On the Stationarity of Sum-of-Cisoids-Based Mobile Fading Channel Simulators

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1 On the Stationarity of Sum-of-Cisoids-Based Mobile Fading Channel Simulators Bjørn Olav Hogstad and Matthias Pätzold Department of Information and Communication Technology Faculty of Engineering and Science, University of Agder P.O. Box 59, NO-4898 Grimstad, Norway Abstract A finite sum of complex harmonic functions, also known as a sum-of-cisoids (SOC), has been widely applied in the literature for the modeling and simulation of mobile fading channels. In this paper, we focus on the analysis of the stationarity of SOC-based mobile fading channel simulators. An SOC model depends on three kinds of parameters, namely, the gains, the frequencies, and the phases, where each of which can be a collection of random variables or constants. Hence, there exist altogether eight classes of SOC models. Each individual class will systematically be investigated with respect to its stationary properties. Keywords Mobile fading channels, flat fading channels, channel simulators, sum-of-cisoids, stochastic processes, deterministic processes. I. INTRODUCTION Mobile fading channel simulators are commonly used in the laboratory because they allow system tests and evaluations which are less expensive and more reproducible than field trials. In the literature, many different approach have been used for the modeling and simulation of mobile fading channels [1 [4. Among them, the sum-of-sinusoids (SOS) principle has been widely used for the design of efficient and flexible mobile fading channel simulators. This principle was first introduced in Rice s seminal work [5, [6 as a method to model Gaussian noise processes with given temporal correlation properties. The SOS principle has been used in [1, [2 for the development of some simple time-variant Rayleigh fading channels, and in [3 for Nakagami-m channels. Furthermore, the SOS principle has been used for the development of frequency-selective channels [7, narrowband MIMO channels [8, and wideband MIMO channels [9, [1. Recently, it has been shown in [11, [12 that the SOS principle together with set partition [13 can be used to generate multiple uncorrelated Rayleigh fading waveforms. This procedure can be used in performance studies of MIMO systems and diversity schemes. An SOC model can be interpreted as a special case of the SOS model. A detailed description of the differences between the SOS model and the SOC model can be found in [14. Therein, some basic statistical properties of SOC models, such as the autocorrelation function, cross-correlation function, and density functions of the envelope and phase, have been investigated. Furthermore, the level-crossing rate, the average duration of fades, and the cumulative distribution function of the envelope of an SOC model have been analyzed in [15. In [14, it has been demonstrated that in case of nonisotropic scattering conditions, the SOC model can be used for the development of efficient simulation models, while the SOS model is the better choice if isotropic scattering conditions are assumed. It is important to know the conditions on which an SOS model results in a stationary and ergodic process. If a stochastic channel simulator is ergodic, then each sample function contains the same statistical information. In this case, a single sample function is sufficient to characterize the channel. Hence, an averaging over several realizations of sample functions can be avoided. For simulation purposes, the overall simulation time can be reduced drastically by employing ergodic channel simulators. A detailed analyses of the stationary and ergodic properties of different classes of SOS Rayleigh fading channel simulators can be found in [16, [17. Therein, the proposed concept has also been applied to the most important parameter computation methods. The analyses of the stationarity of different classes of channel simulators by using the SOC model instead of SOS model will be presented in this paper. The obtained density functions of the various classes of SOC-based Rayleigh fading channel simulators are completely different from what is obtained in [16, [17. The rest of the paper is organized as follows. Section II reviews briefly the SOC model. In Section III, we review the characteristics of stationary processes. Section IV introduces eight classes of SOC-based simulation models and investigates their stationary properties. A summary of the results obtained in Section IV can be found in Section V. Therein, the proposed concept has also been applied to some important parameter computation methods. Finally, the conclusion is drawn in Section VI. II. THE SOC MODEL A Rayleigh process ζ(t) is defined as ζ(t) = µ 1 (t)+jµ 2 (t) (1) where µ 1 (t) and µ 2 (t) are zero mean real Gaussian processes. For clarity, we will use bold letters to indicate stochastic processes as well as random variables, and normal letters are used for deterministic processes (sample functions) and realizations (outcomes) of random variables. The complex process µ(t) =µ 1 (t)+µ 2 (t) can be modeled as µ(t) = lim N c n e j(2πfnt+θn) (2) /8/$ IEEE 4

2 where N denotes the number of cisoids, c n is called the gain, f n is the Doppler frequency, and θ n is the phase of the nth cisoids of µ(t). The phases θ n are independent and identically distributed (i.i.d.) random variables with a uniform distribution over [, 2π). The formula in (2) is non-realizable for simulation purposes. From (2), a stochastic simulation model is obtained by using only a finite number of cisoids N. The underlying stochastic process will be expressed as ˆµ(t) = c n e j(2πfnt+θn) (3) where the gains c n and frequencies f n are still constants, and the phases θ n are again i.i.d. random variables with a uniform distribution over (, 2π. Obviously, the stochastic process ˆµ(t) tends to the Gaussian process µ(t) as N. If the phases θ n are outcomes (realizations) of a random generator with a uniform distribution on the interval (, 2π, the stochastic process ˆµ(t) results in a sample function denoted by µ(t) = c n e j(2πfnt+θn). (4) The SOC model depends on three types of parameters, namely, the gains c n, the frequencies f n, and the phases θ n. We can choose each of these parameters as random variables or constants. Hence, altogether 2 3 =8classes of SOC-based simulation models for Rayleigh fading channels can be defined. If all parameters are constants, we obtain a completely deterministic process, denoted by ζ(t) = µ 1 (t) +j µ 2 (t). Therefore, at least one random variable is required to obtain a stochastic process, denoted by ˆζ(t) = ˆµ 1 (t)+j ˆµ 2 (t). For the performance evaluation of the simulation model defined in (3), it is important to know the conditions for which the stochastic process ˆζ(t) is stationary. In the next section, we will review briefly the term stationarity. III. STATIONARITY A stochastic process ˆζ(t) is called strict-sense stationary [18, p. 387 if its statistical properties are invariant to a shift of the origin. This means that the stochastic processes ˆζ(t) and ˆζ(t+c) have the same statistics for any c R. It follows that the mth-order density function must satisfy pˆζ(x 1,...,x m ; t 1,...,t m )=pˆζ(x 1,...,x m ; t 1 +c,...,t m +c) (5) for all values of t 1,...,t n and c. A stochastic process ˆζ(t) is called first-order stationary [18, p. 392 if the density of ˆζ(t) satisfy pˆζ(x; t) =pˆζ(x; t + c) (6) for all values of t and c. This implies that the mean and the variance of ˆζ(t) are independent of time. If a stochastic process is strict-sense stationary, it is also a first-order stationary process. The inverse statement is not always true. For simplicity, in this paper we only consider the first-order stationary properties. IV. CLASSIFICATION OF CHANNEL SIMULATORS In the following, 8 classes of SOC-based simulation models for Rayleigh fading channels will be introduced, seven of which are stochastic simulation models and one is completely deterministic. Before presenting a detailed analysis of the various classes of simulation models, some constraints are imposed on the three types of parameters. These constraints are as follows. In cases, where the gains c n, frequencies f n, and phases θ n are random variables, it is reasonable to assume that they are mutually independent. Also, it is assumed that the gains c 1, c 2,...,c N are i.i.d. random variables. The same is assumed for the sequences of random frequencies f 1, f 2,...,f N and phases θ 1, θ 2,...,θ N. Whenever the gains c n and frequencies f n are constant quantities, it is assumed that they are different from zero, so that c n and f n hold for all values of n =1, 2,...,N. Further constraints might also be imposed on the SOC model. For example, it is required that the absolute values of all frequencies, f n, are different. This latter condition is introduced to avoid correlations within the terms of µ(t). A. Class I Channel Simulators The channel simulators of Class I are defined by the set of deterministic processes ζ(t) with constant gains c n, constant frequencies f n, and constant phases θ n. The concept of stationary processes can only be applied on stochastic processes. Since all model parameters are constants, there is no meaning to examine the stationary properties of this class of channel simulators. B. Class II Channel Simulators The channel simulators of Class II are defined by the set of stochastic processes ˆζ(t) with constant gains c n, constant frequencies f n, and random phases θ n, which are uniformly distributed in the interval (, 2π. In this case, the complex process ˆµ(t) = ˆµ 1 (t)+jˆµ 2 (t) can be written as ˆµ(t) = c n e j(2πfnt+θn). (7) In [14, it has been shown that the density pˆζ(z) of the stochastic process ˆζ(t) = ˆµ(t) can be represented as [ N pˆζ(z) =(2π) 2 z J (2π c n y) J (2πzy)ydy. (8) From (8), we can conclude that the density pˆζ(z) is independent of time. Hence, the stochastic process ˆζ(t) is first-order stationary. C. Class III Channel Simulators The channel simulators of Class III are defined by the set of stochastic processes ˆζ(t) with constant gains c n, random frequencies f n, and constant phases θ n. Hence, the complex process ˆµ(t) = ˆµ 1 (t)+jˆµ 2 (t) has the following form ˆµ(t) = c n e j(2πfnt+θn). (9) 41

3 To obtain the density pˆζ(z; t) of the stochastic process ˆζ(t), we continue as follows. In the first step, we consider a single complex sinusoid at the time instant t = t. Thus, ˆµ n (t )=c n e j(2πfnt+θn) (1) describes a complex random variable. From [16, in the limit t, the density pˆµ1,n (x 1 ) of ˆµ 1,n (t ) = Re{ ˆµ n (t )} = c n cos(2πf n t + θ n ) can be expressed as { 1 pˆµ1,n (x 1 )=, x 1 <c n π c n 1 (x 1/c n) 2 (11), x 1 c n. Now, by following exactly the same procedure as described in [14, we can conclude that the density pˆζ(z; ± ) is given by (8). If t is finite, we must substitute the expression in (11) with [16, Eq. (32). In this case, it turns out that the density pˆζ(z; t) depends on time t, so that the stochastic process ˆζ(t) is not first-order stationary. Thus, we can conclude that a Class III channel simulator is first-order stationary only in the limit t ±. D. Class IV Channel Simulators The channel simulators of Class IV are defined by the set of stochastic processes ˆζ(t) with constant gains c n, random frequencies f n, and random phases θ n, which are uniformly distributed in the interval (, 2π. Since the random frequencies f n have no effect on the density pˆζ(z) in (8), it follows that the density of ˆζ(t) of the Class IV channel simulators is also given by (8). Hence, the stochastic process ˆζ(t) is first-order stationary. E. Class V Channel Simulators A Class V channel simulator is based on a stochastic process ˆζ(t) with random gains c n, constant frequencies f n, and constant phases θ n. Thus, the complex process ˆµ(t) = ˆµ 1 (t)+jˆµ 2 (t) has the following form ˆµ(t) = c n e j(2πfnt+θn). (12) The starting point for the derivation of the density of the stochastic process ˆζ(t) = ˆµ(t) is a single complex sinusoid of the form ˆµ n (t) =c n e j(2πfnt+θn). (13) For fixed values of t = t, ˆµ 1,n (t ) = Re{ ˆµ n (t)} = c n cos(2πf n t + θ n ) and ˆµ 2,n (t ) = Im{ ˆµ n (t)} = c n sin(2πf n t + θ n ) represent two dependent random variables. From [18, Eq. (5-18), the density pˆµ1,n (x 1 ; t ) of ˆµ 1,n (t ) can be expressed as pˆµ1,n (x 1 ; t )= p c(x 1 / cos(2πf n t + θ n )) (14) cos(2πf n t + θ n ) where p c ( ) denotes the common density of the gains c n.by using [19, Eq. (3.15), the joint density pˆµ1,n ˆµ 2,n (x 1,x 2 ; t ) of the dependent random variables ˆµ 1,n (t ) and ˆµ 2,n (t ) can be obtained as pˆµ1,n ˆµ 2,n (x 1,x 2 ; t )=pˆµ1,n (x 1 ; t )δ(x 2 g(x 1 ; t )) (15) where g(x 1 ; t )=x 1 tan(2πf n t + θ n ). The joint characteristic function (ν 1,ν 2 ; t ) of the random variables ˆµ 1,n (t ) and ˆµ 2,n (t ) is defined by the Fourier transformation (ν 1,ν 2 ; t ) = pˆµ1,n ˆµ 2,n (x 1,x 2 ; t ) e j2π(ν1x1+ν2x2) dx 1 dx 2. (16) After substituting (15) in (16) and carrying out some algebraic computations, we find (ν 1,ν 2 ; t ) = j2πν1x1 cos(2πfnt+θn) p c (x 1 )e e j2πν2x1 sin(2πfnt+θn) dx 1. (17) Since we have assumed that the gains c n are i.i.d. random variables, it follows that the quantities µ n (t ) are also i.i.d. random variables. Hence, the joint characteristic function Ψˆµ1 ˆµ 2 (ν 1,ν 2 ; t ) of ˆµ 1 (t ) and ˆµ 2 (t ) can be expressed as the product of the joint characteristic functions (ν 1,ν 2 ; t ) of ˆµ 1,n (t ) and ˆµ 2,n (t ), i.e., Ψˆµ1 ˆµ 2 (ν 1,ν 2 ; t )= (ν 1,ν 2 ; t ). (18) From this and the two-dimensional inversion formula for Fourier transforms, it follows that the joint density pˆµ1 ˆµ 2 (x 1,x 2 ; t ) of the statistically dependent random variables ˆµ 1 (t ) and ˆµ 2 (t ) can be expressed as pˆµ1 ˆµ 2 (x 1,x 2 ; t )= j2π(ν1y cos(2πfnt+θn) p c (y)e e j2πν2y sin(2πfnt+θn) dy e j2π(ν1x1+ν2x2) dν 1 dν 2. (19) If we now transform the Cartesian coordinates (x 1,x 2 ) into polar coordinates (z,θ) by means of x 1 = z cos θ and x 2 = z sin θ, then we obtain the joint density pˆζ ˆϑ(z,θ; t ) of the envelope ˆζ(t )= ˆµ(t ) and the phase ˆϑ(t )=arg{ ˆµ(t )} as follows pˆζ ˆϑ(z,θ; t )=zpˆµ1 ˆµ 2 (z cos θ, z sin θ; t ) = z j2πν1y cos(2πfnt+θn) p c (y)e e j2πν2y sin(2πfnt+θn) dy e j2πz(ν1 cos θ+ν2 sin θ) dν 1 dν 2 (2) for z and θ ( π, π). Finally, the density pˆζ(z; t ) of the envelope ˆζ(t ) can be obtained from the joint density pˆζ ˆϑ(z,θ; t ) by integrating over θ. By following this procedure, using [2, Eq. ( ), and transforming the Cartesian coordinates (ν 1,ν 2 ) into polar coordinates (r, θ) we finally 42

4 obtain pˆζ(z; t ) = 2πz 2π p c (y) e j2πry cos(2πfnt+θn θ) dy J (2πzr)rdrdθ. (21) Note that this expression holds for all values of t R. Therefore, we can replace t by t. In this case, the density pˆζ(z; t) is given by (21). From (21), we realize that the density pˆζ(z; t) is a function of time. Hence, the stochastic process ˆζ(t) is not first-order stationary. F. Class VI Channel Simulators The channel simulators of Class VI are defined by the set of stochastic processes ˆζ(t) with random gains c n, constant frequencies f n, and random phases θ n. In this case, the complex process ˆµ(t) = ˆµ 1 (t)+jˆµ 2 (t) is of type ˆµ(t) = c n e j(2πfnt+θn). (22) To find the density pˆζ(z) of ˆζ(t) = ˆµ(t) of the Class VI channel simulators, we make use of the conditional density pˆζ(z c n = c n ) of the Class II channel simulators. The density pˆζ(z) of the Class VI channel simulators can then be obtained by averaging the conditional density pˆζ(z c n = c n ) over the distribution p c (y) of the N i.i.d. random variables c 1, c 2,...,c N, i.e., pˆζ(z) = pˆζ(z c 1 = c 1,...,c N = c N ) [ N p c (y n ) dy 1 dy N = (2π) 2 z p c (y)j (2π y ν)νdy J (2πzν) dν. (23) From (23), we can conclude that the density pˆζ(z) is independent of time. Hence, the stochastic process ˆζ(t) is first-order stationary. G. Class VII Channel Simulators This class of channel simulators involves all stochastic processes ˆζ(t) = ˆµ(t) with random gains c n, random frequencies f n, and constant phases θ n, i.e., ˆµ(t) = c n e j(2πfnt+θn). (24) To obtain the density pˆζ(z) of the stochastic process ˆζ(t), we consider for reasons of simplicity the case t ±.The density pˆζ(z) can be considered as the conditional density pˆζ(z c n = c n ) obtained for the Class III channel simulators. By following exactly the same procedure described in Subsection IV F, we can conclude that the density pˆζ(z) is given by N (23). If t is finite, we have to repeat the procedure described in Subsection IV F by using [16, Eq. (32) instead of (11). In this case, it turns out that the density pˆζ(z; t) is a function of t. Hence, the stochastic processes ˆζ(t) of the Class VII channel simulators are first-order stationary only in the limit t ±. H. Class VIII Channel Simulators The channel simulators of Class VIII are defined by the set of stochastic processes ˆζ(t) = ˆµ(t) with random gains c n, random frequencies f n, and random phases θ n, i.e., ˆµ(t) = c n e j(2πfnt+θn). (25) The density pˆζ(z) of the stochastic process ˆζ(t) is given by (23), because the random behavior of the frequencies f n has no influence on pˆζ(z) in (23). Hence, a Class VIII channel simulator is first-order stationary. V. APPLICATION OF THE CONCEPT From the previous sections, it is clear that the SOC model depends on three types of parameters, namely, the gains, the Doppler frequencies, and the phases. We can choose each of these parameters as random variables or constants. From Section IV, it is obvious that eight classes of channel simulators can be defined. The results obtained in Section IV are summarized in Table 1. This table illustrates also the relationships between the eight classes of simulators. For example, the Class VIII simulator is a superset of all the other classes and a Class I simulator can belong to any of the other classes. Table 1 Classes of SOC Models and Their Stationary Properties. Class Gains Frequ. Phases First-order c n f n θ n stationary I const. const. const. II const. const. RV yes III const. RV const. no/yes a IV const. RV RV yes V RV const. const. no VI RV const. RV yes VII RV RV const. no/yes a VIII RV RV RV yes a Only in the limit t ±. In the literature, several parameter computation methods for SOS-based simulation models have been developed [2, [12, [21 [23. For a detailed discussion of well-known methods, we refer to [7. It is important to mention that most of the parameter computation methods developed for SOS models cannot be adopted to SOC models without substantial modifications. In [24, a parameter computation method for SOC models that can be applied on any given 43

5 asymmetrical Doppler power spectrum has been proposed. If this proposed method is applied, then the gains c n and frequencies f n are constant quantities and the phases θ n are random variables. Consequently, the resulting channel simulator belongs to the Class II channel simulators. For the important case of isotropic scattering, the method of exact Doppler spread with set partitioning (MEDS-SP) has been modified in [14 to determine the parameters of SOC models. By applying the modified MEDS-SP, the resulting channel simulator belongs also to the Class II. In [7, the L p -norm method (LPNM) has been introduced as a high performance parameter computation method for SOS models under both isotropic and non-isotropic scattering conditions. The LPNM can directly be applied to SOC-based simulation models. In this case, the channel simulator also belongs to the Class II. VI. CONCLUSION In this paper, the stationary properties of Rayleigh fading channel simulators based on the SOC model have been analyzed. The SOC model depends on three types of parameters, namely, the gains, the frequencies, and the phases. Each of these parameters can be a random variable or a constant. Hence, it is obvious that 2 3 1=7classes of stochastic Rayleigh fading channel simulators and one class of deterministic Rayleigh fading channel simulators can be defined. A detailed analysis of the stationary properties of these 8 classes of Rayleigh fading channel simulators has been presented. Exact analytical expressions for the density function of the envelope of each class of channel simulators have been derived. It has been shown that, under some conditions, six of the eight different classes of SOC-based channel simulators are stationary. If we consider the case when the gains are random variables and the frequencies and phases are constant quantities, a non-stationary stochastic channel simulator is obtained. In case that all model parameters are constant, there is no meaning to examine the stationary properties of the resulting deterministic process, because the concept of stationary processes can only be applied on stochastic processes. The results presented here give guidance to researchers for the development of new parameter computation methods, because our investigation of different classes of channel simulators has shown that only if we consider the gains as random variables and the other parameters are constant, then the resulting channel simulator is always non-stationary. REFERENCES [1 W. C. Jakes, Ed., Microwave Mobile Communications. Piscataway, NJ, : IEEE Press classic reissue ed., [2 M. Pätzold, U. Killat, F. Laue, and Y. 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