MATTHIAS P ATZOLD, MEMBER, IEEE, ULRICH KILLAT, MEMBER, IEEE, FRANK LAUE, YINGCHUN LI. Technical University of Hamburg-Harburg.

Transcription

1 On the Statistical Properties of Deterministic Simulation Models for Mobile Fading Channels MATTHIAS P ATZOLD, MEMBER, IEEE, ULRICH KILLAT, MEMBER, IEEE, FRANK LAUE, YINGCHUN LI Technical University of Hamburg-Harburg Department of Digital Communication Systems Postfach 9 5 Denickestr. 7 D-7 Hamburg Tel. : Fax : paetzold@tu-harburg.d4.de Abstract Rice's sum of sinusoids can be used for an ecient approximation of coloured Gaussian noise processes and is therefore of great importance to the software and hardware realization of mobile fading channel models. Although several methods can be found in the literature for the computation of the parameters characterizing a proper set of sinusoids, only less is reported about the statistical properties of the resulting (deterministic) simulation model. This is the topic of the present paper; whereby not only the simulation model's amplitude and phase probability density function will be investigated, but also higher order statistics (e.g. level-crossing rate and average duration of fades). It will be shown that due to the deterministic nature of the simulation model analytical expressions for the probability density function of amplitude and phase, autocorrelation function, level-crossing rate, and average duration of fades can be derived. We also propose a new procedure for the determination of an optimal set of sinusoids, i.e., the method results for a given number of sinusoids in an optimal approximation of Gaussian, Rayleigh, and Rice processes with given Doppler power spectral density properties. It will be shown that the new method can easily be applied to the approximation of various other kinds of distribution functions, such as the Nakagami and Weibull distributions. Moreover, a quasi optimal parameter computation method is presented. Submitted to IEEE Transactions on Vehicular Technology September 995 Final Version

2 I. INTRODUCTION In general, simulation models for mobile radio channels are realized by employing at least two or more coloured Gaussian noise processes. For instance, for the realization of a Rayleigh or Rice process two real coloured Gaussian noise processes are required; whereas the realization of a Suzuki process [], which is a product process of a Rayleigh and a lognormal process, is based on three real coloured Gaussian noise processes. Such processes (Rayleigh, Rice, Suzuki) are often used as appropriate stochastical models for describing the fading behaviour of the envelope of the signal received over land mobile radio channels, which are often classied as frequency non-selective channels. A further example is given by modelling a (n-path) frequency selective mobile radio propagation channel by using the n-tap delay line model []. This requires the realization of n real coloured Gaussian noise processes. All these examples show us that an ecient design method for the realization of coloured Gaussian noise processes is of particular importance in the area of mobile radio channel modelling. A well known method for the design of coloured Gaussian noise processes is to shape a white 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 fading process (see Fig. (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 is approximated by a nite sum of weighted and properly designed sinusoids. The resulting signal ow diagram of such an approximated Gaussian noise process is shown in Fig. (b), where the quantities c n, f n ; and n are called Doppler coecients, discrete Doppler frequencies, and Doppler phases, respectively, and N denotes the number of sinusoids. In the meanwhile, diverse methods have been developed for the derivation of the relevant model parameters (Doppler coecients c n and discrete Doppler frequencies f n ). For example, the method of equal distances [5, 6] and the mean square error method [6]. The characteristic of both methods is just the same as Rice's original method [3, 4], namely, the dierence between two adjacent discrete Doppler frequencies is equidistant; but the Doppler coecients are adapted to a given Doppler power spectral density in a method specic manner. The disadvantage of these methods is that (due to the equidistant discrete Doppler frequencies) the approximated coloured Gaussian noise process ~(t) is periodical. Another method is known as the method of equal areas [5,6]. That procedure results in a relativly satisfactory approximation of the desired statistics for the Jakes Doppler power spectral density even for a small number of sinusoids but fails or requires a comparatively large number of sinusoids for other types of Doppler power spectral densities such as those with Gaussian shapes. Widespread in use is the so called Monte Carlo method [7,8]. But unfortunately the performance of that procedure is poor [9]. The reason is that the discrete Doppler frequencies are random variables and therefore, the moments of the Doppler power spectral density of the designed Gaussian noise process ~(t) are also random variables, which can dier widely from the desired (ideal) moments even for a large number of sinusoids N, say N = [6,]. A further design method was derived by Jakes []. For the Jakes method an ecient hardware implementation, using the Intel 888 microprocessor, is described in []. Although all the parameter computation methods can be classied in deterministic and statistic procedures, the resulting simulation model is in any case a deterministic one; because all prameters of the simulation model have to be determined during the simulation set up phase and thus, they are known and constant quantities during the total simula-

3 tion run phase. In this paper, the statistical properties of such deterministic simulation models will be investigated. The paper recapitulates in Section II the statistical properties of Rice processes with given spectral characteristics for the underlying Gaussian processes. Section III introduces the concept of deterministic simulation models and discusses the performance of three dierent parameter computation methods, whereby two of them are new. One of the new procedures is for a given number of sinusoids optimal in the following sense: First, the Doppler coecients c n are calculated such that the probability density of the approximated coloured Gaussian process ~(t) approximates optimally the aspired Gaussian probability density function; and second, the discrete Doppler frequencies f n are optimized so that the autocorrelation function of ~(t) approximates optimally the desired autocorrelation function of a given coloured Gaussian noise process. Moreover, analytical expressions for the deterministic simulation model's amplitude and phase probability density function, autocorrelation function, and cross-correlation function will be derived. Section IV investigates higher order statistical properties of deterministic simulation models such as the level-crossing rate, average duration of fades, and the probability density function of the fading intervals. It is shown analytically and experimentally that the level-crossing rate of properly designed and dimensioned deterministic simulation models for Rayleigh and Rice processes is extremely close to the theory even if the number of sinusoids is small, say seven or eight. Section V concludes the paper. 3

4 II. A STOCHASTIC ANALYTICAL MODEL FOR RICE PROCESSES In order to compare the statistical properties of the deterministic simulation model as will be introduced in Section III with the desired (exact) statistics, a reference model is required. As reference model, we consider in this section an analytical model for Rice processes. In cases, where the inuence of a direct line of sight component can not be neglected, Rice processes are often used as suitable stochastical models for characterizing the fading behaviour of the received signal level of land mobile terrestrial channels as well as of land mobile satellite channels. A. Description of the Analytical Model A Rice process, (t), is dened by taking the absolute value of a non-zero mean complex Gaussian process (t) = (t) + m(t) () according to (t) = j (t)j : () In () all the scattered components in the received signal are represented by a zero mean complex Gaussian noise process (t) = (t) + j (t) (3) with uncorrelated real components i (t); i = ; ; and variance V arf(t)g = V arf i (t)g =, whereas the inuence of a direct (line of sight) component is taken into account here by a time variant mean value of the form m(t) = m (t) + jm (t) = e j(ft+) : (4) Amplitude, Doppler frequency, and phase of the direct component are denoted by ; f, and, respectively. Observe that for f = the mean value m(t) = m = e j is obviously time invariant, which directly corresponds to an orthogonality between the direction of the incoming direct wave component at the receiving antenna of the vehicle and the direction of the vehicle movement. Throughout the paper, we consider the more general case where f 6= and hence, the direct component m(t) depends on time. A typical and often assumed shape for the Doppler power spectral density (psd) of the complex Gaussian noise process (t); S (f), is for mobile fading channel models given by the Jakes psd [] S (f) = 8 < : p ; jfj f max ; f max?(f =f max) ; jfj > f max ; (5) 4

5 where f max is the maximum Doppler frequency. From this equation it follows for the corresponding autocorrelation function (acf) r (t), i.e. the inverse Fourier transform of S (f), the following relation r (t) = J (f max t) ; (6) where J o () denotes the zeroth order Bessel function of the rst kind. The resulting structure of the analytical model for Rice processes (t) with underlying Jakes psd characteristic is shown in Fig. (a). Another important shape for the Doppler psd S (f) is given by the frequency shifted Gaussian psd S (f) = f c r ln e? f?fs fc= p ln ; (7) where f s and f c are the frequency shift and the 3-dB cuto frequency, respectively. Superpositions of two Gaussian psd functions with dierent values for ; f s, and f c have been specied for example in [3] for describing the typical Doppler psd of large echo delays, which occur in the frequency selective mobile radio channel of the pan-european cellular GSM system. Furthermore, frequency non-shifted Gaussian psd functions, i.e. f s = in (7), have been introduced for describing the Doppler psd of aeronautical channels [4, 5]. The acf of the Gaussian psd S (f) is given by r (t) = e? p fc t ln e jfst : (8) Observe that the acf r (t) is complex if f s 6=. Thus, it follows from the properties of the Fourier transform that in such cases the real Gaussian processes (t) and (t) are correlated. Fig. (b) shows the resulting structure of the analytical model for Rice processes (t) with underlying frequency shifted Gaussian psd characteristic. B. Statistical Properties of the Analytical Model In 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 given spectral shapes for the underlying Gaussian processes. We refer to these (ideal) statistical properties in Section III, where we investigate the statistics of deterministic simulation models. The probability density function (pdf) of the Rice process (t), p (x), is given by [6] p (x) = 8 < : x o e? x + x o I o ; x ; o ; x < ; where I o () designates the zeroth order modied Bessel function of the rst kind, and o represents the mean power of the Gaussian process i (t), i.e. o = r i i () = r ()= ; i = ;. From (6) and (8) it follows that o is for the Jakes and Gaussian psd identical with o =. 5 (9)

6 Obviously, the exact form of the Doppler psd of (t) has no inuence on the behaviour of the pdf p (x). Such an independence of the Doppler psd S (f) holds also true for the phase pdf of the complex Gaussian process (t), p # (), which is given by [7] p # () = p # (; t) = e? + erf r + cos(? f t? ) e cos(? f t? ) p cos (?ft?) ; () where erf() is the error function, and is within the interval [; ). By taking notice of (9) and (), we see that in the limit for! the amplitude pdf p (x) tends to the Rayleigh distribution p (x) = 8 < : x o e? x ; x ; ; x < ; whereas the phase pdf p # () tends to the uniform distribution p # () =, [; ), respectively. The level-crossing rate (lcr), denoted here by N (r), provides us with a measure of the average number of crossings per second at which the envelope (t) = j (t)j crosses a specied signal level r with positive slope. We quote for the lcr N (r) the result from [7], where N (r) has been derived for Rice processes with cross-correlated inphase and quadrature components: N (r) = rp 3= e? r+ Z = r cosh cos where the quantities and are given by n e?( sin ) + p sin() erf( sin ) o () d; () = f? _! p ; (3) =?? _ o : (4) Thereby, the above notations and _ have been introduced for denoting the curvature of the autocorrelation function r (t) and the gradient of the cross-correlation function r (t) at t =, i.e. = d dt r (t) _ = d dt r (t) t= t= = r () ; (5) = _r () ; (6) 6

7 respectively. If the Doppler frequency of the direct component f is zero, and if the real Gaussian noise processes (t) and (t) are uncorrelated, i.e. _ =, then it follows from (3) that = and consequently, the lcr N (r) according to () reduces to the expression N (r) = = q q r e? r p (r) ; + r I where =?. In such cases, the lcr N (r) is simply the Rice distribution p (r) multiplied by a constant factor the acf r (t) at t =. q (7), which represents the inuence of the curvature of The average duration of fades (adf), denoted here by T?(r), is the mean value for the length of all time intervals over which the signal envelope (t) = j (t)j is below a specied level r. The adf T?(r) is dened by (cf. [] on p. 36) T?(r) = P?(r) N (r) ; (8) where P?(r) designates the probability that the signal envelope (t) is found below the level r, i.e. P?(r) = P rf(t) rg = Z r p (x)dx = e? Z r xe? x x I dx : (9) The equations () and (8) show us that the lcr N (r) and the adf T?(r) are both functions of the characteristic quantities and (see eqs. (3) and (4)), which are completely dened by the Doppler psd S (f) and the Doppler frequency of the direct component f. For the Jakes and frequency shifted Gaussian psd we obtain for and by using (6), (8), and (3) - (6) in connection with the relation r (t) = (r (t) + j r (t)) the following result: = = 8 >< >: ( p p f ; Jakes psd; (a) f f? f s. pln A ; Gaussian psd; (b) f c (f max ) ; Jakes psd; (a) (f c =p ln ) ; Gaussian psd; (b) 7

8 where =. 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 of the direct component, i.e. and f, as well as by the shape of the acf r (t) and its time derivative of rst and second order at t =. 8

9 III. A DETERMINISTIC SIMULATION MODEL FOR RICE PROCESSES In this section, we investigate the statistical properties of deterministic simulation models for Rice processes with desired spectral characteristics for the underlying Gaussian processes. Therefore, we rst of all introduce the reader to the concept of deterministic simulation models and subsequently, we present a new and optimal method for the computation of the simulation model parameters. For reasons of comparison, we also investigate the statistical properties of the simulation system for two other parameter computation methods. Let us proceed by considering the two real functions ~ (t) and ~ (t) which will be expressed as follows ~ i (t) = XN i n= c i;n cos(f i;n t + i;n ) ; i = ; ; () where N i denotes the number of sinusoids of the function ~ i (t). The quantities c i;n, f i;n, and i;n are simulation model parameters, which are adapted to the desired Doppler psd function and are therefore called Doppler coecients, discrete Doppler frequencies, and Doppler phases, respectively. It is worth mentioning that the simulation model parameters c i;n, f i;n, and i;n have to be computed during the simulation set up phase, e.g. by one of the methods described in the next subsections. Afterwards these parameters are known quantities and are kept constant during the whole simulation run phase. From the fact that all parameters are known quantities, it follows that ~ i (t) itself is determined for all time and thus, ~ i (t) can be considered as a deterministic function. We will see that the statistical properties of ~ i (t) approximate closely the statistical properties of coloured Gaussian stochastic processes. In order to emphasize that property, we will denote henceforth the deterministic function ~ i (t) as deterministic (Gaussian) process. Consequently, the interpretation of ~ i (t) as a deterministic process (function) allows us to compute an analytical expression for the III.3 corresponding acf ~r i i (t) via the denition ~r i i (t) := lim T o! T o Z T o?t o ~ i ()~ i (t + )d : (3) Substituting () in (3) and taking thereupon the Fourier transform, we nd the following analytical expressions for the acf ~r i i (t) and the corresponding psd ~ S i i (f) ~r i i (t) = ~S i i (f) = for i = ;. N X i n= N X i n= c i;n cos(f i;n t) ; (4a) c i;n 4 [(f? f i;n) + (f + f i;n )] ; (4b) 9

10 Notice, that the deterministic processes ~ (t) and ~ (t) are uncorrelated if and only if f ;n 6= f ;m for all n = ; ; : : : ; N and m = ; ; : : : ; N. But if f ;n = f ;m is valid for some or all n; m, then ~ (t) and ~ (t) are correlated and the following cross-correlation function, ~r (t), can be derived ~r (t) = 8 < : NP n= c ;n c ;m cos(f ;n t? ;n ;m ) ; if f ;n = f ;m ; ; if f ;n 6= f ;m ; where N indicates the minimum value of N and N, i.e. N = minfn ; N g. Fig. 3 shows the resulting general structure of the deterministic simulation model for Rice processes in its continuous-time representation. Therefrom, the discrete-time fading simulation 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 c i;n and f i;n appearing in () we will present in the next two subsections a new computation procedure, which is for a given number of sinusoids N i optimal in the following two senses: i) The pdf of the deterministic process ~ i (t), named by ~p i (x), is according to the L -norm an optimal approximation of the ideal (desired) pdf p i (x) of the stochastic process i (t). (5) ii) The acf of the deterministic process ~ i (t), named by ~r i i (t), is according to the L -norm an optimal approximation of the ideal (desired) acf r i i (t) of the stochastic process i (t) within a distinct time interval. A. Determination of Optimal Doppler Coecients In this subsection we compute the Doppler coecients c i;n in such a way that the pdf ~p i (x) of the deterministic process ~ i (t) results for a given number of sinusoids N i in an optimal 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 ()). For that purpose let us consider a single sinusoid ~ i;n (t) = c i;n cos(f i;n t + i;n ) : (6) Furthermore, let us assume in this subsection that the time t is a random variable, which is uniformly distributed in the interval (; f? ) (suppose i;n f i;n 6= ). Then ~ i;n (t) is also a random variable and the pdf of ~ i;n (t) is given by (cf. [6] on p.98) ~p i;n (x) = ( p ; jxj < c i;n ; c i;n?(x=c i;n ) ; jxj c i;n : (7) If the random variables ~ i;n (t) are independent with respective densities ~p i;n (x), then the density ~p i (x) of their sum ~ i (t) = ~ i; (t) + ~ i; (t) + : : : + ~ i;ni (t) (8)

11 equals the convolution of their corresponding densities ~p i (x) = ~p i; (x) ~p i; (x) : : : ~p i ;N i (x) : (9) Although (9) provides us with a rule for the computation of the density ~p i (x), it is more appropriate to apply the concept of the characteristic function. The characteristic function ~ i;n () of the random variable ~ i;n (t) is obtained by taking the Fourier transform of (7) ~ i;n () = J o (c i;n ) : (3) The properties of characteristic functions are essentially the same as the properties of Fourier transforms. Thus, the convolution of the densities (9) reduces to the product of their characteristic functions, i.e. ~ i () = ~ i; () ~ i; () : : : ~ i;ni () (3a) = N Y i n= J o (c i;n ) : (3b) Finally, from the above equation the pdf ~p i (x) can be obtained by taking the inverse Fourier transform, i.e. Z Z ~p i (x) = ~ i ()e?jx d =? N Y i n= J o (c i;n ) cos(x)d : (3) p Let f i;n 6= and c i;n = =Ni for all n = ; ; : : : ; N i and i = ;, then the sum ~ i (t) (dened by (8)) is a random variable with zero mean and variance ~ = (c i; + : : : + c i;n i )= =. The central limit theorem states that in the limit N i! the random variable ~ i (t) tends to a Gaussian distributed random variable i (t) with zero mean and variance, i.e. lim ~p i (x) = p i (x) = N i! p? x e : (33) By taking the Fourier transform of the equation given above, we obtain lim ~ i () = i () = e?() ; (34) N i! and it follows by making use of (3b) the noteworthy result r Ni?! J o N i! e?( ) : (35) N i The characteristic function of a random variable X is dened as the statistical average () = Efe jx g = R? e jx p X (x)dx, where is a real variable. In spite of the positive sign in the exponential () is often called the Fourier transform of p X (x).

12 Of course, for a limited number of sinusoids N i, 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 c i;n = o p =Ni, we see that the approximation error is small and can be neglected for most practical applications if N i 7. Now, the following question arises: Does there exist an optimal set of Doppler coecients fc i;n g in such a sense that the pdf of the simulation system ~p i (x) results for a given number of sinusoids N i in an optimal approximation of the ideal Gaussian pdf p i (x)? This question can be answered by considering the following L -norm, which is also known as the square root of the mean square error, i.e. E () p i := 8 < : Z p i (x)? ~p i (x) dx? 9 = ; = : (36) After substituting (3) and (33) in (36), the above error function can be minimized numerically by using a suitable optimization procedure such as the method described in [8]. The minimization of (36) gives us an optimized set for the Doppler coecients c (opt). i;n Fig. 4(b) shows the resulting pdf of the simulation system ~p i (x) by using the optimized quantities c i;n = c (opt) i;n. A proper choice of the initial values of the Doppler coecients is p given by choosing c i;n = o =Ni. For these values the evaluation of the error function E () p i is shown in Fig. 5(a). The same gure shows also the resulting error function E () p i for the optimized Doppler coecients c i;n = c (opt) i;n, which are for a given number of sinusoids N i all identical after the minimization of (36). A further result worth mentioning is that the optimization p procedure always decreases the initial values for the Doppler coecients, i.e. c (opt) i;n =Ni, and therefore, the variance of the deterministic process ~ i (t) is for nite N i always smaller than the variance of the stochastic process i (t) as depicted in Fig. 5(b). But this eect diminishes if N i increases and can completely be neglected for reasonably values of N i, say N i 7. The consequence from the above is that the improvements made by the optimization procedure are negligible for most practical applications if N i 7, and thus, we can say for simplicity that c (opt) i;n is approximately given by c i;n = p =Ni if N i 7 (37) for all n = ; ; : : : ; N i. Next, let us impose on this optimization approach the power constraint, i.e. ~ =. The power constraint is nothing else but a boundary condition, which can easily be included in the optimization procedure by optimizing the rst N i? Doppler coecients c i; ; ; c i;ni?, whereas the remaining Doppler coecient c i;ni will be computed so that ~ = yields. It follows that the result (after applying the optimization procedure) is for all N i identical with (37) even if the initial values are choosen arbitrarily. Thus, (37) ensures for a given number of sinusoids N i an optimal approximation of ideal Gaussian pdfs p i (x) given the power constraint. Henceforth, we will make use of (37) if we consider the approximation of stochastic processes with Gaussian distribution. In the following, we are interested in the amplitude and phase pdf of the deterministic simulation system for Rice processes as represented in Fig. 3. We only consider the timeinvariant direct component m = m + jm (see (4) with f = ), where the pdf of the

13 real and imaginary part are given by p mi (x i ) = (x i? m i ); i = ;. Thus, the pdf of the deterministic process ~ i (t) can be expressed by ~p i (x i ) = ~p i (x i ) p mi (x i ) = ~p i (x i? m i ) : (38) Let us assume that the deterministic processes ~ (t) and ~ (t) are statistically independent, i.e. f ;n 6= f ;m for all n = ; ; : : : ; N and m = ; ; : : : ; N, then we can express the joint pdf of ~ (t) and ~ (t), ~p (x ; x ), as follows ~p (x ; x ) = ~p (x ) ~p (x ) : (39) Finally, the transformation of the cartesian coordinates (x ; x ) to polar coordinates (z; ) by using x = z cos ; x = z sin ; (4a,b) allows us to derive the joint pdf of the amplitude ~ (t) and phase ~ #(t), ~p # (z; ), in the following way ~p # (z; ) = z ~p (z cos ; z sin ) ; (4a) ~p # (z; ) = z ~p (z cos ) ~p (z sin ) ; (4b) ~p # (z; ) = z ~p (z cos? cos ) ~p (z sin? sin ) : (4c) From (4c) we can directly compute the desired amplitude pdf ~p (z) and phase pdf ~p # () of the deterministic simulation system according to Z ~p (z) = z ~p (z cos? cos ) ~p (z sin? sin ) d ; (4a)? Z ~p # () = z~p (z cos? cos ) ~p (z sin? sin ) dz : (4b) The result of the evaluation of (4a) and (4b) by using (3) is shown in the Figs. 6(a) and 6(b), respectively. Thereby, the Doppler coecients c i;n are computed from (37), and the number of sinusoids N i has been restricted to N = N = 7. For the reasons of comparison, we present also in the same gures the behaviour of the ideal probability density functions p (z) and p # () according to the eqs. (9) and (), respectively. The proposed method not only applies to the approximation of Gaussian processes or stochastic processes derived from Gaussian processes (e.g. Rayleigh processes, Rice processes, and lognormal processes) but can also be used for the approximation of other kinds of distribution functions such as the Nakagami distribution. 3

14 The Nakagami distribution [9], also denoted as m-distribution, oers a greater exibility and often tends to t the pdf of the received signal amplitude computed from experimental data of certain urban channels better than the common used Rayleigh or Rice distribution []. The Nakagami distribution with parameters and m is dened by p (z) = mm z m? e?(m=)z?(m) m ; m = ; z ; (43) where?() is the Gamma function, =< (t) > denotes the time average power of the received signal strength, and m =< (t) > = < ((t)? < (t) >) > is the inverse of the normalized variance of (t). We remark that for m = = and m = the Nakagami distribution becomes a one-sided Gaussian and Rayleigh distribution, respectively; and nally, the Rice and lognormal distribution also can be approximated by the Nakagami distribution [9, ]. For further details on the derivation and simulation of Nakagami fading channel models we refer to [, ]. In order to nd a suitable set for the Doppler coecients fc i;n g, so that the amplitude pdf of the proposed simulation system ~p (z) ts the Nakagami distribution (43), we proceed in a similar way as we have successfully done for the minimization of the L -norm (36) in connection with Gaussian distributions. We only have to replace in (36) the Gaussian distribution p i (x) by the Nakagami distribution p (z) as dened by (43), and ~p i (x) has to be replaced by ~p (z) as introduced by (4a). The optimization results for the Nakagami distribution are shown for various values of m in Fig. 7, whereby in each case N = N = sinusoids have been used. Weibull distributions can be derived from uniform distributions, and on the other hand uniform distributions can be derived from Gaussian distributions. Consequently, the derivation of deterministic Weibull processes is straightforward and will not be investigated in detail. B. Determination of Optimal Discrete Doppler Frequencies In the previous subsection, we have discussed a method which enables us to compute the Doppler coecients c i;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 stochastic process i (t). In the present subsection, we will see that the discrete Doppler frequencies f i;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 modication of the L p -norm method []. The proposed (modied) L p -norm method assumes that the Doppler coecients c i;n are xed and given by (37), whereas optimal values for the discrete Doppler frequencies f i;n can be found numerically by minimizing the L p -norm for p =, i.e. the square root of the mean square 4

15 error norm, E () r i i := 4 T o Z T o jr i i (t)? ~r i i (t)j dt 3 5 = ; (44) where ~r i i (t) is given by (4a), r i i (t) can be any desired acf (e.g. resulting from "snap shot" measurements of real world mobile channels), and T o is dened below and denotes an appropriate time interval [; T o ] 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 to both Doppler power spectral densities S (f) as introduced in Section II (i.e. Jakes and Gaussian). 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 expression r i i (t) = r (t)= = 8 >< >: J o(f max t) (Jakes) ; (45a) p fc t ln e? (Gauss) : (45b) Numerical investigations have shown that for these acfs appropriate values for T o, i.e. the upper bound of the integral in (44), are given if T o is adapted to the number of sinusoids N i according to T o = 8 < : N i f max (Jakes) ; q N i ln 4f c (Gauss) : The evaluation of the error function E () r i i (see (44)) with the resulting optimized discrete Doppler frequencies f i;n = f (opt) i;n is shown for the Jakes psd and the Gaussian psd in the Figs. 8(a) and 8(b), respectively. For the sake of comparison, we also have shown in these gures the results obtained by applying two other ecient parameter computation techniques, 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 ff i;n g in such a manner that in the range of f i;n? f < f i;n the area under the (symmetrical) Doppler psd S i i (f) is equal to =(N i) for all n = ; ; : : : ; N i with f i; =. The application of the method of equal areas to the Jakes psd S i i (f) = S (f)= (see (5)) gives us for the discrete Doppler frequencies n f i;n = f max sin ; (47) N i for all n = ; ; : : : ; N i and i = ;. Whereas for the frequency non-shifted Gaussian psd (see (7) with f s = ) no closedform expression for the discrete Doppler frequencies f i;n exists, and we must calculate the discrete Doppler frequencies f i;n by nding the zeros of n N i? erf (46) fi;n f c p ln = (48) 5

16 for all n = ; ; : : : ; N i and i = ;. The Doppler coecients are exactly the same as introduced by (37). For a comprehensive derivation and discussion of the method of equal areas, we refer to [6], where also some applications and simulation results can be found. Method of Exact Doppler Spread: This procedure is due to its simplicity and high performance predestinated to the approximation of the acf r i i (t) as given by (45a), i.e. when the Jakes psd is of interest. The method can easily be derived by using the integral representation [3] J o (z) = Z = cos(z sin )d ; (49) and replacing the above integral by a series expansion as follows J o (z) = lim N i! XN i n= cos(z sin n ) ; (5) where n := (n? )=(4N i ) and := =(N i ). Hence, we can write by using (45a) and (5) where r i i (t) = o J o(f max t) = lim N i! f i;n = f max sin N i (n? ) o N i XN i n= cos(f i;n t) ; (5) : (5) Finally, as for nite values of N i we require r i i (t) ~r i i (t), a comparison of (5) with (4a) leads to ~r i i (t) = o N i N X i n= cos(f i;n t) : (53) A comparison of (53) with (4a) shows that the discrete Doppler frequencies f i;n are identical with (5), and the Doppler coecients c i;n can immediately be identied with those given by (37). The Doppler psd S i i (f) is often characterized by the Doppler spread B i, that is the square root of the second central moment (variance) of S i i (f). In [6] it has been shown that the Doppler spread of deterministic simulation models, B ~ i, depends directly on the discrete Doppler frequencies f i;n and Doppler coecients c i;n. Consequently, the approximation quality B i Bi ~ is mainly aected by the employed parameter computation method. In Section IV, we will see that for the Jakes psd the proposed method always results in a Doppler spread of the simulation model B ~ i that is identical with the Doppler spread of the analytical model B i, i.e. B i = B ~ i. Therefore, we have designated the 6

17 procedure as "method of exact Doppler spread". Clearly, the method of exact Doppler spread is adapted to the Jakes psd, but a comparison of (47) and (5) motivates us - as a heuristic approach - to replace in (48) n by n? =. This enables the computation of the discrete Doppler frequencies f i;n for the frequency non-shifted Gaussian psd by nding the zeros of n?? erf N i fi;n f c p ln = (54) for all n = ; ; : : : ; N i and i = ;. The Doppler coecients c i;n are remain unchanged and are further on given by (37). After the calculation of the discrete Doppler frequencies f i;n and the Doppler coecients c i;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 (4a). The result for ~r i i (t) that has been obtained by applying the L -norm method, the method of equal areas, and the method of exact Doppler spread are presented for the Jakes psd in Fig. 9(a) and for the frequency nonshifted Gaussian psd in Fig. 9(b). In all cases, the number of sinusoids N i was determined by N i = 7. Observe that we have also depicted in Fig. 9(a) and Fig. 9(b) the corresponding acfs of the analytical model r i i (t) = o J o(f max t) and r i i (t) = o exp[?(f c t) = ln ], respectively. This allows us directly to compare and criticize the performance of the various parameter computation methods. The L -norm method is more complex and requires higher numerical eort than the method of exact Doppler spread, but surprisingly, we realize from Fig. 8(a) and Fig. 9(a) that these two methods are yielding nearly equivalent and excellent approximation results in the interval t T o. Thus, due to its simplicity and high approximation quality, we prefer for the Jakes psd the method of exact Doppler spread. We gather from the Figs. 8(b) and 9(b) that for the Gaussian psd the method of exact Doppler spread results in essential smaller approximation errors than the method of equal areas. A comparison of the performance of these two methods with the L -norm method reveals undoubtedly the advantage of that optimization procedure. Therefore, we prefer for the Gaussian psd the L -norm method. Finally, we mention that the period of the deterministic process ~ i (t) (see ()) is inversely proportional to the greatest common devisor of the discrete Doppler frequencies f i;n. For all discussed methods, the greatest common devisor can be made small enough for realistic values of N i, say N i 7, simply by increasing the word length of the discrete Doppler frequencies f i;n ; and thus, the period of ~ i (t) can be made extremely large. C. Determination of the Doppler Phases The Doppler phases i;n are uniformly distributed random variables, which can be obtained simply by means of a random generator with uniform distribution over the interval (; ]. But for deterministic simulation models, where the Doppler coecients c i;n and the discrete Doppler frequencies f i;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 a deterministic approach. 7

18 Therefore, we consider the following standard phase vector ~ i with N i deterministic phase components ~ i = N i + ; N i + ; : : : ; N i N i + ; (55) and we group the Doppler phases i;n of the N i sinusoids to a so called Doppler phase vector ~ i in accordance with ~ i = ( i; ; i; ; : : : ; i;ni ) (56) for i = ;. Now, the components of the Doppler phase vector ~ i can be identied with the components of the standard phase vector ~ i after permutating the components of ~ i. In this way, for a given number of sinusoids N i one can construct N i! dierent Doppler phase vectors ~ i with equally distributed components. Let us assume that the real deterministic processes ~ (t) and ~ (t) are uncorrelated, then it can be shown that the corresponding Doppler phases ;n and ;n have no inuence on the statistical properties of the complex deterministic process ~(t) = ~ (t) + j ~ (t). Consequently, for one and the same sets of Doppler coecients fc i;n g and discrete Doppler frequencies ff i;n g it is possible to construct N i! dierent sets of Doppler phases f i;n g, and thus N! N! dierent complex deterministic processes ~(t) = ~ (t) + j ~ (t) with dierent time behaviour but identical statistics can be realized. This is of great advantage for 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 ~ (t) and ~ (t) are correlated, then the Doppler phases i;n have an inuence on the cross-correlation function ~r (t), as can easily be seen by considering (5). In this case, the Doppler phases i;n can be designed such that they control the higher order statistics of ~ (t) = j~(t) + m(t)j, e.g. the simulation model's level-crossing rate and the average duration of fades [4]. 8

19 IV. HIGHER ORDER STATISTICAL PROPERTIES OF DETERMINISTIC SIMULATION MODELS In 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), but also enable us to study the degradation eects due to the selected parameter computation method and the limited number of sinusoids N i. Moreover, we investigate the pdf of fading 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), if the number of sinusoids N i is suciently large, say N i 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 dened by (); one merely has to replace the quantities of the analytical model and (consider therefore (3) and (4)) by the corresponding quantities of the simulation model ~ and ~. Thus ~ = f? _~ ~!q ~ ; (57) ~ =? ~? _~ ; ~ (58) where ~, _ ~ are related by the acf ~r (t) (see (4a)) and ccf ~r (t) (see (5)) of the simulation model as follows ~ = d dt ~r (t) _~ = d dt ~r (t) t= t= = ~r () ; (59) = _~r () ; (6) and ~ = ~r () = P N i n= c ;n =. Let the Doppler coecients c i;n are as given by (37), then ~ = yields. The quality of the approximations ~ and ~ depends on the underlying Doppler psd and will be investigated separatly for the Jakes and frequency shifted Gaussian psds. A. Higher Order Statistical Properties by Using the Jakes PSD The Jakes psd (5) is a symmetrical function and thus, it follows from the properties of the Fourier transform that the ccf r (t) is zero, i.e. (t) and (t) are uncorrelated. 9

20 Now, we impose the same requirement on the simulation system. From (5) it follows that ~r (t) = can easily be achieved for all design methods discussed here by selecting the number of sinusoids such that N = N? yields. Then N is unequal to N and per denition the discrete Doppler frequencies are designed in such a way that f ;n 6= f ;m holds for all n = ; ; : : : ; N and m = ; ; : : : ; N. Thus, it follows from (57) - (6) in connection with (4a) that ~ = f = q ~ (6) and X ~ =? N ~ = (c ;n f ;n ) : (6) n= The evaluation of the above equation by using the method of exact Doppler spread can be performed by substituting (37) and (5) in (6), i.e. ~ = ( f max ) XN N n= By taking notice of = XN N n= sin N (n? ) and making use of the relation : (63) sin N (n? ) = ; (64) we nally obtain for ~ and ~ the result ~ = p f f max ; (65) ~ = (f max ) : (66) A comparison of (a) and (a) 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 =() and B i = p = o =(). In a similar way, we can compute ~ by using the method of equal areas. In this case, we obtain ~ = ( + =N i ) and ~ tends to when N i!. Let us for a moment assume that the Doppler frequency of the direct component f is equal to zero, i.e. f =, then = ~ = and the lcr N (r) reduces to (7). Now the relative error of ~ N (r) can be expressed by " ~N = ~ N (r)? N (r) N (r) = q ~ p? p : (67)

21 q The substitution of ~ = ( + =N i ) in (67) and the use of the approximation + N i + N i allows us to express the relative error " ~N directly as function of the number of sinusoids " ~N : (68) N i The preceding equation shows that by using the method of equal areas about 5 sinusoids are required for the design of ~ (t) and ~ (t) in order to reduce the relative error of ~ N (r) to %. It shoud be noted that for the L -norm method no closed-form expression for ~ can be derived. The use of that method leads after the numerical evaluation of (6) to the results for the normalized quantity =f ~ max as depicted in Fig. (a). This gure also shows the behaviour of =f ~ max as function of the number of sinusoids N i by using the method of equal areas and the method of exact Doppler spread. Next, we consider the deterministic simulation model of Fig. 3(a), and we calculate directly the lcr N ~ (r) by counting the number of crossings per second at which the simulated channel output signal (t) ~ crosses a specied signal level r with positive slope. Therefore, we have chosen a maximum Doppler frequency f max (see (5)) of f max = 9:73 Hz, that corresponds to a mobile speed of km/h and a carrier frequency of 9 MHz. The variance was normalized to = =. The parameters of the deterministic simulation system have been designed according to the method of exact Doppler spread with N = 7 and N = 8 sinusoids, so that the cross-correlation between the real part ~ (t) and the imaginary part ~ (t) is zero. For the realization of the discrete-time deterministic simulation system we have used a sampling interval T of T = 3?4 s and altogether K = 4 5 samples of the deterministic process (kt ~ ), k = ; ; : : : ; K?, have been simulated for the evaluation of the lcr N ~ (r). The result for the normalized lcr of the deterministic simulation model N ~ (r)=f max is presented in Fig. (a). This gure demonstrates for a deterministic Rice process with = the inuence of the Doppler frequency of the direct component f on the behaviour of N ~ (r). For the reasons of comparison, we also have shown in Fig. (a) the lcr of the (ideal) analytical model N (r) as dened by (). It can be seen that the lcr of the simulation model N ~ (r) coincides extremely closely with the lcr of the theoretical model N (r) by applying the method of exact Doppler spread. B. Higher Order Statistical Properties by Using the Gaussian PSD The Gaussian psd (7) is for f s 6= an unsymmetrical function. Consequently, the ccf r (t) is unequal to zero and thus (t) and (t) are correlated. Let the stochastic processes (t) and (t) (see Fig. (b)) are uncorrelated, then we yield after some simple computations the following expressions for the acf r (t) and ccf r (t) r (t) = [r (t) + r (t)] cos(f s t) ; (69a) r (t) = [r (t) + r (t)] sin(f s t) ; (69b)

22 respectively. Starting with the above equations and using r i i () = =, we can show that (5) and (6) are resulting for the frequency shifted Gaussian psd in p i =? o h(f c = ln ) + (f s ) ; (7a) _ = f s ; (7b) respectively. In a similar way, the corresponding quantities of the deterministic simulation model can be derived. The result is X ~ =? " N n= (c ;n f ;n ) + XN n= (c ;n f ;n ) #? (f s ) ; (7a) _~ = f s ; (7b) where we have used ~r i i () = ~ =, i.e., the Doppler coecients are given by c i;n = p =Ni. A comparison of (7b) with (7b) shows that _~ = _. Now, we are able to compute the quantities ~ and, ~ which determine the approximation quality of the deterministic simulation model's level-crossing rate. Therefore, we substitute (7a), (7b) in (57), (58) and obtain nally ~ = s ~ = " N ~ (f? f s ) ; (7) X n= (c ;n f ;n ) + XN n= (c ;n f ;n ) # : (73) Eq. (7) allows further observations: when ~!, ~ tends to and consequently ~ N (r) tends to N (r). Obviously, the quantity ~ is responsible for the approximation quality of the higher order statistical properties of the deterministic simulation model. For all parameter computation methods discussed in Section III, ~ can not be expressed in a closed-form for the Gaussian psd, so that (73) has to be evaluated numerically. Fig. (b) shows the numerical evaluation results for the normalized quantity ~ =f c, which have been obtained by employing all parameter computation methods investigated in the present paper. Now, let us consider Fig. 3(b) and let us determine for the frequency shifted Gaussian psd the lcr N ~ (r) from the simulated sequence (kt ~ ), k = ; ; : : : ; K?, in the same way as we have already done for the Jakes psd. Therefore, we have selected the following model parameters: T = 3?4 s, K = 4 5, f c = p ln f max, f max = 9:73 Hz, =, f =, and in order to study the inuence of the frequency shift f s, we have used the values f s ; 3 f max; f max. Observe, by considering of (), that the 3-dB cuto frequency f c = p ln f max was selected such that the quantity is identical for both power spectral densities (Jakes

23 and Gaussian). The discrete Doppler frequencies f i;n and Doppler coecients c i;n have been determined by the method of exact Doppler spread with N = 5 and N = 6. The results for the normalized level crossing rates of the deterministic simulation model ~N (r)=f max as well as of the analytical model N (r)=f max are shown in Fig. (b). A comparison of Fig. (a) with Fig. (b) shows the same result for both the analytical and the simulation model: identical level-crossing rates are observed as it was expected by the theory although the underlying Doppler power spectral densities are quite dierent. 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 (8) N (r) by ~ N (r) and P?(r) by ~P?(r). Thus, the analytical and numerical investigations of ~ T?(r) are straightforward and will not be presented here. C. Distribution of the Fading Time Intervals The performance of mobile communication systems can mainly be improved by an ecient design of the interleaver/deinterleaver and the channel coding/decoding unit. For the specication of these units, not only the knowledge of the level-crossing rate and the average duration of fades is of special interest, but also the distribution of the fading time intervals. The rest of this section is focused on the probability density function of the fading time intervals that will be denoted here by p?(?; r). This pdf p?(?; r) is the conditional probability density that the stochastic process (t) crosses a certain level r for the rst time at t = t +? with positive slope given a crossing with negative slope at time t. An exact analytical evaluation of p?(?; r) is still an unsolved problem even for Rayleigh and Rice processes. So we will determine ~p?(?; r) experimentally by employing deterministic simulation systems. For that purpose, we simulate (kt ~ ) with a sampling interval of T = :5?4 s, and we x =, i.e. (kt ~ ) is a deterministic Rayleigh process. Typical measurement results of ~p?(?; r) for r = : and r = by evaluating 4 fading time intervals? are presented in Figs. (a) and (b) for the Jakes psd and in Figs. (c) and (d) for the Gaussian psd, respectively, whereby we have applied the method of exact Doppler spread by using various values for thepnumber of sinusoids N i. The remaining model parameters were f max = 9 Hz, f c = f max ln ; o = =, and f s =. Approximative solutions for p?(?; r) can be found for example in [5, 6, 7, 8, 9]. The rst approximation to p?(?; r) is p?(?; r), where p?(?; r) is the conditional probability density that the stochastic process (t) passes upwards through a certain level r at time t = t +? given that (t) has passed downward through r at time t. Note, that nothing is said about the behaviour of (t) between t = t and t = t. In [5] an analytical solution for p?(?; r) can be found that tends for deep fades to p?(?; r)!? d du ut? I ( u )e? u ; if r! ; (74) where u =?=T?(r) and I () denotes the modied Bessel function of rst order. The graph of this function also is shown for the Rayleigh process (t) with underlying Jakes and Gaussian psd characteristics in Figs. (a) and (c), respectively. These gures impressivly show the excellent accordance between the theoretical approximation p?(?; r) 3