Classes of Sum-of-Sinusoids Rayleigh Fading Channel Simulators ...

Classes of Sum-of-Sinusoids Rayleigh Fading Channel Simulators and Their Stationary and Ergodic Properties — Part II ¨ MATTHIAS PATZOLD, and BJØRN OLAV HOGSTAD Faculty of Engineering and Science Agder University College Grooseveien 36, NO-4876 Grimstad NORWAY {matthias.paetzold, bjorn.o.hogstad}@hia.no http://www.ikt.hia.no/mobilecommunications Abstract: – In this second part of our paper, we introduce four further classes of Rayleigh fading channel simulators and continue with the analysis of the stationary and ergodic properties of the resulting sumof-sinusoids models used to design Rayleigh fading channels simulators. Just as in Part I, each individual class will systematically be investigated with respect to its stationary and ergodic properties. Our study allows the establishment of general conditions under which the sum-of-sinusoids procedure results in stationary and ergodic channel simulators. Moreover, with the help of the proposed classification scheme, several popular parameter computation methods are investigated concerning their usability to design efficient fading channel simulators. The treatment of the problem shows that if and only if the gains and frequencies are constant quantities and the phases are random variables, then the sum-of-sinusoids defines a stationary and ergodic stochastic process. Keywords: – Mobile fading channels, channel modelling, propagation, channel simulators, Rice’s sumof-sinusoids, stochastic processes, deterministic processes.

1 Introduction In present days, the application of the sumof-sinusoids principle ranges from the development of channel simulators for relatively simple time-variant Rayleigh fading channels [1–3] and Nakagami channels [4, 5] over frequency-selective channels [6–10] up to elaborated space-time narrowband [11,12] and space-time wideband [13–16] channels. Further applications can be found in research areas dealing with the design of multiple cross-correlated [17] and multiple uncorrelated [18,19] Rayleigh fading channels. Such channel models are of special interest, e.g., in system performance studies of multiple-input multipleoutput systems [20, 21] and diversity schemes [22, Chap. 6], [23]. Moreover, it has been shown that the sum-of-sinusoids principle enables the design of fast fading channel simulators [24] and facilitates the development of perfect channel models [25]. A perfect channel model is a model, whose scattering function can perfectly be fitted to any given measured scattering function obtained from snap-shot measurements carried out in real-world

environments. Finally, it should be mentioned that the sum-of-sinusoids principle has successfully been applied recently to the design of burst error models with excellent burst error statistics [26, 27] and to the development of frequency hopping channel simulators [28, 29]. In Part I [30], we have introduced four different classes of sum-of-sinusoids models having in common that the gains are constant. In this second part, we focus on sum-of-sinusoids models characterized by random gains, while the frequencies and phases can either be constants or random variables or a combination of random variables and constants. The organization of the paper is as follows. Section 2 introduces another four different classes of sum-of-sinusoids-based simulation models and investigates their stationary and ergodic properties. Section 3 summarizes the results and applies the proposed concept to the most important parameter computation methods, which are often used in practice. Finally, the conclusion is drawn in Section 4.

2 Classification of Channel Simulators In the following, we study the stationary and ergodic properties of the remaining four classes of sum-of-sinusoids determined by random gains, constant or random frequencies, and constant or random phases. We will use the same notation for the mathematical symbols introduced in Part I of our paper [30]. Especially, all random variables and stochastic processes will be written in boldface letters, and constants and deterministic processes are denoted by normal letters.

are i.i.d. random variables with zero mean and variance σc2 , i.e., E{ci,n } = 0 and Var{ci,n } = E{c2i,n } = σc2 . Then, the mean of µ ˜ i (t) is constant and equal to zero, because mµ˜i

= E{˜ µi (t)} (N ) i X = E ci,n cos(2πfi,n t + θi,n ) n=1

=

Ni X

E{ci,n } cos(2πfi,n t + θi,n )

n=1

2.1 Class V Channel Simulators The channel simulators of Class V are defined by the set of stochastic processes µ ˜ i (t) with random gains ci,n , constant frequencies fi,n , and constant phases θi,n , i.e., µ ˜ i (t) =

Ni X

ci,n cos(2πfi,n t + θi,n ) .

= 0.

The autocorrelation function rµ˜i µ˜i (t1 , t2 ) of µ ˜ i (t) is obtained by substituting (1) in [30, Eq. (12)]. Taking into account that the gains ci,1 , ci,2 , . . . , ci,Ni are i.i.d. random variables of zero mean, we find

(1)

n=1

The derivation of the density pµ˜i (x; t) of µ ˜ i (t) is similar to the procedure described in the Appendix of Part I [30]. For reasons of brevity, we only present here the final result, which can be read as follows

Z∞

 

pµ˜i (x; t) = −∞

Ni Z∞ Y

pc (y)ej2πyν cos(2πfi,n t+θi,n )

n=1−∞ −j2πνx

· dy] e

dν

(3)

(2)

where pc (·) denotes the common density function of the gains ci,n . The above expression can be interpreted as follows. The inner integral represents the characteristic function of a single sinusoid µ ˜ i,n (t) = ci,n cos(2πfi,n t + θi,n ), where the gains ci,n are i.i.d. random variables described by the density pc (·). The product form of the characteristic function Ψµ˜i (ν) of the sum-of-sinusoids µ ˜ i (t) is a result of (1), and the outer integral represents the inverse transform of Ψµ˜i (ν). From (2), we realize that the density pµ˜i (x; t) is a function of time. Hence, the stochastic process µ ˜ i (t) is not FOS. In the following, we impose on the stochastic channel simulator the condition that the gains ci,n

rµ˜i µ˜i (t1 , t2 ) =

Ni σc2 X [cos(2πfi,n (t1 − t2 )) 2 n=1

+ cos(2πfi,n (t1 + t2 ) + 2θi,n )] . (4) Without imposing any specific distribution on ci,n , we can realize that rµ˜i µ˜i (t1 , t2 ) is not only a function of τ = t1 − t2 but also of t1 + t2 . Hence, µ ˜ i (t) is not even WSS. The condition mµ˜i = mµ˜i is fulfilled if E{ci,n } = 0, so that µ ˜ i (t) is mean-ergodic. A comparison of (4) and [30, Eq. (16)] shows that rµ˜i µ˜i (τ ) 6= rµ˜i µ˜i (τ ). Therefore, the stochastic process µ ˜ i (t) is non-autocorrelation-ergodic. In the following, we study the density pµ˜i (x; t) in (2) for three specific distributions pc (y) of the gains ci,n . Case 1: In the first case, we derive the distribution of µ ˜ i (t) under the condition that the gains are given by [30, Eq. (24a)], i.e., the gains ci,n are no random variables anymore but constant quantities. This implies that the density function of ci,n is equal to pc (y) = δ(y − ci,n )

(5)

where p δ(·) being the delta function and ci,n = σ0 2/Ni . Inserting (5) in (2) gives us

pµ˜i (x; t) =

Z∞ " Y Ni

each with zero mean and variance σc2 , i.e.,

# e

j2πνci,n cos(2πfi,n t+θi,n )

1 pc (y) = √ e 2πσc

n=1

−∞ −j2πνx

·e Z∞

e j2πν

=

dν PNi

n=1 ci,n

y2 2σc2 .

(9)

Then, after substituting (9) in (2) and using [32, Eq. (7.4.6)], we get the following density function pµ˜i (x; t) of µ ˜ i (t)  Z∞  Y Z∞ Ni 2 2 2 √ pµ˜i (x; t) = · e −y /(2σc )  2πσc n=1

cos(2πfi,n t+θi,n )

−∞

· e −j2πνx dν Z∞ = e −j2πν(x−˜µi (t)) dν

−∞

0

· cos[2πyν cos(2πfi,n t + θi,n )] dy}

−∞

= δ(x − µ ˜i (t)) .

(6)

This result describes the instantaneous density of the deterministic process µ ˜i (t). Case 2: In the second case, we assume that the gains ci,n are given by ci,n = sin (2πui,n ), where ui,n denotes again a random variable with a uniform distribution in the interval (0, 1]. Here, the gains are allowed to take on negative values. Typically, gains are positive, with the negative sign being attributed to a phase in the interval (0, 2π]. From a transformation of random variables [31], we find the density pc (y) of ci,n in the form   p1 , |y| < 1 pc (y) = (7) π 1 − y2  0, |y| ≥ 1 . The equation above defines the well-known bathtub shape, like the classical Doppler spectrum for Clarke’s isotropic scattering model. This is not surprising, as the definition of the gains is closely related to the Doppler frequencies of the isotropic scattering model. Substituting (7) in (2) and solving the inner integral by using [32, Eq. (9.1.18)] enables us to express the first-order density of µ ˜ i (t) as # Z∞ " Y Ni pµ˜i (x; t) = 2 J0 (2π cos(2πfi,n t + θi,n )ν) 0

−

n=1

· cos(2πνx) dν .

(8)

Note that (8) is identical to the density function in (17) of Part I [30], when substituting the quantities ci,n by cos(2πfi,n t + θi,n ). Case 3: Finally, we consider the gains ci,n as independent Gaussian distributed random variables,

· e −j2πνx dν # Z∞ " Y Ni 2 =2 e−2[πσc ν cos(2πfi,n t + θi,n )] 0

n=1

· cos(2πνx) dν Ni X cos2 (2πfi,n t + θi,n ) Z∞ −2(πσc ν)2 n=1 =2 e 0

· cos(2πνx) dν .

(10)

The remaining integral in the above expression can be solved analytically by using [32, Eq. (7.4.6)] once again. This enables us to express pµ˜i (x; t) in the following closed form x2 2 e 2σ (t) −

1 pµ˜i (x; t) = √ (11) 2πσ(t) qP Ni 2 where σ(t) = σc n=1 cos (2πfi,n t + θi,n ). The result in (11) reveals that the stochastic process µ ˜ i (t) behaves like a zero-mean Gaussian 2 process PNi with2 a time-variant variance σ (t) = 2 σc n=1 cos (2πfi,n t + θi,n ). Note that the frequencies fi,n determine the rate of change of the time-variant variance σ 2 (t), and that its time average, denoted by σ ¯ 2 , equals σ ¯ 2 = Ni σc2 /2.

2.2 Class VI Channel Simulators The channel simulators of Class VI are defined by the set of stochastic processes µ ˜ i (t) with random gains ci,n , constant frequencies fi,n , and random phases θ i,n with a uniform distribution in the interval (0, 2π], i.e., µ ˜ i (t) =

Ni X n=1

ci,n cos(2πfi,n t + θ i,n ) .

(12)

By taking into account that random variables ci,n and θ i,n are independent, it is shown in the Appendix that the density pµ˜i (x) of µ ˜ i (t) can be expressed as

∞ ·Z ∞

Z pµ˜i (x) = 2

0

−∞

¸Ni

µ ˜ i (t) =

pc (y)J0 (2πyν) dy

· cos(2πνx) dν

(13)

= E{˜ µi (t)} (N ) i X = E ci,n cos(2πfi,n t + θ i,n ) n=1

=

Ni X

E{ci,n }E{cos(2πfi,n t + θ i,n )}

n=1

= 0

(14)

where we have made use of the fact that E{cos(2πfi,n t + θ i,n )} = 0. The autocorrelation function rµ˜i µ˜i (τ ) of µ ˜ i (t) is obtained by averaging rµ˜i µ˜i (τ ), as given in [30, Eq. (19)], with respect to the distribution of the gains ci,n , i.e.,

rµ˜i µ˜i (τ ) =

Ni X E{c2i,n } n=1

=

2

Ni X

ci,n cos(2πfi,n t + θi,n )

(16)

n=1

where pc (·) denotes again the density function of the gains ci,n . The mean mµ˜i of µ ˜ i (t) is obtained as follows mµ˜i

2.3 Class VII Channel Simulators This class of channel simulators involves all stochastic processes µ ˜ i (t) with random gains ci,n , random frequencies fi,n , and constant phases θi,n , i.e.,

where, with regards to practical situations, it is assumed that the gains ci,n and frequencies fi,n are independent, as stated above. To find the density pµ˜i (x) of the Class VII channel simulators, we consider—for reasons of simplicity—the case t → ±∞ and we exploit the fact that the density of the Class III channel simulators can be considered as the conditional density pµ˜i (x|ci,n = ci,n ). Since this density is given by [30, Eq. (23)] for t → ±∞, the desired density pµ˜i (x) of the Class VII channel simulators can be obtained by averaging pµ˜i (x|ci,n = ci,n ) over the distribution pc (y) of the Ni i.i.d. random variables ci,1 , ci,2 , . . . , ci,Ni , i.e., Z∞ Z∞ pµ˜i (x) = · · · pµ˜i (x|ci,1 = y1 , . . . , ci,Ni = yNi ) −∞ −∞

·

"N Yi

 Ni Z∞ Z∞ = 2  pc (y)J0 (2πyν) dy  −∞

0

n=1

where σc2 and mc are the variance and the mean of ci,n , respectively. Since the conditions in [30, Eqs. (10)–(12)] are fulfilled, we have proved that µ ˜ i (t) is a FOS and WSS process. Also the condition mµ˜i = mµ˜i is fulfilled, so that µ ˜ i (t) is mean-ergodic. Without imposing any specific distribution on ci,n , we can say that the autocorrelation function rµ˜i µ˜i (τ ) of the stochastic process µ ˜ i (t) is different from the autocorrelation function rµ˜i µ˜i (τ ) of a sample function µ ˜i (t), i.e., rµ˜i µ˜i (τ ) 6= rµ˜i µ˜i (τ ). In this context, the process µ ˜ i (t) is in general non-autocorrelationergodic.

pc (yn ) dy1 · · · dyNi

n=1

cos(2πfi,n τ )

Ni σc2 + m2c X cos(2πfi,n τ ) (15) 2

#

· cos(2πνx) dν

as t → ±∞ .

(17)

Note that this result is identical to the density in (13). If t is finite, then we have to repeat the above procedure by using [30, Eq. (21)] instead of [30, Eq. (23)]. In this case, it turns out that the density pµ˜i (x; t) depends on time t, so that µ ˜ i (t) is not FOS. The mean mµ˜i (t) of µ ˜ i (t) is obtained as follows mµ˜i (t) = E{˜ µi (t)} (N ) i X = E ci,n cos(2πfi,n t + θi,n ) n=1

=

Ni X

E{ci,n }E{cos(2πfi,n t + θi,n )}

n=1

= mc

Ni X n=1

E{cos(2πfi,n t + θi,n )}

(18)

where mc denotes the mean value of the i.i.d. random variables ci,n . Hence, mµ˜i (t) is only zero, if mc = 0 and/or the distribution P i of the random frequencies fi,n is such that N n=1 E{cos(2πfi,n t + θi,n )} can be forced to zero. The latter condition is fulfilled, if the distribution of fi,n is an even PNi function and if n=1 cos(θi,n ) = 0. Note that this statement has also been made below (25) in Part I [30]. The autocorrelation function rµ˜i µ˜i (t1 , t2 ) of a Class VII channel simulator is obtained by averaging the right-hand side of (4) with respect to fi,n , i.e., rµ˜i µ˜i (t1 , t2 ) =

Ni σc2 X [E{cos(2πfi,n (t1 − t2 ))} 2 n=1

+E{cos(2πfi,n (t1 + t2 ) + 2θi,n )}] (19) where σc2 denotes the variance of ci,n . Obviously, the autocorrelation function rµ˜i µ˜i (t1 , t2 ) depends on t1 − t2 and t1 + t2 . However, if the conditions: (i) E{ci,n } = mc = 0, (ii) the density of fi,n is P i even, and (iii) N n=1 cos(2θi,n ) = 0 are fulfilled, then rµ˜i µ˜i (t1 , t2 ) is only a function of τ = t1 − t2 . In this case, we obtain Ni σc2 X rµ˜i µ˜i (τ ) = E{cos(2πfi,n τ )} . 2

(20)

n=1

Now, let fi,n be given as in [30, Eq. (24b)] and σc2 by σc2 = 2σ02 /Ni . Then the autocorrelation function rµ˜i µ˜i (τ ) in (20) equals the autocorrelation function rµi µi (τ ) of the reference model in [30, Eq. (3)], i.e., rµ˜i µ˜i (τ ) = σ02 J0 (2πfmax τ ) .

(21)

From the investigations in this subsection, it turns out that µ ˜ i (t) is WSS and mean-ergodic, if the above mentioned boundary conditions are fulfilled. If the boundary conditions are fulfilled and t → ±∞, then µ ˜ i (t) tends to a FOS process. In no case, the stochastic process µ ˜ i (t) is autocorrelation-ergodic, since the condition rµ˜i µ˜i (τ ) = rµ˜i µ˜i (τ ) is violated.

2.4 Class VIII Channel Simulators The Class VIII channel simulators are defined by the set of stochastic processes µ ˜ i (t) with random gains ci,n , random frequencies fi,n , and random

phases θ i,n , which are uniformly distributed in the interval (0, 2π]. In this case, the stochastic process µ ˜ i (t) is of type µ ˜ i (t) =

Ni X

ci,n cos(2πfi,n t + θ i,n )

(22)

n=1

where it is assumed that all random variables are statistically independent. Concerning the density function pµ˜i (x), we can resort to the result in (13), which was obtained for a sum-of-sinusoids with random gains ci,n and random phases θ i,n . This result is still valid, because the random behavior of the frequencies fi,n has no influence on pµ˜i (x) in (13). Also the mean mµ˜i is identical to [30, Eq. (18)], i.e., mµ˜i = 0. To ease the derivation of the autocorrelation function rµ˜i µ˜i (τ ) of µ ˜ i (t), we average rµ˜i µ˜i (τ ) (see [30, Eq. (19)]) with respect to the distributions of ci,n and fi,n . This results in rµ˜i µ˜i (τ ) =

Ni σc2 + m2c X E{cos(2πfi,n τ )} . (23) 2 n=1

Without assuming any specific distribution of ci,n and fi,n , we can say that µ ˜ i (t) is both FOS and WSS, since the conditions in [30, Eqs. (10)– (12)] are fulfilled. Also, we can easily see that the condition mµ˜i = mµ˜i is fulfilled, proving the fact that µ ˜ i (t) is mean-ergodic. But µ ˜ i (t) is nonautocorrelation-ergodic, since rµ˜i µ˜i (τ ) 6= rµ˜i µ˜i (τ ).

3 Application of the Concept

For any given number Ni > 0, we have seen that the sum-of-sinusoids depends on three types of parameters (gains, frequencies, and phases), each of which can be a random variable or a constant. However, at least one random variable is required to obtain a stochastic process µ ˜ i (t)—otherwise the sum-of-sinusoids defines a deterministic process µ ˜i (t). From the discussions in Section 5, it is obvious that altogether 23 − 1 = 7 classes of stochastic Rayleigh fading channel simulators and one class of deterministic Rayleigh fading channel simulators can be defined. The analysis of the various classes with respect to their stationary and ergodic properties has been given in the previous section. The results 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. To the best of the authors knowledge, the Classes III, V, VI, and VII have never been introduced before. The new Class VI—and under certain conditions also the new Classes III and VII—have the same stationary and ergodic properties as the well-known Class IV, which is often used in practice.

Table 1 Classes of Deterministic and Stochastic Channel Simulators and Their Statistical Properties. Class

Gains Frequ. Phases ci,n fi,n θi,n

Firstorder stat.

Widesense stat.

Meanergodic

Autocorr. erg.

I

const. const.

const.

–

–

–

–

II

const. const.

RV

yes

yes

yes

yes

III

const.

no/

no/

no/

no

RV

const.

yesa , b , c , d yes a,b,c

yes a,b

const.

RV

RV

yes

yes

yes

no

V

RV

const.

const.

no

no

no/

no

VI

RV

const.

RV

VII

RV

RV

const.

no/ yes a,b,c,d

VIII

RV

RV

RV

yes

yes

yes

no/ no/ yes a,b,c yes e or a,b yes

yes

Class

First- Wide- Mean- Autoorder sense erg. correl.stat. stat. erg.

Rice method [33, 34]

II

yes

yes

yes

Monte Carlo method [6, 7]

IV

yes

yes

yes

no

no

Jakes method [1]

II

yes

yes

yes

yes

no

(with random phases) II

yes

yes

yes

yes

Method of equal distances [35]

II

yes

yes

yes

yes

Method of equal areas [35]

II

yes

yes

yes

yes

Mean-square-error method [35]

II

yes

yes

yes

yes

Method of exact Doppler

II

yes

yes

yes

yes

Lp-norm method [2]

II

yes

yes

yes

yes

Method proposed by Zheng

IV

yes

yes

yes

no

VIII

yes

yes

yes

no

yes e yes

Table 2 Overview of Parameter Computation Methods and the Statistical Properties of the Resulting Stochastic Channel Simulators. Parameter computation method

IV

.

is FOS and mean-ergodic, but unfortunately nonautocorrelation-ergodic. For a given parameter computation method, the stationary and ergodic properties of the resulting channel simulator can directly be examined with the help of Table 1. It goes without saying that the above concept is easy to handle and can be applied to any given parameter computation method. For some selected methods [1–3,6,7,9,18,33–35], the obtained results are presented in Table 2.

Harmonic decomposition no

a

If the density of fi,n is an even function. P i b If the boundary condition N n=1 cos(θi,n ) = 0 is fulfilled. P c i If the boundary condition N n=1 cos(2θi,n ) = 0 is fulfilled. d Only in the limit t → ±∞. e If the gains ci,n have zero mean, i.e., E{ci,n } = 0.

yes

technique [9]

spread [2]

and Xiao [18] Improved method by Zheng and Xiao [3]

In the following, we apply the above concept to some selected parameter computation methods. Starting with the original Rice method [33, 34], we realize — by considering (5) and (6) in Part I [30] — that the gains ci,n and frequencies fi,n are constant quantities. Due to the fact that the phases θ i,n are random variables, it follows from Table 1 that the resulting channel simulator belongs to Class II. Such a channel simulator enables the generation of stochastic processes which are not only FOS but also mean- and autocorrelation-ergodic. On the other hand, if the Monte Carlo method [6, 7] or the method due to Zheng and Xiao [18] is applied, then the gains ci,n are constant quantities and the frequencies fi,n and phases θ i,n are random variables. Consequently, the resulting channel simulator can be identified as a Class IV channel simulator, which 1

[36].

Of great popularity is the Jakes method [1]. When this method is used, then the gains ci,n , frequencies fi,n , and phases θi,n are constant quantities. Consequently, the Jakes channel simulator is per definition completely deterministic. To obtain the underlying stochastic channel simulator, it is advisable to replace the constant phases1 θi,n by random phases θ i,n . According to Table 1, the mapping θi,n 7→ θ i,n transforms a Class I type channel simulator into a Class II type one, which is FOS, mean-ergodic, and autocorrelationergodic. It should be pointed out here that our result is in contrast with the analysis in [37], where it has been claimed that Jakes’ simulator is nonstationary. However, this is not surprising when we take into account that the proof in [37] is based

It should be noted that the phases θi,n in Jakes’ channel simulator are equal to 0 for all i = 1, 2 and n = 1, 2, . . . , Ni

on the assumption that the gains ci,n are random variables. But this assumption cannot be justified in the sense of the original Jakes method, where all model parameters are constant quantities. Nevertheless, the Jakes simulator has some disadvantages [36], which can be avoided, e.g., by using the method of exact Doppler spread [2] or the even more powerful Lp -norm method [2]. Both methods enable the design of deterministic channel simulators, where all parameters are fixed including the phases θi,n . The investigation of the stationary and ergodic properties of deterministic processes makes no sense, since the concept of stationarity and ergodicity is only applicable to stochastic processes. A deterministic channel simulator can be interpreted as an emulator for sample functions of the underlying stochastic channel simulator. According to the concept of deterministic channel modelling, the corresponding stochastic simulation model is obtained by replacing the constant phases θi,n by random phases θ i,n (see Fig. 1). Important is now to realize that all stochastic channel simulators derived in this way from deterministic channels simulators are Class II channel simulators with the known statistical properties as they are listed in Table 1. This must be taken into account when considering the results shown in Table 2.

clusions, it was necessary to investigate all eight classes of channel simulators. The results presented here are also of fundamental importance for the performance assessment of existing design methods. Moreover, the results give strategic guidance to engineers for the development of new parameter computation methods enabling the design of stationary and ergodic channel simulators, because the proposed scheme might prevent them from introducing random variables for the gains and/or frequencies. Although we have restricted our investigations to Rayleigh fading channel simulators, the obtained results are of general importance to all types of channel simulators, wherever the sum-of-sinusoids principle is employed.

Appendix In this appendix, we derive the density function pµ˜i (x) of a sum-of-sinusoids with random gains ci,n , constant frequencies fi,n , and random phases θ i,n , which are uniformly distributed in the interval (0, 2π]. It is assumed that the gains ci,n and the phases θ i,n are statistically independent. Our starting point is the solution of the Problem 6-38 in [31, p. 239]. In this reference, we can find the probability density function of a single sinusoid µ ˜ i,n (t) = ci,n cos(2πfi,n t + θ i,n ) in the following form

4 Conclusion In this paper, the stationary and ergodic properties of Rayleigh fading channel simulators using the sum-of-sinusoids principle have been analyzed. Depending on whether the model parameters are random variables or constant quantities, altogether one class of deterministic channel simulators and seven classes of stochastic channel simulators have been defined, where four of them have never been studied before. Our analysis has shown that if and only if the phases are random variables and the gains and frequencies are constant quantities, then the resulting channel simulator is both stationary and ergodic. If the frequencies are random variables, then the stochastic channel simulator is always non-autocorrelation-ergodic, but stationary if certain boundary conditions are fulfilled. The worst case, however, is given when the gains are random variables and the other parameters are constants. Then, a non-stationary channel simulator is obtained. To arrive at these con-

pµ˜i,n (x) =

1 π +

−|x| Z

p (y) p c dy y 2 − x2

−∞ Z∞

1 π

|x|

p (y) p c dy y 2 − x2

(24)

where pc (·) is the common density function of the gains ci,n . Let us denote the characteristic func(1) (2) tion of pµ˜i,n (x) by Ψµ˜i,n (ν) = Ψµ˜i,n (ν) + Ψµ˜i,n (ν), (1)

(2)

where Ψµ˜i,n (ν) and Ψµ˜i,n (ν) are the characteristic functions of the density defined by the first and second integral in (24), respectively. The first (1) characteristic function Ψµ˜i,n (ν) can be expressed as

function Ψµ˜i (ν) of pµ˜i (x) is given by 1 (1) Ψµ˜i,n (ν) = π =

1 π

1 = π

Z∞ −|x| Z −∞ −∞ Z0 Zy

−∞ −y

p (y) p c ej2πxν dydx y 2 − x2

Z0 pc (y)

−∞

1 π

−y



pc (y) −∞

0

2 = π

Z0

Finally, the density function pµ˜i (x) is obtained by computing the inverse transformation of the characteristic function Ψµ˜i (ν) given above, i.e., Z∞

cos(2πxν) p dxdy . (25) y 2 − x2

Zπ/2 pc (y) cos(2πyν cos θ) dθdy(26)

−∞

0

from which, by using the integral representation of the zeroth-order Bessel function of the first kind [38, Eq. (3.715.19)] 2 J0 (z) = π

Zπ/2 cos(z cos θ) dθ

(27)

0

the result Z0 (1)

Ψµ˜i,n (ν) =

pc (y)J0 (2πyν) dy

(28)

−∞

immediately follows. Similarly, it can be shown (2) that the second characteristic function Ψµ˜i,n (ν) is given by Z∞ (2) Ψµ˜i,n (ν)

=

pc (y)J0 (2πyν) dy  . (31)

−∞

In the first term of (25), we substitute x by −y cos(θ), and in the second term, we replace x by y cos(θ). This leads to the expression (1) Ψµ˜i,n (ν)

Ni

Z∞

= 

cos(2πxν) p dxdy y 2 − x2

Zy

Z0

Ψµ˜i,n (ν)

n=1

p (y) p c ej2πxν dxdy 2 2 y −x

Z0

+

Ψµ˜i (ν) =

Ni Y

pc (y)J0 (2πyν) dy .

(29)

0

Hence, it turns out that

Ψµ˜i (ν) e−j2πxν dν

pµ˜i (x) = −∞

 Ni Z∞ Z∞ = 2  pc (y)J0 (2πyν) dy  0

−∞

· cos(2πνx) dν .

(32)

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