Abstract
Abstract-This paper deals with the design of a set of multiple uncorrelated Rayleigh fading waveforms. The Rayleigh fading waveforms are mutually uncorrelated, but each waveform is correlated in time. The waveforms are generated by using the deterministic sum-of-sinusoids (SOS) channel modeling principle. Two new closed-form solutions are presented for the computation of the model parameters. Analytical and numerical results show that the resulting deterministic SOS-based channel simulator fulfills all main requirements imposed by the reference model with given correlation properties derived under two-dimensional isotropic scattering conditions. The proposed methods are useful for the design of simulation models for diversity-combined Rayleigh fading channels, relay fading channels, frequencyselective channels, and multiple-input multiple-output (MIMO) channels.
Abstract
Index Terms-Uncorrelated Rayleigh fading waveforms, deterministic sum-of-sinusoids channel simulator, parameter computation method, statistics.

been investigated. In general, the SOS channel simulators can be classified as deterministic [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] , ergodic stochastic, or non-ergodic stochastic [12] [13] [14] [15] [16] depending on the underlying parameter computation methods. A deterministic SOS channel simulator has constant model parameters (gains, frequencies, and phases) for all simulation trials. An ergodic stochastic SOS channel simulator has constant gains and frequencies but random phases [17] . Due to the ergodic property, it needs only one simulation trial to represent its complete statistical properties. A sample function, i.e., a single simulation trial of a stochastic SOS channel simulator actually results in a deterministic process (waveform). Thus, we can also say that a deterministic channel simulator can be used to generate sample functions of a stochastic process. A non-ergodic stochastic SOS channel simulator assumes that the frequencies and/or gains are random variables. The statistical properties of a non-ergodic stochastic SOS channel simulator vary for each simulation trial and have to be calculated by averaging over a large number of simulation trials. Both ergodic stochastic (deterministic) and non-ergodic stochastic SOS channel simulators have pros and cons, which have been discussed, e.g., in [11] , [12] . Deterministic SOS channel simulators generally have a higher efficiency compared to non-ergodic stochastic SOS channel simulators [18] .

Although Jakes' deterministic SOS channel simulator [3] is widely in use, it has some undesirable properties. One of them comes from the non-zero cross-correlation function (CCF) of the inphase and quadrature components of the generated complex waveforms. In [3] , Jakes proposed also an extension of his approach aiming to generate K multiple uncorrelated waveforms, but it was shown in [4] that the CCF between any pair of generated complex waveforms can be quite large. Dent et al. [5] suggested a modification to Jakes' method by using orthogonal Walsh-Hadamard matrices to decorrelate the generated waveforms. This reduces the CCFs but they are still not exactly zero. The same problem of non-zero CCFs between different waveforms is retained for the deterministic method proposed in [6] . Another deterministic method that enables the generation of a set of K mutually uncorrelated Rayleigh fading waveforms was introduced in [7] . Using this method, the temporal autocorrelation function (ACF) of each of the K underlying complex waveforms is very close to the specified one. Unfortunately, this is not the case for the ACFs of the inphase and quadrature parts of the designed complex waveforms. In [12] , both a deterministic and a stochastic 1536-1276/09$25.00 c 2009 IEEE method were suggested aiming to tackle the problem of designing multiple uncorrelated Rayleigh fading waveforms. However, when applying the deterministic method, the ACFs of the inphase and quadrature parts of the generated complex waveforms are quite different from the corresponding ACFs of the reference model-even if the number of sinusoidal terms tends to infinity-and the proposed stochastic method results in a non-ergodic channel simulator. The L p -norm method [8] [9] [10] is very powerful and not limited to isotropic channels, but it lacks a simple closed-form solution and requires professional experience in numerical optimization techniques to achieve the expected results. The usefulness of the method of exact Doppler spread (MEDS) [19] concerning the generation of multiple uncorrelated Rayleigh fading processes with a deterministic SOS channel simulator was revisited in [9] , [10] . There it was shown that all the main requirements can be fulfilled, but unfortunately the complexity of the resulting channel simulator increases almost exponentially with the increase of the number of uncorrelated waveforms. This makes the original MEDS less efficient if the number of uncorrelated waveforms is large.

Non-ergodic stochastic methods, such as those proposed in [12] [13] [14] [15] [16] , can be used to guarantee that the CCFs of different waveforms are zero, but the temporal ACF of the waveform obtained from a single simulation trial is generally not sufficiently close to the desired ACF of the reference model. This problem can only be solved by averaging over many simulation trials, which reduces the efficiency of the approach. Considering all the pros and cons of the existing methods in [3] [4] [5] [6] [7] [8] [9] [10] , [12] [13] [14] [15] [16] that have been proposed by many researchers over several decades, one must come to the conclusion that a better solution to the problem of designing multiple uncorrelated Rayleigh fading waveforms with a better tradeoff between the model accuracy and simulation efficiency is still desirable.

In this paper, we present two deterministic solutions of the problem. For the first time, we introduce a generalized version of the MEDS. This generalized version can be interpreted as a class of parameter computation methods, which includes many other well-known approaches as special cases. Two new special cases are introduced here, each of which enables the efficient and accurate design of multiple uncorrelated Rayleigh fading waveforms using deterministic concepts. Our proposed methods can fulfill all main requirements imposed on the correlation properties of the resulting channel simulator. Also, they keep the model complexity low and provide simple closed-form solutions for the computation of the model parameters.

The rest of this paper is organized as follows. Section II describes the problem and the conditions that must be fulfilled to obtain K mutually uncorrelated Rayleigh fading waveforms using an SOS channel simulator. In Section III, a new class of parameter computation methods is introduced, which then provides two closed-form solutions of the problem under the conditions of isotropic scattering. In Section IV, it is shown that the new class of parameter computation methods includes several other well-known methods as special cases. Section V provides a comparison and discussion of the proposed new methods. Finally, the conclusions are drawn in Section VI.
II. PROBLEM DESCRIPTION
We want to simulate K mutually uncorrelated Rayleigh fading waveforms
II. PROBLEM DESCRIPTION
by using an SOS channel simulator, which generates the waveforms
II. PROBLEM DESCRIPTION
where j = √ −1, N i denotes the number of sinusoids, f 
The phases θ (k)
i,n are considered as outcomes of independent and identically distributed (i.i.d.) random variables θ (k) i,n , each having a uniform distribution over the interval (0, 2π]. For increased clarity, the structure of the SOS channel simulator is shown in Fig. 1 .
The phases θ (k)
For given sets of constant model parameters {f
The phases θ (k)
for i = 1, 2 and k = 1, 2, . . . , K. Analogously, the timeaveraged CCFr
The phases θ (k)
λ (t) can be obtained as in (4) [8] , [9] for i, λ = 1, 2 and k, l = 1, 2, . . . , K. Thẽ
The phases θ (k)
. . .
The phases θ (k)
cos(2πf
The phases θ (k)
Structure of the SOS channel simulator for Rayleigh fading waveforms.
The phases θ (k)
problem is to find proper values for the discrete Doppler frequencies f (k) i,n in such a way that the following two conditions are fulfilled:
The phases θ (k)
(i) The ACFr
The phases θ (k)
μiμi (τ ) of the simulation model must be as close as possible to the ACF r μiμi (τ ) of a given reference model over a certain domain, i.e.,
The phases θ (k)
where τ max denotes the maximum time lag that determines the domain over which the quality of the approximation is of interest. To measure the quality of the approximationr
The phases θ (k)
we use the following L 2 -norm
The phases θ (k)
(ii) The CCFsr
The phases θ (k)
λ (t) must be equal to zero, i.e.,r
The phases θ (k)
where i, λ = 1, 2 and k, l = 1, 2, . . . , K.
The phases θ (k)
It is worth noting that the above conditions should be fulfilled by keeping the complexity of the simulation model to a minimum, which means that N i must be as small as possible.
The phases θ (k)
By considering the CCFr (4), we may conclude that (7a) and (7b) can be guaranteed if and only if
The phases θ (k)
hold, respectively, for all n = 1, 2, . . . , N i and m = 1, 2, . . . , N λ . The above two equations state that the sets of discrete Doppler frequencies of different waveformsμ
The phases θ (k)
where {∅} denotes the empty set.
The phases θ (k)
Since it follows from (7a) that the CCFsr
The phases θ (k)
μ2μ1 (τ ) are zero, we can express the ACFr
The phases θ (k)
In this paper, we will restrict our reference model to the Rayleigh model under two-dimensional isotropic scattering conditions [3] , [20] . Recall that the ACF r μi μi (τ ) of the reference model is given by
The phases θ (k)
where J 0 (·) denotes the zeroth-order Bessel function of the first kind and f max is the maximum Doppler frequency. In this case, the ACF of the complex process μ(t) = μ 1 (t) + jμ 2 (t) equals r μμ (τ ) = 2r μiμi (τ ) = 2J 0 (2πf max τ ), since the CCFs r μ1μ2 (τ ) and r μ2μ1 (τ ) are zero.
III. PROBLEM SOLUTIONS
To solve the problem described in Section II, we propose the generalized MEDS (GMEDS q ). According to this method, the discrete Doppler frequencies f
III. PROBLEM SOLUTIONS
where α
III. PROBLEM SOLUTIONS
i,0 is called the angle of rotation that will be defined subsequently and q ∈ {0, 1, 2}. Note that the quantity q mainly determines the range of values for the angles of arrival α (k) i,n . Empirical studies have shown that for the GMEDS q , the quantity τ max in (6) is given by τ max = N i /(2qf max ) when q = 1, 2. From (11), it is clear that the GMEDS q reduces to the original MEDS if q = 1 and α
III. PROBLEM SOLUTIONS
A. The GMEDS 1 In the following, we show how the GMEDS 1 can be used to generate any number K of multiple uncorrelated Rayleigh fading waveforms. According to the GMEDS 1 , the discrete Doppler frequencies f (k) i,n are obtained from (11) by setting q to 1 and defining the angle of rotation α
III. PROBLEM SOLUTIONS
where i = 1, 2 and k = 1, 2, . . . , K. It follows an analysis of the GMEDS 1 with some background information that clarifies the motivation for introducing (12) . For this purpose, we start from (11) with q = 1 and substitute (3) and (10) in (6), which allows us to study the influence of the angle of rotation α (k)
III. PROBLEM SOLUTIONS
i,0 on the quality of the approximation in (5) . The behavior of the error function Fig. 2 for various values of N i . Clearly, with an increase of
III. PROBLEM SOLUTIONS
It can also be observed that the error function E 2 (α i,0 , so that (8a) and (8b) can be satisfied.
III. PROBLEM SOLUTIONS
Since 1 ≤ n ≤ N i , we may conclude that
III. PROBLEM SOLUTIONS
. Within this range, it follows from (11) that the discrete Doppler frequencies f (k) i,n are monotonically decreasing quantities over the interval [0, f max ] with increasing values of n, i.e., 0 ≤ f
III. PROBLEM SOLUTIONS
For simplicity, let N 1 = N 2 . Then, (8a) and (8b) can always be fulfilled if and only if
III. PROBLEM SOLUTIONS
hold, respectively, for all n, m = 1, 2, . . . , N i , where k, l = 1, 2, . . . , K and i, λ = 1, 2. The above conditions are sufficient to guarantee that the CCFsr
III. PROBLEM SOLUTIONS
λ (t) are equal to 0, i.e., (7a) and (7b) will follow.
III. PROBLEM SOLUTIONS
There are a number of ways to satisfy the conditions (13a) and (13b) and to design a set of K mutually uncorrelated Rayleigh fading waveforms. In this paper, we define the angle of rotation α (k) i,0 as in (12) . Note that α
III. PROBLEM SOLUTIONS
For the GMEDS 1 , the worst approximation result of the ACFr
III. PROBLEM SOLUTIONS
, no matter of the value chosen for K. In contrast to the MEDS, the introduction of the angle of rotation α (k) i,0 in (11) guarantees that the conditions in (8a) and (8b) can be satisfied without choosing different values for N i . In this sense, the GMEDS 1 removes the constraint on N i . Consequently, a very large number (theoretically infinite) K of mutually uncorrelated Rayleigh fading waveformsζ (k) (t) (k = 1, 2, . . . , K) can be designed by using (12) without increasing the model complexity determined by N i .
III. PROBLEM SOLUTIONS
Following the above discussion, the inequality (11) is always smaller for the GMEDS 1 than the corresponding f Fig. 2 . Consequently, we obtain the following important properties for the ACFr
III. PROBLEM SOLUTIONS
. These results can be phrased as follows. A positive (or negative) angle of rotation α
III. PROBLEM SOLUTIONS
always results in non-negative (or non-positive) errors of the ACFr The above property is confirmed by the results shown in Fig. 3 , where plots of the ACFsr i,0 according to (12) with N i = 20, and k = K = 4 are presented. For comparison, this figure also illustrates the behavior of the ACF r μiμi (τ ) of the reference model. Obviously, the best fitting is obtained when the original MEDS is used. In this case, the approximatioñ r Fig. 3 shows the worst approximation result of the ACFr i,0 = ±π/120, which follows from (12) if k = K. For k < K, the approximation results are much better, which is also clear from Fig. 2 . A comparative study has revealed that even the worst case of the GMEDS 1 provides better approximation results regarding the ACF r μiμi (τ ) than many other methods, including Jakes' method in [3] , the deterministic methods in [5] [6] [7] , and the random methods in [12] [13] [14] [15] with respect to single trials. From (12) and (14), we can further conclude that the inphase ACFr μ2μ2 (τ ), respectively, compensate each other in such a way that the approximation ofr
III. PROBLEM SOLUTIONS
. Similar to (6), the quality of the approximationr according to (12) and Fig. 4 . The even symmetry property is self-evident. The ACFr i,0 = 0) and the ACF r μμ (τ ) for the reference model are also shown for reasons of comparison. Notice that the performance of the MEDS is worse than the performance of the GMEDS 1 in the worst case in terms of the fitting to r μμ (τ ). However, even in the worst case of the GMEDS 1 , the approximation to r 
B. The GMEDS 2
In the following, we present another solution to the problem described in Section II by using the GMEDS 2 . As in the previous subsection, the reference model is again an ideal Rayleigh channel model under isotropic scattering conditions. By using the GMEDS 2 , we can design a set of K mutually uncorrelated Rayleigh fading waveformsζ (k) (t) if the angle of rotation α
B. The GMEDS 2
for k = 1, 2, . . . , K and i = 1, 2, where α per = π/(2N i ) and
B. The GMEDS 2
In the following, we analyze the GMEDS 2 and give some reasons for choosing (16) . First, we choose q = 2 in (11) and study the influence of α i,0 ) when using the GMEDS 1 . Moreover, the even symmetry property is kept for the error function when using the GMEDS 2 , i.e., E 2 (α i,0 = 0, it turns out that the GMEDS 2 with N i terms has exactly the same performance as the original MEDS with N i /2 terms. However, the original MEDS is not suitable for designing efficiently a large number K of uncorrelated waveforms, but the GMEDS 2 is, as we will see in the following.
B. The GMEDS 2
Choosing α (k)
B. The GMEDS 2
i,0 as in (16) ensures that the conditions in (8a) and (8b) are fulfilled, so that (7a) and (7b) follow, respectively. An interesting property of the GMEDS 2 is that the approximation in (5) i,0 = 0. This is consistent with the conclusions drawn from the results shown in Fig. 6 . In Fig. 8 , we show the ACF r μμ (τ ) of the reference model and the ACFr i,0 as in Fig. 7. Finally, Fig. 9 illustrates exemplarily for K = 3 the temporal behavior of the resulting three uncorrelated fading envelopesζ (k) (t) (k = 1, 2, 3).
IV. RELATED PARAMETER COMPUTATION METHODS
DERIVED FROM THE GMEDS q The GMEDS q actually represents a class of parameter computation methods, which includes the following five well- known methods as special cases.
1) MEDS:
The original MEDS [19] results from (11) if q = 1 and α
1) MEDS:
in order to generate multiple uncorrelated processes [9] , [10] . The discrete frequencies are therefore given by f
1) MEDS:
. It was shown in [9] , [10] that the only way to fulfill (8a) and (8b) with the MEDS is to guarantee that N {8, 9, 10, 12, 16, 32, 64, 128} [10] . This obvious drawback limits the usefulness of the MEDS if a large number of uncorrelated processes is required.
2) MEDS with Set Partitioning (MEDS-SP):
, which results in the so-called MEDS-SP [18] . This method actually belongs to the GMEDS 1 , but with a different expression for α (k) i,0 . The purpose of the MEDS-SP is to take advantage of averaging over multiple sample functions (trials) -a technique that is unavoidable when non-ergodic stochastic methods [12] [13] [14] [15] are used. In [18] , it was shown that the MEDS-SP outperforms the non-ergodic method proposed in [14] with respect to both single trials and multiple trials. The use of the MEDS-SP, without further modifications, is not recommended for the design of multiple uncorrelated Rayleigh fading waveforms.
3) Method of Equal Areas (MEA):
The MEA [21] ensues if q = 1 and α
3) Method of Equal Areas (MEA):
Note that the MEA was originally proposed for the design of a single Rayleigh fading waveform. To generate multiple uncorrelated Rayleigh fading waveforms by using the MEA, N i in (11) has to be replaced by N (k) i . Similar to the MEDS, the only way to fulfill (8a) and (8b) with the MEA is to change the number of sinusoids for different waveforms, which will greatly increase the simulator complexity if a large number of uncorrelated waveforms is required.
4) Randomized MEDS (R-MEDS):
Furthermore, the R-MEDS [14] is obtained if q = 1 and α
4) Randomized MEDS (R-MEDS):
are i.i.d. random variables, each having a uniform distribution over [−π, π). The R-MEDS and the following MCM can be used to generate a large number of uncorrelated Rayleigh fading waveforms without increasing the simulator's complexity. However, due to the non-ergodicity property of the methods, the resulting simulators are not efficient, as already discussed in [10] .
5) Monte Carlo Method (MCM):
Finally, if q = 0 and α
5) Monte Carlo Method (MCM):
i,n being i.i.d. random variables uniformly distributed over (0, π/2], then we obtain the MCM, which was originally proposed in [22] and further developed in [13] .
5) Monte Carlo Method (MCM):
V. COMPARISON AND DISCUSSION OF THE GMEDS 1 AND THE GMEDS 2 Compared with the MEDS, one obvious advantage of both the GMEDS 1 and the GMEDS 2 is that they can be used to design a very large (theoretically an infinite) number of mutually uncorrelated Rayleigh fading waveforms by keeping the number of sinusoids constant. In fact, the complexity of the resulting channel simulators designed using the proposed two methods is low and independent of the number of generated uncorrelated fading waveforms. The drawback of the GMEDS 1 is that small non-negative (non-positive) errors have to be accepted concerning the fitting of the ACF of the inphase (quadrature) component of the generated complex waveforms. However, these errors compensate each other over the domain of interest when considering the ACF of the resulting complex waveform. Note that the performance of many communication systems, such as differential phase shift keying (DPSK) modulation [3] and orthogonal frequency division multiplexing (OFDM) systems [4] , depends only on the ACF of the complex waveform, rather than the ACFs of the quadrature components. Therefore, the GMEDS 1 is applicable to such systems.
5) Monte Carlo Method (MCM):
At the price of increased complexity, the GMEDS 2 enables an excellent fitting to both the quadrature ACFs and the ACF of the complex waveform over the domain of interest. Note that for the GMEDS 2 , the domain of interest is only one half of that for the GMEDS 1 . However, this does not diminish the usefulness of the resulting channel simulator because the performance of most mobile communication systems is only sensitive to errors of the ACF if the time lag is small, meaning τf max ≤ 0.3 [23] . For such systems, the GMEDS 2 should be preferred to the GMEDS 1 .
5) Monte Carlo Method (MCM):
The results of a detailed comparative performance study of the GMEDS 1 and the GMEDS 2 are shown in Tables I and II.  Table I presents the values of the error function E 2 (α Table I , we can clearly conclude that the GMEDS 2 outperforms the GMEDS 1 with respect to the fitting of the ACF of the inphase (quadrature) component if τ max is set to N i /(4f max ). If τ max = N i /(2f max ), then it turns out that the GMEDS 1 performs better than the GMEDS 2 except for the case when k = K. Table II shows that the GMEDS 1 in general outperforms the GMEDS 2 with respect to the error function E 2 (α (k) i,0 ), which measures the accuracy of the channel simulator's ACF of the complex generated waveform.
5) Monte Carlo Method (MCM):
In addition, we have compared the performance of the GMEDS 1 and GMEDS 2 with the modified MEDS (MMEDS), which has recently been proposed in [11] . According to the results shown in Tables I and II, we can conclude that the MMEDS is in general better than the GMEDS 1 and GMEDS 2 in terms of the fitting of the ACF of the inphase (quadrature) component for both τ max = N i /(2f max ) and τ max = N i /(4f max ). For the fitting of the ACF of the complex waveform, both the GMEDS 1 and MMEDS are in general superior to the GMEDS 2 , while the GMEDS 1 has a slightly better performance than the MMEDS.
5) Monte Carlo Method (MCM):
Next, we study the performance of the proposed methods under limited simulation time constraints. That is, we assume that the simulation ofμ λ (t) are correlated even when the unequalities in (8a) and (8b) are fulfilled. To study the effect of limited simulation time, we define the following correlation coefficient
5) Monte Carlo Method (MCM):
Notice that the correlation coefficientc
5) Monte Carlo Method (MCM):
Obviously, the quantityc
5) Monte Carlo Method (MCM):
μiμ λ depends on all model parameters, including the phases θ (k) i,n and T . For our purpose, it is sufficient to focus on the upper limit ofc 
5) Monte Carlo Method (MCM):
It is apparent thatĉ
5) Monte Carlo Method (MCM):
In practise, however, the simulation time T sim = 2T is limited, and thus the theoretical uncorrelated processesμ Table III for the GMEDS 1 , GMEDS 2 , and MMEDS. The results clearly show that under practical aspects the GMEDS 1 and GMEDS 2 are superior to the MMEDS, as the latter method results in deterministic processesμ i,n tightly around the optimum (original MEDS [19] ) for k = 1, 2, . . . , K. This keeps the frequency differences in the denominator of (19) very small and results thus in large values forĉ (k,l) μiμ λ . This drawback is avoided by the GMEDS 1 and GMEDS 2 , where the spread of the discrete Doppler frequencies f (k) i,n is much larger. Finally, we mention that for both the GMEDS 1 and GMEDS 2 , the domain of interest increases linearly with the number of sinusoids and can thus easily be controlled. It should be highlighted that both methods result in closed-form expressions [see (11) in combination with (12) and (16)] for the model parameters, which have been derived in our paper by assuming isotropic scattering conditions. The extension of the proposed techniques to wireless non-isotropic channels might be a topic for future research. For some applications, it is desirable to generate multiple fading envelopes with specified cross-correlation functions. This is important, e.g., for studying the effects caused by correlated sub-channels on the capacity of MIMO systems. One straightforward method is to employ a linear combination of multiple uncorrelated processes [4] , [24] . In line with the idea in [24] , the deterministic SOS channel simulator with the newly developed methods can be extended to generate multiple cross-correlated Rayleigh fading waveforms for the simulation of more realistic MIMO channels. Finally, it should be mentioned that a detailed comparison of the MEDS with the L p -norm method and various random methods in [14] [15] [16] can be found in [10] . (N 1 = N 2 = 20), THE GMEDS 2 (N 1 = 20, N 2 = 21) , AND THE MMEDS [11] (N 1 = N 2 = 20) WITH K = 3 AND fmax = 91 HZ. 

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