IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 5, NO. 12, DECEMBER 2006 3667 Novel Sum-of-Sinusoids Simulation Models for Rayleigh and Rician Channels Chengshan Xiao, Senior Member, IEEE, Yahong Rosa Zheng, Member, IEEE, and Norman C. Beaulieu, Fellow, IEEE

Abstract— The statistical properties of Clarke’s fading model [16]-[19] and improvements [22], [24], [25] of Jakes’ simu- with a finite number of sinusoids are analyzed, and an improved lator for generating multiple uncorrelated fading waveforms reference model is proposed for the simulation of needed for modeling frequency selective fading channels and channels. A novel statistical simulation model for Rician fading channels is examined. The new Rician fading simulation model multiple-input multiple-output (MIMO) channels have been employs a zero-mean stochastic sinusoid as the specular (line-of- reported. Since Jakes’ simulator needs only one fourth the sight) component, in contrast to existing Rician fading simulators number of low-frequency oscillators as needed in Clarke’s that utilize a non-zero deterministic specular component. The sta- model, it is commonly perceived that Jakes’ simulator (and its tistical properties of the proposed Rician fading simulation model modifications) is more computationally efficient than Clarke’s are analyzed in detail. It is shown that the probability density function of the Rician fading phase is not only independent of model. However, it was recently established by Pop and time but also uniformly distributed over [−π,π). This property Beaulieu [19] that Jakes’ simulator and its variants (e.g., is different from that of existing Rician fading simulators. [3] and [16]) are not wide sense stationary (WSS) and that The statistical properties of the new simulators are confirmed “reduction in the number of simulator oscillators based on by extensive simulation results, showing good agreement with azimuthal symmetries is meritless”. They proposed a Clarke’s theoretical analysis in all cases. An explicit formula for the level-crossing rate is derived for general Rician fading when the model-based simulator design having the WSS property in specular component has non-zero Doppler frequency. [19], [21]. The Pop-Beaulieu simulator has been employed in a number of diverse applications [26]-[29]. In the first part of Index Terms— Fading channel simulator, Rayleigh fading, Rician fading, statistics. this paper, we give a statistical analysis of Clarke’s model with a finite number of sinusoids and show that the Pop- Beaulieu simulator has deficiencies in some of its higher-order I. INTRODUCTION statistics (as warned in [19, Section III.B]). We then propose an improved version of the Pop-Beaulieu simulator based on OBILE channel simulators are commonly used Clarke’s model for Rayleigh fading channels. in the laboratory because they make system tests and M All the existing Rician channel simulation models in the evaluations less expensive and more reproducible than field literature assume that the specular (line-of-sight) component trials. Many different techniques have been proposed for the is either constant and non-zero [13], or time-varying and modeling and simulation of mobile radio channels [1]-[25]. deterministic [4], [16]. These assumptions may not reflect the Among them, the well known Jakes’ model [3], which is physical nature of specular components, particularly when a a simplified simulation model of Clarke’s model [1], has specular component is random, changing from time to time been widely used for frequency nonselective Rayleigh fading and from mobile to mobile. Furthermore, according to [4], channels for about three decades. Various modifications [9], all these Rician fading models are nonstationary in the wide Manuscript received February 3, 2005; revised October 6, 2005; accepted sense and the probability density function (PDF) of the fading November 27, 2005. The editor coordinating the review of this paper and phase is a function of time [4], [16]. In the second part of this approving it for publication is X. Shen. The work of C. Xiao was supported paper, a novel statistical simulation model will be proposed for in part by the National Science Foundation under Grant CCF-0514770 and the University of Missouri-Columbia Research Council under Grant URC-05-064. Rician fading channels. The specular component will employ The work of Y. R. Zheng was supported in part by the University of Missouri a zero-mean stochastic sinusoid with a pre-chosen angle of System Research Board. The work of N. C. Beaulieu was supported in part arrival and a random initial phase. This assumption implies by the Alberta Informatics Circle of Research Excellence (iCORE). Parts of this paper were previously presented at the IEEE Wireless Communications that different specular components in different channels may and Networking Conference (WCNC) 2003, New Orleans, LA, and the IEEE have different initial phases. International Conference on Communications (ICC) 2003, Anchorage, AK. C. Xiao is with the Department of Electrical and Computer Engi- The remainder of this paper is organized as follows. In neering, University of Missouri, Columbia, MO 65211 USA (e-mail: Section II, we present the statistical properties of Clarke’s [email protected]; http://www.missouri.edu/∼xiaoc/). model with a finite number of sinusoids and show that the Y. R. Zheng is with the Department of Electrical and Computer Engineering, University of Missouri, Rolla, MO 65409 USA (e-mail: Pop-Beaulieu simulator has limitations in its higher-order [email protected]; http://web.umr.edu/∼zhengyr/). statistics. An improved simulator for Rayleigh fading channels N. C. Beaulieu is with the Department of Electrical and Computer is proposed. In Section III, we present a novel statistical Engineering, University of Alberta, Edmonton, Canada T6G 2G7 (e-mail: [email protected]; http://www.ee.ualberta.ca/∼beaulieu/). simulation model for Rician fading channels, and analyze Digital Object Identifier 10.1109/TWC.2006.05068 the statistical properties of the new Rician fading model. 1536-1276/06$20.00 c 2006 IEEE 3668 IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 5, NO. 12, DECEMBER 2006

Section IV gives extensive performance evaluations of the new Proof: The autocorrelation function of the real part of Rayleigh and Rician fading simulators. Section V concludes the fading g(t) is proved as follows the paper. ( )= [ ( ) ( + )] Rgcgc τ Eα,φ gc t gc t τ N N II. AN IMPROVED RAYLEIGH FADING SIMULATOR 1 = E {cos(w t cos α + φ ) N α,φ d n n Clarke’s Rayleigh fading model is sometimes referred to as n=1 i=1 · cos[ ( + )cos + ]} a mathematical reference model, and is commonly considered wd t τ αi φi as a computationally inefficient model compared to Jakes’ 1 N = E [cos(w τ cos α )] Rayleigh fading simulator. In this section, we show that 2N α d n n=1 Clarke’s model with a finite number of sinusoids can be directly used for Rayleigh fading simulation, and that its 1 N π = cos [ cos ] dαn computational efficiency and second-order statistics are as 2 wdτ αn 2 N =1 −π π good as those of improved Jakes’ simulators. We then briefly n N show that the Pop-Beaulieu simulator has some higher-order 1 1 = J0(w τ)= J0(w τ). statistical deficiencies and improve the model by introducing 2N d 2 d n=1 randomness to the angle of arrival, which leads to improved higher-order statistics. Similarly, one can prove the second part of (2a) and equations (2b)-(2c). The proof of equation (2d) is lengthy and can be treated as a special case of the proof of equation (8d) given in A. Clarke’s Rayleigh Fading Model the next subsection. The details are omitted here for brevity. The baseband signal of the normalized Clarke’s two- dimensional (2-D) isotropic scattering Rayleigh fading model It is noted here that when N approaches infinity, all the is given by [1], [30] derived statistical properties in equations (2) become identical to the desired ones of Clarke’s reference model given in [30]. N 1 In simulation practice, time-averaging is often used in place g(t)=√ exp[j(w t cos α + φ )], (1) d n n of ensemble averaging. For example, the autocorrelation of N n=1 the real part of the fading signal for one trial (sample of the where N is the number of propagation paths, wd is the process) is given by maximum radian Doppler frequency and αn and φn are, respectively, the angle of arrival and initial phase of the nth 1 T ˆ ( ) = lim ( ) ( + ) Rgcgc τ gc t gc t τ dt propagation path. Both αn and φn are uniformly distributed T →∞ T 0 over [−π, π) for all n and they are mutually independent. 1 N The central limit theorem justifies that the real part, gc(t)= = cos(w τ cos α ). 2N d n Re[g(t)], and the imaginary part, gs(t)=Im[g(t)], of the fad- n=1 ing g(t) can be approximated as Gaussian random processes for large N. Some desired second-order statistics for fading Clearly, this time averaged autocorrelation changes from simulators are manifested in the autocorrelation and cross- trial to trial due to the random angle of arrival. Note { ˆ ( )} = correlation functions which are given in [30] for the case when that the variance of the time average, Var Rgcgc τ | ˆ ( )−0 5 ( )|2 N approaches infinity. However, the statistical properties of E Rgcgc τ . J0 wdτ , carries important information Clarke’s model with a finite value of N (number of sinusoids) indicating the closeness between a single trial with finite N are not available in the literature. These properties are very and the ideal case with N = ∞. We now present the time- important for justifying the suitability of Clarke’s model as averaged variances of the aforementioned correlation statistics. a valid Rayleigh fading simulator. Thus, we present some of Theorem 2: The variances of the autocorrelation and cross- these key statistics here. correlation of the quadrature components, and the variance Theorem 1: The autocorrelation and cross-correlation func- of the autocorrelation of the complex envelope of the fading tions of the quadrature components, and the autocorrelation signal g(t) are given by functions of the complex envelope and the squared envelope of fading signal g(t) are given by { ˆ ( )} = { ˆ ( )} Var Rgcgc τ Var Rgsgs τ 1+ (2 ) − 2 2( ) 1 = J0 wdτ J0 wdτ R (τ)=R (τ)= J0(w τ) (3a) gcgc gsgs 2 d (2a) 8N ( )= ( )=0 { ˆ ( )} = { ˆ ( )} Rgcgs τ Rgsgc τ (2b) Var Rgcgs τ Var Rgsgc τ ∗ R (τ)=E [g (t)g(t + τ)] = J0(w τ) (2c) 1 − J0(2wdτ) gg α,φ d = (3b) 2 8 2 J0 (wdτ) N 2 2 ( )=1+ ( ) − 2 R|g| |g| τ J0 wdτ , (2d) 1 − J (w τ) N Var{Rˆ (τ)} = 0 d . (3c) gg N where Eα,φ[·] denotes expectation w.r.t. α and φ,andJ0(·) is Proof: We start with the first equality of eqns. (3a) and the zero-order Bessel function of the first kind [31]. (3b) and derive XIAO et al.: NOVEL SUM-OF-SINUSOIDS SIMULATION MODELS FOR RAYLEIGH AND RICIAN FADING CHANNELS 3669

{ ˆ ( )} Var Rgcgc τ The results given in Theorems 1 and 2 show that those 2 statistics considered that depend on N, depend on N exclu- ( ) −1 = ˆ ( ) − J0 wdτ sively as N . Therefore, the dependence on N is reduced E Rgcgc τ 2 by increasing N. We shall see later that Clarke’s model using ≥ 8 2 J 2(w τ) a number of sinusoids, N , can be usefully employed as = E Rˆ (τ) − 0 d gcgc 4 a Rayleigh fading simulator, in some applications (typically short simulation runs). In applications where the asymptotic N N 1 variance must be small (typically for long simulation runs), = E cos(w τ cos α )cos(w τ cos α ) 4N 2 d n d m larger values of N (say, 40) can be used for greater simulation n=1 m=1 2 accuracy. Its computational efficiency and statistics are similar J (w τ) − 0 d to those of the recently improved Jakes’ models [22], [24], 4 [25], which have removed some statistical deficiencies of 2( ) 1 N J0 wdτ 2 Jakes’ original model [3] and various modified Jakes’ models = − + E cos (wdτ cos αn) 4 4N 2 proposed in [9], [16], [17] and [19]. n=1 ⎫ Before proceeding to further discussion, we make a remark ⎪ N N ⎬ to acknowledge and correct a mistake in [25], which was + E [cos(wdτ cos αn)] E [cos(wdτ cos αm)] originally discovered by Sun, Ye and Choi [32]. Specifically, ⎪ n=1 m=1 ⎭ the complex fading process defined by eqn. (14) of [25] may m=n not be a Gaussian random process when the duration of time is 1 1+J0(2w τ) = · d +( 2 − ) 2( ) very short, and the autocorrelation of the squared envelope of 4 2 N 2 N N J0 wdτ N this fading process is nonstationary. However, the problems 2( ) −J0 wdτ with this fading process, which arise from a slight over- 4 simplification of earlier results in an associated conference 2 1+J0(2wdτ) − 2J0 (wdτ) version of the paper, can be easily solved by changing φ of = . 8N (14) in [25] to φn with φn being statistically independent and uniformly distributed over [−π, π) for all n. Actually, in the conference version of [25], the complex fading process was { ˆ ( )} Var Rgcgs τ defined correctly; details can be found in (14) of [22]. It is 2 also noted that inspired by Sun et al [32], we revisited and = ˆ ( ) − 0 E Rgcgs τ corrected the expression for the squared envelope correlation function of the complex fading processes we defined. After the N N 1 submission of this paper, we also noticed that Patel, Stuber and = E sin(w τ cos α )sin(w τ cos α ) 4N 2 d n d m Pratt [33] have independently discovered the aforementioned n=1 m=1 mistake in [25]. 1 N = sin2( cos ) 4 2 E wdτ αn N =1 n ⎫ B. The Pop-Beaulieu Simulator ⎪ N N ⎬ Based on Clarke’s model given by (1), Pop and Beaulieu + E [sin(wdτ cos αn)] E [sin(wdτ cos αm)] ⎪ [19], [21] recently developed a class of wide-sense stationary n=1 m=1 ⎭ 2πn Rayleigh fading simulators by setting αn = in g(t). Thus, m=n N 1 1 − (2 ) the lowpass fading process becomes = · J0 wdτ +0 4 2 N 2 N X(t)=Xc(t)+jXs(t) (4a)   1 − J0(2wdτ) N = . 1 2πn 8N Xc(t)=√ cos wdt cos + φn (4b) N N n=1 Similarly, we can validate the second equality of eqns. (3a)   1 N 2 and (3b). Thus, we have ( )=√ sin cos πn + Xs t wdt φn . (4c) 2 N N ˆ ˆ n=1 Var{Rgg(τ)} = E Rgg(τ) − J0(wdτ) They warned, however, that while their improved simulator 2 ˆ ˆ is wide sense stationary (contrary to previous sum-of-sinusoids = E 2R (τ)+j2R (τ) − J0(w τ) gcgc gcgs d simulators such as, for example, [3], [16]), it may not model 2 2 some higher-order statistical properties accurately. Reference =4 ˆ ( ) +4 ˆ ( ) E Rgcgc τ E Rgcgs τ [26] reported outstanding agreement between results obtained 2 from one implementation of the Pop-Beaulieu simulator and −J (w τ) 0 d theory in some turbo decoding applications. However, in 1 − J 2(w τ) = 0 d . general, the quality required of a simulator will depend on N the application and some higher-order behaviors may not be This completes the proof of Theorem 2. accurately modeled using this simulator. To further reveal the 3670 IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 5, NO. 12, DECEMBER 2006 statistical properties of this model, we present the following and uniformly distributed inside this sector, in contrast to correlation statistics of this model. being fixed as it is in Jakes’ model and in the Pop-Beaulieu ( )= ( ) simulator. Clarke’s model and a simulator proposed by Hoeher RXcXc τ RXsXs τ (5a)   [7], assume independent αn, each uniformly distributed on N 1 2πn [−π, π). Although our simulator design requires generating = cos wdτ cos (5b) 2N N the same number of random αn, it ensures a more uniform n=1 ( )=− ( ) empirical distribution of αn, particularly for small values RXcXs τ RXsXc τ (5c)   of N, (but does not fix the values of αn). We shall see N 1 2πn subsequently that this modification reduces the variances of = sin w τ cos (5d) 2N d N the empirical simulator statistics. It can be shown that the n=1 first-order statistics of this improved model are the same as ( )=2 ( )+ 2 ( ) RXX τ RXcXc τ j RXcXs τ (5e) those of the Pop-Beaulieu simulator. However, some second- 2 2 1 order statistics of this improved model are different, and they R| |2| |2 (τ)=1+4R (τ)+4R (τ) − . X X XcXc XcXs N are presented below. (5f) Theorem 3: The autocorrelation and cross-correlation func- tions of the quadrature components, and the autocorrelation The proof of these statistics shown above is a special case functions of the complex envelope and the squared envelope of the proof of Theorem 3 given in the next subsection. The of fading signal Y (t) are given by details are omitted here for brevity. 1 We make three remarks based on (5): 1) These second- ( )= ( )= ( ) RYcYc τ RYsYs τ J0 wdτ (8a) order statistics of this modified model with N = ∞ are 2 ( )= ( )=0 the same as the desired ones of the original Clarke’s model. RYcYs τ RYsYc τ (8b) However, when N is finite, the statistics of this model are RYY(τ)=J0(wdτ) (8c) different from the desired ones derived from Clarke’s model; 2 R|Y |2|Y |2 (τ)=1+J0 (wdτ) − fc(wdτ,N) 2) the statistics of this model do not converge asymptotically −f (w τ,N), (8d) to the desired ones when N increases as was discussed in s d [21] for the real part of R (τ);3)whenN is finite and where XX ( ) ( ) 2πk+π 2 odd, the imaginary part of RXX τ , along with RXcXs τ N ( ) 1 N and RXsXc τ , can significantly deviate from zero (the desired fc(wdτ,N)= cos(wdτ cos γ)dγ (9a) 2π 2πk−π value), which implies that the quadrature components of this k=1 N 2 model are statistically correlated when N is odd. N 2πk+π 1 N fs(wdτ,N)= sin(wdτ cos γ)dγ . (9b) 2π 2πk−π C. An Improved Rayleigh Fading Channel Simulator k=1 N The proof of this theorem is lengthy; a proof is outlined in Based on the statistical analyses of Clarke’s model and the Appendix I. Pop-Beaulieu simulator, we propose an improved simulation We now present the time-averaged variances of some key model as follows. correlation statistics of Y (t) in Theorem 4. Definition 1: The normalized lowpass fading process of Theorem 4: The variances of the autocorrelation and cross- an improved sum-of-sinusoids statistical simulation model is correlation of the quadrature components, and the variance defined by of the autocorrelation of the complex envelope of the fading signal Y (t) are given by Y (t)=Yc(t)+jYs(t) (6a) N { ˆ ( )} = { ˆ ( )} 1 Var RYcYc τ Var RYsYs τ Yc(t)=√ cos(wdt cos αn + φn) (6b) 1+J0(2wdτ) fc(wdτ,N) N =1 = − (10a) n 8N 4 N 1 { ˆ ( )} = { ˆ ( )} Var RYcYs τ Var RYsYc τ Ys(t)=√ sin(wdt cos αn + φn) (6c) N 1 − J0(2w τ) f (w τ,N) n=1 = d − s d (10b) 8N 4 with 1 { ˆ ( )} = − ( ) − ( ) 2πn + θn Var RYY τ fc wdτ,N fs wdτ,N . αn = ,n=1, 2, ··· ,N (7) N N (10c) where φn and θn are statistically independent and uniformly Proof: The proof of this theorem is similar to that of distributed over [−π, π) for all n. It is noted that the difference Theorem 2; details are omitted for brevity. between this improved model and the Pop-Beaulieu simulator As can be seen from Theorems 1 and 3, the correlation is the introduction of random variables θn to the angle of statistics, except the autocorrelation of the squared envelope, arrival. Randomizing θn slightly decreases the efficiency of of the improved model are the same as those of Clarke’s the simulator, but significantly improves the statistical quality model when both models have the same number of sinusoids. of the simulator. This model differs from Clarke’s model in Fig. 1 shows that the autocorrelations of the squared envelope that it forces the angle of arrival, αn, to have a value restricted for Clarke’s model (2d) and for the new model (8d) are 2nπ−π 2nπ+π =8 to the interval N , N . The angle of arrival is random similar, and that this statistic for N is closer to the ideal XIAO et al.: NOVEL SUM-OF-SINUSOIDS SIMULATION MODELS FOR RAYLEIGH AND RICIAN FADING CHANNELS 3671

Squared envelope autocorrelation by Clarke’s model with N = 8 1 Clarke’s model with N = ∞ Z(t)=Zc(t)+jZs(t) (11a) Improved model with N = 8 √ √ ) τ (

2 ( )= ( )+ cos( cos + ) 1+ 0.9 Zc t Yc t K wdt θ0 φ0 / K |Y| 2

|Y| (11b) R √ √ 0.8 Zs(t)= Ys(t)+ K sin(wdt cos θ0 + φ0) / 1+K ) and τ ( 2

|g| (11c) 2 0.7 |g|

R where K is the ratio of the specular power to scattered power, 0.6 θ0 and φ0 are the angle of arrival and the initial phase, respectively, of the specular component, and φ0 is a random Normalized 0.5 variable uniformly distributed over [−π, π). A Rician fading simulator having a specular component 0.4 with a non-zero Doppler frequency was studied in [16]. Our 0 2 4 6 8 10 Normalized time: f τ d simulator model (11) is different from the simulator in [16] be- cause in our model the initial phase of the specular component Fig. 1. Theoretical autocorrelations of the squared envelopes of Clarke’s is considered a random variable uniformly distributed over model and our improved model. [−π, π), while the initial phase of the specular component in [16] is assumed to be constant. This is an important difference Variance of autocorrelation of the complex envelope, N = 8 0.15 since it results in a wide-sense stationary model for our case, Clarke’s model whereas the model in [16] is nonstationary. We present the ensemble correlation statistics of the fading signal, Z(t), in the following theorem. )}

τ Theorem 5: The autocorrelation and cross-correlation func- ( 0.1 YY tions of the quadrature components, and the autocorrelation functions of the complex envelope and the squared envelope Improved model of fading signal Z(t) are given by )} or Var{R τ (

gg ( )= ( ) RZcZc τ RZsZs τ 0.05

Var{R =[J0(wdτ)+K cos(wdτ cos θ0)] /(2 + 2K) (12a) ( )=− ( ) Simulation RZcZs τ RZsZc τ Theory = sin( cos 0) (2 + 2 ) 0 2 4 6 8 10 K wdτ θ / K (12b) Normalized time: f τ d RZZ(τ)=[J0(wdτ)+K cos(wdτ cos θ0) + sin( cos )] (1 + ) Fig. 2. Variances of autocorrelations of the complex envelope of Clarke’s jK wdτ θ0 / K (12c) model and our improved model. 2 2 R|Z|2|Z|2 (τ)= 1+J0 (wdτ)+K −fc(wdτ,N)−fs(wdτ,N) 2 +2K [1+J0(wdτ)cos(wdτ cos θ0)]} /(1+K) . value (N = ∞) for the improved simulator than for Clarke’s (12d) model. However, the variances of the empirical correlations The proof of Theorem 5 is given in Appendix II. of the improved model are smaller than the empirical corre- Based on Definition 2 and Theorems 4 and 5, we present lation variances of Clarke’s model. Using Theorems 2 and 4, the following corollary omitting the proof. Fig. 2 shows, as an example, some theoretical results and the Corollary: The variances of the autocorrelation and cross- corresponding simulation results for the correlation variances correlation of the quadrature components, and the variance of Clarke’s model and the improved model. Obviously, the of the autocorrelation of the complex envelope of the fading variances of the autocorrelation of the complex envelope of signal Z(t) are given by our improved model are smaller than those of Clarke’s model. Var{Rˆ (τ)} = Var{Rˆ (τ)} This implies that the improved simulator converges faster than ZcZc ZsZs 1+J0(2w τ) f (w τ,N) Clarke’s model (and Hoeher’s simulator) to an average value = d − c d (1+ )2 8 4 / K for a finite number of simulation trials. N (13a) { ˆ ( )} = { ˆ ( )} Var RZcZs τ Var RZsZc τ III. A NOVEL RICIAN FADING SIMULATOR 1−J0(2w τ) f (w τ,N) = d − s d /(1+K)2 In this section, we present a statistical Rician fading simu- 8N 4 lation model and its statistical properties. (13b) Definition 2: The normalized lowpass fading process of a [1/N −f (w τ,N)−f (w τ,N)] Var{Rˆ (τ)} = c d s d (13c) new statistical simulation model for Rician fading is defined ZZ (1+K)2 3672 IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 5, NO. 12, DECEMBER 2006

where fc(wdτ,N) and fs(wdτ,N) are given by (9). Note that PDF of the envelope |Z| and the phase Ψ = arctan(zc,zs) is when the number of sinusoids, N, is fixed, the variances of given by the aforementioned correlation statistics tends to be smaller as the Rice factor, K, increases. (1 + ) ( )= K z · exp − − (1 + ) 2 We now present the PDF’s of the fading envelope |Z(t)| f|Z|,Ψ z,ψ K K z 1 π and phase Ψ(t)=arctan [Zc(t),Zs(t)] . ·I0 2z K(1 + K) ,z≥ 0,ψ∈ [−π, π). Theorem 6: When N approaches infinity, the envelope |Z(t)| is Rician distributed and the phase Ψ(t) is uniformly [− ) distributed over π, π , and their PDF’s are given by Then, the marginal PDF’s of the envelope and the phase f (z)=2(1+K)z · exp −K − (1 + K)z2 can be obtained by the following two integrations |Z|  ·I0 2z K(1 + K) ,z≥ 0 (14a) π ( )= ( ) 1 f|Z| z f|Z|,Ψ z,ψ dψ ( )= ∈ [− ) −π fΨ ψ ,ψπ, π (14b) 2π =2(1+K)z · exp −K − (1 + K)z2 respectively, where I0(·) is the zero-order modified Bessel  function of the first kind [31]. ·I0 2z K(1 + K) ,z≥ 0 ( ) Proof: Since the random sinusoids in the sums of Yc t ∞ 1 ( ) ( )= ( ) = ∈ [− ) and Ys t are statistically independent and identically distrib- fΨ ψ f|Z|,Ψ z,ψ dz ,ψ π, π 0 2π uted, Yc(t) and Ys(t) tend to Gaussian random processes as the number of sinusoids, N, increases without limit, according to a ( )=0 where the last equality utilizes  the identity central limit theorem [34]. Moreover, since RYcYs τ and  ∞ 2 1 b2 ( )=0 ( ) ( ) exp(− ) 0( ) = exp RYsYc τ , Yc t and Ys t are uncorrelated and asymptot- 0 x ax I bx dx 2a 4a [31, p.699]. K ically independent. Let m (t)= cos(w t cos θ0 + φ0) This completes the proof.  c 1+K d K We now highlight Theorem 6 with three remarks. First, and m (t)= sin(w t cos θ0 + φ0). Then, [Z (t) − s 1+K d c both the fading envelope and the phase are stationary because ( )] [ ( )− ( )] mc t and Zs t ms t are uncorrelated and asymptotically their PDF’s are independent of time t.Thisisverydifferent independent. from the previous Rician models [4], [16], where the PDF Given an initial phase φ0 of the specular component, the of the fading phase is a very complicated function of time ( ) ( ) conditional joint PDF of Zc t and Zs t can be derived as t, and therefore the fading phase is not stationary as pointed follows out in [4]. Here, the fading phase of our new model is not f (z ,z |φ0) Zc,Zs c s only stationary but also uniformly distributed over [−π, π).   1+K 2 2 Second, the fading envelope and phase of our new Rician = exp −(1+K)[z −m ] −(1+K)[z −m ] π c c s s model are independent. As usual, the PDF’s of the envelope  1+K 2 2 and the phase of our Rician channel model include Rayleigh = exp −(1 + K)(z + z ) − K =0 π c s fading (K ) as a special case. Third, the PDF of the fading +2(1 + K)[zcmc + zsms]} . envelope of our Rician model can be derived by using the theory of two-dimensional random walks described in [35] Since the initial phase φ0 is uniformly distributed over and [36]. Details are omitted. [−π, π), the joint PDF of Zc(t) and Zs(t) is given by Two other important properties associated with the fading π 1 envelope are the level-crossing rate (LCR) and the average f (z ,z )= f (z ,z |φ0) · · dφ0 Zc,Zs c s Zc,Zs c s 2 fade duration (AFD). Both of these represent higher-order −π π 1+ behaviors that a high quality simulator should emulate accu- = K exp −(1 + )( 2 + 2) − K zc zs K rately. The LCR is defined as the rate at which the envelope π π crosses a specified level with positive slope. The AFD is the · exp {2(1+ )[ + ]} dφ0 K zcmc zsms 2 average time duration that the fading envelope remains below −π π 1+ a specified level after crossing below that level. Both the = K exp −(1 + )( 2 + 2) − K zc zs K LCR and AFD provide important information for the statistics π  of burst errors [37], [38], which facilitates the design and · 2 (1 + )( 2 + 2) I0 K K zc zs selection of error correction techniques. Also, both represent practical behaviors of the simulator that depend on the higher- where the last step uses the√ identity π 2 2 order statistics of the simulator. We now present explicit exp [a cos(t + x)+b sin(t + x)] dx =2πI0( a + b ) −π formulas for the LCR and AFD for a general Rician fading [31, p.336]. channel whose specular component has non-zero Doppler ( ) Transforming the Cartesian coordinates zc,zs to polar frequency. The following result (15a) is original while result ( ) = · cos = · sin coordinates z,ψ with zc z ψ and zs z ψ,we (15b) represents a minor extension of a known result [30, p.66] = obtain the transformation’s Jacobian J z; therefore, the joint for the case when the specular component is a constant.

1The function arctan(x, y) maps the arguments (x, y) into a phase in the Theorem 7: When N approaches infinity, the level-crossing correct quadrant in [−π, π). rate L|Z| and the average fade duration T|Z| of the new XIAO et al.: NOVEL SUM-OF-SINUSOIDS SIMULATION MODELS FOR RAYLEIGH AND RICIAN FADING CHANNELS 3673 simulator output are given by Before concluding this section, it is important to point out  that the new simulation model can be directly used to generate 2(1 + K) 2 L| | = ρf · exp −K − (1 + K)ρ multiple uncorrelated fading sample sequences for simulating Z π d  frequency selective Rayleigh and/or Rician channels, MIMO π 2 K 2 channels, and diversity combining techniques. Let Z (t) be the · 1+ cos θ0 · cos α k 0 ρ 1+K kth Rician (or Rayleigh with K =0) fading sample sequence k  given by · exp 2 (1 + )cos − 2 cos2 · sin2 ρ K K α K θ0 α dα     1 1 N 2 + (15a) ( )= exp cos πn θn,k √  Zk t jwd,kt 1+Kk N N 1 − Q 2K, 2(1 + K)ρ2  n=1 = T|Z| (15b) Kk L|Z| · exp (jφn,k)+ exp [j (wd,kt cos θ0,k +φ0,k)] 1+Kk where ρ is the normalized fading envelope level given by (18) |Z|/|Z|rms with |Z|rms being the root-mean-square envelope level, and Q(·) is the first-order Marcum Q-function [39]. where wd,k, Kk and θ0,k are, respectively, the maximum Proof: When N approaches infinity, the fading envelope radian Doppler frequency, the Rice factor and the specular is Rician distributed as shown in Theorem 6. Therefore, we component’s angle of arrival of the kth Rician fading sample can use the formula provided in [40] to obtain the LCR, L|Z|, sequence, and where θn,k, φn,k and φ0,k are mutually inde- pendent and uniformly distributed over [−π, π) for all n and k. viz ∞ Then, Z (t) retains all the statistical properties of Z(t) defined = ˙ (| | ˙) ˙ k L|Z| rf Z , r dr, by eqn. (11). Furthermore, Z (t) and Z (t) are statistically 0 k l independent for all k = l, due to the mutual independence of where r˙ is the envelope slope, f(r, r˙) is the joint PDF of the θ , φ , φ0 , θ , φ and φ0 when k = l. envelope r and its slope r˙ given by [40], [30] n,k n,k ,k n,l n,l ,l   2 + 2 ( ˙)= r exp −r s IV. EMPIRICAL TESTING f r, r 3 (2π) Bb0 2b0 Verification of the proposed fading simulator is carried out π 2 cos ( 0 ˙ + 1 sin ) by comparing the corresponding simulation results for finite N · exp rs α − b r b s α 2 dα with those of the theoretical limit when N approaches infinity. −π b0 Bb0 2 Throughout the following discussions, the newly proposed where, for our model defined in Definition 2, s, B = b0b2−b , 1 statistical simulators have been implemented by choosing b0, b1 and b2 are given by  N =8unless otherwise specified. It is noted that if we K 1 choose a larger value for N, then the statistical accuracy of s = ,b0 = 1+ 2(1 + ) the simulator will be increased. K K π dα b1 =2πb0 (f cos α − f cos θ0) = −2πb0f cos θ0 d d 2 d A. Correlation Statistics −π π π 2 2 dα We have conducted extensive simulations of the autocorrela- b2 =(2π) b0 (f cos α − f cos θ0) d d 2 tions and cross-correlations of the quadrature components, and −π  π 2 2 2 the autocorrelation of the complex envelope of both Rayleigh =2π b0f 1+2cos θ0 d and Rician (with various Rice factors) fading signals. The =2 2 2 2 B π b0fd . simulation results of these correlation statistics match the Using the procedure provided in [40] for deriving the LCR, theoretically calculated results with high accuracy even for we can validate (15a). Employing the procedure proposed in small N. For example, Figs. 3 and 4 show the good agreement [30] for the AFD, we can obtain (15b). Details are omitted for the real part and imaginary part of the autocorrelation of here for brevity. the complex envelope of the fading. The simulation results and π π It is noted here that if θ0 = 2 or θ0 = − 2 , which means the theoretically calculated results for the autocorrelation of that the specular component has zero Doppler frequency, then the squared envelope of the fading signals are slightly different the LCR given by (15a) has a closed-form solution as follows when N =8as can be seen from Fig. 5. The differences  2 decrease if we increase the value of N, as expected. L| | = 2π(1 + K)ρf exp −K − (1 + K)ρ Z  d ·I0 2ρ K(1 + K) . (16) B. Envelope and Phase PDF’s This is the same solution as that given in [40] and [30] for the Figs. 6 and 7 show that the PDF’s of the fading envelope and case of the specular component being deterministic. If K =0, phase of the simulator with N =8are in very good agreement Z(t)=Y (t) becomes a Rayleigh fading process; then both with the theoretical ones. It is noted that when N>8,these the LCR and the AFD have closed-form solutions given by PDF’s will have even better agreement with the theoretically √ −ρ2 desired ones. It is also noted that the more random samples L| | = 2πρf e (17a) Y d used for the ensemble average in the simulations, the smaller ρ2 − 1 e √ the difference between the simulated curves and the desired T|Y | = . (17b) ρfd 2π reference curves for the phase PDF. 3674 IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 5, NO. 12, DECEMBER 2006

Real part of R (τ), N = 8 Squared envelope autocorrelation, N = 8 ZZ Simulation Simulation 1 Theory 1 Theory K = 1 K = 3 0.9 ) τ

0.5 ( 2 K = 3

|Z| 0.8 2 |Z| )] τ ( ZZ 0.7 0 Re[R K = 1 Normalized R 0.6

−0.5 0.5 K = 0 (Rayleigh) K = 0 (Rayleigh) −1 0.4 0 2 4 6 8 10 0 2 4 6 8 10 Normalized time: f τ Normalized time: f τ d d

2 Fig. 3. The real part of the autocorrelation of the complex envelope Z(t); Fig. 5. The autocorrelation of the squared envelope |Z(t)| with θ0 = π/4 θ0 = π/4 for K =1and K =3Rician cases. for K =1and K =3Rician cases.

Imaginary part of R (τ), N =8 PDF of the fading envelope ZZ 2 Simulation (N=8) Simulation K = 10 Theory (N=∞) 1 K = 1 K = 3 Theory

1.5 K = 5

0.5 K = 3 )] τ ( 1 (z) K = 1 ZZ |Z| 0 f Im[R

0.5 K = 0 −0.5 (Rayleigh)

K = 0 (Rayleigh) 0 −1 0 2 4 6 8 10 0 0.5 1 1.5 2 2.5 3 Normalized time: f τ d z

Fig. 4. The imaginary part of the autocorrelation of the complex envelope Fig. 6. The PDF of the fading envelope |Z(t)|. Z(t); θ0 = π/4 for K =1and K =3Rician cases.

the value of K is increased, the specular component becomes C. LCR and AFD more dominant over the Rayleigh scatter components, and the The simulation results for the normalized level-crossing rate AFD tends to be even smaller. However, when θ0 = π/2,the (LCR), L|Z| , and the normalized average fade duration (AFD), Doppler frequency of the specular component is zero, for each fd single trial and the AFD becomes larger when the value of K fdT|Z|, of the new simulators are shown in Figs. 8 and 9, respectively, where the theoretically calculated LCR and AFD is increased. for N = ∞ are also included in the figures for comparison, indicating generally good agreement in both cases. Again, if V. C ONCLUSION we increase the number of sinusoids, N, the simulation results In this paper, it was shown that Clarke’s model with a for the case of finite N approach the theoretical N = ∞ finite number of sinusoids can be directly used for simulating results. Rayleigh fading channels, and its computational efficiency and For the region of ρ<0 dB, it is interesting to note that second-order statistics are better than those of Jakes’ original the average fade duration for θ0 =0(or θ0 <π/4) tends model [3] and as good as those of the recently improved to be smaller for larger values of the Rice factor K.This Jakes’ Rayleigh fading simulators [22], [24] and [25]. An is different from the AFD for θ0 = π/2, which tends to be improved Clarke’s model was proposed to reduce the variance larger with larger Rice factors [30]. The main reason for this of the time averaged correlations of a fading realization from phenomenon is that when θ0 =0, the Doppler frequency of a single trial. A novel simulation model employing a random the specular component is equal to the maximum Doppler specular component was proposed for Rician fading channels. frequency, fd.Foragivenρ<0 dB and K>0,theLCRis The specular (line-of-sight) component of this Rician fading at its largest value and the AFD is at its smallest value. When model is a zero-mean stochastic sinusoid with a pre-chosen XIAO et al.: NOVEL SUM-OF-SINUSOIDS SIMULATION MODELS FOR RAYLEIGH AND RICIAN FADING CHANNELS 3675

PDF of the fading phase AFD of the fading, θ = 0 0 1 0.18 10 ∞ Theory (N= ) Simulation (N=8) Simulation: K=0 Theory (N=∞) Simulation: K=1 Simulation: K=3 Simulation: K=5 0.17 Simulation: K=10 N=8 for all simulations

0 K = 0 ) 10 ψ

( 0.16 Ψ f K = 1

K = 3

0.15 K = 5 Normalized average fade duration

−1 K = 10 10

0.14 −1 −0.5 0 0.5 1 ψ × π −20 −15 −10 −5 0 5 , ( ) Normalized fading envelope level ρ (dB)

Fig. 7. The PDF of the fading phase Ψ(t). Fig. 9. The normalized AFD of the fading envelope |Z(t)|,whereθ0 =0 for all K>0 Rician fading.

LCR of the fading, θ = π/4 0

Simulation (N=8) Theory (N=∞) ACKNOWLEDGMENT 0 10 K = 0 The first author, C. Xiao, is grateful to Drs. Q. Sun, H. Ye and W. Choi of Atheros Communications Inc. for pointing K = 1 out an error in an earlier version of the manuscript. He is also −1 10 indebted to Dr. F. Santucci and Professor G. L. Stuber for

K = 3 helpful discussions at the early stage of this work.

−2 K = 5 10 K = 10

Normalized level crossing rate APPENDIX I PROOF OF THEOREM 3

−3 10 Proof: The autocorrelation function of the real part of −25 −20 −15 −10 −5 0 5 10 the fading is proved first. One has Normalized fading envelope level ρ (dB)

Fig. 8. The normalized LCR of the fading envelope |Z(t)|,whereθ0 = π/4 ( )= [ ( ) ( + )] RYcYc τ Eα,φ Yc t Yc t τ for all K>0 Rician fading. 1 N N = E {cos(w t cos α + φ ) N d n n n=1 i=1 · cos[ ( + )cos + ]} Doppler frequency and a random initial phase. Compared to wd t τ αi φi all the existing Rician fading simulation models, which have 1 N = E[cos(w τ cos α )] a non-zero deterministic specular component, the new model 2N d n n=1 better reflects the fact that the specular component is random   from ensemble sample to ensemble sample and from mobile 1 N π 2 + = cos cos πn θn dθn to mobile. Additionally and importantly, the fading phase PDF 2 wdτ 2 N −π N π of the new Rician fading model is independent of time and n=1 N 2πn+π uniformly distributed over [−π, π). 1 N N = cos (wdτ cos γn) dγn 2N 2πn−π 2π This paper has also analyzed the statistical properties of the n=1 N 2 + π new simulation models. Mathematical formulas were derived 1 π N for the autocorrelation and cross-correlation of the quadrature = cos (wdτ cos γ) dγ 4π π components, the autocorrelation of the complex envelope and N 2π the squared envelope, the PDF’s of the fading envelope and 1 = cos(wdτ cos γ)dγ phase, the level-crossing rate and the average fade duration. It 4π 0 has been shown that all these statistics of the new simulators 1 = J0(w τ). either exactly match or quickly converge to the desired ones. 2 d Good convergence can be reached even when the number of sinusoids is as small as 8. Similarly, we can obtain the autocorrelation of the imagi- 3676 IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 5, NO. 12, DECEMBER 2006

2( ) 2( + ) nary part of the fading signal in (8a) as E Yc t Yc t τ 1st 1 N  R (τ)=E [Y (t)Y (t + τ)] = E cos2(w t cos α + φ ) YsYs s s N 2 d n n 1 2π n=1 = cos(w τ cos γ)dγ 2 4 d · cos [wd(t + τ)cosαn + φn] π 0 1 1 N 1+cos(2 cos +2 ) = ( ) wdt αn φn J0 wdτ . = E 2 N 2 2 n=1 ! 1+cos[2 ( + )cos +2 ] · wd t τ αn φn We are now in a position to prove equation (8b) starting 2 from 1 1 N = N + [cos(2 cos )] 2 E wdτ αn ( )= [ ( ) ( + )] N 4 8 RYcYs τ E Yc t Ys t τ n=1 1 1 1 N N = + J0(2wdτ). = E {cos(wdt cos αn + φn) 4N 8N N =1 =1 n i For the second case, n = i, p = q,andn = p,wehave · sin[wd(t + τ)cosαi + φi]} 2( ) 2( + ) E Yc t Yc t τ 2nd 1 N ⎧ = E [sin(w τ cos α )] ⎪ 2N d n ⎨N N n=1 1 2 2 = E cos (wdt cos αn + φn) 1 π N 2 ⎪ ⎩n=1 p=1 = sin(wdτ cos γ)dγ =0. 4 = π 0  p n  · E cos2[w (t + τ)cosα + φ ] d p p The second part of eqns. (8b) and (8c) are proved in a similar 1 N 2 − N 1 1 = = − . manner. The proof of equation (8d) is different and lengthy. N 2 4 4 4N A brief outline with some salient details is given below. One = = = has For the third case, n p, i q,andn i,wehave 2( ) 2( + ) E Yc t Yc t τ 3rd 2 2 2 2 N N R|Y |2|Y |2 (τ)=E Y (t)Y (t + τ) + E Y (t)Y (t + τ) 1 c c s s = E {cos(w t cos α + φ ) +E Y 2(t)Y 2(t + τ) + E Y 2(t)Y 2(t + τ) . N 2 d n n c s s c n=1 i=1 (19) i=n · cos[wd(t + τ)cosαn + φn]} ·E {cos(w t cos α + φ ) The derivation of the first term on the right side of (19) in d i i · cos[ ( + )cos + ]} detail starts as wd t τ αi φi 2 E Y 2(t)Y 2(t + τ) 1 N 1 c c = E[cos(w τ cos α )] N 2 2 d n n=1 N ! 1 1 N 1 2 = · E cos(wdt cos αn + φn) − [cos( cos )] 2 2 E wdτ αn N =1 N 2 n n=1 N 1 f (w τ,N) · cos( cos + ) = 2( ) − c d wdt αi φi 4J0 wdτ 4 . i=1 N · cos[wd(t + τ)cosαp + φp] For the fourth case, n = q, i = p,andn = i; in a manner =1 p similar to that used in the third case, one can prove N 2 2 1 2 f (w τ,N) · cos[w (t + τ)cosα + φ ] . E Y (t)Y (t + τ) = J (w τ) − c d . d q q (20) c c 4th 4 0 d 4 q=1 Since these four cases are the exclusive and exhaustive 2( ) 2( + ) possibilities for E Yc t Yc t τ being non-zero, adding Since the random phases φ and φ are statistically indepen- k l them together we have dent for all k = l, the right side of (20) is zero except for 2( ) 2( + ) E Yc t Yc t τ four different cases: a) n = i = p = q;b)n = i, p = q,and = = = = = = = E Y 2(t)Y 2(t + τ) + E Y 2(t)Y 2(t + τ) n p;c)n p, i q,and n i; and d) n q, i p,and c c 1st c c 2nd = 2( ) 2( + ) 2 2 2 2 n i. Subsequently, E Yc t Yc t τ is derived for each + ( ) ( + ) + ( ) ( + ) E Yc t Yc t τ 3rd E Yc t Yc t τ 4th of the four cases. 1 1 2 1 fc (wdτ,N) = = = = + J0 (wdτ)+ J0(2wdτ) − . For the first case, n i p q,wehave 4 2 8N 2 XIAO et al.: NOVEL SUM-OF-SINUSOIDS SIMULATION MODELS FOR RAYLEIGH AND RICIAN FADING CHANNELS 3677

2( ) 2( + ) 2( ) 2( + ) This completes the derivation of E Yc t Yc t τ . E Zs t Zs t τ Using the same procedure for the second, third and fourth 1 √ 2 terms on the right side of (19), one obtains = E Y (t)+ K sin (w t cos θ0 + φ0) (1 + K)2 s d 1 1 1   ! 2( ) 2( + ) = + 2( )+ (2 ) √ 2 E Ys t Ys t τ J0 wdτ J0 wdτ 4 2 8N · Ys(t + τ)+ K sin [wd(t + τ)cosθ0 + φ0] fs (wdτ,N) − E Y 2(t)Y 2(t + τ) 2 = s s 1 1 ( ) (1 + K)2 2( ) 2( + ) = − (2 ) − fc wdτ,N E Yc t Ys t τ J0 wdτ 4 8N 2 + K [1 + 2 ( ) · cos( cos )] 2 J0 wdτ wdτ θ0 2 2 1 1 fs (wdτ,N) 2(1 + K) E Y (t)Y (t + τ) = − J0(2w τ) − . s c 4 8N d 2 2 cos(2 cos ) + K 1+ wdτ θ0 Therefore, 4(1 + K)2 2 2 R|Y |2|Y |2 (τ)=1+J0 (wdτ) − fc (wdτ,N) − fs (wdτ,N) . and 2( ) 2( + ) E Zc t Zs t τ This completes the proof of Theorem 3. 1 √ 2 = E Y (t)+ K cos (w t cos θ0 + φ0) (1 + K)2 c d APPENDIX II !  √ 2 PROOF OF THEOREM 5 · Ys(t + τ)+ K sin [wd(t + τ)cosθ0 + φ0] Proof: Based on the assumption that the initial phase of E Y 2(t)Y 2(t + τ) the specular component is uniformly distributed over [−π, π), = c s (1 + )2 and independent of the initial phases of the scattered compo- K  2 2 nents, one can prove eqns. (12a)-(12c) by using the results of K · E Y (t) · E sin [w (t + τ)cosθ0 + φ0] + c d Theorem 3. The details are omitted for brevity. The proof of (1 + )2 K equation (12d) is outlined as follows. One has 2 2 K · E Y (t + τ) · E cos (w t cos θ0 + φ0) + s d 2 2 2 2 2 R| |2| |2 (τ)=E Z (t)Z (t + τ) + E Z (t)Z (t + τ) (1 + ) Z Z c c s s K 2 2 2 2 2  +E Z (t)Z (t + τ) + E Z (t)Z (t + τ) . K 2 c s s c + · E cos (w t cos θ0 + φ0) (1 + K)2 d Then, · sin2 [ ( + )cos + ] 2( ) 2( + ) wd t τ θ0 φ0 E Zc t Zc t τ √ 2 1 2( ) 2( + ) = ( )+ cos ( cos + ) E Yc t Ys t τ K 2 E Yc t K wdt θ0 φ0 = + (1 + K) (1 + )2 2(1 + )2   ! K K √ 2 2 K cos(2w τ cos θ0) · Yc(t + τ)+ K cos [wd(t + τ)cosθ0 + φ0] + 1 − d 4(1 + K)2 2 E Y 2(t)Y 2(t + τ) = c c and (1 + )2 K  E Z2(t)Z2(t + τ) 2 2 s c K · E Y (t) · E cos [wd(t + τ)cosθ0 + φ0] + c 1 √ 2 (1 + K)2 = ( )+ sin ( cos + ) 2 E Ys t K wdt θ0 φ0 2 2 (1 + K) K · E Y (t + τ) · E cos (w t cos θ0 + φ0) ! + c d  √ 2 2 (1 + K) · Yc(t + τ)+ K cos [wd(t + τ)cosθ0 + φ0] 4 · [ ( ) ( + )] K E Yc t Yc t τ 2 2 + ·E {cos (wdt cos θ0 +φ0) E Y (t)Y (t + τ) (1 + K)2 = c s + K (1 + K)2 2(1 + K)2 · cos [w (t + τ)cosθ0 + φ0]} d 2 2  K cos(2wdτ cos θ0) K 2 + 1 − + · cos ( cos + ) 2 . (1 + )2 E wdt θ0 φ0 4(1 + K) 2 K 2 · cos [wd(t + τ)cosθ0 + φ0] Therefore, R| |2| |2 (τ) Z Z 2( ) 2( + ) 2 E Y t Y t τ 2 2 ( )+ +2 [1+ ( )cos( cos )] = c c R|Y | |Y | τ K K J0 wdτ wdτ θ0 (1 + )2 = K (1 + K)2 K 2 2 + [1 + 2 ( ) · cos( cos )] 1+J0 (wdτ)+K +2K [1+J0(wdτ)cos(wdτ cos θ0)] 2(1 + )2 J0 wdτ wdτ θ0 = K (1 + K)2 2 K cos(2wdτ cos θ0) + 1+ fc(wdτ,N)+fs(wdτ,N) 4(1 + )2 2 . − K (1 + K)2 Similarly, we have This completes the proof. 3678 IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 5, NO. 12, DECEMBER 2006

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Babich, “Achievable performance of turbo codes over the correlated Rician channel,” IEEE Trans. Commun., vol. 51, pp. 1-4, Jan. 2003. [27] S. Gazor and H. S. Rad, “Space-time coding ambiguities in joint adap- tive channel estimation and detection,” IEEE Trans. Signal Processing, vol. 52, pp. 372-384, Feb. 2004. [28] F. Babich, “On the performance of efficient coding techniques over fading channels,” IEEE Trans. Wireless Commun., vol. 3, pp. 290-299, Jan. 2004. XIAO et al.: NOVEL SUM-OF-SINUSOIDS SIMULATION MODELS FOR RAYLEIGH AND RICIAN FADING CHANNELS 3679

Yahong Rosa Zheng (S’99-M’03) received the B.S. Norman C. Beaulieu (S’82-M’86-SM’89-F’99) re- degree from the University of Electronic Science ceived the B.A.Sc. (honors), M.A.Sc., and Ph.D de- and Technology of China, Chengdu, China, in 1987, grees in electrical engineering from the University of the M.S. degree from Tsinghua University, Beijing, British Columbia, Vancouver, BC, Canada in 1980, China, in 1989, both in electrical engineering. She 1983, and 1986, respectively. He was awarded the received the Ph.D. degree from the Department of University of British Columbia Special University Systems and Computer Engineering, Carleton Uni- Prize in Applied Science in 1980 as the highest versity, Ottawa, ON, Canada, in 2002. standing graduate in the faculty of Applied Science. From 1989 to 1997, she held Engineer positions He was a Queen’s National Scholar Assistant Pro- in several companies. From 2003 to 2005, she was a fessor with the Department of Electrical Engineer- Natural Science and Engineering Research Council ing, Queen’s University, Kingston, ON, Canada from of Canada (NSERC) Postdoctoral Fellow at the University of Missouri, September 1986 to June 1988, an Associate Professor from July 1988 to June Columbia, MO. Currently, she is an Assistant Professor with the Department 1993, and a Professor from July 1993 to August 2000. In September 2000, he of Electrical and Computer Engineering at the University of Missouri, became the iCORE Research Chair in Broadband Wireless Communications Rolla, MO. Her research interests include array signal processing, wireless at the University of Alberta, Edmonton, AB, Canada and in January 2001, the communications, and wireless sensor networks. Canada Research Chair in Broadband Wireless Communications. His current Dr. Zheng has served as a Technical Program Committee member for the research interests include broadband digital communications systems, ultra- 2004 IEEE International Sensors Conference, the 2005 IEEE Global Telecom- wide bandwidth systems, fading channel modeling and simulation, diversity munications Conference, and the 2006 IEEE International Conference on systems, interference prediction and cancellation, importance sampling and Communications. Dr. Zheng is currently an Editor for the IEEE Transactions semi-analytical methods, and space-time coding. on Wireless Communications. Dr. Beaulieu is a Member of the IEEE Communication Theory Committee and served as its Representative to the Technical Program Committee of the 1991 International Conference on Communications and as Co-Representative to the Technical Program Committee of the 1993 International Conference on Communications and the 1996 International Conference on Communications. He was General Chair of the Sixth Communication Theory Mini-Conference in association with GLOBECOM 97 and Co-Chair of the Canadian Workshop on Information Theory 1999. He has been an Editor for Wireless Communica- tion Theory of the IEEE Transactions on Communications since January 1992, and was Editor-in-Chief from January 2000 to December 2003. He served as an Associate Editor for Wireless Communication Theory of the IEEE Communications Letters from November 1996 to August 2003. He has also served on the Editorial Board of the Proceedings of the IEEE since November 2000. He received the Natural Science and Engineering Research Council of Canada (NSERC) E.W.R. Steacie Memorial Fellowship in 1999. Professor Beaulieu was elected a Fellow of the Engineering Institute of Canada in 2001 and was awarded the Medaille K.Y. Lo Medal of the Institute in 2004. He was elected Fellow of the Royal Society of Canada in 2002 and was awarded the Thomas W. Eadie Medal of the Society in 2005.