Generation of Two Correlated Stationary Gaussian Processes

Abstract

Since correlated stochastic processes are often presented in practical problems, feasible methods to model and generate correlated processes appropriately are needed for analysis and simulation. The present paper systematically presents three methods to generate two correlated stationary Gaussian processes. They are (1) the method of linear filters, (2) the method of series expansion with random amplitudes, and (3) the method of series expansion with random phases. All three methods intend to match the power spectral density for each process but use information of different levels of correlation. The advantages and disadvantages of each method are discussed.

1. Introduction

Stochastic processes are involved in many areas, such as physics, engineering, ecology, biology, medicine, psychology, finance, and other disciplines. For purposes of analysis and simulation, they are required to be properly modeled and generated mathematically, according to their measured or estimated statistical and probabilistic properties. Among various properties, the probability distribution and the power spectral density are the most important. If it is stationary, both properties are invariant with time. Based on these two properties, several methods were developed to model a single stationary process. If it is Gaussian, a process with a given spectral density can be generated [1,2]. If a process is non-Gaussian, methods were proposed for processes defined in an infinite region, a semi-infinite region, or a finite region [2]. In this case, only certain types of spectral density functions can be matched, such as low-pass spectrum, single-peak spectrum, and two-peak spectrum.

In some problems, two stochastic processes are present and play roles in system dynamics. If they are correlated, the above-mentioned modeling methods for single processes are incomplete. The correlation of the two processes should be taken into account. A typical scenario is structure vibration caused by earthquakes. In general, the ground motion in an earthquake has both horizontal and vertical components. Both may play roles in the destruction of structures. One example is the vibration of a water tank supported by a column and subjected to both horizontal and vertical ground motions [3]. Since both motions are in fact the two components of the same ground motion, they must be correlated, not independent. Another typical situation is the ship rolling motion caused by random sea waves, investigated in [4,5]. In the equation of the motion of the ship rolling angle, there are two excitation terms, an external excitation representing the moment changing the rolling angle directly and a parametric excitation caused by the wave level relative to the rolling ship. The two excitations by the same sea waves are naturally correlated.

It is a common practice that many stochastic processes are assumed to be Gaussian, except cases with obvious non-Gaussian distributions. Several reasons support such a practice, including that (i) the bell shape of the Gaussian distribution matches many real processes; (ii) its mathematical treatment is simple; (iii) the Gaussian distribution is commonly used for temporal and spectral smoothing in a time-frequency analysis [6]; and (iv) the generation of a stochastic process with a given non-Gaussian distribution and a given spectral density is much more difficult [7,8,9]. Moreover, the power spectral density describes the energy distribution of the process in the frequency domain, and it is more important than the probability distribution in many situations [10]. Therefore, the Gaussian distribution is usually assumed, and the spectral density is used to generate an individual process. In this investigation, the emphasis is on the correlation of the two processes.

It is known that the correlation of two stochastic processes can be described at different levels, such as their joint moment or correlation coefficient, their cross-correlation function, or their cross-spectral density. To model and generate two correlated stationary stochastic processes, various methods were developed, such as the methods using series expansions [11,12,13], based on an autoregressive moving average model [14,15], using linear filters [16], and based on other methods [17,18]. The present paper is a review paper, summarizing three relatively simple and practical methods to model and generate two correlated stationary Gaussian processes based on different levels of their correlation information. They are the method of linear filters, the method of series expansion with random amplitudes, and the method of series expansion with random phases. The three methods are presented systematically, and simulations are performed to verify their accuracy. The relative advantages and disadvantages of each method are pointed out for the convenience of practical applications.

2. Method of Linear Filters

2.1. Two Correlated Gaussian White Noises

Gaussian white noises are the most commonly used stochastic processes in various fields of science and technology due to the simplicity in mathematical treatment and the relation to the Markov theory. For example, white noises are frequently used for simulations in signal processing since many real-world signals contain white noises. Moreover, they can be used to generate more realistic stochastic processes [2]. Thus, we would develop a method to model two correlated Gaussian white noises W₁(t) and W₂(t). Assume that they have the following correlation functions:

$$E\left[W_i\left(t\right)W_j\left(t+\tau \right)\right]=2\pi K_{ij}\delta \left(\tau \right),\hspace{1em}i,j=1,2$$

(1)

where K₁₁ and K₂₂ are the spectral densities of W₁(t) and W₂(t), respectively; K₁₂ is their cross-spectral density, which is the unique measure of their correlation. The samples of W₁(t) and W₂(t) can be generated as follows:

$$W_1\left(t_k\right)=\sqrt{\frac{2\pi K_{11}}{\mathsf{\Delta }t}}{X}_{1k},W_2\left(t_k\right)=\sqrt{\frac{2\pi K_{22}}{\mathsf{\Delta }t}}{X}_{2k}$$

(2)

where Δt is a small time interval; X_1k and X_2k are samples of two Gaussian random variables with a correlation coefficient

$$\rho =\frac{K_{12}}{\sqrt{K_{11}{K}_{22}}}$$

(3)

The procedure of generating X_1k and X_2k is described in reference [2]. It is noted that each of the X_1k and X_2k sequences is independent for different k values, and X_1i and X_2j are independent for i ≠ j. It is proved that the spectral densities of X_1k and X_2k are [2]

$${\mathsf{\Phi }}_{ii}\left(\omega \right)=\frac{2K_{ii}}{{(\omega \mathsf{\Delta }t)}^2}\left[1-\cos (\omega \mathsf{\Delta }t\right)],\mathit{i} =1,2$$

(4)

and their cross-spectral density is [2]

$${\mathsf{\Phi }}_{12}\left(\omega \right)=\frac{2K_{12}}{{(\omega \mathsf{\Delta }t)}^2}\left[1-\cos (\omega \mathsf{\Delta }t\right)]$$

(5)

In the limiting process ωΔt → ∞, we have

$$\underset{\omega \mathsf{\Delta }t\to 0}{\lim }\frac{2K_{ii}}{{(\omega \mathsf{\Delta }t)}^2}\left[1-\cos (\omega \mathsf{\Delta }t\right)]=K_{ii}, i=1,2$$

(6)

$$\underset{\omega \mathsf{\Delta }t\to 0}{\lim }\frac{2K_{12}}{{(\omega \mathsf{\Delta }t)}^2}\left[1-\cos (\omega \mathsf{\Delta }t\right)]=K_{12}$$

(7)

Equations (6) and (7) show that if $\omega \mathsf{\Delta }t$ is very small, the spectral densities are approximately constants matching the given values. Therefore, the time step Δt is the most important parameter in the simulation, and a smaller Δt leads to broader frequency bands for both noises. It is pointed out that Δt should be much shorter than all important system relaxation times [2].

2.2. Two Correlated Gaussian Processes

It is known that the response of a linear filter excited by a Gaussian white noise is also Gaussian, but its spectrum may have different shapes. A first-order filter generates a process of low-pass spectrum, i.e., its peak is at zero frequency. A second-order filter has a response of a single-peak spectrum with its peak at a non-zero frequency. The response of a higher-order filter has a spectrum of multiple peaks. Moreover, the bandwidth of each peak can be adjusted by appropriately selecting the filter parameters. In the following, an example of two correlated stochastic processes, one of low-pass and another of single peak at non-zero frequency, is given for illustration.

Consider the system consisting of one first-order filter and one second-order filter, described by

$$\begin{gathered} {\dot{X}}_1+\alpha X_1=W_1\left(t\right) \\ {\ddot{X}}_2+2\zeta {\omega }_0{\dot{X}}_2+{\omega }_0^2{X}_2=W_2\left(t\right) \end{gathered}$$

(8)

where W₁(t) and W₂(t) are two correlated Gaussian white noises with the spectral densities K₁₁ and K₁₁, and the cross-spectral density K₁₂. Letting ${\dot{X}}_2=X_3$, Equation (8) is re-written in the state space as

$$\begin{gathered} {\dot{X}}_1+\alpha X_1=W_1\left(t\right) \\ {\dot{X}}_2=X_3 \\ {\dot{X}}_3=-{\omega }_0^2{X}_2-2\zeta {\omega }_0{X}_3+W_2\left(t\right) \end{gathered}$$

(9)

Equation (9) generates three stochastic processes, X₁(t), X₂(t), and X₃(t). Since X₃(t) is the derivative of X₂(t), they are uncorrelated; thus, the two pairs of correlated processes are X₁(t) and X₂(t), and X₁(t) and X₃(t).

The auto- and cross-correlation functions can be determined exactly following the method described in [2]. The autocorrelation functions are found to be

$$\begin{gathered} R_{11}\left(\tau \right)=E\left[X_1^2\right]e^{-\alpha \tau }, \tau \ge 0 \\ {R}_{22}\left(\tau \right)=E\left[X_2^2\right]\left[e^{-\zeta {\omega }_0\tau }\left(\cos {\omega }_d\tau +\frac{\zeta {\omega }_0}{{\omega }_d}\sin {\omega }_d\tau \right)\right],\tau \ge 0 \\ {R}_{33}\left(\tau \right)=E\left[X_3^2\right]\left[e^{-\zeta {\omega }_0\tau }\left(\cos {\omega }_d\tau -\frac{\zeta {\omega }_0}{{\omega }_d}\sin {\omega }_d\tau \right)\right],\tau \ge 0 \end{gathered}$$

(10)

and the cross-correlations are

$$R_{12}\left(\tau \right)=\left\{\begin{array}{c}E\left[X_1{X}_2\right]e^{-\zeta {\omega }_0\tau }\left(\cos {\omega }_d\tau +\frac{\alpha +\zeta {\omega }_0}{{\omega }_d}\sin {\omega }_d\tau \right), \tau \ge 0\\ E\left[X_1{X}_2\right]e^{\alpha \tau }, \tau <0\end{array}\right.$$

(11)

$$R_{13}\left(\tau \right)=\left\{\begin{array}{c}E\left[X_1{X}_3\right]e^{-\zeta {\omega }_0\tau }\left[\cos {\omega }_d\tau -\frac{{\omega }_0\left({\omega }_0+\alpha \zeta \right)}{\alpha {\omega }_d}\sin {\omega }_d\tau \right],\tau \ge 0\\ E\left[X_1{X}_3\right]e^{\alpha \tau }, \tau <0\end{array}\right.$$

(12)

where

$${\omega }_d={\omega }_0\sqrt{1-{\zeta }^2}$$

(13)

and

$$\begin{gathered} E[X_1^2]=\frac{\pi K_{11}}{\alpha }, E[X_2^2]=\frac{\pi K_{22}}{2\zeta {\omega }_0^3}, E\left[X_3^2\right]=\frac{\pi K_{22}}{2\zeta {\omega }_0} \\ E[X_1{X}_2]=\frac{2\pi K_{12}}{{\omega }_0^2+\alpha \left(\alpha +2\zeta {\omega }_0\right)}, E\left[X_1{X}_3\right]=\frac{2\pi \alpha K_{12}}{{\omega }_0^2+\alpha \left(\alpha +2\zeta {\omega }_0\right)} \end{gathered}$$

(14)

For the negative range of τ < 0, it is known that $R_{ii}\left(-\tau \right)=R_{ii}\left(\tau \right)$ and $R_{ij}\left(-\tau \right)=R_{ji}\left(\tau \right)$.

Equations (11) and (12) show that the correlation function of X₁(t) and X₂(t) is specified by the joint moment E[X_{1 × 2}], and that of X₁(t) and X₃(t) by E[X₁X₃]. Both joint moments are determined by K₁₂, as shown in the last equation in (14). Thus, the correlation level of either X₁(t) and X₂(t) or X₁(t) and X₃(t) can be matched by adjusting K₁₂.

It is noted that X₃(t) has been used to generate the random wave excitation in the ship rolling motion [19,20].

The cross-correlation functions R₁₂(τ) and R₁₃(τ) are calculated from Equations (11) and (12) and depicted in Figure 1 and Figure 2 for Equation (9) with parameter values of α = 0.6, ζ = 0.3, ${\omega }_0$= 1.3, K₁₁ = 1, K₂₂ = 0.5, and K₁₂ = 0.1. The results from Monte Carlo simulations are also shown in the figures. They agree quite well.

One advantage of using linear filters is that the generated correlated stochastic processes are ergodic after a short transient period. It is important since only one sample function is needed in the simulation provided that the sample function is long enough.

3. Method of Series Expansion with Random Amplitudes

3.1. Simulation of a Single Stationary Gaussian Process

Assume that a stationary Gaussian process X(t) possesses a power spectral density $\mathsf{\Phi }\left(\omega \right)$. For practical situations, $\mathsf{\Phi }\left(\omega \right)$→ 0 when ω → ∞; therefore, there is a cutting frequency ${\omega }_r$, beyond which $\mathsf{\Phi }\left(\omega \right)$ is negligible. Moreover, assume that there is a lower frequency band ${\omega }_l$, below which Φ(ω) can also be neglected. In many cases, ${\omega }_l$ = 0. Divide $\left[{\omega }_l, {\omega }_r\right]$ into N intervals, and denote

$$\mathsf{\Delta }\omega =\frac{1}{N}\left({\omega }_r-{\omega }_l\right),{\omega }_j={\omega }_l+(j-\frac{1}{2})\mathsf{\Delta }\omega ,{\sigma }_j^2=2\mathsf{\Phi }\left({\omega }_j\right)\mathsf{\Delta }\omega , j=1, 2,\cdots ,N$$

(15)

By noting that,

$$E\left[X^2\left(t\right)\right]={{\displaystyle \int }}_{-\infty }^{\infty }\mathsf{\Phi }\left(\omega \right)d\omega$$

(16)

${\sigma }_j^2$ in Equation (15) specifies the contribution of the spectral density in the jth interval toward a total mean-square value of X(t) [2]. Denote X_s(t) with the subscript which represents s as a sample function of X(t). The following series can be used to generate X_s(t),

$$X_s\left(t\right)={\displaystyle \sum }_{j=1}^N{\sigma }_j\left(U_j\cos {\omega }_{j}t+V_j\sin {\omega }_{j}t\right)$$

(17)

where each of U_j and V_j is a series of Gaussian random variables independent for different j, and U_j and V_j are also independent for any j, ${\sigma }_j$ and ${\omega }_j$ which are determined from (15).

The correlation function and spectral density of X_s(t) are, respectively, [2]

$$R_{X_s{X}_s}\left(\tau \right)=E\left[X_s\left(t\right)X_s\left(t+\tau \right)\right]={\displaystyle \sum }_{j=1}^N{\sigma }_j^2\cos \left({\omega }_j\tau \right)$$

(18)

$${\mathsf{\Phi }}_{X_s{X}_s}\left(\omega \right)={\displaystyle \sum }_{j=1}^N\left[{\mathsf{\Phi }}_{XX}\left({\omega }_j\right)\delta \left(\omega -{\omega }_j\right)+{\mathsf{\Phi }}_{XX}\left(-{\omega }_j\right)\delta \left(\omega +{\omega }_j\right)\right]\mathsf{\Delta }\omega$$

(19)

The summation in (19) becomes an integration when N → ∞ and Δω → 0, i.e.,

$$\underset{N\to \infty }{\lim }{\mathsf{\Phi }}_{X_s{X}_s}\left(\omega \right)={\mathsf{\Phi }}_{XX}\left(\omega \right)$$

(20)

indicating that X_s(t) is a good sample function of X(t) if the interval number N is large and the frequency interval Δω is small.

3.2. Generation of Two Correlated Stationary Gaussian Processes

Now we try to model two stationary Gaussian processes X₁(t) and X₂(t) with power spectral densities Φ₁₁($\omega$) and Φ₂₂($\omega$), respectively. First, we have to determine a common frequency range $\left[{\omega }_l, {\omega }_r\right]$ for both processes and apply Equation (15) for each process. Then, the sample function X_is(t) (i = 1 or 2) of each process can be generated from (17) as follows:

$$X_{is}\left(t\right)={\displaystyle \sum }_{j=1}^N{\sigma }_{ij}\left(U_{ij}\cos {\omega }_{j}t+V_{ij}\sin {\omega }_{j}t\right), i=1, 2$$

(21)

In order to make the two processes correlated, U_1j and U_2j in (21) are generated with a correlation coefficient ρ, as well as V_1j and V_2j. It is shown in [13] that

$$R_{X_{1s}{X}_{2s}}\left(\tau \right)=E\left[X_{1s}\left(t\right)X_{2s}\left(t+\tau \right)\right]={\displaystyle \sum }_{j=1}^N\rho {\sigma }_{1j}{\sigma }_{2j}\cos ({\omega }_j\tau )$$

(22)

and

$$E\left[X_{1s}\left(t\right)X_{2s}\left(t\right)\right]={\displaystyle \sum }_{j=1}^N\rho {\sigma }_{1j}{\sigma }_{2j}$$

(23)

Replacing ${\sigma }_{1j}$ and ${\sigma }_{2j}$ according to Equation (15),

$$E\left[X_{1s}\left(t\right)X_{2s}\left(t\right)\right]={\displaystyle \sum }_{j=1}^{N}2\rho \sqrt{{\mathsf{\Phi }}_{11}\left({\omega }_j\right){\mathsf{\Phi }}_{22}\left({\omega }_j\right)}\mathsf{\Delta }\omega$$

(24)

As Δω → 0 and N → ∞, Equations (22) and (24) are transferred to

$$R_{X_{1s}{X}_{2s}}\left(\tau \right)\to R_{12}\left(\tau \right)={{\displaystyle \int }}_0^{\infty }2\rho \sqrt{{\mathsf{\Phi }}_{11}\left(\omega \right){\mathsf{\Phi }}_{22}\left(\omega \right)}\cos (\omega \tau )d\omega$$

(25)

$$E\left[X_{1s}\left(t\right)X_{2s}\left(t\right)\right]\to {{\displaystyle \int }}_0^{\infty }2\rho \sqrt{{\mathsf{\Phi }}_{11}\left(\omega \right){\mathsf{\Phi }}_{22}\left(\omega \right)}d\omega$$

(26)

Equation (26) indicates that the correlation coefficient of X₁ (t) and X₂ (t), denoted as ρ₁₂, is obtained as follows

$${\rho }_{12}=\frac{{{\displaystyle \int }}_0^{\infty }2\rho \sqrt{{\mathsf{\Phi }}_{11}\left(\omega \right){\mathsf{\Phi }}_{22}\left(\omega \right)}d\omega }{\sqrt{E\left[X_1^2\left]E\right[X_2^2\right]}}$$

(27)

Then, for a given ρ₁₂, ρ can be calculated from (27) and used to generate U_ij and V_ij in (21).

The advantage of the above method is that the arbitrary spectral densities of the two processes can be matched exactly, and that only the correlation coefficient is needed regarding the correlation information of the two processes. However, the generated processes are not ergodic, and a large number of samples is required in simulations.

The above method was used to model two correlated ocean wave forces on the ship rolling motion. One of the commonly used spectral densities for the ocean wave forces is known as the Pierson–Moskowits spectrum [21] which is used in [5,19,20,22,23,24,25,26,27]. The spectral density is controlled by two parameters, the peak frequency ω_p and the mean-square value σ² specifying the intensity of the process. It is represented by

$$\mathsf{\Phi }\left(\omega \right)=\frac{\mathsf{\Phi }\left({\omega }_p\right)e^{5/4}}{{({\omega }^{*})}^5}\exp \left[-\frac{5}{4{({\omega }^{*})}^4}\right],{\omega }^{*}=\frac{\omega }{{\omega }_p}, \omega >0$$

(28)

where the peak value of the spectrum is determined by

$$\mathsf{\Phi }\left({\omega }_p\right)=0.7163{\sigma }^2/{\omega }_p$$

(29)

The two correlated ocean wave forces with the Pierson–Moskowits spectrum of a different peak frequency ${\omega }_p$ and/or different intensity σ² can be modeled using the series expansion Equation (17). Some numerical results are shown in Figure 3 for two processes with the same ${\omega }_p$ and σ², and in Figure 4 for those of a different peak ${\omega }_p$ and the same σ². The analytical results are obtained from Equation (25), and the simulation results are calculated from Equation (17).

4. Method of Series Expansion with Random Phases

Simulations of stochastic processes using a series with random phases based on their spectral representations were proposed and developed in [11,12].

4.1. Simulation of a Single Stationary Gaussian Process

A Gaussian process X(t) can be simulated from

$$X_s\left(t\right)=2\sqrt{\mathsf{\Delta }\omega }{\displaystyle \sum }_{j=1}^N\sqrt{{\mathsf{\Phi }}_{XX}\left({\omega }_j\right)}\cos ({\omega }_{j}t+{\varphi }_j)$$

(30)

where ω_j and Δω are determined from Equation (15), and ϕ_j are independent random variables uniformly distributed in [0, 2π). In each term in Equation (30), $\cos ({\omega }_{j}t+{\varphi }_j)$ represents a random variable for a given t. For different j, these random variables are independent and have an identical probability distribution. According to the central limit theorem, the probability distribution of X_s(t) in (30) converges to a Gaussian distribution as the number of terms N approaches infinity. The correlation function of X_s(t) is obtained as

$$\begin{array}{c}{R}_{X_s{X}_s}\left(\tau \right)=E\left[X_s\left(t\right)X_s\left(t+\tau \right)\right]\hfill \\ =4\mathsf{\Delta }\omega {\displaystyle \sum }_{j=1}^N{\mathsf{\Phi }}_{XX}\left({\omega }_j\right)E\left\{\cos ({\omega }_{j}t+{\varphi }_j)\cos [{\omega }_j\left(t+\tau \right)+{\varphi }_j]\right\}\hfill \\ =2\mathsf{\Delta }\omega {\displaystyle \sum }_{j=1}^N{\mathsf{\Phi }}_{XX}\left({\omega }_j\right)\cos {\omega }_j\tau \hfill \end{array}$$

(31)

Equation (31) is the same as Equation (18), and the spectral density of X_s(t) is the same as that given in Equations (19) and (20), respectively.

4.2. Generation of Two Correlated Stationary Gaussian Processes

Assume that the spectral density functions of two correlated stochastic processes X₁(t) and X₂(t) are given as Φ₁₁($\omega$), Φ₁₂($\omega$), Φ₂₁($\omega$), and Φ₂₂($\omega$). In matrix form, we have

$$\mathsf{\Phi }\left(\omega \right)=\left[\begin{array}{cc}{\mathsf{\Phi }}_{11}\left(\omega \right)& {\mathsf{\Phi }}_{12}\left(\omega \right)\\ {\mathsf{\Phi }}_{21}\left(\omega \right)& {\mathsf{\Phi }}_{22}\left(\omega \right)\end{array}\right]$$

(32)

It is known that for i, j = 1, 2,

$$\begin{gathered} {\mathsf{\Phi }}_{ii}\left(-\omega \right)={\mathsf{\Phi }}_{ii}\left(\omega \right)\ge 0 \\ {\mathsf{\Phi }}_{ij}\left(\omega \right)={\mathsf{\Phi }}_{ji}^{*}\left(\omega \right), {\mathsf{\Phi }}_{ij}\left(-\omega \right)={\mathsf{\Phi }}_{ij}^{*}\left(\omega \right) \end{gathered}$$

(33)

where the asterisk denotes the complex conjugate. Equation (33) indicates that the power spectral density functions are real, nonnegative, and even, and the cross-spectral densities are generally complex, with even real parts and odd imaginary parts. Denote a high triangular matrix H($\omega$) as

$$H\left(\omega \right)=\left[\begin{array}{cc}{H}_{11}\left(\omega \right)& H_{12}\left(\omega \right)\\ 0& H_{22}\left(\omega \right)\end{array}\right]$$

(34)

which satisfies

$$H\left(\omega \right){[H^{*}\left(\omega \right)]}^{\mathrm{T}}=\Phi \left(\omega \right)$$

(35)

where the superscript T denotes the transpose of the matrix. It is found from (35) that

$$H_{11}\left(\omega \right)=\sqrt{{\mathsf{\Phi }}_{11}\left(\omega \right)-{\left|H_{12}\left(\omega \right)\right|}^2}, H_{22}\left(\omega \right)=\sqrt{{\mathsf{\Phi }}_{22}\left(\omega \right)},H_{12}\left(\omega \right)=\frac{{\mathsf{\Phi }}_{12}\left(\omega \right)}{H_{22}\left(\omega \right)}$$

(36)

The two processes can then be simulated from

$$\begin{gathered} X_{1s}\left(t\right)=2\sqrt{\mathsf{\Delta }\omega }{\displaystyle \sum }_{j=1}^N{H}_{11}\left({\omega }_j\right)\cos ({\omega }_{j}t+{\varphi }_{1j}) \\ +2\sqrt{\mathsf{\Delta }\omega }{{\displaystyle \sum }}_{j=1}^N\left|H_{12}\left({\omega }_j\right)\right|\cos \left[{\omega }_{j}t-\theta \left({\omega }_j\right)+{\varphi }_{2j}\right] \end{gathered}$$

(37)

$$X_{2s}\left(t\right)=2\sqrt{\mathsf{\Delta }\omega }{\displaystyle \sum }_{j=1}^N{H}_{22}\left({\omega }_j\right)\cos ({\omega }_{j}t+{\varphi }_{2j})$$

(38)

where ${\omega }_j$ and Δ$\omega$ are determined from Equation (15); ${\varphi }_{1j}$ and ${\varphi }_{2j}$ are mutually independent random variables uniformly distributed in [0, 2π), and they are also independent for different j, and θ(ω) is determined from $H_{12}\left(\omega \right)=\left|H_{12}\left(\omega \right)\right|\exp [i\theta \left(\omega \right)]$, namely,

$$\sin \theta \left(\omega \right)=\frac{Im[H_{12}\left(\omega \right)]}{\left|H_{12}\left(\omega \right)\right|},\cos \theta \left(\omega \right)=\frac{\mathrm{Re}[H_{12}\left(\omega \right)]}{\left|H_{12}\left(\omega \right)\right|}$$

(39)

It is known from Equations (32), (36), and (39) that function θ($\omega$) is an odd function, i.e.,

$$\theta \left(-\omega \right)=-\theta \left(\omega \right)$$

(40)

Equation (38) has the same form as Equation (30). The correlation function of X_1s(t) is obtained as

$$\begin{array}{c}{R}_{X_{1s}{X}_{1s}}\left(\tau \right)=E\left[X_{1s}\left(t\right)X_{1s}\left(t+\tau \right)\right]\hfill \\ =4\mathsf{\Delta }\omega {\displaystyle \sum }_{j=1}^N{[H_{11}\left({\omega }_j\right)]}^{2}E\left\{\cos ({\omega }_{j}t+{\varphi }_{1j})\cos [{\omega }_j\left(t+\tau \right)+{\varphi }_{1j}]\right\}\hfill \\ +4\mathsf{\Delta }\omega {\displaystyle \sum }_{j=1}^N{\left|H_{12}\left({\omega }_j\right)\right|}^{2}E\left\{\cos [{\omega }_{j}t-\theta \left({\omega }_j\right)+{\varphi }_{2j}\left]\cos [{\omega }_j\left(t+\tau \right)-\theta \left({\omega }_j\right)+{\varphi }_{2j}\right]\right\}\hfill \\ =2\mathsf{\Delta }\omega {\displaystyle \sum }_{j=1}^N\left[{[H_{11}\left({\omega }_j\right)]}^2+{\left|H_{12}\left({\omega }_j\right)\right|}^2\right]\cos {\omega }_j\tau =2\mathsf{\Delta }\omega {\displaystyle \sum }_{j=1}^N{\mathsf{\Phi }}_{11}\left({\omega }_j\right)\cos {\omega }_j\tau \hfill \end{array}$$

(41)

which has the same form as Equation (31). The cross-correlation functions of X_1s(t) and X_2s(t) are obtained as

$$\begin{gathered} R_{X_{1s}{X}_{2s}}\left(\tau \right)=E\left[X_{1s}\left(t\right)X_{2s}\left(t+\tau \right)\right] \\ =4\mathsf{\Delta }\omega {{\displaystyle \sum }}_{j=1}^N\left|H_{12}\left({\omega }_j\right)\right|H_{22}\left({\omega }_j\right)E\left\{\cos [{\omega }_{j}t-\theta \left({\omega }_j\right)+{\varphi }_{2j}\left]\cos [{\omega }_j\left(t+\tau \right)+{\varphi }_{2j}\right]\right\} \\ =2\mathsf{\Delta }\omega {{\displaystyle \sum }}_{j=1}^N\left|H_{12}\left({\omega }_j\right)\right|H_{22}\left({\omega }_j\right)\cos \left[{\omega }_j\tau +\theta \left({\omega }_j\right)\right] \end{gathered}$$

(42)

When Δω → 0 and N → ∞, Equation (42) tends to

$$R_{X_{1s}{X}_{2s}}\left(\tau \right)\to {{\displaystyle \int }}_{-\infty }^{\infty }\left|H_{12}\left(\omega \right)\right|H_{22}\left({\omega }_j\right)\cos [\omega \tau +\theta \left(\omega \right)]d\omega$$

(43)

Since θ($\omega$) is an odd function, we have

$${{\displaystyle \int }}_{-\infty }^{\infty }\left|H_{12}\left(\omega \right)\right|H_{22}\left({\omega }_j\right)\sin [\omega \tau +\theta \left(\omega \right)]d\omega =0$$

(44)

Taking (44) into account, Equation (43) can be rewritten as

$$\begin{gathered} R_{X_{1s}{X}_{2s}}\left(\tau \right)\to {{\displaystyle \int }}_{-\infty }^{\infty }\left|H_{12}\left(\omega \right)\right|H_{22}\left({\omega }_j\right)\exp \left\{i\left[\omega \tau +\theta \left(\omega \right)\right]\right\}d\omega \\ ={{\displaystyle \int }}_{-\infty }^{\infty }{H}_{12}\left(\omega \right)H_{22}\left(\omega \right)\exp (i\omega \tau )d\omega ={{\displaystyle \int }}_{-\infty }^{\infty }{\mathsf{\Phi }}_{12}\left(\omega \right)\exp (i\omega \tau )d\omega =R_{12}\left(\tau \right) \end{gathered}$$

(45)

In deriving (45), Equations (36) and (39) are used. Equation (45) indicates that the cross-correlation function of the processes given by (37) and (38) approaches that of the target processes.

As an example to illustrate the procedure, take the three processes X₁(t), X₂(t), and X₃(t) in system (9) for numerical calculation. Their cross-correlations R₁₂(τ) and R₁₃(τ) are given in Equations (11) and (12). The spectral densities can be obtained by using the Fourier transform, or by using the method described in [2], as follows:

$${\Phi }_{11}\left(\omega \right)=\frac{K_{11}}{{\alpha }^2+{\omega }^2},{ \mathsf{\Phi }}_{22}\left(\omega \right)=\frac{K_{22}}{\mathsf{\Delta }},{ \mathsf{\Phi }}_{33}\left(\omega \right)=\frac{K_{22}{\omega }^2}{\mathsf{\Delta }}$$

(46)

$$\begin{gathered} {\Phi }_{12}\left(\omega \right)=\frac{E\left[X_1{X}_2\right]}{2\pi }\left\{\frac{\alpha }{{\alpha }^2+{\omega }^2}+\frac{1}{\mathsf{\Delta }}\left[\alpha \left({\omega }_0^2-{\omega }^2\right)+2\zeta {\omega }_0^3\right]\right. \\ +i\left.\left(\frac{\omega }{{\alpha }^2+{\omega }^2}+\frac{1}{\mathsf{\Delta }}\left[\omega \left({\omega }_0^2-{\omega }^2\right)-2\zeta \omega {\omega }_0\left(\alpha +2\zeta {\omega }_0\right)\right]\right)\right\} \end{gathered}$$

(47)

$$\begin{gathered} {\Phi }_{13}\left(\omega \right)=\frac{E\left[X_1{X}_2\right]}{2\pi }\left\{\frac{{\alpha }^2}{{\alpha }^2+{\omega }^2}+\frac{1}{\mathsf{\Delta }}\left[2\alpha \zeta {\omega }_0{\omega }^2-{\omega }_0^2\left({\omega }_0^2-{\omega }^2\right)\right]\right. \\ +i\left.\left(\frac{\alpha \omega }{{\alpha }^2+{\omega }^2}+\frac{1}{\mathsf{\Delta }}\left[\alpha \omega \left({\omega }_0^2-{\omega }^2\right)+2\zeta \omega {\omega }_0^3\right]\right)\right\} \end{gathered}$$

(48)

where E[X₁X₂] is given in Equation (14), and Δ is given by

$$\mathsf{\Delta }={({\omega }_0^2-{\omega }^2)}^2+4{\zeta }^2{\omega }^2{\omega }_0^2$$

(49)

Numerical calculations are carried out for the same parameter values as those used in Figure 1 and Figure 2, i.e., α = 0.6, ζ = 0.3, ω₀ = 1.3, K₁₁ = 1, K₂₂ = 0.5, and K₁₂ = 0.1. The cross-correlation functions R₁₂($\tau$) and R₁₃($\tau$) are depicted in Figure 5 and Figure 6, respectively. The analytical results are calculated from Equations (11) and (12), and the simulation results are obtained using Equations (37) and (38) with the spectral densities given by Equations (46)–(48).

It is clear that the method of the series expansion is the most sophisticated method if the cross-correlation function or the cross-spectral density of the two processes is known. The same disadvantage is presented as that of the method of the series expansion with random amplitudes, specifically, that the generated processes are not ergodic.

5. Conclusions

Three methods to generate two correlated stationary Gaussian processes are reviewed and presented systematically. Their accuracy is substantiated by simulations. The three methods have their own advantages and disadvantages, and they need to be applied in different situations.

The method of linear filters offers a simple way to simulate two processes with a given power spectral density for each process and with their joint moment or correlation coefficient. The generated processes are ergodic after a short period of time. This is a valuable advantage for simulations since only a single sample function is needed for calculation. For low-pass processes and processes of a single spectrum peak, the method is especially feasible by simply using first-order and second-order filters. However, if processes with more complicated shapes of spectra, such as multiple spectrum peaks, higher-order filters are required, and it is more difficult to determine the parameters in the filters.

Similar to the method of linear filters, the method of series expansion with random amplitudes is also based on the power spectra of the two processes and their correlation coefficient. The advantage of the method is the exact match of the power spectral density for each process. However, the modeled stochastic processes are not ergodic, and large numbers of sample functions must be generated for simulations.

The above two methods only use the correlation coefficient of the two correlated processes, which is incomplete information of the correlation. If the correlation function, as the complete information of the correlation of the two processes, is known, the method of series expansion with random phases is the most accurate. It generates the two processes with exactly matched power spectral densities and cross-spectral density. Similar to the series expansion method with random magnitudes, the generated processes are not ergodic for an arbitrary length of time period even it is very long.

In practical problems, the correlation function may not be known, and only the extent of the correlation can be estimated; thus, the first two methods are more feasible.