Efficient Simulation of Rayleigh Fading with ...

Appears in IEEE Transactions on Wireless Communications, vol.5, pp.1866-1875, July 2006.

Efficient Simulation of Rayleigh Fading with Enhanced De-Correlation Properties

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Alenka G. Zajić, Student Member, IEEE, and Gordon L. Stüber, Fellow, IEEE

Abstract— New sum-of-sinusoids simulation models are proposed for Rayleigh 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, Rayleigh fading, sum-of-sinusoids.

I. I NTRODUCTION IMULATION 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. Many different approaches have been used to model and simulate Rayleigh 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. Many approaches have been suggested for SoS modeling of Rayleigh fading channels. Clarke [5] was among the first to propose a mathematical reference model for Rayleigh fading channels. A simplified version of Clarke’s model, proposed

S

Paper approved by Hao Xu, the Editor for IEEE Transactions on Wireless Communications. Manuscript received August 27, 2004; revised June 28, 2005. This work was prepared through collaborative participation in the Collaborative Technology Alliance for Communications & Networks sponsored by the U.S. Army Research Laboratory under Cooperative Agreement DAAD19-01-2-0011. 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 Research 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 30332 USA.

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) [12]. 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 100 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 Rayleigh 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 [12]. 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 II presents the mathematical reference model. Section III reviews existing simulation models for Rayleigh fading channels [6]-[16]. Section IV describes our new sum-of-sinusoids simulation models for Rayleigh fading channels and analyzes

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their statistical properties. Section V evaluates the new models and compares them to previously reported models. Section VI provides some concluding remarks. II. M ATHEMATICAL R EFERENCE M ODEL Clarke’s reference model [5] defines the complex faded envelope as g(t) =

N −1 X

Cn ej(ωm t cos αn +φn ) ,

(1)

n=0

where N , ωm , Cn , αn , and φn are the number of propagation paths, the maximum angular Doppler frequency, the path gain, the angle of arrival, and the phase associated with the nth propagation path, respectively. It is assumed that Cn , αn , and φn are mutually independent and that αn and φn are uniformly distributed on the interval [−π, π). Invoking the Central Limit Theorem [18], the real part gi (t) = < {g(t)} and the imaginary part gq (t) = = {g(t)} of the complex faded envelope are Gaussian random processes as N → ∞. Therefore, the envelope |g(t)| is Rayleigh distributed and the phase Θg (t) is uniformly distributed. The auto- and crosscorrelation functions of the reference model, assuming a two dimensional (2-D) scattering environment, are summarized below [6], [18] £ ¤ Rgi/q gi/q (τ ) = E gi/q (t + τ )gi/q (t) = J0 (ωm τ ),

(2)

(N − 2)/4. The complex faded envelope is defined as g(t) = gi (t) + jgq (t), where r 4 cos (β0 ) cos (ωm t) gi (t) = N r · µ ¶¸ M 8 X 2πn + cos (βn ) cos ωm t cos , (6) N n=1 N r

gq (t)

4 sin (β0 ) cos (ωm t) N r · µ ¶¸ M 2πn 8 X sin (βn ) cos ωm t cos , (7) + N n=1 N

=

and where ωm is the maximum angular Doppler frequency. The parameter βn is defined as βn = πn/M for n = 0, . . . , M . Often it is desirable to generate multiple uncorrelated faded envelopes, something that Model I cannot do. Dent et al. [8] modified Model I by using orthogonal Walsh-Hadamard code words to de-correlate the multiple faded envelopes. The k th complex faded envelope is defined as gk (t) = gik (t)+jgqk (t), where µ ¶ M X πn gik (t) = Ak (n) cos (8) M +1 n=1 ½ µ ¶ ¾ 2πn πn 2πn(k − 1) cos ωm t cos + + , N M +1 M +1

£ ¤ Rgi gq (τ ) = Rgq gi (τ ) = E gi/q (t + τ )gq/i (t) = 0,

1 Rgk gl (τ ) = E [gk (t + τ )gl∗ (t)] = 2

½

(3)

J0 (ωm τ ) l = k , 0 l= 6 k (4)

h i 2 2 R|gk |2 |gk |2 (τ ) = E |gk (t + τ )| |gk (t)| = 4 + 4J0 (ωm τ ), (5) where E [·] is the statistical expectation operator, J0 (·) is the zero-order Bessel function of the first kind, and ωm 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. E XISTING S IMULATION M ODELS

µ

¶ πn gqk (t) = Ak (n) sin (9) M +1 n=1 ½ µ ¶ ¾ 2πn πn 2πn(k − 1) cos ωm t cos + + , N M +1 M +1 M X

M = (N −2)/4, k = 1, . . . , M , and Ak (n) is the nth element of the k th row of a Hadamard matrix HM of dimension M × M. By reducing the number of sinusoidal components, Jakes simplified simulation of Rayleigh fading channels. However, Model I does not satisfy most of the statistical properties of the reference model [7] and it is not wide-sense stationary (WSS) [12]. Model I satisfies only the following properties: the I and Q components of the complex faded envelope are Gaussian random processes for N → ∞ and the autocorrelation function of the complex faded envelope is equal to J0 (ωm 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 Model I have been proposed in the literature [10]-[12].

A. Jakes’ Model–Model I

B. Pop and Beaulieu’s Model–Model II

Jakes derived his well-known deterministic simulation model for Rayleigh fading channels [6] starting from (1) and √ selecting Cn = 1/ N , θn = 2πn/N , and φn = 0 for n = 0, . . . , N . Using the symmetry of the 2-D isotropic scattering environment, Jakes reduced the number of sinusoidal components necessary for simulation from N to M =

Pop and Beaulieu [12] showed that Model I is not WSS and modified Model I to fix this problem. This was done by removing the constraint φn = 0 from Model I and allowing the phases φn to be independent random variables uniformly distributed on the interval [0, 2π). The procedure yields an ergodic statistical (“deterministic”) simulator, since

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the random phases are generated only once for all simulation trials. The k th complex faded envelope is defined as gk (t) = gik (t) + jgqk (t), where r 4 cos (β0 ) cos (ωm t + φ0k ) (10) gik (t) = N r M · µ ¶ ¸ 8X 2πn + cos (βn ) cos ωm t cos + φnk , N n=1 N r gqk (t) = +

4 sin (β0 ) cos (ωm t + φ0k ) (11) N r M · µ ¶ ¸ 2πn 8X sin (βn ) cos ωm t cos + φnk , N n=1 N

M = (N − 2)/4, k = 0, . . . , M − 1, and ωm is the maximum angular Doppler frequency. The parameter βn is defined as βn = πn/M for n = 0, . . . , M . Although WSS, Model II inherits its statistical properties from Model I. Hence, this model does not satisfy most of the other statistical properties required by the reference model. C. Li and Huang’s Model–Model III To improve Model II, Li and Huang [10] proposed an ergodic statistical (“deterministic”) model that generates multiple uncorrelated faded envelopes gk (t). Model III assumes P independent complex faded envelopes, each with M = N/4 sinusoidal terms in the I and Q components. The k th complex faded envelope is gk (t) = gik (t) + jgqk (t), where gik (t) = 2C

M −1 X

¡ ¢ cos ωm t cos αnk + φink ,

(12)

n=0

gqk (t) = 2C

M −1 X

sin (ωm t sin αnk + φqnk ),

(13)

n=0

and where k = 0, .., P − 1, φink and φqnk are independent random phases uniformly distributed on the interval [0, 2π), αnk is the nth angle of arrival in the k th complex faded envelope, C is a constant gain, and ωm is the maximum angular Doppler frequency. The angles of arrivals are αnk = (2πn)/N + (2πk)/(P N ) + α00 for n = 0, . . . , M − 1, k = 0, . . . P − 1, where α00 is an initial angle of arrival, chosen to be 0 < α00 < (2π)/(P N ) and α00 6= π/(P N ). Model III preserves the desirable statistical properties of Model II, while generating multiple uncorrelated faded envelopes. Compared to Model II, the I and Q components of the complex faded envelope in Model III are less correlated. However, Model III fails to satisfy equations (2) and (5) of the reference model. D. MEDS Model–Model IV To resolve the remaining disadvantages of Model II, Pätzold et al. [11] proposed a deterministic model called the Method of Exact Doppler Spreads (MEDS). The auto-correlation functions of the I and Q components of the complex faded envelope

are designed to satisfy equation (2). The k th complex faded envelope is defined as gk (t) = gik (t) + jgqk (t), where s N i/q h i 2 X i/q cos ωm t sin αni/q + φnk (14) g(i/q)k (t) = Ni/q n=1 i/q

for k = 0, . . . , P − 1, and where φnk are independent random phases uniformly distributed on the interval [0, 2π), P is the number of desired faded envelopes, and Ni/q is the number of sinusoidal terms in the I and Q components of gk (t), respectively. The nth angle of arrival is given by i/q αn = π(n − 0.5)/(2Ni/q ) for n = 1, . . . , Ni/q . Model IV preserves the desirable properties of Model II. In addition, the auto-correlation functions of the I and Q components in Model IV satisfy (2). Compared to Model II, the I and Q components of the complex faded envelope in Model IV are less correlated if Nq = Ni + 1. However, Model IV produces multiple faded envelopes that are correlated. E. Zheng and Xiao’s Models 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 100 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 Model V and will compare it with our new models. Model V: The k th complex faded envelope is defined as gk (t) = gik (t) + jgqk (t), where r M £ ¤ 2 X gik (t) = cos ωm t cos αnk + φink , (15) M n=1 r M 2 X gqk (t) = cos [ωm t sin αnk + φqnk ], (16) M n=1 for k = 0, . . . , P − 1, and where φink and φqnk are independent random phases uniformly distributed on the interval [−π, π). Model V assumes P independent complex envelopes, each with M = N/4 sinusoidal terms in the I and Q components. The nth angle of arrival in the k th complex envelope αnk is αnk = (2πn − π + θk )/(4M ) for n = 1, . . . , M , where the θk 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. Model VI: The k th complex faded envelope is defined as gk (t) = gik (t) + jgqk (t), where M 2 X cos (ψnk ) cos [ωm t cos αnk + φk ], gik (t) = √ M n=1

(17)

4 M 2 X sin (ψnk ) cos [ωm t cos αnk + φk ], gqk (t) = √ M n=1

(18)

αnk = (2πn − π + θk )/(4M ), ψnk , θk , and φk are statistically independent random variables, uniformly distributed on the interval [−π, π), and n = 1, . . . , M , k = 0, . . . , P − 1. Patel et al. [17] have shown that the I and Q components of the complex faded envelope in Model 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 φk , which is the same for all sinusoidal terms in the k th complex envelope, with mutually independent random phases φnk , uniformly distributed on the interval [−π, π). The proof is omitted for brevity. We refer to this model as modified Model VI. IV. N EW S IMULATION M 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 gk (t) =

N −1 X

Cn ej(ωm t cos αnk +φnk ) ,

Based on gk (t), we define our new simulation models. A. New “Deterministic” Model 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 gk (t) = gik (t) + jgqk (t), where "M # X 2 gik (t) = √ an cos(ωm t cos αnk + φnk ) , (22) N n=0 2 gqk (t) = √ N

½

(19) bn =

√

where Cn = (2e )/ N , αnk , φnk , and ωm are the random path gain, the angle of arrival, the phase associated with the nth 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 N sinusoidal components. To reduce the number of sinusoidal components needed for simulation, we use a method similar to the one described in [10]. By choosing M = N/4 to be an integer and by taking into account shifts of the angles αnk and φnk in each quadrant of the circle, the sum in (19) can be split into four terms, viz. gk (t)

=

M 2 X jβn j[ωm t cos αnk +φnk ] √ e e N n=1

+

M 2 X jβn j[ωm t cos(αnk +0.5π)+(φnk +0.5π)] √ e e N n=1

+

M 2 X jβn j[ωm t cos(αnk +π)+(φnk +π)] √ e e N n=1

+

M 2 X jβn j[ωm t cos(αnk +1.5π)+(φnk +1.5π)] √ e e N n=1

(20)

Equation (20) can be further simplified to M 2 X √ gk (t) = 2 cos (βn ) cos (ωm t cos αnk + φnk ) (21) N n=1 M 2 X 2 sin (βn ) sin (ωm t sin αnk + φnk ). + j√ N n=1

M X

# bn sin(ωm t sin αnk + φnk ) .

(23)

n=0

The motivation for this model originates in Model II. As in Model II, the phases φnk are chosen to be independent random variables uniformly distributed on the interval [0, 2π), and the path gains βn are defined as βn = πn/M for n = 0, . . . , M . By including β0 = 0 in (21), the total number of propagation paths is increased slightly to N = 4M + 2. Parameters an and bn are defined as follows: ½ 2√cos (βn ) , n = 1, ..., M an = , (24) 2 cos (βn ) , n = 0

n=0 jβn

"

2√sin (βn ) , n = 1, ..., M . 2 sin (βn ) , n = 0

(25)

The angles of arrivals αnk are defined as in Model III: αnk = (2πn)/N +(2πk)/(P N )+α00 for n = 0, . . . , M , 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 α00 (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 α00 = (0.2π)/(P N ). Remark 1: Model II and the new “deterministic” model differ in the selection of the angles of arrivals and the cosine and sine functions. Our choice makes gik (t) and gqk (t) orthogonal and, therefore, uncorrelated functions. Remark 2: Model 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 Model III. Also, the use of fewer random variables makes our model less complex than Model 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 Rgik gik (τ )

N →∞

=

M 4 X a2n cos(ωm cos αnk τ ) N →∞ N 2 n=0

lim

= J0 (ωm τ ) + J4 (ωm τ ),

(26)

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lim

N →∞

=

J0 (ωm τ ) − J4 (ωm τ ),

(27)

Rgik gqk (τ ) = Rgqk gik (τ ) = Rgk gl6=k (τ ) = 0,

(28)

M 4 X a2n lim cos(ωm cos αnk τ ) N →∞ N 2 n=0

lim Rgk gk (τ ) =

N →∞

+

M 4 X b2n lim cos(ωm sin αnk τ ) N →∞ N 2 n=0

=

J0 (ωm τ ),

(29)

1.0

Normalized Correlation Functions

M 4 X b2n cos(ωm sin αnk τ ) N →∞ N 2 n=0

lim Rgqk gqk (τ ) =

Rg g Rg g J0(ωmτ)

0.8

1 1 1 2

0.6 0.4 0.2 0.0 -0.2 -0.4 -0.6 -0.8 -1.0 0

R|gk |2 |gk |2 (τ ) =

M M 8 X 4 8 X 4 a + b + 2Rg2i gi (τ ) k k N 2 n=0 n N 2 n=0 n

+ 2Rg2q

g (τ ) k qk

+ 4Rg2i

g (τ ), k qk

(30)

where J0 (·) is the zero-order Bessel function of the first kind and J4 (·) is the forth-order Bessel function of the first kind. Outlines for derivations of these expressions are presented in Appendix I. Figures 1 and 2 confirm that, for M = P = 8, the auto- and cross-correlations of the quadrature components and the autoand cross-correlation of the multiple faded envelopes approach values given by (26)-(29), 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 (2) and (5), respectively. Rg g J0(ωmτ)+J4(ωmτ)

1.0

Rg g J0(ωmτ)-J4(ωmτ) Rg g


i i

0.8

q q

0.6

i q

0.4 0.2 0.0 -0.2 -0.4 -0.6 -0.8 -1.0 0

2

4

6

8

10

12

14

16

18

20

Normalized time delay [ωmτ] 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 New Statistical Model Our new “deterministic” model can be modified to possess all statistical properties of the reference p model, by letting all three parameters Cnk = (2ejβnk )/ (N ), αnk , and φnk to

2

4

6

8

10

12

14

16

18

20

Normalized time delay [ωmτ] Fig. 2. 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 Model VI. The k th complex faded envelope is defined as gk (t) = gik (t) + jgqk (t), where M

2 X gik (t) = √ 2 cos (βnk ) cos (ωm t cos αnk + φnk ), (31) N n=1 M

2 X 2 sin (βnk ) sin (ωm t sin αnk + φnk ). (32) gqk (t) = √ N n=1 It is assumed that P independent complex envelopes are desired (k = 0, . . . , P − 1), each having M = N/4 sinusoidal terms in the I and Q components. The parameters φnk , βnk , and θ are independent random variables uniformly distributed on the interval [−π, π). The angles of arrivals are chosen as follows: αnk = (2πn)/N + (2πk)/(P N ) + (θ − π)/N , for n = 1, . . . , M , k = 0, . . . , P − 1. The angles of arrivals in the k th complex faded envelope are obtained by rotating the angles of arrivals in the (k − 1)th complex envelope by (2π)/(P N ). Remark 3: Model 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 Model 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 Model VI, our choice of the angles of arrival for the multiple faded envelopes makes them less correlated. Remark 4: The quadrature components of the complex faded envelope in Model 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 Model VI is nonstationary. We will prove that the auto-correlation of the squared envelope of our new model is stationary and satisfies (5) for M → ∞. Remark 5: The modified Model 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

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1.0

Rg g Rg g Rg g J0(ωmτ)

0.8

i i

q q

0.6

i q

0.4 0.2 0.0 -0.2 -0.4 -0.6 -0.8 -1.0 0

2

4

6

8

10

12

14

16

18

20

Normalized Time Delay [ωmτ] Fig. 3. Theoretical and simulated (Nstat = 30) auto-correlation functions and the cross-correlation function of the in-phase and the quadrature component of the new statistical model.

V. P ERFORMANCE E VALUATION This section compares the performance and complexity of our new models with Models I-VI. In all simulations, we use a normalized sampling period fm Ts = 0.05 (fm is the maximum Doppler frequency and Ts is the sampling period) and M = P = 8. However, for Model IV we use Ni = 8 and Nq = 9 to obtain uncorrelated quadrature components of the complex envelope. Note that in Models IIIV and in our new “deterministic” model, the random phases

1.0


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 Model 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 (2)-(5) of the reference model. For brevity, Appendix II outlines only 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 M = P = 8 and Nstat = 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. Note that different sets of 30 simulation trials yield slightly different simulation results. To quantify these differences, variances are computed averaging over 100 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 R = 1.5 · 10−3 , gi/qk gi/qk ¡ ¢ −4 Var¡Rgik gqk ¢ = 1.35 · 10 , Var(Rgk gk ) = 6.81 · 10−4 , Var Rgk gl6=k = 8.4·10−4 . The variances are extremely small.

Rg g Rg g J0(ωmτ)

0.8

1 1

0.6

1 2

0.4 0.2 0.0 -0.2 -0.4 -0.6 -0.8 -1.0 0

2

4

6

8

10

12

14

16

18

20

Normalized time delay [ωmτ] Fig. 4. Theoretical and simulated (Nstat = 30) auto-correlation functions and the cross-correlation function of the first and the second fader in the new statistical model.

associated with the nth 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 Model II we use the following set of uniformly distributed random numbers (in radians): φn,0 =[4.0387, 1.7624, 2.7844, 1.5590, 0.9523, 1.2972, 5.7420, 3.6592, 4.3548] and φn,1 =[5.3798, 3.0556, 2.1528, 2.6296, 0.7457, 3.2572, 6.1027, 2.0670, 2.1304]. The same set of numbers is used in Model IV, for the Q component gq (t), while for the I component gi (t) we use φin,0 =[2.2107, 5.3033, 2.4634, 1.0679, 2.2818, 4.6113, 0.7513, 0.7383] and φin,1 =[3.2627, 4.7036, 0.5824, 2.1097, 4.6264, 5.4790, 0.9391, 0.2017]. Model III uses the same set of numbers as Model IV for the I component gi (t), while for the Q component gq (t) we use: φqn,0 =[2.6372, 4.7339, 4.9865, 5.7784, 5.3059, 2.3099, 3.8994, 4.5933], and φqn,1 =[4.1175, 2.4616, 3.9403, 4.3911, 2.4948, 2.5981, 4.115, 5.2610]. Using these parameters, we calculate the mean square error (MSE) and maximum deviations (MAX) 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. Note 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 103 simulation trials each using a randomly selected phase vector. The mean MSE (µMSE) and variance of the MSE (VMSE) of the cross-correlation are, ¡ ¢ of the I and Q component ¡ ¢ respectively, µMSE Rgik gqk = 1.2 · 10−5 , VMSE Rgik gqk = 3.36 · 10−5 . Likewise, the mean and variance of the MSE of the crosscorrelation of the ¡ ¢ complex faded envelopes ¡ are, ¢respectively µMSE Rgk gl6=k = 1.4 · 10−3 , VMSE Rgk gl6=k = 1.22 · 10−31 . 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

TABLE I M EAN SQUARE ERROR (MSE) AND MAXIMAL DEVIATION (MAX). Simulators

MSE Rg I g Q

Max Rg I g Q

MSE Real( R g1 g 2 )

Max Real( R g1 g 2 )

Model I

2.63 ⋅ 10 −2

42.61 ⋅ 10 −2

5.92 ⋅ 10 −2

52.3 ⋅ 10 −2

Model II

2.47 ⋅ 10 −2

44.88 ⋅ 10 −2

5.39 ⋅ 10 −2

59.74 ⋅ 10 −2

Model III

1.28 ⋅ 10 −2

24.30 ⋅ 10 −2

5.20 ⋅ 10 −3

15.14 ⋅ 10 −2

Model IV

−3

−2

−2

−2

Model V ( N stat = 100 ) Model VI ( N stat = 50 ) Model VI ( N stat = 100 ) Modified Model VI ( N stat = 50 ) Modified Model VI ( N stat = 100 ) New Statistical Model ( N stat = 30 ) New Statistical Model ( N stat = 50 )

17.46 ⋅ 10

4.90 ⋅ 10

61.01 ⋅ 10

4.25 ⋅ 10 −5

1.61 ⋅ 10 −2

1.60 ⋅ 10 −3

5.75 ⋅ 10 −2

−5

−2

−5

−2

9.80 ⋅ 10

2.27 ⋅ 10

6.26 ⋅ 10

2.30 ⋅ 10

4.46 ⋅ 10 −5

2.05 ⋅ 10 −2

3.87 ⋅ 10 −5

6.50 ⋅ 10 −4

5.89 ⋅ 10 −2

1.06 ⋅ 10 −4

2.30 ⋅ 10 −2

2.52 ⋅ 10 −4

3.57 ⋅ 10 −2

7.25 ⋅ 10 −5

1.60 ⋅ 10 −2

3.75 ⋅ 10 −4

3.83 ⋅ 10 −2

6.51 ⋅ 10 −5

1.56 ⋅ 10 −2

2.48 ⋅ 10 −2

4.93 ⋅ 10 −5

1.48 ⋅ 10 −2

−5

−2

−5

1.17 ⋅ 10 −2

0.94 ⋅ 10 −2

6.47 ⋅ 10 −6

0.60 ⋅ 10 −2

−2

−6

0.35 ⋅ 10 −2

1.91 ⋅ 10 −5

New Statistical Model ( N stat = 100 ) 1.62 ⋅ 10 −5

1.44 ⋅ 10 0.69 ⋅ 10

3.31 ⋅ 10 2.38 ⋅ 10

phases which were evenly distributed on the interval [0, 2π) with zero as a starting point. The results are presented in Table II.1 TABLE II M AGNITUDE OF DEVIATIONS IN THE NEW ” DETERMINISTIC ” Value

MODEL .

Phase Vector

φ n0 = [4.8869, 5.5851, 4.1888, 2.7925,

The Highest MSE ( R g I g Q )

1.05 ⋅ 10 −4

1.3963, 3.4907, 0.6981, 0, 2.0944]

The Lowest MSE ( R g I g Q )

2.87 ⋅ 10 −6

φ n0 = [0.6981, 1.3963, 3.4907, 5.5851, 4.1888, 2.0944, 0, 4.8869, 2.7925]

The Highest Max Deviation ( R g I g Q )

2.78 ⋅ 10 −2

φ n0 = [4.1888, 5.5851, 4.8869, 1.3963, 0.6981, 3.4907, 2.0944, 0, 2.7925]

The Lowest Max Deviation ( R g I g Q )

3.78 ⋅ 10

−3

The Highest MSE ( Real( R g1 g 2 ) )

3.3 ⋅ 10 −3

The Lowest MSE ( Real( R g1 g 2 ) )

−3

1.8 ⋅ 10

The Highest Max Deviation ( Real( R g1 g 2 ) )

1.08 ⋅ 10 −1

The Lowest Max Deviation ( Real( R g1 g 2 ) )

1.03 ⋅ 10 −1

Number of simulation trials

1.11 ⋅ 10 −2

1.14 ⋅ 10 −4 5.08 ⋅ 10

Statistical models

Model I Model II Model III Model IV New “Deterministic” Model Model V Model VI Modified Model VI New Statistical Model

1 1 1 1 1

Relative sim. Estimated number of computations needed time to to generate one sample of g k (t ) generate a addition/ number of sample of cosine multiplication random variables g k (t ) w. desired stat. properties None Tx 2M / 2M 4M 1.03Tx 3M / 2 M 4M M +1 1.52Tx 4M / 0 4M 2M 1.52Tx 4M / 0 4M 2M 1.98Tx 6M 4M / 2M M +1 4.85Tx

4M

4M / 0

7.37Tx

4M

3M / 2 M

M +2

100

7.45Tx

4M

3M / 2 M

2M + 1

30

3.15Tx

6M

4M / 2M

2M + 1

Model I Model II Model III Model IV Model V (Nstat=50) New "Deterministic" Model

φ n0 = [4.8869, 0, 2.0944, 5.5851, 0.6981, 4.1888, 3.4907, 3.4907, 1.3963]

0.20

φ n0 = [4.8869, 5.5851, 4.1888, 2.7925, 1.3963, 3.4907, 0.6981, 0, 2.0944] φ n1 = [2.0944, 2.7925, 5.5851, 0.6981 4.8869, 3.4907, 0 , 4.1888, 1.3963] φ n0 = [0.6981, 1.3963, 3.4907, 5.5851, 4.1888, 2.0944, 0, 4.8869, 2.7925] φ n1 = [4.8869, 2.0944, 4.1888, 2.7925, 3.4907, 0, 0.6981, 1.3963, 5.5851] φ n0 = [4.1888, 5.5851, 4.8869, 1.3963, 0.6981, 3.4907, 2.0944, 0, 2.7925] φ n1 = [0.6981, 1.3963, 3.4907, 0, 2.0944, 4.886, 5.5851, 4.1888, 2.7925] φ n0 = [4.8869, 0, 2.0944, 5.5851, 0.6981, 4.1888, 3.4907, 3.4907, 1.3963] φ n1 = [5.5851, 2.0944, 4.1888, 2.7925, 3.4907, 0.6981, 0, 1.3963, 4.8869]

2M + 1

50 100

Figure 5 compares the cross-correlation functions of the I and Q components obtained by our new “deterministic” model and Models I-V. For clarity, we only plot the results for Model V, being the best of Zheng and Xiao’s models. Since Model V is statistical model, we plot the average of Nstat = 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

0.15 0.10

i q

Model V ( N stat = 50 )

8.80 ⋅ 10

TABLE III C OMPLEXITY OF DIFFERENT MODELS .

Rg g (τ)

New “Deterministic” Model

of the complex faded envelope uncorrelated, slightly increases the complexity of our models.

0.05 0.00 -0.05 -0.10 -0.15 -0.20 0

To compare complexity of the new models and Models 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 Matlab 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 Results 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.

5

10

15

20

25

30

Normalized time delay [ωmτ] Fig. 5. The normalized cross-correlation function of the in-phase and quadrature components of the new ”deterministic” model and Models I-V.

compare cross-correlation functions of two faded envelopes for our new “deterministic” model and for Models I-V (for Nstat = 50 trials). The curve plotted for Model 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 Models III and V. Figures 7 and 8 compare the cross-correlation functions of the I and Q components and the cross-correlation functions

8

0.20 0.15

i q

0.01

1 2

0.05 0.00

Model V, Nstat=100 Modified Model VI, Nstat=100 New Statistic Model, Nstat=30 New Statistic Model, Nstat=50

0.02

Rg g (τ)

0.10

Rg g (τ)

0.03

Model I Model II Model III Model IV Model V (Nstat=50) New "Deterministic" Model

0.00

-0.01

-0.05

-0.02 -0.10

-0.03 -0.15

0

5

10

15

20

25

30

Normalized time delay [ωmτ]

-0.20 0

5

10

15

20

25

30

Normalized time delay [ωmτ]

Fig. 8. The normalized cross-correlation function of the first and the second fader of the new statistical model, of Model V, and of modified Model VI.

Fig. 6. The normalized cross-correlation function of the first and the second complex envelope of the new ”deterministic” model and of Models I-V.

of two faded envelopes, respectively, for our new statistical model and Models V and modified VI. For Models V and modified VI, we average over Nstat = 100 trials, while for our new statistical model we average over Nstat = 30 and Nstat = 50 trials. From Figure 7 and Table I, we conclude that our new statistical model with Nstat = 30 has a similar performance as Models V and modified VI with Nstat = 100. An increase of the number of trials to Nstat = 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 Nstat = 50 yields a significantly lower cross-correlation between the I and Q components of the complex faded envelope. Furthermore, with Nstat = 50 trials, the new statistical model achieves a larger de-correlation between different complex envelopes than do Models V and modified Model VI with Nstat = 100 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. 0.03

Model V, Nstat=100 Modified Model VI, Nstat=100 New Statistic Model, Nstat=30 New Statistic Model, Nstat=50

0.02

i q

Rg g (τ)

0.01

0.00

-0.01

-0.02

-0.03 0

5

10

15

20

25

30

Normalized time delay [ωmτ] Fig. 7. The normalized cross-correlation function of the in-phase and quadrature components of the new statistical model, of Model V, and of modified Model VI.


1.00

J0(ωmτ) Model V Nstat=30 Modified Model VI Nstat=30 New Statistical Model Nstat=30

0.75

0.50

0.25

0.00

-0.25

-0.50 0

5

10

15

20

25

30

35

40

45

50

55

60

Normalized Time Delay [ωmτ] Fig. 9. The theoretical and simulated normalized auto-correlation functions of the new statistical model, of Model V, and of modified Model VI.

VI. C ONCLUSION This paper proposed new SoS fading simulators for Rayleigh 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 [12]. The statistical properties of this new model are derived and verified using simulation. Compared to Models 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

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. A PPENDIX I D ERIVATION OF E QUATIONS (26)-(30)

+

M M 8 X X an ai N n=0 i=0 2

E [cos(ωm (t + τ ) cos αik + ωm t cos αnk + φik + φnk )] M M 8 X X an ai (33) N n=0 i=0 2 E [cos(ωm (t + τ ) cos αik − ωm t cos αnk + φik − φnk )] .

Since φnk and φik are independent when n 6= i, and all other terms in the sums are deterministic, we obtain Rgik gik (τ )

=

4 N

M X

a2n

n=0

2

cos(ωm τ cos αnk ).

(34)

Furthermore, Riemann integral theory can be used to show that as N → ∞ the auto-correlation of the in-phase component has the limiting value lim Rgik gik (τ ) =

N →∞

J0 (ωm τ ) + J4 (ωm τ ).

Derivation of the auto-correlation function of the in-phase component is presented below Rgik gik (τ ) =

Derivation of the auto-correlation function of the in-phase component is presented below Rgik gik (τ ) = E [gik (t)gik (t + τ )] =

A PPENDIX II D ERIVATION OF AUTO - CORRELATIONS OF THE I C OMPONENT AND THE S QUARED E NVELOPE

(35)

E [cos(ωm t cos αnk + φnk ) cos(ωm (t + τ ) cos αik + φik )] M 1 X 1 1 = E [cos(ωm cos αnk τ )] = (40) M n=1 M 2π ¶¶ µ µ M Zπ X θ−π 2πn 2πk + + dθ. cos ωm τ cos N PN N n=1 −π

As in [14], the proof can be completed by replacing the variable of integration θ with γnk = 2πn/N +(2πk)/(P N )+ (θ − π)/N and integrating lim Rgik gik (τ )

lim Rgqk gqk (τ ) =

N →∞

=

J0 (ωm τ ) − J4 (ωm τ ),

Rgik gqk (τ ) = Rgqk gik (τ ) = Rgk gl6=k (τ ) = 0,

Rgk gk (τ ) =

M 4 X b2n cos(ωm τ sin αnk ) N n=0 2

+

4 X a2n cos(ωm τ cos αnk ) N n=0 2

=

J0 (ωm τ ),

(36) (37)

(38)

M

R|gk |2 |gk |2 (τ )

=

M M 8 X 4 8 X 4 a + b + 2Rg2i gi (τ ) n k k N 2 n=0 N 2 n=0 n

+

2Rg2q

g k qk

(τ ) + 4Rg2i

g k qk

(τ ).

(39)

(41)

M →∞

M 1 1 X = lim M →∞ M 2π n=1

2πn 2πk N + PN

Z

cos(ωm τ cos γnk )4M dγnk 2πn−2π + 2πk N PN

= J0 (ωm τ ). Derivation of the auto-correlation function of the squared envelope is given below. We follow a procedure similar to the one outlined in [17]. R|gk |2 |gk |2 (τ ) (42) £ 2 ¤ £ 2 ¤ 2 2 = E gik (t)gik (t + τ ) + E gqk (t)gqk (t + τ ) £ 2 ¤ £ 2 ¤ 2 2 + E gik (t)gqk (t + τ ) + E gqk (t)gik (t + τ ) .

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, M 4 X b2n lim cos(ωm τ sin αnk ) N →∞ N 2 n=0

M M 4 XX E [cos βnk cos βik ] M n=1 i=1

The computation of the first term in the right-hand side of (42) is shown below   M M M M X X X X 16 R|gik |2 |gik |2 (τ ) = 2 E  ap bj cq dn  , M p=1 q=1 n=1 j=1 (43) where ap = cos(βpk ) cos(ωm (t + τ ) cos αpk + φpk ), bj = cos(βjk ) cos(ωm t cos αjk + φjk ), cq = cos(βqk ) cos(ωm (t + τ ) cos αqk +φqk ), and dn = cos(βnk ) cos(ωm t cos αnk +φnk ). The mutual independence of the φik ’s ensures that all terms in the above equation are zero, except the four terms with: 1) n = j = p = q; 2) n = j, p = q, j 6= p; 3) n = p, j = q, n 6= j; 4) n = q, j = p, n 6= j. We compute each of these terms individually to derive overall expression. Term 1: n = j = p = q =

16 M2

M P n=1

3 8E

£1

4 wn yn

¤

=

3 2M

+

3 4M J0

(2ωm τ ) ,

(44)

where wn = (1 + cos (2ωm t cos αnk + 2φnk )) and yn = (1 + cos (2ωm (t + τ ) cos αnk + 2φnk )).

10

Term 2: n = j, p = q, j 6= p =

16 M2

M X

M X

£ ¤ E cos2 (βjk ) cos2 (ωm t cos αjk + φjk )

j=1

£ ¤ E cos2 (βpk ) cos2 (ωm (t + τ ) cos αpk + φpk )

p=1,p6=j

=

M 16 X 1 1 · M 2 j=1 2 2

M X p=1,p6=j

1 1 M −1 · = , 2 2 M

(45)

Term 3: n = p, j = q, n 6= j =

M 1 X E [cos (ωm τ cos αnk )] M 2 n=1 M X

E [cos (ωm τ cos αjk )] =

j=1,j6=n

(46)

M −1 2 J0 (ωm τ ) . M

It can be shown that Term 4 is equal to Term 3. Adding all four terms gives £ 2 ¤ M −1 2 M −1 2 E gik (t)gik (t + τ ) = +2 J0 (ωm τ ) M M 3 3 + + J0 (2ωm τ ) . (47) 2M 4M Similarly can be shown that ¤ ¤ £ 2 £ 2 2 2 (t + τ ) , (48) (t)gik (t + τ ) = E gik (t)gqk E gqk £ 2 ¤ £ 2 ¤ 2 2 E gik (t)gqk (t + τ ) = E gqk (t)gik (t + τ ) (49) M −1 1 1 = + + J0 (2ωm τ ) . M 2M 4M Substituting the above terms in (42) and letting M → ∞, gives the desired expression (5). ACKNOWLEDGMENT

[10] Y. X. Li and X. Huang, “The simulation of independent Rayleigh faders,” IEEE Trans. on Communications, vol. 50, September 2002, pp. 1503– 1514. [11] M. 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. 254-269. [12] M. F. Pop and N. C. Beaulieu, “Limitations of sum-of-sinusoids fading channel simulators,” IEEE Trans. on Communications, vol. 49, April 2001, pp. 699–708. [13] C. Xiao, Y. R. Zheng and N. C. Beaulieu, “Second-order statistical properties of the WSS Jakes’ fading channel simulator,” IEEE Trans. on Communications, vol. 50, June 2002, pp. 888–891. [14] Y. R. Zheng and C. Xiao, “Simulation models with correct statistical properties for Rayleigh fading channels,” IEEE Trans. on Communications, vol. 51, June 2003, pp. 920–928. [15] Y. R. Zheng and C. Xiao, “Improved models for the generation of multiple uncorrelated Rayleigh fading waveforms,” Communications Letters, vol. 6, no. 6, June 2002, pp. 256–258. [16] Y. R. Zheng and C. Xiao, “A statistical simulation model for mobile radio fading channels,” Proc. IEEE WCNC’03, New Orleans, USA, March 2003, pp. 144–149. [17] C. S. Patel, G. L. Stuber, and T. G. Pratt, “Comparative analysis of statistical models for the simulation of Rayleigh faded cellular channels,” IEEE Trans. on Communications, vol. 53, June 2005, pp. 1017–1026. [18] G. L. Stuber, Principles of Mobile Communication, 2nd ed. Norwell, MA: Kluwer, 2001.

Alenka G. Zajić received the B.Sc. and M.Sc. degrees form the School of Electrical Engineering, University of Belgrade, in 2001 and 2003, respectively. From 2001 to 2003, she was a design engineer for Skyworks Solutions Inc., Fremont, CA. Since 2004, she has been a Graduate Research 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. Ms. Zajić was recipient of the Dan Noble Fellowship in 2004, awarded by Motorola Inc. and IEEE Vehicular Technology Society for quality impact in the area of vehicular technology.

The authors would like to thank the anonymous reviewers whose feedback helped improve the quality of this paper. R EFERENCES [1] D. J. Young and N. C. Beaulieu, “A quantitative evaluation of generation methods for correlated Rayleigh random variates,” Proc. GLOBECOM ’98, Sydney, Australia, November, 1998, pp. 3332–3337. [2] D. J. Young and N. C. Beaulieu, “The generation of correlated Rayleigh random variates by inverse discrete Fourier transform,” IEEE Trans. on Communications, vol. 48, July 2000, pp. 1114–1127. [3] D. Verdin and T. C. Tozer, “Generating a fading process for the simulation of land-mobile radio communications,” Electronics Letters, vol. 29, no.23, November 1993, pp. 2011–2012. [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. 422–431. [5] R. H. Clarke, “A statistical theory of mobile-radio reception,” Bell Syst. Tech. J., pp. 957–1000, July–Aug. 1968. [6] W. C. Jakes, Microwave Mobile Communications, 2nd ed. Piscataway, NJ: Wiley-IEEE Press, 1994. [7] M. Pätzold and F. Laue,“ Statistical properties of Jakes’ fading channel simulator,” Proc. IEEE VTC’98, Ottawa, Canada, May 1998, pp. 712718. [8] P. Dent, G. E. Bottomley and T. Croft, “Jakes fading model revisited,” Electronics Letters, vol. 29, no. 13, June 1993, pp. 1162–1163. [9] Y. B. Li and Y. L Guan, “Modified Jakes model for simulating multiple uncorrelated fading waveforms,” Proc. IEEE ICC’00, New Orleans, USA, June 2000, pp. 46–49.

¨ Gordon L. Stuber received the B.A.Sc. and Ph.D. degrees in Electrical Engineering from the University of Waterloo, Ontario, Canada, in 1982 and 1986 respectively. In 1986, he joined the School of Electrical and Computer Engineering, Georgia Institute of Technology, where is the Joseph M. Pettit Chair Professor. Dr. Stüber was co-recipient of the IEEE Vehicular Technology Society Jack Neubauer Memorial 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 R. Evans Avant Garde Award in 2003 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, MMT”00, CTW’02, and WPMC’02. He is a past Editor for IEEE Transactions on Communications (1993-1998), and served on the IEEE Communications Society Awards Committee (19992002). He an elected member of the IEEE Vehicular Technology Society Board of Governors (2001-2003, 2004-2006) and received the Outstanding Service Award from the IEEE Vehicular Technology Society.