## 1 Introduction  
With the rapid development of computer technology, Monte Carlo simulation is widely used in the field of engineering to solve complex problems, such as nonlinearity, system randomness, and stochastic stability [1; 2]. An important part of the Monte Carlo simulation is to simulate the samples of random excitations, such as wind, ground motion, and wave [3]. Wind field and ground motion are usually modeled as random processes, while wave field is usually modeled as random field or random wave. Depending on the situation, the above simulations can be classified as one-dimensional or multidimensional [1; 4], single or multi-point [5; 6], ergodic or non-ergodic [6; 7], stationary or nonstationary [8; 9; 10; 11], or Gaussian or non-Gaussian [12; 13; 14; 15; 16]. In this study, the wind field is taken as an example to research the simulation of one-dimensional, multi-point, and Gaussian random processes. The one-dimensional, multi-point, and Gaussian random wind field can be classified as four types: non-ergodic stationary wind field, ergodic stationary wind field, time-invariant coherent nonstationary wind field, and time-varying coherent nonstationary wind field, based on the probabilistic characteristics. The task of wind field simulation is to generate sample functions having prescribed probabilistic characteristics. In wind field simulation, the spectral representation method is more widely used than other approaches, such as the time series method and time-frequency analysis method, because of its accuracy and simplicity. In recent decades, researchers have done a lot of work to optimize and improve the spectral representation method in wind field simulation. In the simulation of the non-ergodic stationary wind field, [PERSON] et al. [7] proposed a closed-form solution of Cholesky decomposition to simplify the spectral matrix decomposition. [PERSON] et al. [17] introduced an improved fast Fourier transform (FFT) method to simulate the short-term wind velocity field. In addition, some methods based on stochastic wave were developed [18; 19]. In terms of the ergodic stationary wind field, [PERSON] et al. [20] introduced the closed-form solution of Cholesky decomposition. [PERSON] et al. [21] proposed an efficient simulation method for vertically distributed wind velocity field, based on approximate piecewise wind spectrum. [PERSON] and [PERSON] [22] proposed an enhanced closed-form solution of Cholesky decomposition for the wind field along an arbitrary axis. [PERSON] et al. [23] established an enhanced spectral representation method with high accuracy and efficiency for the ergodic wind field simulation. Additionally, the interpolation schemes were also developed to optimize the spectral matrix decomposition [24; 25]. For the time-invariant coherent nonstationary wind field, the spectral matrix decomposition is the same as the stationary simulation, and thus most studies focus on the introduction and optimization of FFT [26; 27; 28; 29]. Besides, the stochastic wave approach is also introduced to the nonstationary simulation by proper orthogonal decomposition [30]. The simulation of the time-varying coherent nonstationary wind field is optimized and enhanced mainly based on the interpolation and time-frequency decoupling techniques [31; 32; 33; 34]. However, the classical phase formula in almost all the above-mentioned simulation methods is not rigorous. This may cause misunderstanding for the wind field simulation, considering the wave passage effect. In this study, the influence of the classical phase formula on the simulation of stationary and nonstationary wind fields is analyzed and discussed in depth. Firstly, the theory of the wind field simulation based on spectral representation method are reviewed. Secondly, two treatment schemes of deterministic phases in the simulation formula are introduced. Then, the influence of phase on correlation functions in four types of the wind field simulation is qualitatively discussed, based on the theoretical formula.