SIMULATION of mobile radio channels is commonly used

Transcription

1 IEEE TASACTIOS O WIELESS COUICATIOS, VOL. XX, O. XX, OTH YEA 1 Efficient Simulation of ayleigh Fading with Enhanced De-Correlation Properties Alenka G. Zajić, Student ember, IEEE, and Gordon L. Stüber, Fellow, IEEE Abstract ew sum-of-sinusoids simulation models are proposed for ayleigh fading channels and compared with existing simulation models. First, an ergodic statistical ( deterministic ) model is proposed that, compared to existing models, yields a significantly lower cross-correlation between different complex envelopes and between the quadrature components of each complex envelope. However, the auto-correlation functions of the quadrature components still do not match the theoretical functions. To overcome this disadvantage, we also propose a new statistical simulator that converges faster than existing statistical models, and has lower cross-correlations between different complex envelopes and between the quadrature components of each complex envelope. This new statistical model yields adequate statistics with only 30 simulation runs. Index Terms Channel models, fading channel simulator, ayleigh fading, sum-of-sinusoids. I. ITODUCTIO SIULATIO of mobile radio channels is commonly used as opposed to field trials, because it allows for less expensive and more reproducible system tests and evaluations. any different approaches have been used to model and simulate ayleigh fading channels [1]-[17]. This paper focuses on models that approximate the underlying random processes by the superposition of a finite number of properly selected sinusoids [5]-[17]. Generally, these sum-of-sinusoids (SoS) models can be classified as either statistical or deterministic. Deterministic SoS models have fixed phases, amplitudes, and Doppler frequencies for all simulation trials. In contrast, statistical SoS models leave at least one of the parameter sets (amplitudes, phases, or Doppler frequencies) as random variables that vary with each simulation trial. The statistical properties of the statistical SoS models will also vary for each simulation trial, but converge to the desired properties when averaged over a large number of simulation trials. An ergodic statistical model is one that converges to the desired properties in a single simulation trial. any approaches have been suggested for SoS modeling of ayleigh fading channels. Clarke [5] was among the first to Paper approved by Hao Xu, the Editor for IEEE Transactions on Wireless Communications. anuscript received August 7, 004; revised June 8, 005. This work was prepared through collaborative participation in the Collaborative Technology Alliance for Communications & etworks sponsored by the U.S. Army esearch Laboratory under Cooperative Agreement DAAD The U.S. Government is authorized to reproduce and distribute reprints for Government purposes notwithstanding any copyright notation thereon. The views and conclusions contained in this document are those of the authors and should not be interpreted as representing the official policies, either expressed or implied, of the Army esearch Laboratory or the U. S. Government. Alenka G. Zajić and Gordon L. Stüber are with the School of Electrical and Computer Engineering, Georgia Tech, Atlanta, GA 3033 USA. propose a mathematical reference model for ayleigh fading channels. A simplified version of Clarke s model, proposed by Jakes [6], has been widely used for about three decades. However, Jakes model fails to meet most statistical properties required by the reference model [7] and is not wide-sense stationary (WSS) [1]. Consequently, various modifications of Jakes model have been reported [8]-[13]. Each new model improves some statistical properties, but none of them obtains all the desired statistical properties of Clarke s reference model. To improve on previously reported models, Zheng and Xiao proposed several new statistical models [14]-[16]. By leaving all three parameter sets (amplitudes, phases, and Doppler frequencies) as random variables, Zheng and Xiao s models obtain statistical properties similar to those of the reference model. However, their models are no longer ergodic; their statistical properties vary for each simulation trial, but converge to the desired properties when averaged over 50 to 0 simulation trials. We have observed that most existing models have difficulty in creating uncorrelated in-phase (I) and quadrature (Q) components of the complex faded envelope and in creating multiple uncorrelated faded envelopes. This paper proposes new sum-of-sinusoids simulation models for ayleigh fading channels that address these problems. First, an ergodic statistical ( deterministic ) model is proposed that overcomes the difficulty in creating uncorrelated I and Q components and in creating multiple uncorrelated faded envelopes. This is achieved by using orthogonal functions for the I and Q components and by introducing an asymmetrical arrangement of arrival angles into the model proposed in [1]. The statistical properties of our model are derived and verified by simulation. Compared to existing models, our new model yields a lower cross-correlation between different faded envelopes and between the I and Q components of each complex faded envelope. However, the auto-correlation functions of the I and Q components still do not exactly match the theoretical functions. The proposed deterministic model can be modified by introducing additional randomness to yield a new non-ergodic statistical model having the correct statistical properties. The motivation for this model originates in [14]. The properties of the resulting statistical model are derived and verified by simulation. Compared to Zheng and Xiao s models [14]-[16], our new statistical model converges faster, has less correlated I and Q components, and yields less correlated multiple faded envelopes. The remainder of this paper is organized as follows. Section

2 IEEE TASACTIOS O WIELESS COUICATIOS, VOL. XX, O. XX, OTH YEA II presents the mathematical reference model. Section III reviews existing simulation models for ayleigh fading channels [6]-[16]. Section IV describes our new sum-of-sinusoids simulation models for ayleigh fading channels and analyzes their statistical properties. Section V evaluates the new models and compares them to previously reported models. Section VI provides some concluding remarks. II. ATHEATICAL EFEECE ODEL Clarke s reference model [5] defines the complex faded envelope as g(t) = 1 C n e j(ωmt cos αnφn), (1) where, ω m, C n, α n, and φ n are the number of propagation paths, the maximum angular Doppler frequency, the path gain, the angle of arrival, and the phase associated with the n th propagation path, respectively. It is assumed that C n, α n, and φ n are mutually independent and that α n and φ n are uniformly distributed on the interval [ π, π). Invoking the Central Limit Theorem [18], the real part g i (t) ={g(t)} and the imaginary part g q (t) = I{g(t)} of the complex faded envelope are Gaussian random processes as. Therefore, the envelope g(t) is ayleigh distributed and the phase Θ g (t) is uniformly distributed. The auto- and crosscorrelation functions of the reference model, assuming a two dimensional (-D) scattering environment, are summarized below [6], [18] gi/q g i/q =E [ g i/q (t τ)g i/q (t) ] = J 0 (ω m τ), () gig q = gqg i =E [ g i/q (t τ)g q/i (t) ] =0, (3) gk g l = 1 { E [g k(t τ)gl J0 (ω (t)] = m τ) l = k 0 l k, (4) gk g k =E [ g k (t τ) g k (t) ] =44J 0 (ω m τ), (5) where E [ ] is the statistical expectation operator, J 0 ( ) is the zero-order Bessel function of the first kind, and ω m is the maximum angular Doppler frequency. The objective of the channel simulators discussed in this paper is to reproduce the above reference model properties as faithfully as possible and with reasonable complexity. III. EXISTIG SIULATIO ODELS A. Jakes odel odel I Jakes derived his well-known deterministic simulation model for ayleigh fading channels [6] starting from (1) and selecting C n = 1/, θ n = πn/, and φ n = 0 for n = 1,...,. Using the symmetry of the -D isotropic scattering environment, Jakes reduced the number of sinusoidal components necessary for simulation from to = ( )/4. The complex faded envelope is defined as g(t) = g i (t)jg q (t), where 4 g i (t) = cos (β 0)cos(ω m t) 8 [ ( )] πn cos (β n )cos ω m t cos, (6) g q (t) = 4 sin (β 0)cos(ω m t) 8 sin (β n )cos [ ω m t cos ( )] πn, (7) and where ω m is the maximum angular Doppler frequency. The parameter β n is defined as β n = πn/ for n = 0,...,. Often it is desirable to generate multiple uncorrelated faded envelopes, something that odel I cannot do. Dent et al. [8] modified odel I by using orthogonal Walsh-Hadamard code words to de-correlate the multiple faded envelopes. The k th complex faded envelope is defined as g k (t) =g ik (t)jg qk (t), where ( ) πn g ik (t) = A k (n)cos (8) g qk (t) = cos cos { ω m t cos ( πn ( πn A k (n)sin 1 { ω m t cos ( πn 1 ) πn πn(k 1) 1 1 ) ) πn πn(k 1) 1 1 =( )/4, k =1,...,, and A k (n) is the n th element of the k th row of a Hadamard matrix H of dimension. By reducing the number of sinusoidal components, Jakes simplified simulation of ayleigh fading channels. However, odel I does not satisfy most of the statistical properties of the reference model [7] and it is not wide-sense stationary (WSS) [1]. odel I satisfies only the following properties: the I and Q components of the complex faded envelope are Gaussian random processes for and the autocorrelation function of the complex faded envelope is equal to J 0 (ω m τ). Also, the model of Dent et al. [8] yields a crosscorrelation between different faded envelopes that is strictly zero only for time lag τ =0. As a result, various modifications of odel I have been proposed in the literature []-[1]. B. Pop and Beaulieu s odel odel II Pop and Beaulieu [1] showed that odel I is not WSS and modified odel I to fix this problem. This was done by removing the constraint φ n = 0 from odel I and allowing the phases φ n to be independent random variables uniformly distributed on the interval [0, π). The procedure yields an ergodic statistical ( deterministic ) simulator, since }, (9) },

3 IEEE TASACTIOS O WIELESS COUICATIOS, VOL. XX, O. XX, OTH YEA 3 the random phases are generated only once for all simulation trials. The k th complex faded envelope is defined as g k (t) = g ik (t)jg qk (t), where 4 g ik (t) = cos (β 0)cos(ω m t φ 0k ) () 8 [ ( ) ] πn cos (β n )cos ω m t cos φ nk, g qk (t) = 4 sin (β 0)cos(ω m t φ 0k ) (11) 8 [ ( ) ] πn sin (β n )cos ω m t cos φ nk, =( )/4, k =0,..., 1, and ω m is the maximum angular Doppler frequency. The parameter β n is defined as β n = πn/ for n =0,...,. Although WSS, odel II inherits its statistical properties from odel I. Hence, this model does not satisfy most of the other statistical properties required by the reference model. C. Li and Huang s odel odel III To improve odel II, Li and Huang [] proposed an ergodic statistical ( deterministic ) model that generates multiple uncorrelated faded envelopes g k (t). odel III assumes P independent complex faded envelopes, each with = /4 sinusoidal terms in the I and Q components. The k th complex faded envelope is g k (t) =g ik (t)jg qk (t), where 1 g ik (t) =C cos ( ω m t cos α nk φ i nk), (1) 1 g qk (t) =C sin (ω m t sin α nk φ q nk ), (13) and where k = 0,.., P 1, φ i nk and φq nk are independent random phases uniformly distributed on the interval [0, π), α nk is the n th angle of arrival in the k th complex faded envelope, C is a constant gain, and ω m is the maximum angular Doppler frequency. The angles of arrival are α nk = (πn)/ (πk)/(p)α 00 for n = 0,..., 1, k = 0,...P 1, where α 00 is an initial angle of arrival, chosen to be 0 <α 00 < (π)/(p) and α 00 π/(p). odel III preserves the desirable statistical properties of odel II, while generating multiple uncorrelated faded envelopes. Compared to odel II, the I and Q components of the complex faded envelope in odel III are less correlated. However, odel III fails to satisfy equations () and (5) of the reference model. D. EDS odel odel IV To resolve the remaining disadvantages of odel II, Pätzold et al. [11] proposed a deterministic model called the ethod of Exact Doppler Spreads (EDS). The auto-correlation functions of the I and Q components of the complex faded envelope are designed to satisfy equation (). The k th complex faded envelope is defined as g k (t) =g ik (t)jg qk (t), where i/q [ ] g (i/q)k (t) = cos ω m t sin αn i/q φ i/q nk (14) i/q for k = 0,...,P 1, and where φ i/q nk are independent random phases uniformly distributed on the interval [0, π), P is the number of desired faded envelopes, and i/q is the number of sinusoidal terms in the I and Q components of g k (t), respectively. The n th angle of arrival is given by αn i/q = π(n 0.5)/( i/q ) for n =1,..., i/q. odel IV preserves the desirable properties of odel II. In addition, the auto-correlation functions of the I and Q components in odel IV satisfy (). Compared to odel II, the I and Q components of the complex faded envelope in odel IV are less correlated if q = i 1. However, odel IV produces multiple faded envelopes that are correlated. E. Zheng and Xiao s odels To improve on previously reported models, Zheng and Xiao proposed several new statistical models [14]-[16]. By allowing all three parameter sets (amplitudes, phases, and Doppler frequencies) to be random variables, Zheng and Xiao s models obtain statistical properties similar to ones required by the reference model. However, the models are no longer ergodic. The statistical properties of these models vary for each simulation trial, but they converge to desired properties when averaged over 50 to 0 simulation trials. A detailed comparison of the statistical properties for Zheng and Xiao s models is presented in [17]. It is shown that the model presented in [15] has the statistical properties closest to those of the reference model and requires the fewest simulation trials (50). Hence, we will refer to this model as odel V and will compare it with our new models. odel V: The k th complex faded envelope is defined as g k (t) =g ik (t)jg qk (t), where g ik (t) = cos [ ω m t cos α nk φ i nk], (15) g qk (t) = cos [ω m t sin α nk φ q nk ], (16) for k =0,...,P 1, and where φ i nk and φq nk are independent random phases uniformly distributed on the interval [ π, π). odel V assumes P independent complex envelopes, each with = /4 sinusoidal terms in the I and Q components. The n th angle of arrival in the k th complex envelope α nk is α nk =(πn π θ k )/(4) for n =1,...,, where the θ k are independent random variables uniformly distributed on the interval [ π, π). Since the motivation for our statistical model originates in [14], we will present this model as well. odel VI: The k th complex faded envelope is defined as g k (t) =g ik (t)jg qk (t), where g ik (t) = cos (ψ nk )cos[ω m t cos (α nk )φ k ], (17)

4 IEEE TASACTIOS O WIELESS COUICATIOS, VOL. XX, O. XX, OTH YEA 4 g qk (t) = sin (ψ nk )cos[ω m t cos (α nk )φ k ], (18) α nk =(πn π θ k )/(4), ψ nk, θ k, and φ k are statistically independent random variables, uniformly distributed on the interval [ π, π), and n =1,...,, k =0,...,P 1. Patel et al. [17] have shown that the I and Q components of the complex faded envelope in odel VI are not Gaussian random processes and that the auto-correlation of the squared envelope is nonstationary. These problems can be solved by replacing the random phase φ k, which is the same for all sinusoidal terms in the k th complex envelope, with mutually independent random phases φ nk, uniformly distributed on the interval [ π, π). The proof is omitted for brevity. We refer to this model as modified odel VI. IV. EW SIULATIO ODELS Our evaluation of existing models revealed that most have difficulty in creating uncorrelated I and Q components of each complex faded envelope and in generating multiple uncorrelated faded envelopes. Our goal is to solve these problems by using orthogonal functions for the I and Q components of the complex faded envelope. The following function is considered as the k th complex faded envelope g k (t) = 1 C n e j(ωmt cos α nkφ nk ), (19) where C n =(e jβn )/, α nk, φ nk, and ω m are the random path gain, the angle of arrival, the phase associated with the n th propagation path, and the maximum angular Doppler frequency, respectively. It is assumed that P independent complex faded envelopes are required (k = 0,...,P 1) each consisting of sinusoidal components. To reduce the number of sinusoidal components needed for simulation, we use a method similar to the one described in []. By choosing = /4 to be an integer and by taking into account shifts of the angles α nk and φ nk in each quadrant of the circle, the sum in (19) can be split into four terms, viz. g k (t) = e jβn e j[ωmt cos α nkφ nk ] (0) e jβn e j[ωmt cos(α nk0.5π)(φ nk 0.5π)] e jβn e j[ωmt cos(α nkπ)(φ nk π)] e jβn e j[ωmt cos(α nk1.5π)(φ nk 1.5π)] Equation (0) can be further simplified to g k (t) = cos(β n )cos(ω m t cos α nk φ nk ) (1) j sin(β n )sin(ω m t sin α nk φ nk ). Based on g k (t), we define our new simulation models. A. ew Deterministic odel We first propose an ergodic statistical ( deterministic ) model, which needs only one simulation trial to obtain the desired statistical properties. The k th complex faded envelope is defined as g k (t) =g ik (t)jg qk (t), where [ g ik (t) = ] a n cos(ω m t cos α nk φ nk ), () [ g qk (t) = ] b n sin(ω m t sin α nk φ nk ). (3) The motivation for this model originates in odel II. As in odel II, the phases φ nk are chosen to be independent random variables uniformly distributed on the interval [0, π), and the path gains β n are defined as β n = πn/ for n =0,...,. By including β n = 0 for n = 0, the total number of propagation paths is increased slightly to = 4. Parameters a n and b n are defined as follows: { cos(βn ), n =1,..., a n = cos(βn ), n =0, (4) b n = { sin(βn ), n =1,..., sin(βn ), n =0. (5) The angles of arrival α nk are defined as in odel III: α nk =(πn)/ (πk)/(p)α 00 for n =0,...,, k = 0,...,P 1. This ensures an asymmetrical arrangement of arrival angles, which minimizes the cross-correlation between different faded envelopes. In addition, we optimize the initial angle of arrival α 00 (through an exhaustive search) to minimize the cross-correlation between the I and Q components of each complex faded envelope. As a result, we obtain α 00 = (0.π)/(P). emark 1: odel II and the new deterministic model differ in the selection of the angles of arrival and the cosine and sine functions. Our choice makes g ik (t) and g qk (t) orthogonal and, therefore, uncorrelated functions. emark : odel III and the new model differ in the selection of path gains, cosine and sine functions, and the number of random phases. By choosing the path gains to be random variables instead of being constants, we obtain less correlation in the I and Q components of the complex faded envelope than in odel III. Also, the use of fewer random variables makes our model less complex than odel III. The auto- and cross-correlation functions of the I and Q components, the auto- and cross-correlation functions of the multiple faded envelopes, and the squared envelope autocorrelation of our new model are, respectively, lim 4 g ik g ik = lim a n cos(ω m cos α nk τ) = J 0 (ω m τ)j 4 (ω m τ), (6)

5 IEEE TASACTIOS O WIELESS COUICATIOS, VOL. XX, O. XX, OTH YEA 5 lim 4 g qk g qk = lim b n cos(ω m sin α nk τ) = J 0 (ω m τ) J 4 (ω m τ), (7) gik g qk = gqk g ik = gk g l k =0, (8) lim 4 g k g k = lim gk g k = 8 4 lim a n cos(ω m cos α nk τ) b n cos(ω m sin α nk τ) = J 0 (ω m τ), (9) a 4 n 8 b 4 n g ik g ik g qk g qk 4 g ik g qk, (30) where J 0 ( ) is the zero-order Bessel function of the first kind and J 4 ( ) is the forth-order Bessel function of the first kind. Outlines for derivations of these expressions are presented in Appendix I. Figures 1 and confirm that, for =8, the auto- and cross-correlations of the quadrature components and the autoand cross-correlation of the multiple faded envelopes approach values given by (6)-(9), respectively. Our new model satisfies (3) and (4) of the reference model. However, the auto-correlations of the quadrature components and the auto-correlation of the squared envelope do not satisfy () and (5). ormalized Correlation Functions 1.0 gi g gq i g q J (ω m τ)j 4 (ω m τ) J 0 (ω m τ)-j 4 (ω m τ) gi g q ormalized time delay [ω m Fig. 1. Theoretical and simulated auto-correlation functions and the crosscorrelation function of the in-phase and quadrature components of the new deterministic model. B. A ew Statistical odel Our new deterministic model can be modified to possess all statistical properties of the reference model, by letting all three parameters C nk =(e jβ nk )/ (), α nk, and φ nk to ormalized Correlation Functions g1 g 1 g1 g J0 (ω m τ) ormalized time delay [ω m Fig.. Theoretical and simulated auto-correlation functions and the crosscorrelation function of the first and the second complex envelope of the new deterministic model. be random variables, similar to odel VI. The k th complex faded envelope is defined as g k (t) =g ik (t)jg qk (t), where g ik (t) = cos(β nk )cos(ω m t cos α nk φ nk ), (31) g qk (t) = sin(β nk )sin(ω m t sin α nk φ nk ). (3) It is assumed that P independent complex envelopes are desired (k =0,...,P 1), each having = /4 sinusoidal terms in the I and Q components. The parameters φ nk, β nk, and θ are independent random variables uniformly distributed on the interval [ π, π). The angles of arrival are chosen as follows: α nk =(πn)/ (πk)/(p)(θ π)/, for n =1,...,, k =0,...,P 1. The angles of arrival in the k th complex faded envelope are obtained by rotating the angles of arrivals in the (k 1) th complex envelope by (π)/(p). emark 3: odel VI and the new statistical model differ in the selection of the angles of arrivals for the multiple faded envelopes, and in the selection of the cosine and sine functions and random phases. Compared to odel VI, our choice of the cosine and sine functions makes the I and Q components of the complex faded envelope less correlated. Also, compared to odel VI, our choice of the angles of arrival for the multiple faded envelopes makes them less correlated. emark 4: The quadrature components of the complex faded envelope in odel VI are not Gaussian random processes [17]. Our choice of random phases solves this problem. The proof is omitted for brevity. In [17], it is also shown that the auto-correlation of the squared envelope in odel VI is nonstationary. We will prove that the auto-correlation of the squared envelope of our new model is stationary and satisfies (5) for. emark 5: The modified odel VI and the new statistical model differ only in the combination of cosine and sine functions for the first complex faded envelope (k = 0). Zheng and Xiao [15] mentioned that different combinations

6 IEEE TASACTIOS O WIELESS COUICATIOS, VOL. XX, O. XX, OTH YEA 6 of cosine and sine functions for the quadrature components leads to different statistical models with identical or similar statistical properties. However, the I and Q components of the complex faded envelope in our new statistical model are less correlated compared to modified odel VI. Furthermore, our new model needs fewer simulation trials to obtain the correct statistical properties. If different combinations of cosine and sine functions can improve statistical properties and/or reduce the number of simulation trials, then such combinations merit investigation. It can be shown that our statistical model exhibits properties ()-(5) of the reference model. For brevity, Appendix II outlines derivations of the auto-correlation function for the in-phase component and the auto-correlation function of the squared envelope. Figures 3 and 4 show that, for = P = 8 and stat = 30 trials, the auto- and crosscorrelations of the I and Q components, and the auto- and cross-correlations of the complex faded envelopes approach those of the reference model. ote that different sets of 30 simulation trials yield slightly different simulation results. To quantify these differences, variances are computed averaging over 0 sets of 30 simulation trials. The variances of the auto- and cross-correlations of the I and Q components, and the auto- and cross-correlations of the complex faded envelopes are, respectively, Var ( ) gi/qk g i/qk = 1.5 3, Var ( ) gik g qk = , Var( gk g k ) = , Var ( ) gk g l k = The variances are extremely small. ormalized Correlation Functions gi g i gq g q gi g q J0 (ω m τ) ormalized Time Delay [ω m Fig. 3. Theoretical and simulated ( stat =30) auto-correlation functions and the cross-correlation function of the in-phase and the quadrature component of the new statistical model. V. PEFOACE EVALUATIO This section compares the performance and complexity of our new models with odels I-VI. In all simulations, we use a normalized sampling period f m T s = 0.05 (f m is the maximum Doppler frequency and T s is the sampling period) and = P =8. However, for odel IV we use i = 8 and q = 9 to obtain uncorrelated quadrature components of the complex envelope. ote that in odels II- IV and in our new deterministic model, the random phases ormalized Correlation Functions g1 g 1 g1 g J0 (ω m τ) ormalized time delay [ω m Fig. 4. Theoretical and simulated ( stat =30) auto-correlation functions and the cross-correlation function of the first and the second fader in the new statistical model. associated with the n th propagation path are computed before the actual simulation starts, because an ergodic statistical ( deterministic ) simulator needs only one simulation trial. During the simulations, all parameters are kept constant to provide simulation results that are always the same, i.e., deterministic. In our new deterministic model and in odel II we use the following set of uniformly distributed random numbers (in radians): φ n,0 =[4.0387, 1.764,.7844, , 0.953, 1.97, 5.740, 3.659, ] and φ n,1 =[5.3798, ,.158,.696, , 3.57, 6.7,.0670,.1304]. The same set of numbers is used in odel IV, for the Q component g q (t), while for the I component g i (t) we use φ i n,0 =[.7, ,.4634, ,.818, , , ] and φ i n,1 =[3.67, , 0.584,.97, 4.664, , , 0.017]. odel III uses the same set of numbers as odel IV for the I component g i (t), while for the Q component g q (t) we use: φ q n,0 =[.637, , , , ,.3099, , ], and φ q n,1 =[4.1175,.4616, , ,.4948,.5981, 4.115, 5.6]. Using these parameters, we calculate the mean square error (SE) and maximum deviations (AX) from the theoretical value (zero) for the normalized cross-correlation of the I and Q components, and for the normalized cross-correlation of the first and the second faded envelopes. The results are shown in Table I. ote that different choices of phases yield different simulation results. To quantify these differences, we compute the mean and variance of the mean square error over 3 simulation trials each using a randomly selected phase vector. The mean SE (µse) and variance of the SE (VSE) of the cross-correlation of the I and Q component are, respectively, µse ( ) gik g qk =1., VSE ( ) gik g qk =3.36. Likewise, the mean and variance of the SE of the crosscorrelation of the complex faded envelopes are, respectively µse ( ) gk g l k = 1.4 3, VSE ( ) gk g l k = All quantities are very small. To further estimate the magnitude of differences between simulation results, we ran our deterministic simulator with different permutations of

7 IEEE TASACTIOS O WIELESS COUICATIOS, VOL. XX, O. XX, OTH YEA 7 TABLE I EA SQUAE EO (SE) AD AXIAL DEVIATIO (AX). Simulators SE g I g Q odel I.63 odel II.47 odel III 1.8 odel IV ew Deterministic odel 4.5 odel V ( stat = 50 ) 9.80 odel V ( stat = 0 ) 4.46 odel VI ( stat = 50 ) odel VI ( stat = 0 ) 4.5 odified odel VI ( stat = 50 ) odified odel VI ( stat = 0 ) ew Statistical odel ( stat = 30 ) 5.08 ew Statistical odel ( stat = 50 ) 1.91 ew Statistical odel ( stat = 0 ) 1.6 ax g I g Q SE eal( g ) 1 g ax eal( g ) 1 g phases which were evenly distributed on the interval [0, π) with zero as a starting point. The results are presented in Table II. 1 TABLE II AGITUDE OF DEVIATIOS I THE EW DETEIISTIC ODEL. Value The Highest SE ( g I g ) 4 Q 1.05 The Lowest SE ( g I g ) 6 Q.87 The Highest ax Deviation ( g I g ) Q.78 The Lowest ax Deviation ( g I g ) 3 Q 3.78 The Highest SE ( eal( g ) 1 g ) The Lowest SE ( eal( g ) 1 g ) The Highest ax Deviation ( eal( g ) 1 g ) The Lowest ax Deviation ( eal( g ) 1 g ) Phase Vector φ n0 = [4.8869, , ,.795, , , , 0,.0944] φ n0 = [0.6981, , , , ,.0944, 0, ,.795] φ n0 = [4.1888, , , , , ,.0944, 0,.795] φ n0 = [4.8869, 0,.0944, , , , , , ] φ n0 = [4.8869, , ,.795, , , , 0,.0944] φ n1 = [.0944,.795, , , , 0, , ] φ n0 = [0.6981, , , , ,.0944, 0, ,.795] φ n1 = [4.8869,.0944, ,.795, , 0, , , ] φ n0 = [4.1888, , , , , ,.0944, 0,.795] φ n1 = [0.6981, , , 0,.0944, 4.886, , ,.795] φ n0 = [4.8869, 0,.0944, , , , , , ] φ n1 = [5.5851,.0944, ,.795, , , 0, , ] To compare complexity of the new models and odels I-VI, Table III summarizes the number of simulation trials required to obtain desired statistical properties, the number of operations needed to generate one sample of the complex faded envelope and the relative simulation times needed to generate a sample of the complex faded envelope with desired statistical properties, in atlab on a Pentium III laptop. Here, we count only the frequently executed operations and the number of random variables. Table III shows that our choice of the cosine functions, which makes the I and Q components 1 esults presented in Table II are representative of the magnitude of differences between simulations. Choosing some other set of uniformly distributed random numbers may exceed the presented range of variations. of the complex faded envelope uncorrelated, slightly increases the complexity of our models. TABLE III COPLEXITY OF DIFFEET ODELS. umber of elative sim. time to Estimated number of computations needed to generate one sample of g k (t) Statistical models simulation generate a addition/ number of trials sample of cosine multiplication random variables g k (t) w. desired stat. properties odel I 1 Tx 4 / one odel II Tx 4 3 / 1 odel III 1 1.5Tx 4 4 / 0 odel IV 1 1.5Tx 4 4 / 0 ew Tx 6 4 / 1 Deterministic odel odel V Tx 4 4 / 0 1 odel VI Tx 4 3 / odified odel Tx 4 3 / 1 VI ew Statistical odel Tx 6 4 / 1 Figure 5 compares the cross-correlation functions of the I and Q components obtained by our new deterministic model and odels I-V. For clarity, we only plot the results for odel V, being the best of Zheng and Xiao s models. Since odel V is statistical model, we plot the average of stat = 50 trials. Figure 5 and Table I show that our deterministic model yields a lower cross-correlation between the I and Q components of the complex faded envelope, and also a lower maximum deviation from the theoretical value. In Figure 6, we gi g q odel I odel II odel III odel IV odel V ( stat =50) ew "Deterministic" odel ormalized time delay [ω m Fig. 5. The normalized cross-correlation function of the in-phase and quadrature components of the new deterministic model and odels I-V. compare cross-correlation functions of two faded envelopes for our new deterministic model and for odels I-V (for stat =50trials). The curve plotted for odel I is obtained from a simulation of Jakes modified model presented in [8]. We conclude that our deterministic model yields a low cross-correlation between two different faded envelopes, as do odels III and V. Figures 7 and 8 compare the cross-correlation functions of the I and Q components and the cross-correlation functions

8 IEEE TASACTIOS O WIELESS COUICATIOS, VOL. XX, O. XX, OTH YEA 8 gi g q odel I odel II odel III odel IV odel V ( stat =50) ew "Deterministic" odel ormalized time delay [ω m Fig. 6. The normalized cross-correlation function of the first and the second complex envelope of the new deterministic model and of odels I-V. of two faded envelopes, respectively, for our new statistical model and odels V and modified VI. For odels V and modified VI, we average over stat = 0 trials, while for our new statistical model we average over stat =30and stat =50trials. From Figure 7 and Table I, we conclude that our new statistical model with stat =30has a similar performance as odels V and modified VI with stat = 0. An increase of the number of trials to stat = 50 yields a significantly lower cross-correlation between the I and Q components of the complex faded envelope. An increase of the number of trials to stat = 50 yields a significantly lower cross-correlation between the I and Q components of the complex faded envelope. Furthermore, with stat =50 trials, the new statistical model achieves a larger de-correlation between different complex envelopes than do odels V and modified odel VI with stat = 0 trials. Figures 7, 8, and 9 show that our new statistical model converges faster than the other statistical models. Adequate statistics can be achieved with only 30 trials using our new statistical model. gi g q odel V, stat =0 odified odel VI, stat =0 ew Statistic odel, stat =30 ew Statistic odel, stat = ormalized time delay [ω m Fig. 7. The normalized cross-correlation function of the in-phase and quadrature components of the new statistical model, of odel V, and of modified odel VI. g1 g odel V, stat =0 odified odel VI, stat =0 ew Statistic odel, stat =30 ew Statistic odel, stat = ormalized time delay [ω m Fig. 8. The normalized cross-correlation function of the first and the second fader of the new statistical model, of odel V, and of modified odel VI. ormalized Correlation Functions J 0 (ω m τ) odel V stat =30 odified odel VI stat =30 ew Statistical odel stat = ormalized Time Delay [ω m Fig. 9. The theoretical and simulated normalized auto-correlation functions of the new statistical model, of odel V, and of modified odel VI. VI. COCLUSIO This paper proposed new SoS fading simulators for ayleigh fading channels. We first presented a deterministic (ergodic statistical) simulator in Section IV-A that overcomes the difficulty of creating uncorrelated I and Q components of each complex faded envelope and the difficulty of creating multiple uncorrelated faded envelopes. This is achieved by using orthogonal functions for the I and Q components of the complex faded envelope and by introducing an asymmetrical arrangement of arrival angles into the model proposed in [1]. The statistical properties of this new model are derived and verified using simulation. Compared to odels I-VI, our new deterministic model yields a lower cross-correlation between different faded envelopes, and between the I and Q components of each complex faded envelope. However, our deterministic model still has the disadvantage that the auto-correlation functions of the I and Q components do not match those of the reference model. To overcome this disadvantage, we introduce a new statistical model in Section IV-B. Properties of the resulting new statistical model are

9 IEEE TASACTIOS O WIELESS COUICATIOS, VOL. XX, O. XX, OTH YEA 9 derived and verified using simulation. The new model matches the statistical properties of the reference model and, when compared to [14]-[16], converges faster and has a lower correlation between the I and Q components of the complex faded envelope and between different faded envelopes. APPEDIX I DEIVATIO OF EQUATIOS (6)-(30) Derivation of the auto-correlation function of the in-phase component is presented below gik g ik =E [g ik (t)g ik (t τ)] = 8 a n a i i=0 E [cos(ω m (t τ)cosα ik ω m t cos α nk φ ik φ nk )] 8 a n a i (33) i=0 E [cos(ω m (t τ)cosα ik ω m t cos α nk φ ik φ nk )]. Since φ nk and φ ik are independent when n i, and all other terms in the sums are deterministic, we obtain gik g ik = 4 a n cos(ω mτ cos α nk ). (34) Furthermore, iemann integral theory can be used to show that as the auto-correlation of the in-phase component has the limiting value lim g ik g ik =J 0 (ω m τ)j 4 (ω m τ). (35) Similarly can be shown that the auto-correlation of the Q component, the cross-correlation of the I and Q components, the cross- and the auto-correlation of the faded envelopes, and the auto-correlation of the squared envelope are, respectively, lim 4 g qk g qk = lim b n cos(ω mτ sin α nk ) = J 0 (ω m τ) J 4 (ω m τ), (36) gik g qk = gqk g ik = gk g l k =0, (37) gk g k = 4 gk g k = 8 b n cos(ω mτ sin α nk ) (38) 4 a n cos(ω mτ cos α nk ) = J 0 (ω m τ), a 4 n 8 b 4 n g ik g ik g qk g qk 4 g ik g qk. (39) APPEDIX II DEIVATIO OF AUTO-COELATIOS OF THE I COPOET AD THE SQUAED EVELOPE Derivation of the auto-correlation function of the in-phase component is presented below gik g ik = 4 E [cos β nk cos β ik ] i=1 E [cos(ω m t cos α nk φ nk )cos(ω m (t τ)cosα ik φ ik )] = 1 E [cos(ω m cos α nk τ)] = 1 1 (40) π π ( ( πn cos ω m τ cos πk P θ π )) dθ. π As in [14], the proof can be completed by replacing the variable of integration θ with γ nk =πn/ (πk)/(p) (θ π)/ and integrating lim g ik g ik (41) = lim 1 = J 0 (ω m τ). 1 π πn πk P πnπ πk P cos(ω m τ cos γ nk )4dγ nk Derivation of the auto-correlation function of the squared envelope is given below. We follow a procedure similar to the one outlined in [17]. gk g k (4) = E [ gik(t)g ik(t τ) ] E [ gqk(t)g qk(t τ) ] E [ gik(t)g qk(t τ) ] E [ gqk(t)g ik(t τ) ]. The computation of the first term in the right-hand side of (4) is shown below 16 gik g ik = E a p b j c q d n, p=1 j=1 q=1 (43) where a p = cos(β pk )cos(ω m (t τ)cosα pk φ pk ), b j = cos(β jk )cos(ω m t cos α jk φ jk ), c q =cos(β qk )cos(ω m (t τ)cosα qk φ qk ), and d n =cos(β nk )cos(ω m t cos α nk φ nk ). The mutual independence of the φ ik s ensures that all terms in the above equation are zero, except the four terms with: 1) n = j = p = q; )n = j, p = q, j p; 3)n = p, j = q, n j; 4)n = q, j = p, n j. We compute each of these terms individually to derive overall expression. Term 1: n = j = p = q = E [ 1 4 w ] ny n = J 0 (ω m τ), (44) where w n = (1 cos (ω m t cos α nk φ nk )) and y n = (1 cos (ω m (t τ)cosα nk φ nk )).

10 IEEE TASACTIOS O WIELESS COUICATIOS, VOL. XX, O. XX, OTH YEA Term : n = j, p = q, j p = 16 E [ cos (β jk )cos (ω m t cos α jk φ jk ) ] p=1,p j = 16 j=1 E [ cos (β pk )cos (ω m (t τ)cosα pk φ pk ) ] j=1 1 1 p=1,p j 1 1 = 1, (45) Term 3: n = p, j = q, n j = 1 E [cos (ω m τ cos α nk )] (46) j=1,j n E [cos (ω m τ cos α jk )] = 1 J 0 (ω m τ). It can be shown that Term 4 is equal to Term 3. Adding all four terms gives E [ gik(t)g ik(t τ) ] = 1 1 J 0 (ω m τ) J 0 (ω m τ). (47) Similarly can be shown that E [ g qk(t)g qk(t τ) ] = E [ g ik(t)g ik(t τ) ], (48) E [ gik(t)g qk(t τ) ] = E [ gqk(t)g ik(t τ) ] (49) = J 0 (ω m τ). Substituting the above terms in (4) and letting, gives the desired expression (5). ACKOWLEDGET The authors would like to thank the anonymous reviewers whose feedback helped improve the quality of this paper. [] Y. X. Li and X. Huang, The simulation of independent ayleigh faders, IEEE Trans. on Communications, vol. 50, September 00, pp [11]. Pätzold, U. Killat, F. Laue and Y. Li, On the statistical properties of deterministic simulation models for mobile fading channels, IEEE Trans. on Vehicular Technology, vol. 47, February 1998, pp [1]. F. Pop and. C. Beaulieu, Limitations of sum-of-sinusoids fading channel simulators, IEEE Trans. on Communications, vol. 49, April 001, pp [13] C. Xiao, Y.. Zheng and. C. Beaulieu, Second-order statistical properties of the WSS Jakes fading channel simulator, IEEE Trans. on Communications, vol. 50, June 00, pp [14] Y.. Zheng and C. Xiao, Simulation models with correct statistical properties for ayleigh fading channels, IEEE Trans. on Communications, vol. 51, June 003, pp [15] Y.. Zheng and C. Xiao, Improved models for the generation of multiple uncorrelated ayleigh fading waveforms, Communications Letters, vol. 6, no. 6, June 00, pp [16] Y.. Zheng and C. Xiao, A statistical simulation model for mobile radio fading channels, Proc. IEEE WCC 03, ew Orleans, USA, arch 003, pp [17] C. S. Patel, G. L. Stuber, and T. G. Pratt, Comparative analysis of statistical models for the simulation of ayleigh faded cellular channels, IEEE Trans. on Communications, vol. 53, June 005, pp [18] G. L. Stuber, Principles of obile Communication, nd ed. orwell, A: Kluwer, 001. Alenka G. Zajić received the B.Sc. and.sc. degrees form the School of Electrical Engineering, University of Belgrade, in 001 and 003, respectively. From 001 to 003, she was a design engineer for Skyworks Solutions Inc., Fremont, CA. Since 004, she has been a Graduate esearch Assistant with the Wireless Systems Laboratory, and pursuing the Ph.D. degree in the School of Electrical and Computer Engineering, Georgia Institute of Technology. Her research interests are in wireless communications and applied electromagnetics. s. Zajić was recipient of the Dan oble Fellowship awarded by otorola Inc. and IEEE Vehicular Technology Society for quality impact in the area of vehicular technology. EFEECES [1] D. J. Young and. C. Beaulieu, A quantitative evaluation of generation methods for correlated ayleigh random variates, Proc. GLOBECO 98, Sydney, Australia, ovember, 1998, pp [] D. J. Young and. C. Beaulieu, The generation of correlated ayleigh random variates by inverse discrete Fourier transform, IEEE Trans. on Communications, vol. 48, July 000, pp [3] D. Verdin and T. C. Tozer, Generating a fading process for the simulation of land-mobile radio communications, Electronics Letters, vol. 9, no.3, ovember 1993, pp [4] S. A. Fechtel, A novel approach to modeling and efficient simulation of frequency-selecive fading radio channels, IEEE Journal on Selected Areas in Commun., vol. 11, no. 3, April 1993, pp [5]. H. Clarke, A statistical theory of mobile-radio reception, Bell Syst. Tech. J., pp , July Aug [6] W. C. Jakes, icrowave obile Communications, nd ed. Piscataway, J: Wiley-IEEE Press, [7]. Pätzold and F. Laue, Statistical properties of Jakes fading channel simulator, Proc. IEEE VTC 98, Ottawa, Canada, ay 1998, pp [8] P. Dent, G. E. Bottomley and T. Croft, Jakes fading model revisited, Electronics Letters, vol. 9, no. 13, June 1993, pp [9] Y. B. Li and Y. L Guan, odified Jakes model for simulating multiple uncorrelated fading waveforms, Proc. IEEE ICC 00, ew Orleans, USA, June 000, pp Gordon L. Stüber received the B.A.Sc. and Ph.D. degrees in Electrical Engineering from the University of Waterloo, Ontario, Canada, in 198 and 1986 respectively. In 1986, he joined the School of Electrical and Computer Engineering, Georgia Institute of Technology, where is the Joseph. Pettit Chair Professor. Dr. Stüber was co-recipient of the IEEE Vehicular Technology Society Jack eubauer emorial Award in 1997 for the best systems paper. He became an IEEE Fellow in 1999 for contributions to mobile radio and spread spectrum communications. He received the IEEE Vehicular Technology Society James. Evans Avant Garde Award in 003 for contributions to theoretical research in wireless communications. Dr. Stüber served as General Chair and Program Chair for several conferences, including VTC 96, ICC 98, T 00, CTW 0, and WPC 0. He is a past Editor for IEEE Transactions on Communications ( ), and served on the IEEE Communications Society Awards Committee ( ). He an elected member of the IEEE Vehicular Technology Society Board of Governors ( , ) and received the Outstanding Service Award from the IEEE Vehicular Technology Society.