Press, 1993. [2] M. P¨atzold, U. Killat, F. Laue, “An Extended Suzuki Model for Land Mobile Satellite Channels and Its Statistical Properties,” IEEE Trans. Veh. Technol., submitted for publication, 1995. [3] M. P¨atzold, U. Killat, Y. Li, F. Laue, “Modelling, Analysis, and Simulation of Non-Frequency Selective Mobile Radio Channels with Asymmetrical Doppler Power Spectral Density Shapes” IEEE Trans. Veh. Technol., submitted for publication, 1995. [4] M. P¨atzold, U. Killat, F. Laue, Y. Li, “On the Statistical Properties of Deterministic Simulation Models for Mobile Fading Channels,” IEEE Trans. Veh. Technol., submitted for publication, 1995. [5] M. P¨atzold, U. Killat, F. Laue, “A Deterministic Digital Simulation Model for Suzuki Processes with Application to a Shadowed Rayleigh Land Mobile Radio Channel,” IEEE Trans. Veh. Technol., submitted and accepted for publication, 1994. [6] J.S. Butterworth, E.E. Matt, “The Characterization of Propagation Effects for Land Mobile Satellite Services,” Intern. Conf. on Satellite Systems for Mobile Commun. and Navigations, pp. 51-54, June 1983. [7] C. Loo, “A Statistical Model for a Land Mobile Satellite Link,” IEEE Trans. Veh. Technol., vol. VT-34, no. 3, pp. 122-127, Aug. 1985. [8] H. Suzuki, “A Statistical Model for Urban Radio Propagation,” IEEE Trans. Commun., vol. COM-25, no. 7, pp. 673-680, July 1977. [9] A. Krantzik, D. Wolf, “Distribution of the Fading-Intervals of Modified Suzuki Processes,” SIGNAL PROCESSING V: Theories and Applications, L.,Torres, E. Masgrau, and M.A. Lagunas (eds.), Elsevier Science Publishers B.V, pp. 361-364, 1990.

-1

Heavy Shadowing

10

-2

Light Shadowing -3

10 -30

-25

-20

-15

-10 r/ρ (dB)

Normalized Average Duration of Fades

Fig. 7 Normalized level-crossing rate and heavy shadowing

10

10

-5

0

5

N (r=)=fmax for light

2

Measurement [6] Analytical Model Simulation Model 1

Heavy Shadowing 10

0

Light Shadowing -1

10 -30

-25

-20

-15

-10 r/ρ (dB)

-5

0

5

Fig. 8 Normalized average duration of fades T (r=)
fmax for light and heavy shadowing

5