A Study of a Land Mobile Satellite Channel Model with Asymmetrical Doppler Power Spectrum and Lognormally Distributed Line-of-Sight Component

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1 IEEE TRANSACTIONS ON VEHICULAR TECHNOLOGY, VOL. 47, NO. 1, FEBRUARY A Study of a Land Mobile Satellite Channel Model with Asymmetrical Doppler Power Spectrum and Lognormally Distributed Line-of-Sight Component Matthias Pätzold, Member, IEEE, Yingchun Li, and Frank Laue Abstract In satellite to mobile communications, there is generally, in addition to the diffuse multipath component, a strong lineof-sight (LOS) component present. The spectral and statistical properties of both components are influenced by shadowing caused by obstacles such as trees, houses, or small buildings. In this paper, an analytical model for a land mobile satellite channel is analyzed that takes into consideration various kinds of shadowing situations. For the LOS component, it is assumed that, first, the received frequency shows a Doppler shift. Second, the amplitude variations coincide with a lognormal distribution. Moreover, it is also assumed that shadowing influences the diffuse scattered component, with the consequence that its Doppler power spectrum gets an asymmetrical shape. The resulting spectral and statistical properties of the analytical model are investigated. Exact solutions for the probability density function (pdf) of the amplitude and phase are given, and approximative solutions for the level-crossing rate (LCR) and average duration of fades (ADF s) are derived. Finally, an efficient deterministic simulation model is presented that enables the implementation of the proposed analytical model on a digital computer. Index Terms Fading channel modeling, fading channel simulation, land mobile satellite channel, nonfrequency-selective fading, statistics. I. INTRODUCTION THE TRANSMISSION performance of land mobile satellite services is mainly impaired by rapid amplitude and phase fluctuations of the received signal. Such fluctuations are caused by multipath wave propagation as well as by timevarying attenuation due to shadowing. The statistical properties of these signal variations are closely related to the propagation environment in which the vehicle is located. Many stochastical models with different degrees of sophistication have been developed for modeling the time-varying behavior of the received signal. The famous representative of stochastical models for land mobile (satellite) channels are the classical Rice and Rayleigh processes, where the former process takes into account the presence of a direct signal path and the latter not. On the basis of experimental measurements, it has been shown that the usefulness of these processes is restricted to modeling only different aspects of short-term fading variations. Manuscript received February 27, 1996; revised October 16, The authors are with the Department of Digital Communication Systems, Technical University of Hamburg-Harburg, D Hamburg, Germany ( paetzold@tu-harburg.d400.de). Publisher Item Identifier S (98) Superimposed on the rapid short-term fading variations are slow variations in the local mean of the received signal. These long-term variations, usually referred to as shadow effects, are due to trees, small buildings, and other roadside obstructions. A stationary stochastic model, which takes into account shortterm and long-term variations, is the so-called Suzuki process [1], [2]. This process is obtained by multiplying a Rayleigh process with a lognormal process. Thereby, it is assumed that the inphase and quadrature components generating the Rayleigh part are uncorrelated. But this assumption of statistical independence is often not in agreement with real-world situations in multipath wave propagation. Therefore, modified Suzuki processes have been introduced [3], [4], where the components generating the Rayleigh process are allowed to be cross correlated. Although the Suzuki process and its modified version have originally been proposed as appropriate models for the cellular land mobile channel, they are also suitable for satellite land mobile channels for urban areas with almost complete obstruction of the direct path. But for suburban and rural (open) areas with partial (no) obstruction of the direct wave, further extensions are necessary. A statistical channel model has been proposed in [5] that is suitable for all types of environments (urban, suburban, rural, and open). This model is based on a product process of a Rice and lognormal process, i.e., both components (direct and diffuse) are affected by shadowing. The flexibility according to the higher order statistical properties [level-crossing rate (LCR) and average duration of fades (ADF s)] of this model can considerably be increased if the inphase and quadrature components describing the Rice process are allowed to be cross correlated. Depending on the type of cross correlation, the so-called extended Suzuki process of Types I and II have been introduced in [6] and [7], respectively. Moreover, generalized Suzuki processes have been proposed [8]. A generalized Suzuki process contains the classical Suzuki process [1], [2], the modified Suzuki process [3], [4], and the two types of extended Suzuki processes [6], [7] as a special case and offers therefore a greater flexibility and thus enables a better adaptation to measurement data. Another statistical model was proposed by Loo [9], [10]. Loo s model is suitable for a land mobile satellite channel in a rural environment, where for most of the time, a direct component is available at the receiver. This model assumes that the multipath component is Rayleigh distributed with constant power and that the direct component follows a /98$ IEEE

2 298 IEEE TRANSACTIONS ON VEHICULAR TECHNOLOGY, VOL. 47, NO. 1, FEBRUARY 1998 Fig. 1. Analytical model for a mobile channel with underlying cross-correlated inphase and quadrature components and time-varying LOS component, where H i (f) = S (f), i =1;2. lognormal distribution to account for the effect of shadowing due to foliage. All the above statistical channel models have in common that they are stationary, i.e., they are based on stationary stochastic processes with fixed parameters. A nonstationary model for very large areas has been introduced by Lutz et al. [11]. This model is based on a two-state Markov model, where the fading process is switched between a Rice process (good channel state) and a Rayleigh-lognormal process (bad channel state). A generalization of this procedure can easily be obtained and results in an -state Markov model [12], where each state represents a specific channel model. Thus, a nonstationary channel model can be approximated by stationary channel models [13], [14]. From experimental measurements, it turned out that it would be enough for most channels to use a fourstate Markov model [15]. To reproduce the fluctuations of the received signal, we propose in this paper a channel model, which is similar to Loo s [9], [10]. Our model differs from his in two important aspects. First, we have attempted to make the model correspond more closely to the physical channel phenomena involved by allowing the inphase and quadrature components defining the complex diffuse component to be cross correlated. Note that such a cross correlation corresponds to an asymmetrical Doppler power spectral density shape. Second, the frequency of the direct component is assumed to be Doppler shifted due to the movement of the vehicle. These assumptions greatly increase the flexibility of the LCR and the ADF s and thus enable a better statistical adaption to experimental measurement recordings. The resulting spectral and statistical properties of the received signal will be investigated. The organization of the present paper is as follows. Section II begins with a general description of the analytical model that is proposed for a land mobile satellite channel with a lognormally distributed line-of-sight (LOS) component and asymmetrical Doppler power spectral density characteristics. It follows a discussion of the resulting spectral properties. For that purpose, all relevant Doppler power spectral density functions and the corresponding autocorrelation and crosscorrelation functions are derived. Afterwards, the statistical properties are analyzed. Thereby, the stress is on the investigation of the probability density function (pdf) of the amplitude and phase, LCR, and ADF s. In Section III, we propose for the analytical model an efficient simulation model. The excellent conformity of both models is demonstrated by showing various computer simulation results. Moreover, we demonstrate in Section IV the usefulness of the proposed model by adapting the higher order statistics (LCR) of the analytical model to measurement results of various types of land mobile satellite channels. Finally, Section V concludes the paper with a summary of the main results. II. THE ANALYTICAL CHANNEL MODEL AND ITS STATISTICAL PROPERTIES A new multipath model for land mobile satellite channels that takes into consideration the effects produced by a timevarying LOS path is presented in Fig. 1. The underlying Doppler power spectral density function and the statistical properties of that model will be investigated in this section. Let us begin with a general description of the proposed model. Throughout the paper, we will make use of the complex baseband representation of passband signals. From Fig. 1, we observe that a single colored zero-mean real Gaussian noise process is used to produce a complex Gaussian noise process with cross-correlated inphase and quadrature components and. In our analytical model, a time-varying LOS component is included, which has the following form: (1) (2)

3 PÄTZOLD et al.: STUDY OF LAND MOBILE SATELLITE CHANNEL MODEL 299 where and denote the Doppler frequency and phase of the LOS component, respectively. The time variations of the amplitude of the LOS component is a result of shadowing effects due to foliage, cars, buildings, etc. Measurements have revealed that the amplitude follows usually a lognormal distribution [16]. Such a lognormal process is derived here from a second real Gaussian noise process with zero mean and unit variance according to where and are two quantities, which are strongly dependent on the type of shadowing environment. In comparison with the Doppler power spectrum of the process, it is assumed that the power spectrum of is confined to a relatively narrow band limited region. Thus, the stochastic process is a relatively slowly varying function of time. Furthermore, we assume that the two Gaussian noise processes and are statistically independent, and then it follows immediately that the lognormal process is uncorrelated with the complex Gaussian process. The addition of (1) and (2) gives a complex Gaussian noise process with a slowly time-varying local mean,, from which a further stochastic process can be obtained by taking the absolute value of, i.e., The above-introduced process with underlying crosscorrelated inphase and quadrature components is proposed as an appropriate stochastic model for modeling the statistics of the received signal in the complex baseband of large classes of land mobile satellite channels. For the special case, where is independent of time, i.e.,, our proposed model for the stochastic process reduces to the Rice process with underlying cross-correlated inphase and quadrature components that were recently investigated in [7]. A. The Doppler PSD Function A multipath propagation scenario contains several different paths by which waves travel from the (static) transmitter to the (mobile) receiver. The superposition of all waves arriving the receiver s (omnidirectional) antenna results in a multipath component with environment specific Doppler power spectral density characteristics. One often takes the hypothesis for the mobile fading channel that all the incoming directions of the received waves are equally distributed in the interval 0 2, which results in a symmetrical Doppler power spectral density (PSD) function. A widely accepted Doppler PSD function for mobile fading channel models is the Jakes PSD [17], [18], as shown in Fig. 2(a) (see dashed line). But the equal distribution hypothesis usually does not tally with real fading environments as some of the multipath signals are blocked by obstacles or absorbed by the electromagnetic properties of the physical environment, and then the resulting Doppler PSD of the complex Gaussian noise process,, becomes asymmetrical. In addition, the shape of the Doppler (3) (4) PSD is strongly influenced by the antenna pattern. For example, if the mobile receiver uses a beam antenna directed along vehicle motion, then gets also an asymmetrical shape [18], [28]. How to get for the received multipath signal component a Doppler PSD with asymmetrical shape by taking also into account a slowly time-varying LOS component is the topic of this section. Therefore, we use for the Doppler PSD of the process the following shape [18], [28]: where is the maximum Doppler frequency, is a constant that determines the mean power of the process, and is a real parameter in the interval (0,1]. If, then the classical Jakes Doppler PSD is obtained, and if 0 1, a so-called restricted Jakes Doppler PSD results as is shown in Fig. 2(a) (see solid line). In [6], it has been demonstrated that the classical Jakes Doppler PSD often results in an analytical channel model with a fading rate (LCR) that is much higher than the corresponding measurement results. It should also be noted that the restricted Jakes PSD (5) is rather a heuristical assumption than a theoretical result for the Doppler PSD of land mobile satellite channels, but nevertheless (5) allows an optional reduction of the resulting fading rate simply by reducing the parameter. As we know, the inverse Fourier transform of (5) gives the autocorrelation function of the Gaussian process, which is for the classical Jakes Doppler PSD ( ) given by, but for the restricted Jakes Doppler PSD (0 1), no closed-form expression for exists. In this case, the autocorrelation function is obtained via solving the following integral: by applying numerical integration techniques. The Doppler PSD of the second Gaussian process,, is assumed to be a Gaussian function where is related to the 3-dB cutoff frequency according to. Generally, the cutoff frequency is much smaller than the maximum Doppler frequency, i.e.,. In the following, it will be advantageous to introduce a quantity in order to express by. The autocorrelation function of the process is obtained by taking the inverse Fourier transform of (7). Hence results. Next, let us consider the lognormal process [see (3)]. For that process, it will be advisable to express the corresponding autocorrelation function as function of. (5) (6) (7) (8)

4 300 IEEE TRANSACTIONS ON VEHICULAR TECHNOLOGY, VOL. 47, NO. 1, FEBRUARY 1998 (a) (b) (c) (d) (e) Fig. 2. Diverse Doppler PSD functions: (a) the restricted Jakes PSD S (f ), (b) an asymmetrical PSD S (f ), (c) the Gaussian PSD S (f ), (d) the PSD S (f ) of the lognormal process (t), and (e) the resulting asymmetrical PSD S (f ). Therefore, we write where (9) (10)

5 PÄTZOLD et al.: STUDY OF LAND MOBILE SATELLITE CHANNEL MODEL 301 is the joint pdf of the process at two different times and. After substituting (10) into (9) and solving the double integrals, the autocorrelation function can be expressed in a closed form as follows: (11) From the above equation, it should be observed that the power of the process is given by. The PSD function of the process can be obtained by solving the Fourier integral, i.e., (12) The preceding equation shows us that the Doppler PSD of the lognormal process is composed of a weighted delta function located at and an infinity sum of monotonically decreasing versions of. Note that is obtained from (7) simply after substituting the quantity by. The Doppler PSD s and are shown in Fig. 2(c) and (d), respectively. We proceed by considering the structure of the analytical model as depicted in Fig. 1. From that figure, the following expressions are immediately readable: (13a) (13b) where the notation denotes the corresponding Hilbert transform of the process. A simple computation allows us to express the autocorrelation function ( ) of the process and the cross-correlation function of the processes and as follows: (14a) of the complex process can now be expressed by making use of (14) as follows: (15a) (15b) respectively. Finally, by using the relation sgn, the Doppler PSD function can be written by sgn (16) and thus, it is shown that has an asymmetrical shape. An example of the resulting asymmetrical Doppler PSD function is depicted in Fig. 2(e), where,,,,, and have been chosen. B. Probability Density Function of Amplitude and Phase In this section, we investigate the pdf of the amplitude and phase of the complex process (see Fig. 1). The pdf of the stochastic process, denoted here by, can be derived by applying the methods described in [6]. In [6], the statistics of a process have been investigated, thereby, the inphase and quadrature components of the complex process were also correlated, but for the LOS component, a more simpler expression of the form with const. was assumed. Note that the process is a special case of our proposed process under the restriction that yields. The pdf of such a process with underlying cross-correlated inphase and quadrature components is equal to the conditional pdf and is given by the Rice distribution [25] where (17) (18) is the mean power of the process (c.f. [7]). In our case, the amplitude of the LOS component is lognormally distributed, i.e., the corresponding pdf is given by [19] (19) Hence, the pdf of,, can be derived from the joint pdf of the processes and in the following way: (14b) where and denote the autocorrelation functions of the processes and as given by (6) and (11), respectively, and is the cross-correlation function of the processes and. The autocorrelation function and the corresponding Doppler PSD function (20)

6 302 IEEE TRANSACTIONS ON VEHICULAR TECHNOLOGY, VOL. 47, NO. 1, FEBRUARY 1998 (a) Fig. 3. The pdf p (z) as function of (a) o ( o =1,s=1, and m = 01) and (b) s ( o =1, o =0:5, and m = 0s 2 ). (b) The preceding equation shows us that the pdf of the proposed process depends on three parameters (, and ), whereas the pdf of a Rice process depends on two parameters ( and ). For the special case and, the pdf tends to and, thus, the above pdf reduces to the Rice density as given by (17). Consequently, the derived pdf is a generalization of the Rice density and is, therefore, more flexible than the latter one. It should also be noted that Loo s model [9], [10] and the proposed model have identical expressions for the probability density of the amplitude, but both models have different higher order statistical properties (LCR and ADF s), as will be shown in Section II-C. The influence of the parameters [see (18)] and on the pdf is shown in Fig. 3(a) and (b), respectively. Next, we consider the pdf that describes to us the distribution of the phase of the complex process. This phase pdf can be derived in a similar way as we have done for the computation of the amplitude pdf. Thereby, we make use of the fact that for const., the phase pdf can be expressed by [6] (21) where. Obviously, the above phase pdf is a function of time if the Doppler frequency of the direct component is unequal to zero. After averaging (21) over time, we obtain for a uniform conditional phase pdf (22) Now, the desired pdf of the phase process can be obtained from the joint pdf and as follows: of (23) i.e., the phase is uniformly distributed between 0 2 if. C. LCR and ADF s The performance of mobile communication systems deteriorates rapidly when the received signal falls below some noise-related threshold. The consequence of such situations is that error bursts occur. The LCR and the ADF s are two important measures for the rate of occurrence and the average length of these error bursts, respectively. Especially, the analysis of the ADF s is of particular interest in the connection with an optimal design of the interleaver and forward-error correcting codes. In this section, we analyze the LCR and ADF s of the received signal envelope of the proposed land mobile satellite channel model. The LCR of the process,, is the average number of crossings per second at which crosses a specified signal level with a positive slope. For a wide-sense stationary random process, the LCR is generally defined by [20] (24) where denotes the joint pdf of the process and its corresponding time derivative at the same time. That joint pdf can be derived by taking into account that the amplitude of the LOS component is a slowly timechanging process, i.e.,, then it follows from further investigations that the joint pdf of and its

7 PÄTZOLD et al.: STUDY OF LAND MOBILE SATELLITE CHANNEL MODEL 303 time derivative is approximately given by. Hence, it follows: in time in comparison with the diffuse components. Note that such relations are completely in conformity with real-world multipath propagation situations, where often 1 yields. The ADF s of the process,, is the mean value of the time intervals over which the process is below a specified level. Generally, the ADF s is related to the LCR as follows [17]: (30) (25) where process is the cumulative distribution function of the, which can be obtained by solving the integral can ap- and the LCR After substituting (25) into (24), the LCR proximately be expressed by the pdf as follows: (31) (26) An expression for the LCR has been derived in [6], where the following result can be found: The influence of the parameters,, and on the LCR and ADF s can be studied in Figs. 4 and 5, respectively. For a fair investigation of the influence of the parameter, we identified in Figs. 4(b) and 5(b) the parameter with, which results for all values of in a lognormal process with unit power, i.e.,. III. COMPUTER SIMULATIONS where sinc (27) (28a) (28b) (28c) (28d) Two fundamental methods are generally used for the realization of stochastic processes: the filter method and Rice s sum of sinusoids. By applying the former method, a simulation model is obtained by replacing each ideal filter of the analytical model by a digital filter with an appropriate filter degree. In this paper, we have applied the latter method by means of which the realization of the two low-pass filters as well as of the Hilbert transformer has been circumvented. According to that procedure, an efficient simulation model, which corresponds to the analytical model of Fig. 1, can be derived by approximating the processes and by the following sum of sinusoids Finally, by substituting (27) into (26) and using (19), an approximation of the LCR can be given as follows: (29) In Section III, we will show that such an approximation of is fairly close to the exact LCR of the process if the quantity, i.e., if the amplitude of the LOS component changes relatively slowly (32) where,, and are called Doppler coefficients, 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. 6 results. For shortness, we will not recapitulate the concept of deterministic channel modeling. But for the interested reader, we refer to [21] [24], where a detailed introduction into the theory of deterministic simulation systems can be found. In this paper, we applied the method of exact Doppler spread (c.f. [23] and [24]) to compute the parameters,, and. The application of that method to the restricted Jakes

8 304 IEEE TRANSACTIONS ON VEHICULAR TECHNOLOGY, VOL. 47, NO. 1, FEBRUARY 1998 (a) (b) (c) Fig. 4. The normalized LCR N(r)=f max as function of (a) o ( o =1,s=1,m=01, and f =0:1f max ), (b) s ( o =1, o =0:5,m= 0s 2, and f =0:1f max), and (c) f ( o =0:2, o =1,s=0:5, and m = 00:25). PSD [see (5)] results in the following closed-form expression for the discrete Doppler frequencies : (33) where. For the Gaussian PSD [see (7)], the discrete Doppler frequencies are obtained by finding the zeros of The corresponding Doppler coefficients for. (34) are given by (restricted Jakes PSD) (Gaussian PSD) (35) Finally, the Doppler phases are combined to a vector, where the elements of this vector are identified with a permutation of the elements of the vector for. Thus, all parameters,, and, which define the behavior of (32), are determined by using deterministic methods, and now the simulation procedure can be performed. A plot of the deterministic output signal of the resulting simulation model is shown in Fig. 7 (solid line), where the maximum Doppler frequency is equal to Hz, and for the numbers of sinusoids and, the values have been selected. The resulting deterministic lognormal process is also plotted in Fig. 7 (dotted line). A comparison of the statistics (pdf, LCR, and ADF s) of the analytical model with the simulation model is shown in Figs For the simulation procedure, again sinusoids and a maximum Doppler frequency of Hz have been selected. The sampling interval of the discrete deterministic process was. Altogether, 7 samples of

9 PÄTZOLD et al.: STUDY OF LAND MOBILE SATELLITE CHANNEL MODEL 305 (a) (b) (c) Fig. 5. The normalized ADF s T (r) 1 f max as function of (a) o ( o =1,s=1,m= 01, and f =0:1f max ), (b) s ( o =1, o =0:5,m = 0s 2, and f =0:1f max), and (c) f ( o =0:2, o =1,s=0:5, and m = 00:25). the process have been simulated and evaluated for estimating the pdf, normalized LCR, and normalized ADF s of the simulation system. Fig. 8 reveals that the behavior of the pdf is not influenced by the value of. This result is not surprising because [see (20)] is independent of the parameter. We remark that the observed difference between and is due to the limited selected number of sinusoids. The difference gets smaller the higher the number of sinusoids is chosen. The results of Fig. 9 show us that for small values of ( ), there is a relatively large deviation between the LCR of the analytical model and the simulation model. But already for moderate values of, e.g.,, the LCR of the analytical model is, besides little differences, in a good agreement with the LCR of the simulation model as it was expected by the theory. Fig. 10 allows similar interpretations for the ADF s. IV. APPLICATIONS In this section, we demonstrate the usefulness of the analytical model and the corresponding simulation model by adapting the statistics of the synthetic channel output to measured channel data. Let us therefore consider measurement results of the LCR of an equivalent satellite channel [26] as well as of a real-world satellite channel [27]. In the following, the measured normalized LCR, denoted here as, is used as object function for the optimization of the normalized LCR of the analytical model. Our task is now to find proper values for all those parameters, which influence the behavior of so that the absolute value of the difference between and is sufficiently small. Therefore, we combine these quantities to a parameter vector, and then we minimize the following error norm: (36) where denotes the number of measurement values and represents a weighting function, which is in our case simply defined by the reciprocal of. The minimization of the

10 306 IEEE TRANSACTIONS ON VEHICULAR TECHNOLOGY, VOL. 47, NO. 1, FEBRUARY 1998 Fig. 6. Structure of the deterministic simulation system. Fig. 7. The deterministic process ~(t) ( o =1, o =0:5,s=0:5,m= 00:25, f =0:2f max, c =50, and f max =91Hz). error norm can be performed by applying any elaborate numerical optimization procedure. A. Applications to an Equivalent Land Mobile Satellite Channel The experimental measurement results considered here can be found in [26]. There, a 870-MHz transmitter package was mounted on a helicopter at a fixed elevation angle of 15 with respect to the receiver in order to simulate the effects of a satellite signal source. The test environment was a rural area with about 35% tree cover, and the rest was cleared land. In Fig. 8. Comparison of ~p (z) with p (z) ( o = 0:86, o = 0:28, s = 0:19, m =0:067, and f = 0:2f max ). one situation, the test route was heavily shadowed by trees, thereby, the deciduous trees were in leaf and lightly shadowed in the other one. Fig. 11 shows the measured normalized LCR for the two different situations. Due to the high flexibility of, it is for the present application not necessary to include all elements of the parameter vector in the design. For example, the parameter can be fixed equal to without any appreciable performance loss. The other elements of have been optimized by minimizing (36) numerically, and the obtained results are shown in Table I.

11 PÄTZOLD et al.: STUDY OF LAND MOBILE SATELLITE CHANNEL MODEL 307 TABLE I PARAMETERS OF THE ANALYTICAL MODEL FOR RURAL AREAS WITH LIGHT AND HEAVY SHADOWING Fig. 9. Comparison of ~ N(r)=f max with N (r)=f max ( o =0:86, o = 0:28, s =0:19, m =0:067, and f =0:2f max ). Fig. 10. Comparison of ~ T (r) 1 f max with T (r) 1 f max ( o =0:86, o =0:28, s =0:19, m =0:067, and f =0:2f max). Fig. 11. Normalized LCR ~ N (r)=f max for environments with light and heavy shadowing. For the remaining parameters of the analytical model (,, and ), which have no influence on the behavior of the normalized LCR, we have chosen the values Hz,, and. Thus, all parameters of the analytical model are defined, and the corresponding simulation model can be derived according to the procedure described in Section III. For the number of sinusoids, we propose for the present application to use ( ) harmonic functions. The resulting normalized LCR of the analytical model and simulation model are also shown in Fig. 11. This figure shows us that the results of the analytical and simulation model are in fairly good agreement with the measurement data for both situations (light and heavy shadowing). For completeness, the output signal (envelope) of the simulation model is presented for light shadowing in Fig. 12(a) and for heavy shadowing in (b). The Rice factor, which is defined as the ratio of the power of the LOS component to the total power of the diffuse component, is given here by. Given the parameters listed in Table I and using the expressions (11) and (18), we find for light shadowing db and for heavy shadowing db. B. Applications to a Real-World Land Mobile Satellite Channel Experimental measurement results of a real-world land mobile satellite channel have been obtained by using INMARSAT s MSAT-A satellite in the L band [27]. Fig. 13 shows two measurement results of the normalized LCR. The first curve shown was obtained from a suburban area, where the streets were bordered by one- and two-story homes and the second one from a nonforested rural area, indicated as farmland. As in the previous example, the LCR is used as an object function for the optimization of the parameters of the analytical model by minimizing the error norm (36). On further consideration of the measurement results (see Fig. 13), we can realize that each curve can be interpreted as a composition of two concave functions. Our hypothesis is that such a property is typical for nonstationary channels. Note that the previous curves for, shown in Fig. 11, cannot be interpreted in this way and, thus, the corresponding satellite channels are stationary. Note also that all parameters of the proposed analytical and corresponding simulation model are fixed, i.e., both models are stationary. Therefore, an extension of the proposed model in order to enable the modeling of nonstationary land mobile satellite channels is required. Such an extension can easily be performed when we follow the ideas described in [11]. In that paper, a two-state model is described, where one state represents a Rice process under good propagation conditions and the other state represents a Rayleigh-lognormal process under bad propagation conditions. Here, we also use a two-state model, but each state denoted

12 308 IEEE TRANSACTIONS ON VEHICULAR TECHNOLOGY, VOL. 47, NO. 1, FEBRUARY 1998 Fig. 12. (a) Simulation of the envelope ~(t) for (a) light shadowing and (b) heavy shadowing. (b) by and represents the proposed analytical model with different parameter sets ( ). Now, let the parameter sets are such that the corresponding stochastic processes and are statistically independent, then, it follows that the pdf of the resulting received amplitude can be expressed by (37) where denotes the a priori probability that the channel is in state and is defined by (20) with parameters,, and for ( ). Note that yields and, thus, is now defined by seven independent parameters. For the LCR of the two-state model we can write Fig. 13. areas. Normalized LCR ~ N (r)=f max for suburban and rural (farmland) (38) where denotes the LCR as introduced by (29) if the channel is in state ( ). Appropriate parameters for the analytical two-state model can now be obtained by substituting the expressions (38) in (36) and minimizing the error norm by using the measured LCR (see Fig. 13) as an object function. The execution of the numerical minimization of the error norm (36) is, in this case, a critical point and requires some practical knowledge. The results of our optimized parameters are shown in Table II. Just as in the previous section, we have fixed equal to zero. Due to the fact that the normalized LCR is independent of the parameters,, and, we can select these quantities within reasonable bounds. Here, we have chosen the values Hz,, and ( ). Now, we have determined all parameters of the analytical two-state model and corresponding parameters of the simulation model can be derived by applying the methods TABLE II PARAMETERS OF THE ANALYTICAL TWO-STATE MODEL FOR SUBURBAN AND RURAL (FARMLAND) AREAS described in Section III. For each state, we propose to use ( ) harmonic functions. The resulting normalized LCR s of the analytical two-state model as well as of the corresponding simulation model are also presented in Fig. 13. This figure shows us that for low- and high-signal levels, the measurement results for both areas (suburban and rural) are in excellent agreement with the analytical and simulation results, whereas for medium signal levels, a sufficient agreement can be observed. Finally, the envelope of

13 PÄTZOLD et al.: STUDY OF LAND MOBILE SATELLITE CHANNEL MODEL 309 Fig. 14. (a) Simulation of the envelope ~(t) for (a) suburban and (b) rural (farmland) areas. (b) the two-state simulation model is shown in Fig. 14(a) and (b) for the suburban and rural (farmland) area, respectively. V. CONCLUSION In this paper, an analytical model for nonfrequency-selective land mobile satellite channels with asymmetrical Doppler power spectral density shapes and lognormally distributed LOS components was presented. The model takes into account different kinds of signal shadowing, caused by obstacles such as trees, houses, or small buildings. The spectral and statistical properties of the proposed analytical model have been analyzed. Especially for the pdf of the phase and amplitude, exact solutions could be derived. It turned out that the phase is uniform between 0 2 if the Doppler frequency of the LOS component is unequal to zero. The pdf of the amplitude includes the Rice (Rayleigh) density as special cases, where the LOS component is constant (zero) and is thus more flexible than the latter one. For the LCR and ADF s, approximative solutions have been derived. On the assumption that the bandwidth of the LOS component is significantly lower than the bandwidth of the diffuse component, the presented approximative solutions are fairly close to the corresponding exact solutions. It should be noted that such an assumption generally complies with real-world mobile propagation situations. Finally, for the (stochastic) analytical model, an efficient (deterministic) simulation model has been proposed that avoids the design of any digital filters. Instead of this, the simulation model is derived by employing sums of harmonic functions, whose parameters are determined by simple and closed expressions. By various simulation results, the excellent conformity of the statistical properties of the analytical model with the therefrom-derived simulation model has been demonstrated. The usefulness of the proposed model has been demonstrated by fitting the higher order statistics (LCR) of the analytical model to measurement results of four different categories of measurement data. REFERENCES [1] H. Suzuki, A statistical model for urban radio propagation, IEEE Trans. Commun., vol. COM-25, no. 7, pp , [2] F. Hansen and F. I. Meno, Mobile fading Rayleigh and lognormal superimposed, IEEE Trans. Veh. Technol., vol. VT-26, no. 4, pp , [3] A. Krantzik and D. Wolf, Distribution of the fading-intervals of modified Suzuki processes, in Signal Processing V: Theories and Applications, L. Torres, E. Masgrau, and M. A. Lagunas, Eds. Amsterdam, The Netherlands: Elsevier, 1990, pp [4], Statistical properties of fading processes describing a land mobile radio channel, (in German), Frequenz, vol. 44, no. 6, pp , June [5] G. E. Corazza and F. Vatalaro, A statistical model for land mobile satellite channels and its application to nongeostationary orbit systems, IEEE Trans. Veh. Technol., vol. VT-43, no. 3, pp , [6] M. Pätzold, U. Killat, and F. Laue, An extended Suzuki model for land mobile satellite channels and its statistical properties, IEEE Trans. Veh. Technol., to be published. [7] M. Pätzold, U. Killat, Y. Li, and F. Laue, Modeling, analysis, and simulation of nonfrequency-selective mobile radio channels with asymmetrical Doppler power spectral density shapes, IEEE Trans. Veh. Technol., vol. 46, no. 2, pp , [8] Y. Li, M. Pätzold, U. Killat, and F. Laue, An efficient deterministic simulation model for land mobile satellite channels, in Proc. 46th IEEE Veh. Technol. Conf., Atlanta, GA, Apr./May 1996, pp [9] C. Loo, A statistical model for a land mobile satellite link, IEEE Trans. Veh. Technol., vol. VT-34, no. 3, pp , [10] C. Loo and N. Secord, Computer models for fading channels with applications to digital transmission, IEEE Trans. Veh. Technol., vol. 40, no. 4, pp , [11] E. Lutz, D. Cygan, M. Dippold, F. Dolainsky, and W. Papke, The land mobile satellite communication channel Recording, statistics, and channel model, IEEE Trans. Veh. Technol., vol. 40, no. 2, pp , [12] R. H. McCullough, The binary regenerative channel, Bell Syst. Tech. J., vol. 47, pp , Oct [13] B. Vucetic and J. Du, Channel modeling and simulation in satellite mobile communication systems, IEEE J. Select. Areas Commun., vol. 10, no. 8, pp , [14] M. J. Miller, B. Vucetic, and L. Berry, Eds., Satellite Communications: Mobile and Fixed Services, 3rd ed. Boston, MA: Kluwer, [15] B. Vucetic and J. Du, Channel modeling and simulation in satellite mobile communication systems, in Proc. Int. Conf. Satel. Mobile Commun., Adelaide, Australia, Aug. 1990, pp [16] W. J. Vogel and E. K. Smith, Theory and measurements of propagation for satellite to land mobile communication at UHF, in Proc. IEEE 35th Veh. Technol. Conf., Boulder, CO, 1985, pp [17] W. C. Jakes, Ed., Microwave Mobile Communications. New York: IEEE Press, 1993.

14 310 IEEE TRANSACTIONS ON VEHICULAR TECHNOLOGY, VOL. 47, NO. 1, FEBRUARY 1998 [18] R. H. Clarke, A statistical theory of mobile-radio reception, Bell Syst. Tech. J., vol. 47, pp , July/Aug [19] A. Papoulis, Probability, Random Variables, and Stochastic Processes, 3rd ed. New York: McGraw-Hill, [20] S. O. Rice, Mathematical analysis of random noise, Bell Syst. Tech. J., vol. 23, pp , July 1944 and vol. 24, pp , Jan [21] M. Pätzold, U. Killat, and F. Laue, A deterministic digital simulation model for Suzuki processes with application to a shadowed Rayleigh land mobile radio channel, IEEE Trans. Veh. Technol., vol. 45, no. 2, pp , May [22] M. Pätzold, U. Killat, Y. Shi, and F. Laue, A deterministic method for the derivation of a discrete WSSUS multipath fading channel model, European Transactions on Telecommunications (ETT), vol. ETT-7, no. 2, pp , Mar./Apr [23] M. Pätzold, U. Killat, F. Laue, and Y. Li, On the statistical properties of deterministic simulation models for mobile fading channels, IEEE Trans. Veh. Technol., vol. 47, no. 1, pp , [24], A new and optimal method for the derivation of deterministic simulation models for mobile radio channels, in Proc. IEEE 46th Trans. Veh. Technol. Conf., Atlanta, GA, Apr./May 1996, pp [25] S. O. Rice, Statistical properties of a sine wave plus random noise, Bell Syst. Tech. J., vol. 27, no. 1, pp , [26] J. S. Butterworth and E. E. Matt, The characterization of propagation effects for land mobile satellite services, in Inter. Conf. Satellite Systems for Mobile Commun. Navigations, June 1983, pp [27] R. W. Huck, J. S. Butterworth, and E. E. Matt, Propagation measurements for land mobile satellite services, in Proc. IEEE 33rd Veh. Technol. Conf., Toronto, Canada, 1983, pp [28] M. J. Gans, A power-spectral theory of propagation in the mobile-radio environment, IEEE Trans. Veh. Technol., vol. VT-21, no. 1, pp , Matthias Pätzold (M 94), for a biography, see this issue, p Yingchun Li, for a biography, see this issue, p Frank Laue, for a biography, see this issue, p. 269.