EURASIP Journal on Wireless Communications and Networking 2005:5, 801–815 c ○ 2005 Le Chung Tran et al. A Generalized Algorithm for the Generation of Correlated Rayleigh Fading Envelopes in Wireless Channels

Although generation of correlated Rayleigh fading envelopes has been intensively considered in the literature, all conventional methods have their own shortcomings, which seriously impede their applicability. A very general, straightforward algorithm for the generation of an arbitrary number of Rayleigh envelopes with any desired , equal or unequal power, in wireless channels either with or without Doppler frequency shifts, is proposed. The proposed algorithm can be applied to the case of spatial correlation , such as with multiple antennas in multiple-input multiple-output (MIMO) systems, or spectral correlation between the random processes like in orthogonal frequency-division multiplexing (OFDM) systems. It can also be used for generating correlated Rayleigh fading envelopes in either discrete-time instants or a real-time scenario . Besides being more generalized , our proposed algorithm is more precise , while overcoming all shortcomings of the conventional methods.


INTRODUCTION
In orthogonal frequency-division multiplexing (OFDM) systems, the fading affecting carriers may have cross-correlation due to the small coherence bandwidth of the channel, or due to the inadequate frequency separation between the carriers. In addition, in multiple-input multiple-output (MIMO) systems where multiple antennas are used to transmit and/or receive signals, the fading affecting these antennas may also experience cross-correlation due to the inadequate separation between the antennas. Therefore, a generalized, straightforward and, certainly, correct algorithm to generate correlated Rayleigh fading envelopes is required for the researchers wishing to analyze theoretically and simulate the performance of systems.
Because of that, generation of correlated Rayleigh fading envelopes has been intensively mentioned in the literature, such as [1,2,3,4,5,6,7,8,9,10,11,12,13]. However, besides not being adequately generalized to be able to apply to various scenarios, all conventional methods have their own shortcomings which seriously limit their applicability or even cause failures in generating the desired Rayleigh fading envelopes.
In this paper, we modify existing methods and propose a generalized algorithm for generating correlated Rayleigh fading envelopes. Our modifications are simple, but important and also very efficient. The proposed algorithm thus incorporates the advantages of the existing methods, while overcoming all of their shortcomings. Furthermore, besides being more generalized, the proposed algorithm is more accurate, while providing more useful features than the conventional methods.
The paper is organized as follows. In Section 2, a summary of the shortcomings of conventional methods for generating correlated Rayleigh fading envelopes is derived. In Sections 3.1 and 3.2, we shortly review the discussions on the correlation property between the transmitted signals as functions of time delay and frequency separation, such as in OFDM systems, and as functions of spatial separation between transmission antennas, such as in MIMO systems, respectively. In Section 4, we propose a very general, straightforward algorithm to generate correlated Rayleigh fading envelopes. Section 5 derives an algorithm to generate correlated Rayleigh fading envelopes in a real-time scenario. Simulation results are presented in Section 6. The paper is concluded by Section 7.

SHORTCOMINGS OF CONVENTIONAL METHODS AND AIMS OF THE PROPOSED ALGORITHM
We first analyze the shortcomings of some conventional methods for the generation of correlated Rayleigh fading envelopes.
In [3], the authors derived fading correlation properties in antenna arrays and, then, briefly mentioned the algorithm to generate complex Gaussian random variables (with Rayleigh envelopes) corresponding to a desired correlation coefficient matrix. This algorithm was proposed for generating equal power Rayleigh envelopes only, rather than arbitrary (equal or unequal) power Rayleigh envelopes.
In [4,5], the authors proposed different methods for generating only N = 2 equal power correlated Rayleigh envelopes. In [6], the authors generalized the method of [5] for N ≥ 2. However, in this method, Cholesky decomposition [7] is used, and consequently, the covariance matrix must be positive definite, which is not always realistic. An example, where the covariance matrix is not positive definite, is derived later in Example 1 of Section 4.1 of this paper.
These methods were then more generalized in [8], where one can generate any number of Rayleigh envelopes corresponding to a desired covariance matrix and with any power, that is, even with unequal power. However, again, the covariance matrix must be positive definite in order for Cholesky decomposition to be performable. In addition, the authors in [8] forced the covariances of the complex Gaussian random variables (with Rayleigh fading envelopes) to be real (see [8, (8)]). This limitation prohibits the use of their method in various cases because, in fact, the covariances of the complex Gaussian random variables are more likely to be complex.
In [2], the authors proposed a method for generating any number of Rayleigh envelopes with equal power only. Although the method of [2] works well in various cases, it fails to perform Cholesky decomposition for some complex covariance matrices in Matlab due to the roundoff errors of Matlab. 1 This shortcoming is overcome by some modifications mentioned later in our proposed algorithm.
More importantly, the method proposed in [2] fails to generate Rayleigh fading envelopes corresponding to a desired covariance matrix in a real-time scenario where Doppler frequency shifts are considered. This is because passing Gaussian random variables with variances assumed to be equal to one (for simplicity of explanation) through a Doppler filter changes remarkably the variances of those variables. The variances of the variables at the outputs of Doppler filters are not equal to one any more, but depend on the variance of the variables at the inputs of the filters as well as the characteristics of those filters. The authors in [2] did not realize this variance-changing effect caused by Doppler filters. We will return to this issue later in this paper.
For the aforementioned reasons, a more generalized algorithm is required to generate any number of Rayleigh fading envelopes with any power (equal or unequal power) corresponding to any desired covariance matrix. The algorithm should be applicable to both discrete time instant scenario and real-time scenario. The algorithm is also expected to overcome roundoff errors which may cause the interruption of Matlab programs. In addition, the algorithm should work well, regardless of the positive definiteness of the covariance matrices. Furthermore, the algorithm should provide a straightforward method for the generation of complex Gaussian random variables (with Rayleigh envelopes) with correlation properties as functions of time delay and frequency separation (such as in OFDM systems), or spatial separation between transmission antennas (like with multiple antennas in MIMO systems). This paper proposes such an algorithm.

BRIEF REVIEW OF STUDIES ON FADING CORRELATION CHARACTERISTICS
In this section, we shortly review the discussions on the correlation property between the transmitted signals as func-1 It has been well known that Cholesky decomposition may not work for the matrix having eigenvalues being equal or close to zeros. We consider the following covariance matrix K, for instance: tions of time delay and frequency separation, such as in OFDM systems, and as functions of spatial separation between transmission antennas, such as in MIMO systems. These discussions were originally derived in [3,9], respectively. This review aims at facilitating readers to apply our proposed algorithm in different scenarios (i.e., spectral correlation, such as in OFDM systems, or spatial correlation, such as in MIMO systems) as well as pointing out the condition for the analyses in [3,9] to be applicable to our proposed algorithm (i.e., these analyses are applicable to our algorithm if the powers (variances) of different random processes are assumed to be the same).

Fading correlation as functions of time delay and frequency separation
In [9], Jakes considered the scenario where all complex Gaussian random processes with Rayleigh envelopes have equal powers σ 2 and derived the correlation properties between random processes as functions of both time delay and frequency separation, such as in OFDM systems. Let z k (t) and z j (t) be the two zero-mean complex Gaussian random processes at time instant t, corresponding to frequencies f k and f j , respectively. Denote where τ k, j is the arrival time delay between two signals and Re(·), Im(·) are the real and imaginary parts of the argument, respectively. By definition, the covariances between the real and imaginary parts of z k (t) and z j (t + τ k, j ) are R xxk,j E x k x j , R yy k, j E y k y j , Then, those covariances have been derived in [9, (1.5-20)] as where σ 2 is the variance (power) of the complex Gaussian random processes (σ 2 /2 is the variance per dimension); J 0 is the first-kind Bessel function of the zeroth-order; F m is the maximum Doppler frequency F m = v/λ = v f c /c. In this formula, λ is the wavelength of the carrier, f c is the carrier frequency, c is the speed of light, and v is the mobile speed; ∆ω k, j = 2π( f k − f j ) is the angular frequency separation between the two complex Gaussian processes with Rayleigh envelopes at frequencies f k and f j ; σ τ is the rootmean-square (rms) delay spread of the wireless channel. It should be emphasized that, the equalities (3) hold only when the set of multipath channel coefficients, which were denoted as C nm and derived in [9, (1.5-1) and (1.5-2)], as well as the powers are assumed to be the same for different random processes (with different frequencies). Readers may refer to [9, pages 46-49] for an explicit exposition.

Fading correlation as functions of spatial separation in antenna arrays
The fading correlation properties between wireless channels as functions of antenna spacing in multiple antenna systems have been mentioned in [3]. Figure 1 presents a typical model of the channel where all signals from a receiver are assumed to arrive at T x antennas within ±∆ at angle Φ (|Φ| ≤ π). Let λ be the wavelength, D the distance between the two adjacent transmitter antennas, and z = 2π(D/λ). In [3], it is assumed that fading corresponding to different receivers is independent. This is reasonable if receivers are not on top of each other within some wavelengths and they are surrounded by their own scatterers. Consequently, we only need to calculate the correlation properties for a typical receiver. The fading in the channel between a given kth transmitter antenna and the receiver may be considered as a zero-mean, complex Gaussian random variable, which is presented as b (k) = x (k) + iy (k) . Denote the covariances between the real parts as well as the imaginary parts themselves of the fading corresponding to the kth and jth transmitter antennas 2 to be R xxk,j and R yy k, j , while those terms between the real and imaginary parts of the fading to be R xy k, j and R yx k, j . The terms R xxk,j , R yy k, j , R xy k, j , and R yx k, j are similarly defined as (2). Then, it has been proved that the closed-form expressions of these covariances normalized by the variance per dimension (real and imaginary) are (see [3, (A. 19) and (A. 20)]) whereR k, j = 2R k, j /σ 2 . In other words, we have In these equations, J q is the first-kind Bessel function of the integer order q, and σ 2 /2 is the variance per dimension of the received signal at each transmitter antenna, that is, it is assumed in [3] that the signals corresponding to different transmitter antennas have equal variances σ 2 . Similarly to Section 3.1, the equalities (4) and (5) hold only when the set of multipath channel coefficients, which were denoted as g n and derived in [3, (A-1)], and the powers are assumed to be the same for different random processes. Readers may refer to [3, pages 1054-1056] for an explicit exposition.

Covariance matrix of complex Gaussian random variables with Rayleigh fading envelopes
It is known that Rayleigh fading envelopes can be generated from zero-mean, complex Gaussian random variables. We consider here a column vector Z of N zero-mean, complex Gaussian random variables with variances (or powers) The modulus of z j is r j = x 2 j + y 2 j . It is assumed that the phases θ j 's are independent, identically uniformly distributed random variables. As a result, the real and imaginary parts of each z j are independent (but z j 's are not necessarily independent), that is, the covariances E(x j y j ) = 0 for for all j and therefore, r j 's are Rayleigh envelopes.
Let σ 2 gxj and σ 2 gyj be the variances per dimension (real and imaginary), that is, σ 2 gxj =E(x 2 j ), σ 2 g y j =E(y 2 j ). Clearly, σ 2 gyj , then σ 2 gxj =σ 2 gyj = σ 2 gj /2. Note that we consider a very general scenario where the variances (powers) of the real parts are not necessarily equal to those of the imaginary parts. Also, the powers of Rayleigh envelopes denoted as σ 2 rj are not necessarily equal to one another. Therefore, the scenario where the variances of the Rayleigh envelopes are equal to one another and the powers of real parts are equal to those of imaginary parts, such as the scenario mentioned in either Section 3.1 or Section 3.2, is considered as a particular case.
For k = j, we define the covariances R xxk,j , R yy k, j , R xy k, j , and R yx k, j between the real as well as imaginary parts of z k and z j , similarly to those mentioned in (2).
By definition, the covariance matrix K of Z is where (·) H denotes the Hermitian transposition operation and In reality, the covariance matrix K is not always positive semidefinite. An example where the covariance matrix K is not positive semidefinite is derived as follows.
Example 1. We examine an antenna array comprising 3 transmitter antennas. Let D k j , for k, j = 1, . . . , 3, be the distance between the kth antenna and the jth antenna. The distance D jk between jth antenna and the kth antenna is then D jk = −D k j . Specifically, we consider the case D 21 = 0.0385λ, where λ is the wavelength. Clearly, these antennas are neither equally spaced, nor positioned in a straight line. Instead, they are positioned at the 3 peaks of a triangle.
If the receiver antenna is far enough from the transmitter antennas, we can assume that all signals from the receiver arrive at the transmitter antennas within ±∆ at angle Φ (see Figure 1 for the illustration of these notations). As a result, the analytical results mentioned in Section 3.2 with small modifications can still be applied to this case. In particular, covariance matrix K can still be calculated following (4), (5), (6), (8), and (9), provided that, in (4) and (5), the products z(k − j) (or 2πD(k − j)/λ) are replaced by 2πD k j /λ. This is because, in our considered case, D k j are the actual distances between the kth transmitter antenna and the jth transmitter antenna, for k, j = 1, . . . , 3.
Further, we assume that the variance σ 2 of the received signals at each transmitter antenna in (6) is unit, that is, σ 2 = 1. We also assume that Φ = 0.1114π rad and ∆ = 0.1114π rad.
In order to examine the performance of the considered system, the Rayleigh fading envelopes are required to be simulated. In turn, the covariance matrix of the complex Gaussian random variables corresponding to these Rayleigh envelopes must be calculated. Based on the aforementioned assumptions, from the theoretically analytical equations (4), (5), and (6), and the definition equations (8) and (9), we have the following desired covariance matrix for the considered configuration of transmitter antennas: Performing eigen decomposition, we have the following eigenvalues: −0.0092; 0.0360; and 2.9733. Therefore, K is not positive semidefinite. This also means that K is not positive definite.
It is important to emphasize that, from the mathematical point of view, covariance matrices are always positive semidefinite by definition (8), that is, the eigenvalues of the covariance matrices are either zero or positive. However, this does not contradict the above example where the covariance matrix K has a negative eigenvalue. The main reason why the desired covariance matrix K is not positive semidefinite is due to the approximation and the simplifications of the model mentioned in Figure 1 in calculating the covariance values, that is, due to the preciseness of (4) and (5), compared to the true covariance values. In other words, errors in estimating covariance values may exist in the calculation. Those errors may result in a covariance matrix being not positive semidefinite.
A question that could be raised here is why the covariance matrix of complex Gaussian random variables (with Rayleigh fading envelopes), rather than the covariance matrix of Rayleigh envelopes, is of particular interest. This is due to the two following reasons.
From the physical point of view, in the covariance matrix of Rayleigh envelopes, the correlation properties R xx , R yy of the real components (inphase components) as well as the imaginary components (quadrature phase components) themselves and the correlation properties R xy , R yx between the real and imaginary components of random variables are not directly present (these correlation properties are defined in (2)). On the contrary, those correlation properties are clearly present in the covariance matrix of complex Gaussian random variables with the desired Rayleigh envelopes. In other words, the physical significance of the correlation properties of random variables is not present as detailed in the covariance matrix of Rayleigh envelopes as in the covariance matrix of complex Gaussian random variables with the desired Rayleigh envelopes.
Further, from the mathematical point of view, it is possible to have one-to-one mapping from the cross-correlation coefficients ρ gi j (between the ith and jth complex Gaussian random variables) to the cross-correlation coefficients ρ ri j (between Rayleigh fading envelopes) as follows (see [9, (1.5-26)]): where E int (·) is the complete elliptic integral of the second kind. Some good approximations of this relationship between ρ ri j and ρ gi j are presented in the mapping [4, Table  II], the look-up [8, Table I and Figure 1]. However, the reversed mapping, that is, the mapping from ρ ri j to ρ gi j , is multivalent. It means that, for a given ρ ri j , we have to somehow determine ρ gi j in order to generate Rayleigh fading envelopes and the possible values of ρ gi j may be significantly different from each other depending on how ρ gi j is determined from ρ ri j . It is noted that ρ ri j is always real, but ρ gi j may be complex.
For the two aforementioned reasons, the covariance matrix of complex Gaussian random variables (with Rayleigh envelopes), as opposed to the covariance matrix of Rayleigh envelopes, is of particular interest in this paper.

Forced positive semidefiniteness of the covariance matrix
First, we need to define the coloring matrix L corresponding to a covariance matrix K. The coloring matrix L is defined to be the N × N matrix satisfying It is noted that the coloring matrix is not necessarily a lower triangular matrix. Particularly, to determine the coloring matrix L corresponding to a covariance matrix K, we can use either Cholesky decomposition [7] as mentioned in a number of papers, which have been reviewed in Section 2 of this paper, or eigen decomposition which is mentioned in the next section of this paper. The former yields a lower triangular coloring matrix, while the later yields a square coloring matrix. Unlike Cholesky decomposition, where the covariance matrix K must be positive definite, eigen decomposition requires that K is at least positive semidefinite, that is, the eigenvalues of K are either zeros or positive. We will explain later why the covariance matrix must be positive semidefinite even in the case where eigen decomposition is used to calculate the coloring matrix. The covariance matrix K, in fact, may not be positive semidefinite, that is, K may have negative eigenvalues, as the case mentioned in Example 1 of Section 4.1.
To overcome this obstacle, similarly to (but not exactly as) the method in [2], we approximate the given covariance matrix by a matrix that can be decomposed into K = LL H . While the method in [2] does this by replacing all negative and zero eigenvalues by a small, positive real number, we only replace the negative ones by zeros. This is possible, because we base our decomposition on an eigen analysis instead of a Cholesky decomposition as in [2], which can only be carried out if all eigenvalues are positive. Our procedure is presented as follows.
Assuming that K is the desired covariance matrix, which is not positive semidefinite, perform the eigen decomposition K = VGV H , where V is the matrix of eigenvectors and G is a diagonal matrix of eigenvalues of the matrix K.
We now compare our approximation procedure to the approximation procedure mentioned in [2]. The authors in [2] used the following approximation: where ε is a small, positive real number. Clearly, besides overcoming the disadvantage of Cholesky decomposition, our approximation procedure is more precise under realistic assumptions like finite precision arithmetic than the one mentioned in [2], since the matrix Λ in our algorithm approximates to the matrix G better than the one mentioned in [2]. Therefore, the desired covariance matrix K is well approximated by the positive semidefinite matrix K = VΛV H from Frobenius point of view [2].

Determine the coloring matrix using eigen decomposition
In most of the conventional methods, Cholesky decomposition was used to determine the coloring matrix. As analyzed earlier in Section 2, Cholesky decomposition may not work for the covariance matrix which has eigenvalues being equal or close to zeros.
To overcome this disadvantage, we use eigen decomposition, instead of Cholesky decomposition, to calculate the coloring matrix. Comparison of the computational efforts between the two methods (eigen decomposition versus Cholesky decomposition) is mentioned later in this paper. The coloring matrix is calculated as follows.
At this stage, we have the forced positive semidefinite covariance matrix K, which is equal to the desired covariance matrix K if K is positive semidefinite, or approximates to K otherwise. Further, as mentioned earlier, we have K = VΛV H , where Λ = diag(λ 1 , . . . ,λ N ) is the matrix of eigenvalues of K. Since K is a positive semidefinite matrix, it follows that {λ j } N j=1 are real and nonnegative.
We now calculate a new matrixΛ as Clearly,Λ is a real, diagonal matrix that results in If we denote L VΛ, then it follows that It means that the coloring matrix L corresponding to the covariance matrix K can be computed without using Cholesky decomposition. Thereby, the shortcoming of [2], which is related to roundoff errors in Matlab caused by Cholesky decomposition and is pointed out in Section 2, can be overcome.
We now explain why the covariance matrix must be positive semidefinite even when eigen decomposition is used to compute the coloring matrix. It is easy to realize that, if K is not positive semidefinite covariance matrix, thenΛ calculated by (16) is a complex matrix. As a result, (17) and (18) are not satisfied.

Proposed algorithm
In Section 2, we have shown that the method proposed in [2] fails to generate Rayleigh fading envelopes corresponding to a desired covariance matrix in a real-time scenario where Doppler frequency shifts are considered. This is because the authors in [2] did not realize the variance-changing effect caused by Doppler filters.
To surmount this shortcoming, the two following simple, but important modifications must be carried out.
(1) Unlike step 6 of the method in [2], where N independent, complex Gaussian random variables (with Rayleigh fading envelopes) are generated with unit variances, in our algorithm, this step is modified in order to be able to generate independent, complex Gaussian random variables with arbitrary variances σ 2 g . Correspondingly, step 7 of the method in [2] must also be modified. Besides being more generalized, the modification of our algorithm in steps 6 and 7 allows us to combine correctly the outputs of Doppler filters in the method proposed in [10] and our algorithm.
(2) The variance-changing effect of Doppler filters must be considered. It means that, we have to calculate the variance of the outputs of Doppler filters, which may have an arbitrary value depending on the variance of the complex Gaussian random variables at the inputs of Doppler filters as well as the characteristics of those filters. The variance value of the outputs is then input into the step 6 which has been modified as mentioned above.
The modification (1) can be carried out in the algorithm generating Rayleigh fading envelopes in a discrete-time scenario (see the algorithm mentioned in this section). The modification (2) can be carried out in the algorithm generating Rayleigh fading envelopes in a real-time scenario where Doppler frequency shifts are considered (see the algorithm mentioned in Section 5). From the above observations, we propose here a generalized algorithm to generate N correlated Rayleigh envelopes in a single time instant as given below.
(1) In a general case, the desired variances (powers) (2) From the desired correlation properties of correlated complex Gaussian random variables with Rayleigh envelopes, determine the covariances R xxk,j , R yy k, j , R xy k, j and R yx k, j , for k, j = 1, . . . , N and k = j. In other words, in a general case, those covariances must be known. Specially, in the case where the powers of all random processes are equal and other conditions hold as mentioned in Sections 3.1 and 3.2, we can follow (3) in the case of time delay and frequency separation, such as in OFDM systems, or (4), (5), and (6) in the case of spatial separation like with multiple antennas in MIMO systems to calculate the covariances R xxk,j , R yy k, j , R xy k, j , and R yx k, j . The values {σ 2 gj } N j=1 , R xxk,j , R yy k, j , R xy k, j , and R yx k, j (k, j = 1, . . . , N; k = j) are the input data of our proposed algorithm.
(3) Create the N × N-sized covariance matrix K: where The covariance matrix of complex Gaussian random variables is considered here, as opposed to the covariance matrix of Rayleigh fading envelopes like in the conventional methods. (4) Perform the eigen decomposition: Denote G diag(λ 1 , . . . , λ N ). Then, calculate a new 3 Note that σ 2 gj is the variance of complex Gaussian random variables, rather than the variance per dimension (real or imaginary). Hence, there is no factor of 2 in the denominator. diagonal matrix: Thereby, we have a diagonal matrix Λ with all elements in the main diagonal being real and definitely nonnegative.
We can see that the modification (1) takes place in this step of our algorithm and proceeds in the next step. (7) Generate a column vector Z of N correlated complex Gaussian random samples as follows:

Statistical properties of the resultant envelopes
In this section, we check the covariance matrix and the variances (powers) of the resultant correlated complex Gaussian random samples as well as the variances (powers) of the resultant Rayleigh fading envelopes.
It is easy to check that E(WW H ) = σ 2 g I N , and therefore It means that the generated Rayleigh envelopes are corresponding to the forced positive semidefinite covariance matrix K, which is, in turn, equal to the desired covariance matrix K in case K is positive semidefinite, or well approximates to K otherwise. In other words, the desired covariance matrix K of complex Gaussian random variables (with Rayleigh fading envelopes) is achieved.
In addition, note that the variance of the jth Gaussian random variable in Z is the jth element on the main diagonal of K. Because K approximates to K, the elements on the main diagonal of K are thus equal (or close) to σ 2 gj 's (see (20) and (21) in Z as given below (see [11, (5.51) and (5.52)] and [12, (2.1-131)]): From (19) and (28), it is clear that Therefore, the desired variances (powers) {σ 2 rj } N j=1 of Rayleigh envelopes are achieved.

GENERATION OF CORRELATED RAYLEIGH ENVELOPES IN A REAL-TIME SCENARIO
In Section 4.4, we have proposed the algorithm for generating N correlated Rayleigh fading envelopes in multipath, flat fading channels in a single time instant. We can repeat steps 6 and 7 of this algorithm to generate Rayleigh envelopes in the continuous time interval. It is noted that, the discretetime samples of each Rayleigh fading process generated by this algorithm in different time instants are independent of each other. It has been known that the discrete-time samples of each realistic Rayleigh fading process may have autocorrelation properties, which are the functions of the Doppler frequency corresponding to the motion of receivers as well as other factors such as the sampling frequency of transmitted signals. It is because the band-limited communication channels not only limit the bandwidth of transmitted signals, but also limit the bandwidth of fading. This filtering effect limits the rate of changes of fading in time domain, and consequently, results in the autocorrelation properties of fading. Therefore, the algorithm generating Rayleigh fading envelopes in realistic conditions must consider the autocorrelation properties of Rayleigh fading envelopes.
To simulate a multipath fading channel, Doppler filters are normally used [11]. The analysis of Doppler spectrum spread was first derived by Gans [13], based on Clarke's model [14]. Motivated by these works, Smith [15] developed a computer-assisted model generating an individual Rayleigh fading envelope in flat fading channels corresponding to a given normalized autocorrelation function. This model was then modified by Young [10,16] to provide more accurate channel realization.
It should be emphasized that, in [10,16], the models are aimed at generating an individual Rayleigh envelope corresponding to a certain normalized autocorrelation function of itself, rather than generating different Rayleigh envelopes corresponding to a desired covariance matrix (autocorrelation and cross-correlation properties between those envelopes). Therefore, the model for generating N correlated Rayleigh fading envelopes in realistic fading channels (each individual envelope is corresponding to a desired normalized autocorrelation property) can be created by associating the model proposed in [10] with our algorithm mentioned in Section 4.4 in such a way that, the resultant Rayleigh fading envelopes are corresponding to the desired covariance matrix.
This combination must overcome the main shortcoming of the method proposed in [2] as analyzed in Section 2. In other words, the modification (2) mentioned in Section 4.4 must be carried out. This is an easy task in our algorithm. The key for the success of this task is the modification in steps 6 and 7 of our algorithm (see Section 4.4), where the variances of N complex Gaussian random variables are not fixed as in [2], but can be arbitrary in our algorithm. Again, besides being more generalized, our modification in these steps allows the accurate combination of the method proposed in [10] and our algorithm, that is, guaranteeing that the generated Rayleigh envelopes are exactly corresponding to the desired covariance matrix.
The model of a Rayleigh fading generator for generating an individual baseband Rayleigh fading envelope proposed in [10,16] is shown in Figure 2. This model generates a Rayleigh fading envelope using inverse discrete Fourier transform (IDFT), based on independent zero-mean Gaussian random variables weighted by appropriate Doppler filter coefficients. The sequence {u j [l]} M−1 l=0 of the complex Gaussian random samples at the output of the jth Rayleigh generator ( Figure 2) can be expressed as where (i) M denotes the number of points with which the IDFT is carried out; (ii) l is the discrete-time sample index (l = 0, . . . , M − 1); where d l − m is the sample lag, σ 2 orig is the variance of the real, independent zero-mean Gaussian random sequences {A Similarly, the correlation property between the real part u R [l] and the imaginary part u I [m] is calculated as (see [10, (8) The mean value of the output sequence {u[l]} has been proved to be zero (see [ where * denotes the complex conjugate operation.
Let r nor be that is, let r nor be the autocorrelation function in (31) normalized by the variance σ 2 g in (35). r nor is called the normalized autocorrelation function.
To achieve a desired normalized autocorrelation function r nor = J 0 (2π f m d), where f m is the maximum Doppler frequency F m normalized by the sampling frequency F s of the transmitted signals (i.e., f m = F m /F s ), the Doppler filter {F[k]} is determined in Young's model [10,16] as follows (see [10, (21)]): In (37), k m f m M , where · indicates the biggest rounded integer being less or equal to the argument.
It has been proved in [10] that the (real) filter coefficients in (37) will produce a complex Gaussian sequence with the normalized autocorrelation function J 0 (2π f m d), and with the expected independence between the real and imaginary parts of Gaussian samples, that is, the correlation property in (33) is zero. The zero-correlation property between the real and imaginary parts is necessary in order that the resultant envelopes are Rayleigh distributed.   depending on the variances of the Gaussian random variables at the inputs of Doppler filters as well as the characteristics of those filters (see (35) for more details). We now return to the main shortcoming of the method proposed in [2], which is mentioned earlier in Section 2. In [2, Section 6], the authors generated Rayleigh envelopes corresponding to a desired covariance matrix in a real-time scenario, where Doppler frequency shifts were considered, by combining their proposed method with the method proposed in [10]. Specifically, the authors took the outputs of the method in [10] and simply input them into step 6 in their method.
However, the step 6 in the method in [2] was proposed for generating complex Gaussian random variables with a fixed (unit) variance. Meanwhile, as presented earlier, the variances of the complex Gaussian random variables at the output of the Rayleigh simulator may have arbitrary values, depending on the variances of the Gaussian random variables at the inputs of Doppler filters as well as the characteristics of those filters. Consequently, if the outputs of the method in [10] are simply input into the step 6 as mentioned in the algorithm in [2], the covariance matrix of the resultant correlated Gaussian random variables is not equal to the de-sired covariance matrix due to the variance-changing effect of Doppler filters being not considered. In other words, the method proposed in [2] fails to generate Rayleigh fading envelopes corresponding to a desired covariance matrix in a real-time scenario where Doppler frequency shifts are taken into account.
Our model for generating N correlated Rayleigh fading envelopes corresponding to a desired covariance matrix in a real-time scenario where Doppler frequency shifts are considered is presented in Figure 3. In this model, N Rayleigh generators, each of which is presented in Figure 2, are simultaneously used. To generate N correlated Rayleigh envelopes corresponding to a desired covariance matrix at an observed discrete-time instant l (l = 0, . . . , M − 1), similarly to the method in [2], we take the output u j [l] of the jth Rayleigh simulator, for j = 1, . . . , N, and input it as the element u j into step 6 of our algorithm proposed in Section 4.4. However, as opposed to the method in [2], the variance σ 2 g of complex Gaussian samples u j in step 6 of our method is calculated following (35). This value is used as the input parameter for steps 6 and 7 of our algorithm (see Figure 3). Thereby, the variance-changing effect caused by Doppler filters is taken into consideration in our algorithm, and consequently, our proposed algorithm overcomes the main shortcoming of the method in [2].
The algorithm for generating N correlated Rayleigh envelopes (when Doppler frequency shifts are considered) at a discrete-time instant l, for l = 0, . . . , M − 1, can be summarized as follows.
(1) Perform the steps 1 to 5 mentioned in Section 4.4.
(2) From the desired autocorrelation properties (31) and (36) of each of the complex Gaussian random sequences (with Rayleigh fading envelopes), determine the values M and σ 2 orig . These values can be arbitrarily selected, provided that they bring about the desired autocorrelation properties. The value of M is also the number of points with which IDFT is carried out.
(3) For each Rayleigh generator presented in Figure 2 Figure 3. . It is noted that σ 2 g is the same for N Rayleigh generators. We also emphasize that, by this calculation, the modification (2) mentioned in Section 4.4 has been performed in this step. Steps (7) and (8) are repeated for different time instants l (l = 0, . . . , M − 1), and therefore, the algorithm can be used for a real-time scenario.

SIMULATION RESULTS
In this section, first, we simulate N = 3 frequency-correlated Rayleigh fading envelopes corresponding to the complex Gaussian random variables with equal powers σ g 2 j = 1 ( j = 1, . . . , 3) in the flat fading channels. Parameters considered here include M = 2 14 (the number of IDFT points), σ 2 orig = 1/2 (variances per dimension in Young's model), F s = 8 kHz, F m = 50 Hz (corresponding to a carrier frequency 900 MHz and a mobile speed v = 60 km/h). Frequency separation between two adjacent carrier frequencies considered here is ∆ f = 200 kHz (e.g., in GSM 900) and we assume that f 1 > f 2 > f 3 . Also, we consider the rms delay spread σ τ = 1 microsecond and time delays between three envelopes are τ 1,2 = 1 millisecond, τ 2,3 = 3 milliseconds, τ 1,3 = 4 milliseconds. From (3), (20), and (21), we have the desired covariance matrix K as given below: It is easy to check that K in (38) is positive definite. Using the proposed algorithm in Section 5, we have the simulation result presented in Figure 4a. Next, we simulate N = 3 spatially-correlated Rayleigh fading envelopes. We consider an antenna array comprising three transmitter antennas, which are equally separated by a distance D. Assume that D/λ = 1, that is, D = 33.3 cm for GSM 900. Additionally, we assume that ∆ = π/18 rad (or ∆ = 10 • ) and Φ = 0 rad. The parameters M, σ 2 gj , σ 2 orig , F s , and F m are the same as in the previous case. From (4), (5) Since Φ = 0 rad, the covariances R xy k, j and R yx k, j between the real and imaginary components of any pair of the complex Gaussian random processes (with Rayleigh fading envelopes) are zeros, and consequently, K is a real matrix. Readers may refer to (5) and (6) for more details. It is easy to realize that K in (39) is positive definite. The simulation result is presented in Figure 4b.
In Figure 6, we plot the histograms of the resultant Rayleigh fading envelopes produced by our algorithm in the four aforementioned examples. Without loss of generality, we plot the histograms for one of three Rayleigh fading envelopes, such as the first Rayleigh fading envelope. To compare the accuracy of our algorithm, we also plot the theoretical probability density function (PDF) of a typical Rayleigh   fading envelope by solid curves. In this figure, the parameter σ 2 gj of the PDF is the variance of the complex Gaussian random process corresponding to the considered typical Rayleigh fading envelope. It can be observed from Figure 6 that, the resultant envelopes produced by our algorithm in the four examples follow accurately the theoretical PDF of the typical Rayleigh fading envelope.
Finally, in Figure 7, we compare the computational efforts between our algorithm and the one mentioned in [2] by comparing the average computational time required for both algorithms to simulate N = 2, 4, 8, 16, 32, 64 or 128 Rayleigh envelopes in a real-time scenario over 10 000 trials. It can be realized from Figure 7 that, for N = 64 and N = 128, our algorithm is slightly more complex, while it is almost as computationally efficient as the method in [2] for a smaller N.

CONCLUSIONS
In this paper, we have derived a more generalized algorithm to generate correlated Rayleigh fading envelopes. Using the presented algorithm, one can generate an arbitrary number N of either Rayleigh envelopes with any desired power σ 2 rj , j = 1, . . . , N, or those envelopes corresponding to any desired power σ 2 gj of Gaussian random variables. This algorithm also facilitates to generate equal as well as unequal power Rayleigh envelopes. It is applicable to both scenarios of spatial correlation and spectral correlation between the random processes. The coloring matrix is determined by a positive semidefiniteness forcing procedure and an eigen decomposi-tion procedure without using Cholesky decomposition. Consequently, the restriction on the positive definiteness of the covariance matrix is relaxed and the algorithm works well without being impeded by the roundoff errors of Matlab. The proposed algorithm can be used to generate Rayleigh envelopes corresponding to any desired covariance matrix, no matter whether or not it is positive definite. In comparison with the conventional methods, besides being more generalized, our proposed algorithm (with or without Doppler spectrum spread) is more precise, while overcoming all shortcomings of the conventional methods.  She is especially interested in authentication and privacy. In 1987, at University College, ADFA, she founded the Centre for Computer and Communications Security Research which proved to be a reservoir of expertise for the Australian community. Her studies of the application of discrete mathematics and combinatorial computing via bent functions, S-box design, has led to the design of secure cryptoalgorithms and strong hashing algorithms for secure and reliable information transfer in networks and telecommunications. Her studies of Hadamard matrices and orthogonal designs are applied in CDMA technologies. In 1990 she founded the AUSCRYPT/ASIACRYPT series of International Cryptologic Conferences in the Asia/Oceania area. She has supervised 25 successful Ph.D. candidates, has over 350 scholarly papers and six books. She is a Senior Member of IEEE.

Call for Papers
Wireless mobile ad hoc networks (MANETs), due to their dynamic nature, pose many unique challenges compared to traditional wired or cellular wireless networks. MANETs must be self-organized without any requirement for base stations. Their topologies are unpredictable due to mobility and change with the number and distribution of active nodes in the network. Fading and channel variations also induce changes in the network topology and introduce additional complexities in these networks. Given power and energy constraints, as well as the shared nature of the wireless medium, communications may be expected to be multihop. In such a harsh environment, robustness and quality of service (QoS) are essential factors to be considered. MANETs may consist of a heterogeneous mixture of nodes with variety of traffic types and different QoS requirements. Scaling laws for these networks are not fully understood. Many tradeoff studies related to capacity, delay, bandwidth, and energy consumption are currently under intense investigations.
The goal of this special issue is to collect cutting-edge research results in the field of wireless MANETs. We solicit papers that deal with pressing problems unique to wireless MANETs. The scope of this issue includes all aspects of MANETs, including scaling laws, tradeoff studies, coding, interference management, protocol design, cross-layer design, and, more importantly, fundamental limits of MANETs under different conditions. We seek original and unpublished work. The potential list of topics is not necessarily exhaustive and other appropriate subjects will be considered.
Topics of interest include (but are not limited to): •

Call for Papers
Multiple-input multiple-output (MIMO) wireless communication systems exploit the spatial channel created by transmit and receive antenna arrays to increase quality, capacity, and coverage in wireless communication systems. A wide variety of techniques, including space-time coding, spatial multiplexing schemes, transmitter-receiver architecture, precoding methods together with established physical layer algorithms have been proposed or adopted for MIMO communication systems. Channel modeling plays an important part in validating and proving these systems. Despite rapid progress in MIMO communication systems, lack of understanding and modeling of spatial aspects of wireless channels is a critical obstacle to the further development of this technology.
Topics of interest include (but are not limited to): •

Call for Papers
Some modern applications require an extraordinary large amount of complexity in signal processing algorithms. For example, the 3rd generation of wireless cellular systems is expected to require 1000 times more complexity when compared to its 2nd generation predecessors, and future 3GPP standards will aim for even more number-crunching applications. Video and multimedia applications do not only drive the complexity to new peaks in wired and wireless systems but also in personal and home devices. Also in acoustics, modern hearing aids or algorithms for de-reverberation of rooms, blind source separation, and multichannel echo cancelation are complexity hungry. At the same time, the anticipated products also put on additional constraints like size and power consumption when mobile and thus battery powered. Furthermore, due to new developments in electroacoustic transducer design, it is possible to design very small and effective loudspeakers. Unfortunately, the linearity assumption does not hold any more for this kind of loudspeakers, leading to computationally demanding nonlinear cancelation and equalization algorithms.
Since standard design techniques would either consume too much time or do not result in solutions satisfying all constraints, more efficient development techniques are required to speed up this crucial phase. In general, such developments are rather expensive due to the required extraordinary high complexity. Thus, de-risking of a future product based on rapid prototyping is often an alternative approach. However, since prototyping would delay the development, it often makes only sense when it is well embedded in the product design process. Rapid prototyping has thus evolved by applying new design techniques more suitable to support a quick time to market requirement.
This special issue focuses on new development methods for applications with high complexity in signal processing and on showing the improved design obtained by such methods. Examples of such methods are virtual prototyping, HW/SW partitioning, automatic design flows, float to fix conversions, automatic testing and verification, and power aware designs.

Special Issue on
Formal Methods for GALS Design

Call for Papers
As chips grow in speed and complexity, global control of an entire chip using a single clock is becoming increasingly challenging. In the future, multicore and large-scale systems-onchip (SoC) designs are therefore likely to be composed of several timing domains.
Global Asynchrony and Local Synchrony (GALS) is emerging as the paradigm of choice for SoC design with multiple timing domains. In GALS systems, each timing domain is locally clocked, and asynchronous communication schemes are used to glue all of the domains together. Thus, unlike purely asynchronous design, GALS design is able to make use of the significant industrial investment in synchronous design tools.
There is an urgent need for the development of sound models and formal methods for GALS systems. In synchronous designs, formal methods and design automation have played an enabling role in the continuing quest for chips with ever greater complexity. Due to the inherent subtleties of the asynchronous circuit design, formal methods are likely to be vital to the success of the GALS paradigm.
We invite original articles for a special issue of the journal to be published in 2006. Articles may cover every aspect related to formal modeling and formal methods for GALS systems and/or target any type of embedded applications and/or architectures combining synchronous and asynchronous notions of timing: Vision systems allow computers to understand images, and to take appropriate actions, often under hard real-time constraints.
Most vision systems need high computer performance. The decisive constraint to develop pattern recognition or monitoring systems was therefore to consider computer hardware with excellent key features to fulfill the high requirements. This causality has several disadvantages. The costs of the final products are high, the size of the hardware becomes voluminous, the electromagnetic capability is reduced, and the energy consumption is often a problem. Therefore, the pressure to realize vision systems on the base of Embedded Systems was and is still increasing dramatically. Meanwhile, the number of possible applications has exploded since several disadvantages of classic systems can be avoided. The history of mobile phones evolution is one of the best examples. It would not have been possible without Embedded Systems, and especially not in such an affordable way. However, it is not necessary to consider only the mass market where Embedded Vision Systems can improve the current situation dramatically. If many cameras are installed to watch a scene, one is able to define a virtual camera, which always shows the most important angle of a view. If a bank note should be checked under the conditions of high accuracy, high probability of error recognition, and high throughput, the realization is only feasible, if the computer is assisted by a network of special parallelized chips. Usually, the algorithms can be divided into three areas, the prestage, where data is compressed, the specialized computational phase, and the interpretation stage. With this setup, the bandwidth and the data throughput may be improved in an amazing way.
Many other ideas could be presented. The main issues are the parallelization of processes, as well as the communications between them, which are based on networked chip sets. The challenge for the research work is to find optimal structures concerning real-time problems, energy consumption, low-price solutions, and so forth. However, not all algorithms for vision systems are suitable to be implemented in Embedded Systems; better solutions have to be discovered. In this sense many tasks and problems in the research field have to be solved, and many application areas are concerned.   The enormous available bandwidth, the wide scope of the data rate/range trade-off, as well as the potential for very-lowcost operation leading to pervasive usage, all present a unique opportunity for UWB systems to impact the way people and intelligent machines communicate and interact with their environment.
The aim of this book is to provide an overview of the state of the art of UWB systems from theory to applications.
Due to the rapid progress of multidisciplinary UWB research, such an overview can only be achieved by combining the areas of expertise of several scientists in the field.
More than 30 leading UWB researchers and practitioners have contributed to this book covering the major topics relevant to UWB. These topics include UWB signal processing, UWB channel measurement and modeling, higher-layer protocol issues, spatial aspects of UWB signaling, UWB regulation and standardization, implementation issues, and UWB applications as well as positioning.
The book is targeted at advanced academic researchers, wireless designers, and graduate students wishing to greatly enhance their knowledge of all aspects of UWB systems. For any inquiries on how to order this title please contact books.orders@hindawi.com