A Study Of A Land Mobile Satellite Channel Model With

IEEE TRANSACTIONS ON VEHICULAR TECHNOLOGY, VOL. 47, NO. 1, FEBRUARY 1998 297

A Study of a Land Mobile Satellite Channel Modelwith Asymmetrical Doppler Power Spectrum and

Lognormally Distributed Line-of-Sight ComponentMatthias Patzold, Member, IEEE,Yingchun Li, and Frank Laue

Abstract—In satellite to mobile communications, there is gener-ally, in addition to the diffuse multipath component, a strong line-of-sight (LOS) component present. The spectral and statisticalproperties of both components are influenced by shadowingcaused by obstacles such as trees, houses, or small buildings.In this paper, an analytical model for a land mobile satellitechannel is analyzed that takes into consideration various kindsof shadowing situations. For the LOS component, it is assumedthat, first, the received frequency shows a Doppler shift. Second,the amplitude variations coincide with a lognormal distribu-tion. Moreover, it is also assumed that shadowing influencesthe diffuse scattered component, with the consequence that itsDoppler power spectrum gets an asymmetrical shape. The result-ing spectral and statistical properties of the analytical model areinvestigated. Exact solutions for the probability density function(pdf) of the amplitude and phase are given, and approximativesolutions for the level-crossing rate (LCR) and average durationof fades (ADF’s) are derived. Finally, an efficient deterministicsimulation model is presented that enables the implementation ofthe proposed analytical model on a digital computer.

Index Terms—Fading channel modeling, fading channel simu-lation, land mobile satellite channel, nonfrequency-selective fad-ing, statistics.

I. INTRODUCTION

T HE TRANSMISSION performance of land mobile satel-lite services is mainly impaired by rapid amplitude and

phase fluctuations of the received signal. Such fluctuations arecaused by multipath wave propagation as well as by time-varying attenuation due to shadowing. The statistical propertiesof these signal variations are closely related to the propagationenvironment in which the vehicle is located.

Many stochastical models with different degrees of sophis-tication have been developed for modeling the time-varyingbehavior of the received signal. The famous representative ofstochastical models for land mobile (satellite) channels arethe classical Rice and Rayleigh processes, where the formerprocess takes into account the presence of a direct signal pathand the latter not. On the basis of experimental measurements,it has been shown that the usefulness of these processes isrestricted to modeling only different aspects of short-termfading variations.

Manuscript received February 27, 1996; revised October 16, 1996.The authors are with the Department of Digital Communication Systems,

Technical University of Hamburg-Harburg, D-21071 Hamburg, Germany (e-mail: [email protected]).

Publisher Item Identifier S 0018-9545(98)00692-6.

Superimposed on the rapid short-term fading variations areslow variations in the local mean of the received signal. Theselong-term variations, usually referred to as shadow effects, aredue to trees, small buildings, and other roadside obstructions.A stationary stochastic model, which takes into account short-term and long-term variations, is the so-called Suzuki process[1], [2]. This process is obtained by multiplying a Rayleighprocess with a lognormal process. Thereby, it is assumedthat the inphase and quadrature components generating theRayleigh part are uncorrelated. But this assumption of statis-tical independence is often not in agreement with real-worldsituations in multipath wave propagation. Therefore, modifiedSuzuki processes have been introduced [3], [4], where thecomponents generating the Rayleigh process are allowed to becross correlated. Although the Suzuki process and its modifiedversion have originally been proposed as appropriate modelsfor the cellular land mobile channel, they are also suitablefor satellite land mobile channels for urban areas with almostcomplete obstruction of the direct path. But for suburbanand rural (open) areas with partial (no) obstruction of thedirect wave, further extensions are necessary. A statisticalchannel model has been proposed in [5] that is suitable for alltypes of environments (urban, suburban, rural, and open). Thismodel is based on a product process of a Rice and lognormalprocess, i.e., both components (direct and diffuse) are affectedby shadowing. The flexibility according to the higher orderstatistical properties [level-crossing rate (LCR) and averageduration of fades (ADF’s)] of this model can considerably beincreased if the inphase and quadrature components describingthe Rice process are allowed to be cross correlated. Dependingon the type of cross correlation, the so-called extended Suzukiprocess of Types I and II have been introduced in [6] and[7], respectively. Moreover, generalized Suzuki processes havebeen proposed [8]. A generalized Suzuki process contains theclassical 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 greaterflexibility and thus enables a better adaptation to measurementdata.

Another statistical model was proposed by Loo [9], [10].Loo’s model is suitable for a land mobile satellite channelin a rural environment, where for most of the time, a directcomponent is available at the receiver. This model assumesthat the multipath component is Rayleigh distributed withconstant power and that the direct component follows a

0018–9545/98$10.00 1998 IEEE

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 Hi(f) = S� � (f), i = 1; 2.

lognormal distribution to account for the effect of shadowingdue to foliage.

All the above statistical channel models have in commonthat they are stationary, i.e., they are based on stationarystochastic processes with fixed parameters. A nonstationarymodel for very large areas has been introduced by Lutzet al.[11]. This model is based on a two-state Markov model, wherethe fading process is switched between a Rice process (goodchannel state) and a Rayleigh-lognormal process (bad channelstate). A generalization of this procedure can easily be obtainedand results in an -state Markov model [12], where eachstate represents a specific channel model. Thus, a nonstationarychannel model can be approximated bystationary channelmodels [13], [14]. From experimental measurements, it turnedout that it would be enough for most channels to use a four-state Markov model [15].

To reproduce the fluctuations of the received signal, we pro-pose 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 correspondmore closely to the physical channel phenomena involvedby allowing the inphase and quadrature components definingthe complex diffuse component to be cross correlated. Notethat such a cross correlation corresponds to an asymmetricalDoppler power spectral density shape. Second, the frequencyof the direct component is assumed to be Doppler shifteddue to the movement of the vehicle. These assumptionsgreatly increase the flexibility of the LCR and the ADF’sand thus enable a better statistical adaption to experimentalmeasurement recordings. The resulting spectral and statisticalproperties 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 analyticalmodel that is proposed for a land mobile satellite channel witha lognormally distributed line-of-sight (LOS) component andasymmetrical Doppler power spectral density characteristics.

It follows a discussion of the resulting spectral properties.For that purpose, all relevant Doppler power spectral densityfunctions and the corresponding autocorrelation and cross-correlation functions are derived. Afterwards, the statisticalproperties are analyzed. Thereby, the stress is on theinvestigation of the probability density function (pdf) of theamplitude and phase, LCR, and ADF’s. In Section III, wepropose for the analytical model an efficient simulation model.The excellent conformity of both models is demonstrated byshowing various computer simulation results. Moreover, wedemonstrate in Section IV the usefulness of the proposedmodel by adapting the higher order statistics (LCR) of theanalytical model to measurement results of various types ofland mobile satellite channels. Finally, Section V concludesthe 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 channelsthat takes into consideration the effects produced by a time-varying LOS path is presented in Fig. 1. The underlyingDoppler power spectral density function and the statisticalproperties 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 complexbaseband representation of passband signals. From Fig. 1, weobserve that a single colored zero-mean real Gaussian noiseprocess is used to produce a complex Gaussian noiseprocess

(1)

with cross-correlated inphase and quadrature componentsand . In our analytical model, a time-varying LOS

component is included, which has the following form:

(2)

PATZOLD et al.: STUDY OF LAND MOBILE SATELLITE CHANNEL MODEL 299

where and denote the Doppler frequency and phase ofthe LOS component, respectively. The time variations of theamplitude of the LOS component is a result of shadowingeffects due to foliage, cars, buildings, etc. Measurements haverevealed that the amplitude follows usually a lognormaldistribution [16]. Such a lognormal process is derivedhere from a second real Gaussian noise process withzero mean and unit variance according to

(3)

where and are two quantities, which are strongly depen-dent on the type of shadowing environment. In comparisonwith the Doppler power spectrum of the process , itis assumed that the power spectrum of is confined toa relatively narrow band limited region. Thus, the stochas-tic process is a relatively slowly varying function oftime. Furthermore, we assume that the two Gaussian noiseprocesses and are statistically independent, andthen it follows immediately that the lognormal process isuncorrelated with the complex Gaussian process.

The addition of (1) and (2) gives a complex Gaussiannoise process with a slowly time-varying local mean,

, from which a further stochastic process canbe obtained by taking the absolute value of , i.e.,

(4)

The above-introduced process with underlying cross-correlated inphase and quadrature components is proposed asan appropriate stochastic model for modeling the statisticsof the received signal in the complex baseband of largeclasses of land mobile satellite channels. For the special case,where is independent of time, i.e., , ourproposed model for the stochastic process reduces tothe Rice process with underlying cross-correlated inphase andquadrature components that were recently investigated in [7].

A. The Doppler PSD Function

A multipath propagation scenario contains several differentpaths by which waves travel from the (static) transmitter tothe (mobile) receiver. The superposition of all waves arrivingthe receiver’s (omnidirectional) antenna results in a multipathcomponent with environment specific Doppler power spectraldensity characteristics. One often takes the hypothesis for themobile fading channel that all the incoming directions of thereceived waves are equally distributed in the interval0 2 ,which results in a symmetrical Doppler power spectral density(PSD) function. A widely accepted Doppler PSD functionfor mobile fading channel models is the Jakes PSD [17],[18], as shown in Fig. 2(a) (see dashed line). But the equaldistribution hypothesis usually does not tally with real fadingenvironments as some of the multipath signals are blockedby obstacles or absorbed by the electromagnetic propertiesof the physical environment, and then the resulting DopplerPSD of the complex Gaussian noise process , ,becomes asymmetrical. In addition, the shape of the Doppler

PSD is strongly influenced by the antenna pattern. Forexample, if the mobile receiver uses a beam antenna directedalong vehicle motion, then gets also an asymmetricalshape [18], [28]. How to get for the received multipath signalcomponent a Doppler PSD with asymmetrical shape by takingalso into account a slowly time-varying LOS component isthe topic of this section.

Therefore, we use for the Doppler PSD of the processthe following shape [18], [28]:

(5)

where is the maximum Doppler frequency, is aconstant that determines the mean power of the process,and is a real parameter in the interval (0,1]. If , thenthe classical Jakes Doppler PSD is obtained, and if 0 1,a so-called restricted Jakes Doppler PSD results as is shown inFig. 2(a) (see solid line). In [6], it has been demonstrated thatthe classical Jakes Doppler PSD often results in an analyticalchannel model with a fading rate (LCR) that is much higherthan the corresponding measurement results. It should also benoted that the restricted Jakes PSD (5) is rather a heuristicalassumption than a theoretical result for the Doppler PSD ofland mobile satellite channels, but nevertheless (5) allowsan optional reduction of the resulting fading rate simply byreducing the parameter . As we know, the inverse Fouriertransform of (5) gives the autocorrelation function ofthe Gaussian process , which is for the classical JakesDoppler PSD ( ) given by ,but for the restricted Jakes Doppler PSD (0 1), noclosed-form expression for exists. In this case, theautocorrelation function is obtained via solving thefollowing integral:

(6)

by applying numerical integration techniques.The Doppler PSD of the second Gaussian process ,

, is assumed to be a Gaussian function

(7)

where is related to the 3-dB cutoff frequency accordingto . Generally, the cutoff frequency is muchsmaller than the maximum Doppler frequency , i.e.,

. In the following, it will be advantageous to introducea quantity in order to express by .The autocorrelation function of the process isobtained by taking the inverse Fourier transform of (7). Hence

(8)

results.Next, let us consider the lognormal process [see (3)].

For that process, it will be advisable to express the correspond-ing autocorrelation function as function of .


(a) (b)

(c) (d)

(e)

Fig. 2. Diverse Doppler PSD functions: (a) the restricted Jakes PSDS� � (f), (b) an asymmetrical PSDS��(f), (c) the Gaussian PSDS� � (f), (d)the PSDS��(f) of the lognormal process�(t), and (e) the resulting asymmetrical PSDS� � (f).

Therefore, we write

(9)

where

(10)


is the joint pdf of the process at two different timesand . After substituting (10) into (9) and solvingthe double integrals, the autocorrelation function canbe expressed in a closed form as follows:

(11)

From the above equation, it should be observed that the powerof the process is given by .

The PSD function of the process can beobtained by solving the Fourier integral, i.e.,

(12)

The preceding equation shows us that the Doppler PSDof the lognormal process is composed of a weighteddelta function located at and an infinity sum ofmonotonically decreasing versions of . Notethat is obtained from (7) simply aftersubstituting the quantity by . The Doppler PSD’s

and are shown in Fig. 2(c) and (d), respec-tively.

We proceed by considering the structure of the analyticalmodel as depicted in Fig. 1. From that figure, the followingexpressions are immediately readable:

(13a)

(13b)

where the notation denotes the corresponding Hilberttransform of the process . A simple computation allowsus to express the autocorrelation function (

) of the process and the cross-correlation functionof the processes and as follows:

(14a)

(14b)

where and denote the autocorrelation functionsof the processes and as given by (6) and (11),respectively, and is the cross-correlation function ofthe processes and .

The autocorrelation functionand the corresponding Doppler PSD function

of the complex process cannow be expressed by making use of (14) as follows:

(15a)

(15b)

respectively. Finally, by using the relationsgn , 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 andphase of the complex process (seeFig. 1).

The pdf of the stochastic process ,denoted here by , can be derived by applying the methodsdescribed in [6]. In [6], the statistics of a process

have been investigated, thereby, the inphase andquadrature components of the complex process were alsocorrelated, but for the LOS component , a more simplerexpression of the form with const.

was assumed. Note that the process is a specialcase of our proposed process under the restriction that

yields. The pdf of such a process with underlyingcross-correlated inphase and quadrature components is equalto the conditional pdf and is given by the Ricedistribution [25]

(17)

where

(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 isgiven by [19]

(19)

Hence, the pdf of , , can be derived from the joint pdfof the processes and in the following way:

(20)


(a) (b)

Fig. 3. The pdfp�(z) as function of (a)�o (�o = 1, s = 1, andm = �1) and (b)s (�o = 1, �o = 0:5, andm = �s2).

The preceding equation shows us that the pdf of theproposed process depends on three parameters ( ,and ), whereas the pdf of a Rice process depends ontwo 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 generalizationof the Rice density and is, therefore, more flexible than thelatter one. It should also be noted that Loo’s model [9], [10]and the proposed model have identical expressions for theprobability density of the amplitude, but both models havedifferent 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 thepdf is shown in Fig. 3(a) and (b), respectively.

Next, we consider the pdf that describes to us thedistribution of the phase of the complex process

. This phase pdf can be derived in a similar wayas we have done for the computation of the amplitude pdf

. Thereby, we make use of the fact that forconst. , the phase pdf can be expressed by [6]

(21)

where . Obviously, the above phase pdf is a func-tion of time if the Doppler frequency of the direct component

is unequal to zero. After averaging (21) over time, we obtainfor a uniform conditional phase pdf

(22)

Now, the desired pdf of the phase processcan be obtained from the joint pdf of

and as follows:

(23)

i.e., the phase is uniformly distributed between 0–2if.

C. LCR and ADF’s

The performance of mobile communication systems dete-riorates rapidly when the received signal falls below somenoise-related threshold. The consequence of such situationsis that error bursts occur. The LCR and the ADF’s aretwo important measures for the rate of occurrence and theaverage length of these error bursts, respectively. Especially,the analysis of the ADF’s is of particular interest in theconnection with an optimal design of the interleaver andforward-error correcting codes. In this section, we analyzethe LCR and ADF’s of the received signal envelope of theproposed land mobile satellite channel model.

The LCR of the process , , is the average numberof crossings per second at which crosses a specified signallevel with a positive slope. For a wide-sense stationaryrandom process, the LCR is generally defined by [20]

(24)

where denotes the joint pdf of the processand its corresponding time derivative at the same time.That joint pdf can be derived by taking into account thatthe amplitude of the LOS component is a slowly time-changing process, i.e., , then it follows from furtherinvestigations that the joint pdf of and its


time derivative is approximately given by. Hence, it follows:

(25)

After substituting (25) into (24), the LCR can ap-proximately be expressed by the pdf and the LCR

as follows:

(26)

An expression for the LCR hasbeen derived in [6], where the following result can be found:

(27)

where

(28a)

(28b)

sinc (28c)

(28d)

Finally, by substituting (27) into (26) and using (19), anapproximation of the LCR can be given as follows:

(29)

In Section III, we will show that such an approximationof is fairly close to the exact LCR of the process

if the quantity , i.e., if theamplitude of the LOS component changes relatively slowly

in time in comparison with the diffuse components. Note thatsuch relations are completely in conformity with real-worldmultipath propagation situations, where often 1 yields.

The ADF’s of the process , , is the mean valueof the time intervals over which the process is below aspecified level . Generally, the ADF’s is related tothe LCR as follows [17]:

(30)

where is the cumulative distribution function of theprocess , which can be obtained by solving the integral

(31)

The influence of the parameters, , and on the LCRand ADF’s can be studied in Figs. 4 and 5,

respectively. For a fair investigation of the influence of theparameter , we identified in Figs. 4(b) and 5(b) the parameter

with , which results for all values of in alognormal process with unit power, i.e.,

.

III. COMPUTER SIMULATIONS

Two fundamental methods are generally used for the re-alization of stochastic processes: the filter method and Rice’ssum of sinusoids. By applying the former method, a simulationmodel is obtained by replacing each ideal filter of the analyticalmodel by a digital filter with an appropriate filter degree. Inthis paper, we have applied the latter method by means ofwhich the realization of the two low-pass filters as well asof the Hilbert transformer has been circumvented. Accordingto that procedure, an efficient simulation model, which corre-sponds to the analytical model of Fig. 1, can be derived byapproximating the processes and by the followingsum of sinusoids

(32)

where , , and are called Doppler coefficients, dis-crete Doppler frequencies, and Doppler phases, respectively.After carrying out some additional network transformations,the structure of the (deterministic) simulation model as shownin Fig. 6 results. For shortness, we will not recapitulate theconcept of deterministic channel modeling. But for the inter-ested reader, we refer to [21]–[24], where a detailed introduc-tion into the theory of deterministic simulation systems canbe 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


(a) (b)

(c)

Fig. 4. The normalized LCRN�(r)=fmax as function of (a)�o ( o = 1, s = 1, m = �1, and f� = 0:1fmax), (b) s (�o = 1, �o = 0:5, m =�s

2, and f� = 0:1fmax), and (c) f� (�o = 0:2, �o = 1, s = 0:5, andm = �0:25).

PSD [see (5)] results in the following closed-form expressionfor the discrete Doppler frequencies :

(33)

where .For the Gaussian PSD [see (7)], the discrete Doppler fre-

quencies are obtained by finding the zeros of

(34)

The corresponding Doppler coefficients are given by

(restricted Jakes PSD)

(Gaussian PSD)

(35)

for .

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 usingdeterministic methods, and now the simulation procedure canbe performed.

A plot of the deterministic output signal of the resultingsimulation model is shown in Fig. 7 (solid line), where themaximum Doppler frequency is equal toHz, and for the numbers of sinusoids and , the values

have been selected. The resulting deterministiclognormal process is also plotted in Fig. 7 (dotted line).

A comparison of the statistics (pdf, LCR, and ADF’s) ofthe analytical model with the simulation model is shown inFigs. 8–10. For the simulation procedure, againsinusoids and a maximum Doppler frequency of

Hz have been selected. The sampling intervalof thediscrete deterministic process was

. Altogether, 7 samples of


(a) (b)

(c)

Fig. 5. The normalized ADF’sT� (r) � fmax as function of (a)�o ( o = 1, s = 1, m = �1, and f� = 0:1fmax), (b) s (�o = 1, �o = 0:5, m= �s

2, and f� = 0:1fmax), and (c) f� (�o = 0:2, �o = 1, s = 0:5, andm = �0:25).

the process have been simulatedand evaluated for estimating the pdf , normalized LCR

, and normalized ADF’s of thesimulation system.

Fig. 8 reveals that the behavior of the pdf is notinfluenced by the value of . This result is not surprisingbecause [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 sinusoidsis chosen.

The results of Fig. 9 show us that for small values of( ), there is a relatively large deviation between theLCR of the analytical model and the simulation model. Butalready for moderate values of , e.g., , the LCR ofthe analytical model is, besides little differences, in a goodagreement with the LCR of the simulation model as it wasexpected by the theory. Fig. 10 allows similar interpretationsfor the ADF’s.

IV. A PPLICATIONS

In this section, we demonstrate the usefulness of the analyti-cal model and the corresponding simulation model by adapting

the statistics of the synthetic channel output to measuredchannel data. Let us therefore consider measurement resultsof the LCR of an equivalent satellite channel [26] as well asof a real-world satellite channel [27].

In the following, the measured normalized LCR, denotedhere as , is used as object function for the opti-mization of the normalized LCR of the analyticalmodel. Our task is now to find proper values for all thoseparameters , which influence the behaviorof so that the absolute value of the differencebetween and is sufficiently small.Therefore, we combine these quantities to a parameter vector

, and then we minimize the followingerror norm:

(36)

where denotes the number of measurement values andrepresents a weighting function, which is in our case simplydefined by the reciprocal of . The minimization of the


Fig. 6. Structure of the deterministic simulation system.

Fig. 7. The deterministic process~�(t) (�o = 1, �o = 0:5, s = 0:5, m =�0:25, f� = 0:2fmax, �c= 50, andfmax = 91 Hz).

error norm can be performed by applying any elaboratenumerical optimization procedure.

A. Applications to an Equivalent LandMobile Satellite Channel

The experimental measurement results considered here canbe found in [26]. There, a 870-MHz transmitter package wasmounted on a helicopter at a fixed elevation angle of 15withrespect to the receiver in order to simulate the effects of asatellite signal source. The test environment was a rural areawith 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� = 0:2fmax).

one situation, the test route was heavily shadowed by trees,thereby, the deciduous trees were in leaf and lightly shadowedin 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 pa-rameter vector in the design. For example, the parameter

can be fixed equal to without any appreciable per-formance loss. The other elements ofhave been optimizedby minimizing (36) numerically, and the obtained results areshown in Table I.


Fig. 9. Comparison of~N�(r)=fmax with N�(r)=fmax (�o = 0:86, �o =0:28, s = 0:19, m = 0:067, andf� = 0:2fmax).

Fig. 10. Comparison of~T� (r) � fmax with T� (r) � fmax (�o = 0:86,�o = 0:28, s = 0:19, m = 0:067, andf� = 0:2fmax).

Fig. 11. Normalized LCR ~N�(r)=fmax for environments with light andheavy shadowing.

For the remaining parameters of the analytical model (,, and ), which have no influence on the behavior of the

TABLE IPARAMETERS OF THE ANALYTICAL MODEL FOR

RURAL AREAS WITH LIGHT AND HEAVY SHADOWING

normalized LCR , we have chosen the valuesHz, , and . Thus, all parameters of the

analytical model are defined, and the corresponding simulationmodel can be derived according to the procedure described inSection III. For the number of sinusoids , we propose forthe present application to use ( ) harmonicfunctions. The resulting normalized LCR of the analyticalmodel and simulation model are also shown in Fig. 11. Thisfigure shows us that the results of the analytical and simulationmodel are in fairly good agreement with the measurementdata for both situations (light and heavy shadowing). Forcompleteness, the output signal (envelope) of the simulationmodel is presented for light shadowing in Fig. 12(a) andfor heavy shadowing in (b). The Rice factor , which isdefined as the ratio of the power of the LOS component tothe total power of the diffuse component, is given here by

. Given the parameters listed in Table Iand using the expressions (11) and (18), we find for lightshadowing dB and for heavy shadowing

dB.

B. Applications to a Real-World Land Mobile Satellite Channel

Experimental measurement results of a real-world landmobile satellite channel have been obtained by usingINMARSAT’s MSAT-A satellite in the L band [27]. Fig. 13shows two measurement results of the normalized LCR

. The first curve shown was obtained from asuburban area, where the streets were bordered by one- andtwo-story homes and the second one from a nonforested ruralarea, indicated as farmland.

As in the previous example, the LCR is used asan object function for the optimization of the parameters of theanalytical model by minimizing the error norm (36). On furtherconsideration of the measurement results (see Fig. 13), we canrealize that each curve can be interpreted as acomposition of two concave functions. Our hypothesis is thatsuch a property is typical for nonstationary channels. Note thatthe previous curves for , shown in Fig. 11, cannotbe interpreted in this way and, thus, the corresponding satellitechannels are stationary. Note also that all parameters of theproposed analytical and corresponding simulation model arefixed, i.e., both models are stationary. Therefore, an extensionof the proposed model in order to enable the modeling ofnonstationary land mobile satellite channels is required. Suchan extension can easily be performed when we follow theideas described in [11]. In that paper, a two-state model isdescribed, where one state represents a Rice process undergood propagation conditions and the other state represents aRayleigh-lognormal process under bad propagation conditions.Here, we also use a two-state model, but each state—denoted


(a) (b)

Fig. 12. Simulation of the envelope~�(t) for (a) light shadowing and (b) heavy shadowing.

by and —represents the proposed analytical model withdifferent parameter sets( ). Now, let the parameter sets are such that the corre-sponding stochastic processes and are statisticallyindependent, then, it follows that the pdf of the resultingreceived amplitude can be expressed by

(37)

where denotes thea priori probability that the channel isin 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

(38)

where denotes the LCR as introduced by (29) if thechannel is in state ( ). Appropriate parametersfor the analytical two-state model can now be obtained bysubstituting the expressions (38) in (36) and minimizing theerror norm by using the measured LCR (seeFig. 13) as an object function. The execution of the numericalminimization of the error norm (36) is, in this case, a criticalpoint and requires some practical knowledge. The results ofour optimized parameters are shown in Table II.

Just as in the previous section, we have fixed equal tozero. Due to the fact that the normalized LCR is

independent of the parameters , , and , we canselect these quantities within reasonable bounds. Here, wehave chosen the values Hz, , and

( ). Now, we have determined all parameters of theanalytical two-state model and corresponding parameters ofthe simulation model can be derived by applying the methods

Fig. 13. Normalized LCR~N�(r)=fmax for suburban and rural (farmland)areas.

TABLE IIPARAMETERS 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 asof the corresponding simulation model are also presented inFig. 13. This figure shows us that for low- and high-signallevels , the measurement results for both areas (suburbanand rural) are in excellent agreement with the analyticaland simulation results, whereas for medium signal levels, asufficient agreement can be observed. Finally, the envelope of


(a) (b)

Fig. 14. Simulation of the envelope~�(t) for (a) suburban and (b) rural (farmland) areas.

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-selectiveland mobile satellite channels with asymmetrical Dopplerpower spectral density shapes and lognormally distributedLOS components was presented. The model takes into accountdifferent kinds of signal shadowing, caused by obstaclessuch as trees, houses, or small buildings. The spectral andstatistical properties of the proposed analytical model havebeen analyzed. Especially for the pdf of the phase and am-plitude, exact solutions could be derived. It turned out thatthe phase is uniform between 0–2if the Doppler frequencyof the LOS component is unequal to zero. The pdf of theamplitude includes the Rice (Rayleigh) density as specialcases, where the LOS component is constant (zero) and isthus more flexible than the latter one. For the LCR andADF’s, approximative solutions have been derived. On theassumption that the bandwidth of the LOS component issignificantly lower than the bandwidth of the diffuse com-ponent, the presented approximative solutions are fairly closeto the corresponding exact solutions. It should be noted thatsuch an assumption generally complies with real-world mobilepropagation situations. Finally, for the (stochastic) analyticalmodel, an efficient (deterministic) simulation model has beenproposed that avoids the design of any digital filters. Insteadof this, the simulation model is derived by employing sumsof harmonic functions, whose parameters are determined bysimple and closed expressions. By various simulation results,the excellent conformity of the statistical properties of theanalytical model with the therefrom-derived simulation modelhas been demonstrated. The usefulness of the proposed modelhas been demonstrated by fitting the higher order statistics(LCR) of the analytical model to measurement results of fourdifferent categories of measurement data.

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Matthias Patzold (M’94), for a biography, see this issue, p. 269.

Yingchun Li , for a biography, see this issue, p. 269.

Frank Laue, for a biography, see this issue, p. 269.

Documents

A Study Of A Land Mobile Satellite Channel Model With