On the Statistical Properties of DeterministicSimulation Models for Mobile Fading ChannelsMATTHIAS P�ATZOLD, MEMBER, IEEE, ULRICH KILLAT, MEMBER, IEEE,FRANK LAUE, YINGCHUN LITechnical University of Hamburg-HarburgDepartment of Digital Communication SystemsPostfach 90 10 52Denickestr. 17D-21071 HamburgTel. : +4940-7718-3149Fax : +4940-7718-2941E-Mail : paetzold@tu-harburg.d400.deAbstractRice's sum of sinusoids can be used for an e�cient approximation of colouredGaussian noise processes and is therefore of great importance to the software andhardware realization of mobile fading channel models. Although several methodscan be found in the literature for the computation of the parameters characterizinga proper set of sinusoids, only less is reported about the statistical properties of theresulting (deterministic) simulation model. This is the topic of the present paper;whereby not only the simulation model's amplitude and phase probability densityfunction will be investigated, but also higher order statistics (e.g. level-crossingrate and average duration of fades). It will be shown that due to the deterministicnature of the simulation model analytical expressions for the probability densityfunction of amplitude and phase, autocorrelation function, level-crossing rate, andaverage duration of fades can be derived. We also propose a new procedure for thedetermination of an optimal set of sinusoids, i.e., the method results for a givennumber of sinusoids in an optimal approximation of Gaussian, Rayleigh, and Riceprocesses with given Doppler power spectral density properties. It will be shownthat the new method can easily be applied to the approximation of various otherkinds of distribution functions, such as the Nakagami and Weibull distributions.Moreover, a quasi optimal parameter computation method is presented.Submitted toIEEE Transactions on Vehicular TechnologySeptember 1995Final Version1

I. INTRODUCTIONIn general, simulation models for mobile radio channels are realized by employing at leasttwo or more coloured Gaussian noise processes. For instance, for the realization of aRayleigh or Rice process two real coloured Gaussian noise processes are required; whereasthe realization of a Suzuki process [1], which is a product process of a Rayleigh and a lo-gnormal process, is based on three real coloured Gaussian noise processes. Such processes(Rayleigh, Rice, Suzuki) are often used as appropriate stochastical models for describingthe fading behaviour of the envelope of the signal received over land mobile radio chan-nels, which are often classi�ed as frequency non-selective channels. A further example isgiven by modelling a (n-path) frequency selective mobile radio propagation channel byusing the n-tap delay line model [2]. This requires the realization of 2n real colouredGaussian noise processes. All these examples show us that an e�cient design methodfor the realization of coloured Gaussian noise processes is of particular importance in thearea of mobile radio channel modelling.A well known method for the design of coloured Gaussian noise processes is to shape awhite Gaussian noise (WGN) process by means of a �lter that has a transfer function,which is equal to the square root of the Doppler power spectral density of the fadingprocess (see Fig. 1(a)). Another method, which is recently coming more and more in use,is based on Rice's sum of sinusoids [3, 4]. In this case a coloured Gaussian noise process isapproximated by a �nite sum of weighted and properly designed sinusoids. The resultingsignal ow diagram of such an approximated Gaussian noise process is shown in Fig. 1(b),where the quantities cn, fn; and �n are called Doppler coe�cients, discrete Doppler fre-quencies, and Doppler phases, respectively, and N denotes the number of sinusoids.In the meanwhile, diverse methods have been developed for the derivation of the relevantmodel parameters (Doppler coe�cients cn and discrete Doppler frequencies fn). For ex-ample, the method of equal distances [5, 6] and the mean square error method [6]. Thecharacteristic of both methods is just the same as Rice's original method [3, 4], name-ly, the di�erence between two adjacent discrete Doppler frequencies is equidistant; butthe Doppler coe�cients are adapted to a given Doppler power spectral density in a me-thod speci�c manner. The disadvantage of these methods is that (due to the equidistantdiscrete Doppler frequencies) the approximated coloured Gaussian noise process ~�(t) isperiodical. Another method is known as the method of equal areas [5,6]. That procedu-re results in a relativly satisfactory approximation of the desired statistics for the JakesDoppler power spectral density even for a small number of sinusoids but fails or requi-res a comparatively large number of sinusoids for other types of Doppler power spectraldensities such as those with Gaussian shapes. Widespread in use is the so called MonteCarlo method [7,8]. But unfortunately the performance of that procedure is poor [9]. Thereason is that the discrete Doppler frequencies are random variables and therefore, themoments of the Doppler power spectral density of the designed Gaussian noise process~�(t) are also random variables, which can di�er widely from the desired (ideal) momentseven for a large number of sinusoids N , say N = 100 [6,10]. A further design method wasderived by Jakes [11]. For the Jakes method an e�cient hardware implementation, usingthe Intel 8088 microprocessor, is described in [12].Although all the parameter computation methods can be classi�ed in deterministic andstatistic procedures, the resulting simulation model is in any case a deterministic one; be-cause all prameters of the simulation model have to be determined during the simulationset up phase and thus, they are known and constant quantities during the total simula-2

tion run phase. In this paper, the statistical properties of such deterministic simulationmodels will be investigated.The paper recapitulates in Section II the statistical properties of Rice processes with givenspectral characteristics for the underlying Gaussian processes. Section III introduces theconcept of deterministic simulationmodels and discusses the performance of three di�erentparameter computation methods, whereby two of them are new. One of the new proce-dures is for a given number of sinusoids optimal in the following sense: First, the Dopplercoe�cients cn are calculated such that the probability density of the approximated co-loured Gaussian process ~�(t) approximates optimally the aspired Gaussian probabilitydensity function; and second, the discrete Doppler frequencies fn are optimized so thatthe autocorrelation function of ~�(t) approximates optimally the desired autocorrelationfunction of a given coloured Gaussian noise process. Moreover, analytical expressions forthe deterministic simulation model's amplitude and phase probability density function,autocorrelation function, and cross-correlation function will be derived. Section IV inve-stigates higher order statistical properties of deterministic simulation models such as thelevel-crossing rate, average duration of fades, and the probability density function of thefading intervals. It is shown analytically and experimentally that the level-crossing rateof properly designed and dimensioned deterministic simulation models for Rayleigh andRice processes is extremely close to the theory even if the number of sinusoids is small,say seven or eight. Section V concludes the paper.

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II. A STOCHASTIC ANALYTICAL MODELFOR RICE PROCESSESIn order to compare the statistical properties of the deterministic simulation model aswill be introduced in Section III with the desired (exact) statistics, a reference model isrequired. As reference model, we consider in this section an analytical model for Riceprocesses. In cases, where the in uence of a direct line of sight component can not beneglected, Rice processes are often used as suitable stochastical models for characterizingthe fading behaviour of the received signal level of land mobile terrestrial channels as wellas of land mobile satellite channels.A. Description of the Analytical ModelA Rice process, �(t), is de�ned by taking the absolute value of a non-zero mean complexGaussian process��(t) = �(t) +m(t) (1)according to�(t) = j��(t)j : (2)In (1) all the scattered components in the received signal are represented by a zero meancomplex Gaussian noise process�(t) = �1(t) + j�2(t) (3)with uncorrelated real components �i(t); i = 1; 2; and variance V arf�(t)g =2V arf�i(t)g = 2�20, whereas the in uence of a direct (line of sight) component is takeninto account here by a time variant mean value of the formm(t) = m1(t) + jm2(t) = �ej(2�f�t+��) : (4)Amplitude, Doppler frequency, and phase of the direct component are denoted by �; f�,and ��, respectively. Observe that for f� = 0 the mean value m(t) = m = �ej�� is obvious-ly time invariant, which directly corresponds to an orthogonality between the directionof the incoming direct wave component at the receiving antenna of the vehicle and thedirection of the vehicle movement. Throughout the paper, we consider the more generalcase where f� 6= 0 and hence, the direct component m(t) depends on time.A typical and often assumed shape for the Doppler power spectral density (psd) of thecomplex Gaussian noise process �(t); S��(f), is for mobile fading channel models givenby the Jakes psd [11]S��(f) =8<: 2�20�fmaxp1�(f=fmax)2 ; jf j � fmax ;0 ; jf j > fmax ; (5)4

where fmax is the maximum Doppler frequency. From this equation it follows for thecorresponding autocorrelation function (acf) r��(t), i.e. the inverse Fourier transform ofS��(f), the following relationr��(t) = 2�20J0(2�fmaxt) ; (6)where Jo(�) denotes the zeroth order Bessel function of the �rst kind. The resultingstructure of the analytical model for Rice processes �(t) with underlying Jakes psd cha-racteristic is shown in Fig. 2(a).Another important shape for the Doppler psd S��(f) is given by the frequency shiftedGaussian psdS��(f) = 2�20fc r ln 2� e�� f�fsfc=pln 2�2 ; (7)where fs and fc are the frequency shift and the 3-dB cuto� frequency, respectively. Super-positions of two Gaussian psd functions with di�erent values for �20; fs, and fc have beenspeci�ed for example in [13] for describing the typical Doppler psd of large echo delays,which occur in the frequency selective mobile radio channel of the pan-European cellularGSM system. Furthermore, frequency non-shifted Gaussian psd functions, i.e. fs = 0 in(7), have been introduced for describing the Doppler psd of aeronautical channels [14, 15].The acf of the Gaussian psd S��(f) is given byr��(t) = 2�20 e��� fcpln 2 t�2ej2�fst : (8)Observe that the acf r��(t) is complex if fs 6= 0. Thus, it follows from the propertiesof the Fourier transform that in such cases the real Gaussian processes �1(t) and �2(t)are correlated. Fig. 2(b) shows the resulting structure of the analytical model for Riceprocesses �(t) with underlying frequency shifted Gaussian psd characteristic.B. Statistical Properties of the Analytical ModelIn this subsection we study besides the amplitude and phase probability density function,also the level-crossing rate, and the average duration of fades of Rice processes with givenspectral shapes for the underlying Gaussian processes. We refer to these (ideal) statisticalproperties in Section III, where we investigate the statistics of deterministic simulationmodels.The probability density function (pdf) of the Rice process �(t), p�(x), is given by [16]p�(x) = 8<: x o e�x2+�22 o Io�x� o� ; x � 0 ;0 ; x < 0 ; (9)where Io(�) designates the zeroth order modi�ed Bessel function of the �rst kind, and o re-presents the mean power of the Gaussian process �i(t), i.e. o = r�i�i(0) = r��(0)=2 ; i =1; 2. From (6) and (8) it follows that o is for the Jakes and Gaussian psd identical with o = �20. 5

Obviously, the exact form of the Doppler psd of �(t) has no in uence on the behaviourof the pdf p�(x). Such an independence of the Doppler psd S��(f) holds also true for thephase pdf of the complex Gaussian process ��(t), p#(�), which is given by [17]p#(�) = p#(�; t) = e� �22 02� �1 +r �2 0� cos(� � 2�f�t� ��) e �2 cos2(��2�f�t���)2 0�1 + erf �� cos(� � 2�f�t� ��)p2 0 ��� ; (10)where erf(�) is the error function, and � is within the interval [0; 2�).By taking notice of (9) and (10), we see that in the limit for � ! 0 the amplitude pdfp�(x) tends to the Rayleigh distributionp�(x) =8<: x o e� x22 0 ; x � 0 ;0 ; x < 0 ; (11)whereas the phase pdf p#(�) tends to the uniform distribution p#(�) = 12� , � 2 [0; 2�),respectively.The level-crossing rate (lcr), denoted here by N�(r), provides us with a measure of theaverage number of crossings per second at which the envelope �(t) = j��(t)j crosses aspeci�ed signal level r with positive slope.We quote for the lcr N�(r) the result from [17], where N�(r) has been derived for Riceprocesses with cross-correlated inphase and quadrature components:N�(r) = rp2��3=2 0e� r2+�22 0 �=2Z0 cosh�r� 0 cos ��ne�(�� sin �)2 +p��� sin(�) � erf(�� sin �)o d�; (12)where the quantities � and � are given by� = 2�f� � _�0 0!�p2� ; (13)� = � � 0 � _�20 o : (14)Thereby, the above notations � 0 and _�0 have been introduced for denoting the curvatureof the autocorrelation function r�1�1(t) and the gradient of the cross-correlation functionr�1�2(t) at t = 0, i.e.� 0 = d2dt2 r�1�1(t)���t=0 = �r�1�1(0) ; (15)_�0 = ddtr�1�2(t)���t=0 = _r�1�2(0) ; (16)6

respectively. If the Doppler frequency of the direct component f� is zero, and if the realGaussian noise processes �1(t) and �2(t) are uncorrelated, i.e. _�0 = 0, then it followsfrom (13) that � = 0 and consequently, the lcr N�(r) according to (12) reduces to theexpressionN�(r) = q �2� � r 0 e� r2+�22 0 I0 � r� 0�= q �2� � p�(r) ; (17)where � = � � 0 � 0. In such cases, the lcr N�(r) is simply the Rice distribution p�(r)multiplied by a constant factor q �2� , which represents the in uence of the curvature ofthe acf r�1�1(t) at t = 0.The average duration of fades (adf), denoted here by T��(r), is the mean value for thelength of all time intervals over which the signal envelope �(t) = j��(t)j is below a speci�edlevel r. The adf T��(r) is de�ned by (cf. [11] on p. 36)T��(r) = P��(r)N�(r) ; (18)where P��(r) designates the probability that the signal envelope �(t) is found below thelevel r, i.e.P��(r) = Prf�(t) � rg = rZ0 p�(x)dx= e� �22 0 0 rZ0 xe� x22 0 I0�x� 0� dx : (19)The equations (12) and (18) show us that the lcr N�(r) and the adf T��(r) are bothfunctions of the characteristic quantities � and � (see eqs. (13) and (14)), which arecompletely de�ned by the Doppler psd S��(f) and the Doppler frequency of the directcomponent f�. For the Jakes and frequency shifted Gaussian psd we obtain for � and �| by using (6), (8), and (13) - (16) in connection with the relation r��(t) = 2(r�1�1(t) +j r�1�2(t)) | the following result:� = 8>>>>><>>>>>: 1p 0 � f�fmax ; Jakes psd; (20a)1p 0 �0@ f� � fsfc.pln 21A ; Gaussian psd; (20b)� = ( 2 0(�fmax)2 ; Jakes psd; (21a)2 0(�fc=pln 2)2 ; Gaussian psd; (21b)7

where 0 = �20.From the above it is clear now that the higher order statistics considered here (lcr N�(r)and adf T��(r)) are completely determined by the amplitude and Doppler frequency ofthe direct component, i.e. � and f�, as well as by the shape of the acf r��(t) and its timederivative of �rst and second order at t = 0.

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III. A DETERMINISTIC SIMULATION MODELFOR RICE PROCESSESIn this section, we investigate the statistical properties of deterministic simulation mo-dels for Rice processes with desired spectral characteristics for the underlying Gaussianprocesses. Therefore, we �rst of all introduce the reader to the concept of deterministicsimulation models and subsequently, we present a new and optimal method for the compu-tation of the simulation model parameters. For reasons of comparison, we also investigatethe statistical properties of the simulation system for two other parameter computationmethods.Let us proceed by considering the two real functions ~�1(t) and ~�2(t) which will be expres-sed as follows~�i(t) = NiXn=1 ci;n cos(2�fi;nt+ �i;n) ; i = 1; 2 ; (22)where Ni denotes the number of sinusoids of the function ~�i(t). The quantities ci;n, fi;n,and �i;n are simulation model parameters, which are adapted to the desired Doppler psdfunction and are therefore called Doppler coe�cients, discrete Doppler frequencies, andDoppler phases, respectively. It is worth mentioning that the simulation model parametersci;n, fi;n, and �i;n have to be computed during the simulation set up phase, e.g. byone of the methods described in the next subsections. Afterwards these parameters areknown quantities and are kept constant during the whole simulation run phase. Fromthe fact that all parameters are known quantities, it follows that ~�i(t) itself is determinedfor all time and thus, ~�i(t) can be considered as a deterministic function. We will seethat the statistical properties of ~�i(t) approximate closely the statistical properties ofcoloured Gaussian stochastic processes. In order to emphasize that property, we willdenote henceforth the deterministic function ~�i(t) as deterministic (Gaussian) process.Consequently, the interpretation of ~�i(t) as a deterministic process (function) allows us tocompute an analytical expression for the III.3 corresponding acf ~r�i�i(t) via the de�nition~r�i�i(t) := limTo!1 12To ToZ�To ~�i(� )~�i(t+ � )d� : (23)Substituting (22) in (23) and taking thereupon the Fourier transform, we �nd the followinganalytical expressions for the acf ~r�i�i(t) and the corresponding psd ~S�i�i(f)~r�i�i(t) = NiXn=1 c2i;n2 cos(2�fi;nt) ; (24a)��~S�i�i(f) = NiXn=1 c2i;n4 [�(f � fi;n) + �(f + fi;n)] ; (24b)for i = 1; 2. 9

Notice, that the deterministic processes ~�1(t) and ~�2(t) are uncorrelated if and only iff1;n 6= �f2;m for all n = 1; 2; : : : ; N1 and m = 1; 2; : : : ; N2. But if f1;n = �f2;m is valid forsome or all n;m, then ~�1(t) and ~�2(t) are correlated and the following cross-correlationfunction, ~r�1�2(t), can be derived~r�1�2(t) = 8<: NPn=1 c1;nc2;m2 cos(2�f1;nt� �1;n � �2;m) ; if f1;n = �f2;m ;0 ; if f1;n 6= �f2;m ; (25)where N indicates the minimum value of N1 and N2, i.e. N = minfN1; N2g.Fig. 3 shows the resulting general structure of the deterministic simulation model for Riceprocesses in its continuous-time representation. Therefrom, the discrete-time fading si-mulation model can immediately be obtained by replacing the time variable t by t = kT ,where k is a natural number and T denotes the sampling interval.For the parameters ci;n and fi;n appearing in (22) we will present in the next two subsec-tions a new computation procedure, which is for a given number of sinusoids Ni optimalin the following two senses:i) The pdf of the deterministic process ~�i(t), named by ~p�i(x), is according tothe L2-norm an optimal approximation of the ideal (desired) pdf p�i(x) of thestochastic process �i(t).ii) The acf of the deterministic process ~�i(t), named by ~r�i�i(t), is according tothe L2-norm an optimal approximation of the ideal (desired) acf r�i�i(t) of thestochastic process �i(t) within a distinct time interval.A. Determination of Optimal Doppler Coe�cientsIn this subsection we compute the Doppler coe�cients ci;n in such a way that the pdf~p�i(x) of the deterministic process ~�i(t) results for a given number of sinusoids Ni in anoptimal approximation of the ideal Gaussian pdf p�i(x) of the stochastic process �i(t).In the �rst step, we derive the pdf of ~�i(t) (see (22)). For that purpose let us consider asingle sinusoid~�i;n(t) = ci;n cos(2�fi;nt+ �i;n) : (26)Furthermore, let us assume in this subsection that the time t is a random variable, whichis uniformly distributed in the interval (0; f�1i;n ) (suppose fi;n 6= 0). Then ~�i;n(t) is also arandom variable and the pdf of ~�i;n(t) is given by (cf. [16] on p.98)~p�i;n (x) = ( 1�ci;np1�(x=ci;n)2 ; jxj < ci;n ;0 ; jxj � ci;n : (27)If the random variables ~�i;n(t) are independent with respective densities ~p�i;n (x), then thedensity ~p�i(x) of their sum~�i(t) = ~�i;1(t) + ~�i;2(t) + : : :+ ~�i;Ni(t) (28)10

equals the convolution of their corresponding densities~p�i(x) = ~p�i;1 (x) � ~p�i;2 (x) � : : : � ~p�i ;Ni(x) : (29)Although (29) provides us with a rule for the computation of the density ~p�i(x), it is moreappropriate to apply the concept of the characteristic function1.The characteristic function ~�i;n(�) of the random variable ~�i;n(t) is obtained by takingthe Fourier transform of (27)~�i;n(�) = Jo(2�ci;n�) : (30)The properties of characteristic functions are essentially the same as the properties ofFourier transforms. Thus, the convolution of the densities (29) reduces to the product oftheir characteristic functions, i.e.~�i(�) = ~�i;1 (�) � ~�i;2(�) � : : : � ~�i;Ni (�) (31a)= NiYn=1 Jo(2�ci;n �) : (31b)Finally, from the above equation the pdf ~p�i(x) can be obtained by taking the inverseFourier transform, i.e.~p�i(x) = 1Z�1 ~�i(�)e�j2��xd� = 2 1Z0 NiYn=1Jo(2�ci;n �) cos(2��x)d� : (32)Let fi;n 6= 0 and ci;n = �0p2=Ni for all n = 1; 2; : : : ; Ni and i = 1; 2, then the sum ~�i(t)(de�ned by (28)) is a random variable with zero mean and variance ~�20 = (c2i;1 + : : : +c2i;Ni)=2 = �20.The central limit theorem states that in the limit Ni ! 1 the random variable ~�i(t)tends to a Gaussian distributed random variable �i(t) with zero mean and variance �20,i.e. limNi!1 ~p�i(x) = p�i(x) = 1p2��0 e� x22�20 : (33)By taking the Fourier transform of the equation given above, we obtainlimNi!1 ~�i(�) = �i(�) = e�2(��0�)2 ; (34)and it follows by making use of (31b) the noteworthy result�Jo�2��0r 2Ni���Ni �!Ni!1 e�2(��0�)2 : (35)1The characteristic function of a random variable X is de�ned as the statistical average (�) =Efej2��Xg = R1�1 ej2��xpX(x)dx, where � is a real variable. In spite of the positive sign in the exponential(�) is often called the Fourier transform of pX (x).11

Of course, for a limited number of sinusoids Ni, we must write ~p�i(x) � p�i(x) and~�i(�) � �i(�). By considering Fig. 4(a), where ~p�i(x) has been evaluated for ci;n =�op2=Ni, we see that the approximation error is small and can be neglected for mostpractical applications if Ni � 7.Now, the following question arises: Does there exist an optimal set of Doppler coe�cientsfci;ng in such a sense that the pdf of the simulation system ~p�i(x) results for a givennumber of sinusoids Ni in an optimal approximation of the ideal Gaussian pdf p�i(x)?This question can be answered by considering the following L2-norm, which is also knownas the square root of the mean square error, i.e.E(2)p�i :=8<: 1Z�1 ���p�i(x)� ~p�i(x)���2 dx9=;1=2 : (36)After substituting (32) and (33) in (36), the above error function can be minimized nu-merically by using a suitable optimization procedure such as the method described in[18]. The minimization of (36) gives us an optimized set for the Doppler coe�cients c(opt)i;n .Fig. 4(b) shows the resulting pdf of the simulation system ~p�i(x) by using the optimizedquantities ci;n = c(opt)i;n . A proper choice of the initial values of the Doppler coe�cients isgiven by choosing ci;n = �op2=Ni. For these values the evaluation of the error functionE(2)p�i is shown in Fig. 5(a). The same �gure shows also the resulting error function E(2)p�i forthe optimized Doppler coe�cients ci;n = c(opt)i;n , which are for a given number of sinusoidsNi all identical after the minimization of (36). A further result worth mentioning is thatthe optimization procedure always decreases the initial values for the Doppler coe�cients,i.e. c(opt)i;n � �0p2=Ni, and therefore, the variance of the deterministic process ~�i(t) is for�nite Ni always smaller than the variance of the stochastic process �i(t) as depicted inFig. 5(b). But this e�ect diminishes if Ni increases and can completely be neglected forreasonably values of Ni, say Ni � 7. The consequence from the above is that the improve-ments made by the optimization procedure are negligible for most practical applicationsif Ni � 7, and thus, we can say for simplicity that c(opt)i;n is approximately given byci;n = �0p2=Ni if Ni � 7 (37)for all n = 1; 2; : : : ; Ni.Next, let us impose on this optimization approach the power constraint, i.e. ~�20 = �20.The power constraint is nothing else but a boundary condition, which can easily be in-cluded in the optimization procedure by optimizing the �rst Ni � 1 Doppler coe�cientsci;1; � � � ; ci;Ni�1, whereas the remaining Doppler coe�cient ci;Ni will be computed so that~�20 = �20 yields. It follows that the result (after applying the optimization procedure) isfor all Ni identical with (37) even if the initial values are choosen arbitrarily. Thus, (37)ensures for a given number of sinusoids Ni an optimal approximation of ideal Gaussianpdfs p�i(x) given the power constraint. Henceforth, we will make use of (37) if we considerthe approximation of stochastic processes with Gaussian distribution.In the following, we are interested in the amplitude and phase pdf of the deterministicsimulation system for Rice processes as represented in Fig. 3. We only consider the time-invariant direct component m = m1 + jm2 (see (4) with f� = 0), where the pdf of the12

real and imaginary part are given by pmi(xi) = �(xi �mi); i = 1; 2. Thus, the pdf of thedeterministic process ~��i(t) can be expressed by~p��i (xi) = ~p�i(xi) � pmi(xi) = ~p�i(xi �mi) : (38)Let us assume that the deterministic processes ~��1(t) and ~��2(t) are statistically indepen-dent, i.e. f1;n 6= f2;m for all n = 1; 2; : : : ; N1 and m = 1; 2; : : : ; N2, then we can expressthe joint pdf of ~��1(t) and ~��2(t), ~p��1��2 (x1; x2), as follows~p��1��2 (x1; x2) = ~p��1 (x1) � ~p��2 (x2) : (39)Finally, the transformation of the cartesian coordinates (x1; x2) to polar coordinates (z; �)by usingx1 = z cos � ; x2 = z sin � ; (40a,b)allows us to derive the joint pdf of the amplitude ~�(t) and phase ~#(t), ~p�#(z; �), in thefollowing way~p�#(z; �) = z ~p��1��2 (z cos �; z sin �) ; (41a)~p�#(z; �) = z ~p��1 (z cos �) � ~p��2 (z sin �) ; (41b)~p�#(z; �) = z ~p�1(z cos � � � cos ��) � ~p�2 (z sin � � � sin ��) : (41c)From (41c) we can directly compute the desired amplitude pdf ~p�(z) and phase pdf ~p#(�)of the deterministic simulation system according to~p�(z) = z �Z�� ~p�1(z cos � � � cos ��) � ~p�2(z sin � � � sin ��) d� ; (42a)~p#(�) = 1Z0 z~p�1(z cos � � � cos ��) � ~p�2(z sin � � � sin ��) dz : (42b)The result of the evaluation of (42a) and (42b) by using (32) is shown in the Figs. 6(a)and 6(b), respectively. Thereby, the Doppler coe�cients ci;n are computed from (37),and the number of sinusoids Ni has been restricted to N1 = N2 = 7. For the reasonsof comparison, we present also in the same �gures the behaviour of the ideal probabilitydensity functions p�(z) and p#(�) according to the eqs. (9) and (10), respectively.The proposed method not only applies to the approximation of Gaussian processes orstochastic processes derived from Gaussian processes (e.g. Rayleigh processes, Rice pro-cesses, and lognormal processes) but can also be used for the approximation of other kindsof distribution functions such as the Nakagami distribution.13

The Nakagami distribution [19], also denoted as m-distribution, o�ers a greater exibilityand often tends to �t the pdf of the received signal amplitude computed from experimentaldata of certain urban channels better than the common used Rayleigh or Rice distribution[1]. The Nakagami distribution with parameters and m is de�ned byp�(z) = 2mm z2m�1 e�(m=)z2�(m) m ; m � 1=2 ; z � 0 ; (43)where �(�) is the Gamma function, =< �(t)2 > denotes the time average power of thereceived signal strength, and m =< �(t)2 >2 = < (�(t)2� < �(t)2 >)2 > is the inverse ofthe normalized variance of �(t)2.We remark that for m = 1=2 and m = 1 the Nakagami distribution becomes a one-sidedGaussian and Rayleigh distribution, respectively; and �nally, the Rice and lognormal dis-tribution also can be approximated by the Nakagami distribution [19, 20]. For furtherdetails on the derivation and simulation of Nakagami fading channel models we refer to[21, 22].In order to �nd a suitable set for the Doppler coe�cients fci;ng, so that the amplitude pdfof the proposed simulation system ~p�(z) �ts the Nakagami distribution (43), we proceedin a similar way as we have successfully done for the minimization of the L2-norm (36)in connection with Gaussian distributions. We only have to replace in (36) the Gaussi-an distribution p�i(x) by the Nakagami distribution p�(z) as de�ned by (43), and ~p�i(x)has to be replaced by ~p�(z) as introduced by (42a). The optimization results for theNakagami distribution are shown for various values of m in Fig. 7, whereby in each caseN1 = N2 = 10 sinusoids have been used.Weibull distributions can be derived from uniform distributions, and on the other handuniform distributions can be derived from Gaussian distributions. Consequently, the de-rivation of deterministic Weibull processes is straightforward and will not be investigatedin detail.B. Determination of Optimal Discrete Doppler FrequenciesIn the previous subsection, we have discussed a method which enables us to compute theDoppler coe�cients ci;n in such a way that the pdf ~p�i(x) of the deterministic process~�i(t) results in an optimal approximation of the Gaussian pdf p�i(x) of the stochasticprocess �i(t). In the present subsection, we will see that the discrete Doppler frequenciesfi;n can be determined in such a way that the acf ~r�i�i(t) of the deterministic process ~�i(t)results in an optimal approximation of any desired acf r�i�i(t) of the stochastic process�i(t) within an appropriate time interval.The method discussed here is a modi�cation of the Lp-norm method [10]. The proposed(modi�ed) Lp-norm method assumes that the Doppler coe�cients ci;n are �xed and givenby (37), whereas optimal values for the discrete Doppler frequencies fi;n can be foundnumerically by minimizing the Lp-norm for p = 2, i.e. the square root of the mean square14

error norm,E(2)r�i�i := 24 1To ToZ0 jr�i�i(t)� ~r�i�i(t)j2 dt351=2 ; (44)where ~r�i�i(t) is given by (24a), r�i�i(t) can be any desired acf (e.g. resulting from "snapshot" measurements of real world mobile channels), and To is de�ned below and denotesan appropriate time interval [0; To] over which the approximation of r�i�i(t) is of interest.Our investigations of the performance of the method presented above will be restricted toboth Doppler power spectral densities S��(f) as introduced in Section II (i.e. Jakes andGaussian).For the Jakes psd and the frequency non-shifted Gaussian psd, we obtain | by using (6)and (8) - for the acf r�i�i(t) the expressionr�i�i(t) = r��(t)=2 =8>><>>: �20 Jo(2�fmaxt) (Jakes) ; (45a)�20 e��� fcpln 2 t�2 (Gauss) : (45b)Numerical investigations have shown that for these acfs appropriate values for To, i.e. theupper bound of the integral in (44), are given if To is adapted to the number of sinusoidsNi according toTo = 8<: Ni2fmax (Jakes) ;Ni4fcq ln 22 (Gauss) : (46)The evaluation of the error function E(2)r�i�i (see (44)) with the resulting optimized discreteDoppler frequencies fi;n = f (opt)i;n is shown for the Jakes psd and the Gaussian psd in theFigs. 8(a) and 8(b), respectively. For the sake of comparison, we also have shown inthese �gures the results obtained by applying two other e�cient parameter computationtechniques, which will be consisly described below.Method of Equal Areas [5, 6]:The principle of this method is as follows: Select a set of discrete Doppler frequencies ffi;ngin such a manner that in the range of fi;n�1 � f < fi;n the area under the (symmetrical)Doppler psd S�i�i(f) is equal to �20=(2Ni) for all n = 1; 2; : : : ; Ni with fi;0 = 0. Theapplication of the method of equal areas to the Jakes psd S�i�i(f) = S��(f)=2 (see (5))gives us for the discrete Doppler frequenciesfi;n = fmax sin� �n2Ni� ; (47)for all n = 1; 2; : : : ; Ni and i = 1; 2.Whereas for the frequency non-shifted Gaussian psd (see (7) with fs = 0) no closed-form expression for the discrete Doppler frequencies fi;n exists, and we must calculate thediscrete Doppler frequencies fi;n by �nding the zeros ofnNi � erf �fi;nfc pln 2� = 0 (48)15

for all n = 1; 2; : : : ; Ni and i = 1; 2.The Doppler coe�cients are exactly the same as introduced by (37). For a comprehensivederivation and discussion of the method of equal areas, we refer to [6], where also someapplications and simulation results can be found.Method of Exact Doppler Spread:This procedure is due to its simplicity and high performance predestinated to the appro-ximation of the acf r�i�i(t) as given by (45a), i.e. when the Jakes psd is of interest. Themethod can easily be derived by using the integral representation [23]Jo(z) = 2� Z �=20 cos(z sin�)d� ; (49)and replacing the above integral by a series expansion as followsJo(z) = limNi!1 2� NiXn=1 cos(z sin�n)�� ; (50)where �n := �(2n � 1)=(4Ni) and �� := �=(2Ni). Hence, we can write by using (45a)and (50)r�i�i(t) = �2o Jo(2�fmaxt) = limNi!1 �2oNi NiXn=1 cos(2�fi;nt) ; (51)wherefi;n = fmax sin � �2Ni (n� 12)� : (52)Finally, as for �nite values of Ni we require r�i�i(t) � ~r�i�i(t) , a comparison of (51) with(24a) leads to~r�i�i(t) = �2oNi NiXn=1 cos(2�fi;nt) : (53)A comparison of (53) with (24a) shows that the discrete Doppler frequencies fi;n are iden-tical with (52), and the Doppler coe�cients ci;n can immediately be identi�ed with thosegiven by (37).The Doppler psd S�i�i(f) is often characterized by the Doppler spread B�i , that is thesquare root of the second central moment (variance) of S�i�i(f). In [6] it has been shownthat the Doppler spread of deterministic simulation models, ~B�i, depends directly on thediscrete Doppler frequencies fi;n and Doppler coe�cients ci;n. Consequently, the appro-ximation quality B�i � ~B�i is mainly a�ected by the employed parameter computationmethod. In Section IV, we will see that for the Jakes psd the proposed method alwaysresults in a Doppler spread of the simulation model ~B�i that is identical with the Dopplerspread of the analytical model B�i, i.e. B�i = ~B�i. Therefore, we have designated the16

procedure as "method of exact Doppler spread".Clearly, the method of exact Doppler spread is adapted to the Jakes psd, but a comparisonof (47) and (52) motivates us - as a heuristic approach - to replace in (48) n by n� 1=2.This enables the computation of the discrete Doppler frequencies fi;n for the frequencynon-shifted Gaussian psd by �nding the zeros of2n� 12Ni � erf �fi;nfc pln 2� = 0 (54)for all n = 1; 2; : : : ; Ni and i = 1; 2. The Doppler coe�cients ci;n are remain unchangedand are further on given by (37).After the calculation of the discrete Doppler frequencies fi;n and the Doppler coe�cientsci;n by applying one of the three methods described above, the acf of the simulation model~r�i�i(t) can be computed by means of (24a). The result for ~r�i�i(t) that has been obtainedby applying the L2-norm method, the method of equal areas, and the method of exactDoppler spread are presented for the Jakes psd in Fig. 9(a) and for the frequency non-shifted Gaussian psd in Fig. 9(b). In all cases, the number of sinusoids Ni was determinedbyNi = 7. Observe that we have also depicted in Fig. 9(a) and Fig. 9(b) the correspondingacfs of the analytical model r�i�i(t) = �2oJo(2�fmaxt) and r�i�i(t) = �2o exp[�(�fct)2= ln 2],respectively. This allows us directly to compare and criticize the performance of thevarious parameter computation methods. The L2-norm method is more complex andrequires higher numerical e�ort than the method of exact Doppler spread, but surprisin-gly, we realize from Fig. 8(a) and Fig. 9(a) that these two methods are yielding nearlyequivalent and excellent approximation results in the interval 0 � t � To. Thus, due toits simplicity and high approximation quality, we prefer for the Jakes psd the method ofexact Doppler spread. We gather from the Figs. 8(b) and 9(b) that for the Gaussian psdthe method of exact Doppler spread results in essential smaller approximation errors thanthe method of equal areas. A comparison of the performance of these two methods withthe L2-norm method reveals undoubtedly the advantage of that optimization procedure.Therefore, we prefer for the Gaussian psd the L2-norm method.Finally, we mention that the period of the deterministic process ~�i(t) (see (22)) is inver-sely proportional to the greatest common devisor of the discrete Doppler frequencies fi;n.For all discussed methods, the greatest common devisor can be made small enough forrealistic values of Ni, say Ni � 7, simply by increasing the word length of the discreteDoppler frequencies fi;n; and thus, the period of ~�i(t) can be made extremely large.C. Determination of the Doppler PhasesThe Doppler phases �i;n are uniformly distributed random variables, which can be obtai-ned simply by means of a random generator with uniform distribution over the interval(0; 2�]. But for deterministic simulation models, where the Doppler coe�cients ci;n andthe discrete Doppler frequencies fi;n are all determined by using deterministic methods,it is an obvious idea to introduce also for the computation of the Doppler phases �i;n adeterministic approach. 17

Therefore, we consider the following standard phase vector ~�i with Ni deterministic phasecomponents~�i = �2� 1Ni + 1 ; 2� 2Ni + 1 ; : : : ; 2� NiNi + 1� ; (55)and we group the Doppler phases �i;n of the Ni sinusoids to a so called Doppler phasevector ~�i in accordance with~�i = (�i;1 ; �i;2 ; : : : ; �i;Ni) (56)for i = 1; 2. Now, the components of the Doppler phase vector ~�i can be identi�ed withthe components of the standard phase vector ~�i after permutating the components of ~�i.In this way, for a given number of sinusoids Ni one can construct Ni! di�erent Dopplerphase vectors ~�i with equally distributed components.Let us assume that the real deterministic processes ~�1(t) and ~�2(t) are uncorrelated, thenit can be shown that the corresponding Doppler phases �1;n and �2;n have no in uenceon the statistical properties of the complex deterministic process ~�(t) = ~�1(t) + j~�2(t).Consequently, for one and the same sets of Doppler coe�cients fci;ng and discrete Dopplerfrequencies ffi;ng it is possible to construct Ni! di�erent sets of Doppler phases f�i;ng,and thus N1! � N2! di�erent complex deterministic processes ~�(t) = ~�1(t) + j~�2(t) withdi�erent time behaviour but identical statistics can be realized. This is of great advantagefor simulating channels for mobile digital transmission systems with frequency hopping.In such cases, a frequency hop necessitates a new permutation of the Doppler phases.On the other side, if ~�1(t) and ~�2(t) are correlated, then the Doppler phases �i;n have anin uence on the cross-correlation function ~r�1�2(t), as can easily be seen by considering(25). In this case, the Doppler phases �i;n can be designed such that they control thehigher order statistics of ~�(t) = j~�(t) + m(t)j, e.g. the simulation model's level-crossingrate and the average duration of fades [24].

18

IV. HIGHER ORDER STATISTICALPROPERTIES OF DETERMINISTICSIMULATION MODELSIn this section, we derive for the deterministic simulation models of Figs. 3(a) and 3(b)analytical expressions for the level-crossing rate ~N�(r) and the average duration of fades~T��(r). The analytical expressions do not only allow us to avoid the determination of~N�(r) and ~T��(r) from time consuming simulations of the channel output signal ~�(t), butalso enable us to study the degradation e�ects due to the selected parameter computati-on method and the limited number of sinusoids Ni. Moreover, we investigate the pdf offading time intervals by using the Jakes and Gaussian psds.In Chapter A of Section III it has been shown that the pdf of the deterministic process~�i(t), ~p�i(x), is nearly identical with the pdf of the stochastic process �i(t), p�i(x), ifthe number of sinusoids Ni is su�ciently large, say Ni � 7. On the assumption that~p�i(x) = p�i(x) yields, the level-crossing rate of the deterministic simulation model ~N�(r)is furthermore de�ned by (12); one merely has to replace the quantities of the analyticalmodel � and � (consider therefore (13) and (14)) by the corresponding quantities of thesimulation model ~� and ~�. Thus~� = 2�f� � _~�0~ 0!�q2~� ; (57)~� = ��~ 0 � _~�20~ 0 ; (58)where �~ 0, _~�0 are related by the acf ~r�1�1(t) (see (24a)) and ccf ~r�1�2(t) (see (25)) of thesimulation model as follows�~ 0 = d2dt2 ~r�1�1(t)���t=0 = �~r�1�1(0) ; (59)_~�0 = ddt~r�1�2(t)���t=0 = _~r�1�2(0) ; (60)and ~ 0 = ~r�1�1(0) = PNin=1 c21;n=2. Let the Doppler coe�cients ci;n are as given by (37),then ~ 0 = 0 yields.The quality of the approximations � � ~� and � � ~� depends on the underlying Dopplerpsd and will be investigated separatly for the Jakes and frequency shifted Gaussian psds.A. Higher Order Statistical Properties by Using the JakesPSDThe Jakes psd (5) is a symmetrical function and thus, it follows from the properties ofthe Fourier transform that the ccf r�1�2(t) is zero, i.e. �1(t) and �2(t) are uncorrelated.19

Now, we impose the same requirement on the simulation system. From (25) it followsthat ~r�1�2(t) = 0 can easily be achieved for all design methods discussed here by selectingthe number of sinusoids such that N1 = N2� 1 yields. Then N1 is unequal to N2 and perde�nition the discrete Doppler frequencies are designed in such a way that f1;n 6= �f2;mholds for all n = 1; 2; : : : ; N1 and m = 1; 2; : : : ; N2. Thus, it follows from (57) - (60) inconnection with (24a) that~� = 2�f�=q2~� (61)and ~� = ��~ 0 = 2�2 N1Xn=1(c1;nf1;n)2 : (62)The evaluation of the above equation by using the method of exact Doppler spread canbe performed by substituting (37) and (52) in (62), i.e.~� = (2��0fmax)2 � 1N1 N1Xn=1 sin2 � �2N1 (n� 12)� : (63)By taking notice of 0 = �20 and making use of the relation1N1 N1Xn=1 sin2 � �2N1 (n� 12)� = 12 ; (64)we �nally obtain for ~� and ~� the result~� = 1p 0 � f�fmax ; (65)~� = 2 0(�fmax)2 : (66)A comparison of (20a) and (21a) with (65) and (66) reveals that ~� = � and ~� = �, i.e.,the approximation error is zero, and the Doppler spreads ~B�i and B�i are identical too,because ~B�i =q~�= o=(2�) and B�i =p�= o=(2�).In a similar way, we can compute ~� by using the method of equal areas. In this case, weobtain ~� = �(1 + 1=Ni) and ~� tends to � when Ni ! 1. Let us for a moment assumethat the Doppler frequency of the direct component f� is equal to zero, i.e. f� = 0, then� = ~� = 0 and the lcr N�(r) reduces to (17). Now the relative error of ~N�(r) can beexpressed by" ~N� = ~N�(r) �N�(r)N�(r) = q~� �p�p� : (67)20

The substitution of ~� = �(1+1=Ni) in (67) and the use of the approximationq1 + 1Ni �1 + 12Ni allows us to express the relative error " ~N� directly as function of the number ofsinusoids" ~N� � 12Ni : (68)The preceding equation shows that by using the method of equal areas about 25 sinusoidsare required for the design of ~�1(t) and ~�2(t) in order to reduce the relative error of ~N�(r)to 2 %.It shoud be noted that for the L2-norm method no closed-form expression for ~� can bederived. The use of that method leads after the numerical evaluation of (62) to the resultsfor the normalized quantity ~�=f2max as depicted in Fig. 10(a). This �gure also shows thebehaviour of ~�=f2max as function of the number of sinusoids Ni by using the method ofequal areas and the method of exact Doppler spread.Next, we consider the deterministic simulation model of Fig. 3(a), and we calculate direct-ly the lcr ~N�(r) by counting the number of crossings per second at which the simulatedchannel output signal ~�(t) crosses a speci�ed signal level r with positive slope. Therefore,we have chosen a maximum Doppler frequency fmax (see (5)) of fmax = 91:73 Hz, thatcorresponds to a mobile speed of 110 km/h and a carrier frequency of 900 MHz. Thevariance �20 was normalized to �20 = 1=2. The parameters of the deterministic simulationsystem have been designed according to the method of exact Doppler spread with N1 = 7and N2 = 8 sinusoids, so that the cross-correlation between the real part ~�1(t) and theimaginary part ~�2(t) is zero. For the realization of the discrete-time deterministic simula-tion system we have used a sampling interval T of T = 3 �10�4s and altogether K = 4 �105samples of the deterministic process ~�(kT ), k = 0; 1; : : : ;K � 1, have been simulated forthe evaluation of the lcr ~N�(r). The result for the normalized lcr of the deterministicsimulation model ~N�(r)=fmax is presented in Fig. 11(a). This �gure demonstrates for adeterministic Rice process with � = 1 the in uence of the Doppler frequency of the directcomponent f� on the behaviour of ~N�(r). For the reasons of comparison, we also haveshown in Fig. 11(a) the lcr of the (ideal) analytical model N�(r) as de�ned by (12). Itcan be seen that the lcr of the simulation model ~N�(r) coincides extremely closely withthe lcr of the theoretical model N�(r) by applying the method of exact Doppler spread.B. Higher Order Statistical Properties by Using the GaussianPSDThe Gaussian psd (7) is for fs 6= 0 an unsymmetrical function. Consequently, the ccfr�1�2(t) is unequal to zero and thus �1(t) and �2(t) are correlated. Let the stochasticprocesses �1(t) and �2(t) (see Fig. 2(b)) are uncorrelated, then we yield after some simplecomputations the following expressions for the acf r�1�1(t) and ccf r�1�2(t)r�1�1(t) = 12 [r�1�1(t) + r�2�2(t)] cos(2�fst) ; (69a)r�1�2(t) = 12 [r�1�1(t) + r�2�2(t)] sin(2�fst) ; (69b)21

respectively. Starting with the above equations and using r�i�i(0) = �20 = 0, we can showthat (15) and (16) are resulting for the frequency shifted Gaussian psd in� 0 = � o h2(�fc=pln 2)2 + (2�fs)2i ; (70a)_�0 = 0 � 2�fs ; (70b)respectively.In a similar way, the corresponding quantities of the deterministic simulation model canbe derived. The result is�~ 0 = ��2 " N1Xn=1(c1;nf1;n)2 + N2Xn=1(c2;nf2;n)2#� 0(2�fs)2 ; (71a)_~�0 = 0 � 2�fs ; (71b)where we have used ~r�i�i(0) = ~ 0 = 0, i.e., the Doppler coe�cients are given by ci;n =�0p2=Ni. A comparison of (70b) with (71b) shows that _~�0 = _�0. Now, we are ableto compute the quantities ~� and ~�, which determine the approximation quality of thedeterministic simulation model's level-crossing rate. Therefore, we substitute (71a), (71b)in (57), (58) and obtain �nally~� = �s2~� (f� � fs) ; (72)~� = �2 " N1Xn=1(c1;nf1;n)2 + N2Xn=1(c2;nf2;n)2# : (73)Eq. (72) allows further observations: when ~� ! �, ~� tends to � and consequently ~N�(r)tends to N�(r). Obviously, the quantity ~� is responsible for the approximation qualityof the higher order statistical properties of the deterministic simulation model. For allparameter computation methods discussed in Section III, ~� can not be expressed in aclosed-form for the Gaussian psd, so that (73) has to be evaluated numerically. Fig. 10(b)shows the numerical evaluation results for the normalized quantity ~�=f2c , which have be-en obtained by employing all parameter computation methods investigated in the presentpaper.Now, let us consider Fig. 3(b) and let us determine for the frequency shifted Gaussianpsd the lcr ~N�(r) from the simulated sequence ~�(kT ), k = 0; 1; : : : ;K � 1, in the sameway as we have already done for the Jakes psd. Therefore, we have selected the followingmodel parameters: T = 3 � 10�4s, K = 4 � 105, fc = pln 2 � fmax, fmax = 91:73 Hz, � = 1,f� = 0, and in order to study the in uence of the frequency shift fs, we have used thevalues fs 2 �0; 23fmax; fmax.Observe, by considering of (21), that the 3-dB cuto� frequency fc = pln 2 � fmax wasselected such that the quantity � is identical for both power spectral densities (Jakes22

and Gaussian). The discrete Doppler frequencies fi;n and Doppler coe�cients ci;n havebeen determined by the method of exact Doppler spread with N1 = 25 and N2 = 26.The results for the normalized level crossing rates of the deterministic simulation model~N�(r)=fmax as well as of the analytical model N�(r)=fmax are shown in Fig. 11(b). A com-parison of Fig. 11(a) with Fig. 11(b) shows the same result for both the analytical andthe simulation model: identical level-crossing rates are observed | as it was expected bythe theory | although the underlying Doppler power spectral densities are quite di�erent.In a similar way, an expression for the average duration of fades of the simulation model,~T��(r), can be derived. One merely has to replace in (18) N�(r) by ~N�(r) and P��(r) by~P��(r). Thus, the analytical and numerical investigations of ~T��(r) are straightforwardand will not be presented here.C. Distribution of the Fading Time IntervalsThe performance of mobile communication systems can mainly be improved by an e�-cient design of the interleaver/deinterleaver and the channel coding/decoding unit. Forthe speci�cation of these units, not only the knowledge of the level-crossing rate and theaverage duration of fades is of special interest, but also the distribution of the fading timeintervals. The rest of this section is focused on the probability density function of thefading time intervals that will be denoted here by p0�(��; r). This pdf p0�(��; r) is theconditional probability density that the stochastic process �(t) crosses a certain level r forthe �rst time at t2 = t1 + �� with positive slope given a crossing with negative slope attime t1. An exact analytical evaluation of p0�(��; r) is still an unsolved problem even forRayleigh and Rice processes. So we will determine ~p0�(��; r) experimentally by employingdeterministic simulation systems. For that purpose, we simulate ~�(kT ) with a samplinginterval of T = 0:5 � 10�4s, and we �x � = 0, i.e. ~�(kT ) is a deterministic Rayleighprocess. Typical measurement results of ~p0�(��; r) for r = 0:1 and r = 1 by evaluating2 � 104 fading time intervals �� are presented in Figs. 12(a) and 12(b) for the Jakes psdand in Figs. 12(c) and 12(d) for the Gaussian psd, respectively, whereby we have appliedthe method of exact Doppler spread by using various values for the number of sinusoidsNi. The remaining model parameters were fmax = 91 Hz, fc = fmaxpln 2; �2o = 1=2, andfs = 0.Approximative solutions for p0�(��; r) can be found for example in [25, 26, 27, 28, 29].The �rst approximation to p0�(��; r) is p1�(��; r), where p1�(��; r) is the conditionalprobability density that the stochastic process �(t) passes upwards through a certain levelr at time t2 = t1 + �� given that �(t) has passed downward through r at time t1. Note,that nothing is said about the behaviour of �(t) between t = t1 and t = t2. In [25] ananalytical solution for p1�(��; r) can be found that tends for deep fades top1�(��; r)! � ddu � 2uT�� I1( 2�u2 )e� 2�u2 � ; if r! 0 ; (74)where u = ��=T��(r) and I1(�) denotes the modi�ed Bessel function of �rst order. Thegraph of this function also is shown for the Rayleigh process �(t) with underlying Jakesand Gaussian psd characteristics in Figs. 12(a) and 12(c), respectively. These �gures im-pressivly show the excellent accordance between the theoretical approximation p1�(��; r)23

and the measurement results ~p0�(��; r) for deep fades, i.e. for small values of r, where theprobability that only short fading time intervals �� occur is high. Moreover, Fig. 12 led usto the hypothesis that stochastic processes �(t) with di�erent Doppler psd characteristicsbut identical values for the quantities � and � have identical probability density functionsof the fading time intervals p0�(��; r) for small values of r. Assuming that the hypothesisis true, then some consequences arise for the modelling of real-world mobile channels onthe basis of measurements of the statistics of the received signal. We conclude from theabove that for the simulation model's Doppler psd it is not necessary to match exactlythe measured Doppler psd, but the approximation quality of the respective autocorrela-tion functions around the time origin is of utmost importance for the modelling of themeasured channel statistics.

24

V. CONCLUSIONIn this paper we have investigated the statistical properties of deterministic simulationmodels for mobile fading channels. Especially for frequency non-selective deterministicsimulation models, we have not only derived analytical expressions for the probabilitydensity function of the amplitude and phase, but also higher order statistics like level-crossing rate and average duration of fades have been investigated analytically. This makesthe time consuming evaluation and estimation of the statistics from simulated channeloutput sequences super uous. Moreover, the derived analytical expressions allow detailedinvestigations of the degradation e�ects due to a limited number of sinusoids or due tothe applied computation method for the simulation model's discrete Doppler frequenciesand Doppler coe�cients, and they provide us with a powerful tool when discussing andcomparing the e�ciency of di�erent parameter computation methods.A performance study of three e�cient parameter computation methods was also a topicof the present paper, whereby two of the considered methods are new.In particular for the often used Jakes and Gaussian Doppler power spectral densities, itturned out by a comparison of the so called method of exact Doppler spread with the me-thod of equal areas that the former allows a drastic reduction in the number of sinusoids,without producing noteworthy deviations from the desired (ideal) statistics. The thirdintroduced method, which we have named L2-norm method, requires a greater numericalexpenditure than the other two methods. This method develops its full performance bythe approximation of measured autocorrelation functions obtained from real-world mobileradio channels, where other more conventional parameter computation methods tend tofail.

25

References[1] H. Suzuki, \A statistical model for urban radio propagation," IEEE Trans. Com-mun., vol. COM-25, no. 7, pp. 673{680, July 1977.[2] J.D. Parsons, The Mobile Radio Propagation Channel. London: Pentech Press, 1992,p. 166.[3] S.O. Rice, \Mathematical analysis of random noise," Bell Syst. Tech. Journal, vol. 23,pp. 282{332, July 1944.[4] S.O. Rice, \Mathematical analysis of random noise," Bell Syst. Tech. Journal, vol. 24,pp. 46{156, Jan. 1945.[5] M. P�atzold, U. Killat, F. Laue, \A deterministic model for a shadowed Rayleighland mobile radio channel," in Proc. of the Fifth IEEE International Symposium onPersonal, Indoor, and Mobile Radio Communications (PIMRC'94), pp. 1202{1210,The Hague, Netherlands, Sept. 1994.[6] M. P�atzold, U. Killat, F. Laue, \A deterministic digital simulation model for Suzukiprocesses with application to a shadowed Rayleigh land mobile radio channel," IEEETrans. Veh. Technol., vol. VT-45, no.2, pp. 318-331, May 1996.[7] H. Schulze, \Stochastic models and digital simulation of mobile channels (in Ger-man)," in U.R.S.I/ITG Conf. in Kleinheubach 1988, Germany (FR), vol. 32,pp. 473{483, Proc. Kleinheubacher Berichte by the German PTT, Darmstadt, Ger-many, 1989.[8] P. H�oher, \A statistical discrete-time model for the WSSUS multipath channel,"IEEE Trans. Veh. Technol., vol. VT-41, no. 4, pp. 461{468, Nov. 1992.[9] M. P�atzold, U. Killat, F. Laue, Y. Li, \On the problems of Monte Carlo methodbased simulation models for mobile radio channels," in Proc. IEEE 4th Int. Symp.on Spread Spectrum Techniques & Applications (ISSSTA'96), Mainz, Germany, pp.1214-1220, Sept. 1996.[10] M. P�atzold, U. Killat, Y. Shi, F. Laue, \A deterministic method for the derivationof a discrete WSSUS multipath fading channel model," European Transactions onTelecommunications and Related Technologies (ETT), vol. ETT-7, no. 2, pp. 165{175,March{April 1996.[11] W.C. Jakes, Ed., Microwave Mobile Communications. New York: IEEE Press, 1993.[12] E.F. Casas, C. Leung, \A simple digital fading simulator for mobile Radio," inProceedings of the IEEE Vehicular Technology Conference, pp. 212{217, Sept. 1988.[13] CEPT/COST 207 WG1, \Proposal on channel transfer functions to be used in GSMtests late 1986," COST 207 TD (86)51 Rev. 3, Sept. 1986.[14] A. Neul, J. Hagenauer, W. Papke, F. Dolainsky, F. Edbauer, \Aeronautical channelcharacterization based on measurement ights," IEEE Conf. GLOBECOM'87, pp.1654{1659, Tokyo, Japan, Nov. 1987. 26

[15] P.A. Bello, \Aeronautical channel characterization," IEEE Trans. Commun.,vol. COM-21, pp. 548{563, May 1973.[16] A. Papoulis, Probabilities, Random Variables, and Stochastic Processes. New York:McGraw-Hill, 3 edition, 1991.[17] M. P�atzold, U. Killat, F. Laue, \An extended Suzuki model for land mobile satellitechannels and its statistical properties," IEEE Trans. Veh. Technol., submitted andaccepted for publication, 1995.[18] R. Fletcher, M.J.D. Powell, \A rapidly convergent descent method for minimization,"Computer Journal, vol. CJ-6, no. 2, pp. 163{168, 1963.[19] M. Nakagami, \The m-distribution { A general formula of intensity distribution ofrapid fading," in Statistical Methods in Radio Wave Propagation, W.G. Ho�man(Ed.), Oxford, England: Pergamon Press, 1960.[20] U. Charash, \Reception through Nakagami fading multipath channels with randomdelays," IEEE Trans. Commun., vol. COM-27, no. 4, pp. 657{670, Apr. 1979.[21] W.R. Braun, U. Dersch, \A physical mobile radio channel model," IEEE Trans.Veh. Technol., vol. VT-40, no. 2, pp. 472{482, May 1991.[22] U. Dersch, R.J. R�uegg, \Simulations of the time and frequency selective outdoormobile radio channel," IEEE Trans. Veh. Technol., vol. VT-42, no. 3, pp. 338{344,Aug. 1993.[23] M. Abramowitz, I.A. Stegun, Handbook of Mathematical Functions with Formulas,Graphs, and Mathematical Tables. Washington: National Bureau of Standards, 1972.[24] M. P�atzold, U. Killat, F. Laue, Y. Li, \An e�cient deterministic simulation model forland mobile satellite channels," in Proc. IEEE Veh. Tech. Conf., Atlanta, Georgia,USA, pp. 1028{1032, Apr./May 1996.[25] S.O. Rice, \Distribution of the duration of fades in radio transmission: Gaussiannoise model," Bell Syst. Tech. Journal, vol. 37, pp. 581{635, May 1958.[26] J.A. McFadden, \The axis-crossing intervals of random functions-II," IRE Trans.Inform. Theory, vol. IT-4, pp. 14{24, Mar. 1958.[27] M.S. Longuet-Higgins, \The distribution of intervals between zeros of a stationaryrandom function," Phil. Trans. Royal. Soc., vol. A 254, pp. 557{599, 1962.[28] A.J. Rainal, \Axis-crossing intervals of Rayleigh processes," Bell Syst. Tech. Journal,vol. 44, pp. 1219{1224, 1965.[29] D. Wolf, T. Munakata, J. Wehhofer, \Statistical properties of Rice fading processes,"in Signal Processing II: Theories and Applications, Proc. Eusipco-83 Second EuropeanSignal Processing Conference, H.W. Sch�u�ler (Ed.), Erlangen, Germany, pp. 17{20,Elsevier Science Publishers B.V. (North-Holland), 1983.27

LIST OF FIGURE CAPTIONSFig. 1: Models for the realization of coloured Gaussian noise processes(a) by �ltering a white Gaussian noice process and(b) by a �nite sum of weighted sinusoids.Fig. 2: Stochastic analytical models for Rice processes with given Doppler psd shapeaccording to(a) the Jakes psd, where Hi(f) = �0=[�fmaxp1� (f=fmax)2]1=2, and(b) the frequency shifted Gaussian psd, whereHi(f) = �0pfc � ln 2� �1=4 exp��pln 2f=fc�.Fig. 3: Deterministic simulation models for Rice processes with given Doppler psdshape according to (a) the Jakes psd and (b) the frequency shifted Gaussianpsd.Fig. 4: The approximated Gaussian pdf ~p�i(x) for(a) ci;n = �0p2=Ni and(b) ci;n = c(opt)i;n for Ni = 5; 7;1 and �20 = 1.Fig. 5: (a) Error function E(2)p�i and(b) variance ~�20 of the deterministic process ~�i(t) for ci;n = �0p2=Ni (ooo) andci;n = c(opt)i;n (***) with �20 = 1.Fig. 6: (a) Amplitude pdfs p�(z) (analytical model), ~p�(z) (simulaion model), and(b) phase pdfs p#(�) (analytical model), ~p#(�) (simulation model) for N1 =N2 = 7 (f� = 0; �� = �; �20 = 1=2).Fig. 7: Approximation of the Nakagami distribution with N1 = N2 = 10 ( = 1).Fig. 8: Error function E(2)r�i�i in connection with (a) the Jakes psd and (b) the Gaussianpsd (�20 = 1, fmax = fc = 91Hz; fs = 0).Fig. 9: The ideal autocorrelation functions (a) r�i�i(t) = �20J0(2�fmaxt) (Jakes psd)and (b) r�i�i(t) = �20 exp [�(�fct)2= ln 2] (Gaussian psd) in comparison with therespective approximated autocorrelation functions ~r�i�i(t) for Ni = 7 (�20 =1, fmax = fc = 91Hz, fs = 0).Fig. 10: Characteristic quantities � and ~� of S�i�i(f) and ~S�i�i(f), respectively, for:(a) the Jakes psd and (b) the Gaussian psd.Fig. 11: The normalized level-crossing rate of the analytical modelN�(r)=fmax for �20 =1=2, � = 1 and of the simulation model for (a) the Jakes psd (N1 = 7, N2 = 8)and (b) the Gaussian psd (N1 = 25, N2 = 26) with fc = fmaxpln 2 and f� = 0.Fig. 12: The pdf of the fading time intervals ~p0�(� ; r) of the deterministic Rayleighprocess ~�(t) by using the Jakes psd (�20 = 1=2; fmax = 91 Hz; � = 0) if thesignal level ~�(t) = r is (a) r = 0:1 and (b) r = 1, as well as by using theGaussian psd (�20 = 1=2; fc = fmaxpln 2; � = 0) if (c) r = 0:1 and (d) r = 1.28

(a)(b)Figure 1

29

(a)(b)

Figure 230

(a)(b)

Figure 331

(a)-3 -2 -1 0 1 2 30

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

p̃µi(x)

x →

Ni = ∞

Ni = 7

Ni = 5

ci,n = σo (2/Ni)1/2

(b)-3 -2 -1 0 1 2 30

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

p̃µi(x)

x →

Ni = ∞

Ni = 7

Ni = 5

ci,n = c(opt)i,n

Figure 432

(a)0 5 10 15 20 25 30

0

0.005

0.01

0.015

0.02E(2)

pµi

Ni →

o E(2)pµi

if ci,n = σo (2/Ni)1/2

* E(2)pµi

if ci,n = c(opt)i,n

(b)0 5 10 15 20 25 30

0.5

0.6

0.7

0.8

0.9

1

1.1σ̃2

o

Ni →

o σ̃2o if ci,n = σo (2/Ni)

1/2

* σ̃2o if ci,n = c(opt)

i,n

Figure 533

(a)0 0.5 1 1.5 2 2.5

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

z →

pξ(z), p̃ξ(z)

ρ= 0

ρ= 1/2

ρ= 1

analy. model

simul. model(b) 0 π 2 π0

0.1

0.2

0.3

0.4

0.5 ρ= 1

ρ=1/2

ρ= 0

analy. model

simul. model

pϑ (θ), p̃ϑ (θ)

θ →Figure 634

0 0.5 1 1.5 2 2.5 30

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6Nakagami distribution

z →

pξ(z) (analy. model)

p̃ξ(z) (simul. model)

m=3

m=2

m=1

Figure 7

35

(a)

0 5 10 15 20 25 300

0.02

0.04

0.06

0.08

0.1

0.12

0.14E(2)

rµiµi

Ni →

Method of Exact Doppler Spread

Method of Equal Areas

Lp-Norm Method

(b)0 5 10 15 20 25 30

0

0.1

0.2

0.3

0.4

0.5E(2)

rµiµi

Ni →

Method of Exact Doppler Spread

Method of Equal Areas

Lp-Norm Method

Figure 836

(a) 0 0.025 To 0.05 0.075

-0.5

0

0.5

1

rµiµi(t), r̃µiµi

(t)

t/sec →

Method of Exact Doppler Spread

Method of Equal Areas

L2-Norm Method

Analytical Model

(b) 0 0.002 0.004 0.006 0.008 0.010 To

-0.2

0

0.2

0.4

0.6

0.8

1

rµiµi(t), r̃µiµi

(t)

t/sec →

Method of Exact Doppler Spread

Method of Equal Areas

L2-Norm Method

Analytical Model

Figure 937

(a)0 5 10 15 20 25 30

20

30

β/f2max, β̃/f2

max

Ni →

Method of Exact Doppler SpreadMethod of Equal Areas

L2-Norm Method

Analytical Model-

(b)0 5 10 15 20 25 30

0

50

100

150

200

250

300

350

400

450β/f2

c, β̃/f2c

Ni →

Method of Exact Doppler Spread

Method of Equal Areas

L2-Norm Method

Analytical Model-

Figure 1038

(a)0 1 2 3 4

0

0.2

0.4

0.6

0.8

1

1.2

Nξ(r)/fmax

r

Analytical Model

Simulation (N1=7,

N2=8)

fρ/fmax = 1

fρ/fmax = 2/3

fρ/fmax = 0 (b)0 1 2 3 4

0

0.2

0.4

0.6

0.8

1

1.2

Nξ(r)/fmax

r

Analytical Model

Simulation (N1=25,

N2=26)

fs/fmax = 1

fs/fmax = 2/3

fs/fmax = 0

Figure 1139

(a)0 1 2 3 4

0

0.2

0.4

0.6

0.8

1

N1= 4, N2= 5

N1= 6, N2= 7

N1=24, N2=25

analytical approx.

( p1-(τ -,r) )

ρ = 0

fρ = 0

r = 0.1

p̃0-(τ -,r) ⋅ Tξ -

τ -/Tξ -

(b)0 1 2 3 4

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8p̃0-

(τ -,r) ⋅ Tξ -

τ -/Tξ -

N1= 4, N2= 5N1= 6, N2= 7N1=24, N2=25

ρ = 0fρ = 0r = 1(c)

0 1 2 3 40

0.2

0.4

0.6

0.8

1

N1=5, N2=6

N1=25, N2=26

N1=50, N2=51

analytical approx.

( p1-(τ -,r) )

ρ = 0

fs = 0

r = 0.1

p̃0-(τ -,r) ⋅ Tξ -

τ -/Tξ -

(d)0 1 2 3 4

0

0.2

0.4

0.6

0.8

1

N1=5, N2=6

N1=25, N2=26

N1=50, N2=51

ρ = 0

fs = 0

r = 1

p̃0-(τ -,r) ⋅ Tξ -

τ -/Tξ -Figure 12

40