## 1 Introduction  
Over the past decades, partially coherent beams (PCBs) have attracted wide attention due to their important applications, such as particle trapping/manipulation, wide-field optical coherence tomography, laser material processing, optical metrology, 3D imaging and, so on [1; 2; 3; 4; 5; 6; 7; 8; 9]. Generally, PCBs are characterized by the so-called cross-spectral density (CSD) function in the space-frequency domain or mutual coherence function (MCF) in the space-time domain, and the coherence characteristics of PCBs are described by the degree of coherence (DOC) which can be obtained from the CSD function or the MCF. It is known that the DOC plays a central role in determining the beam propagation characteristics, image resolution, and effects of interaction with matter [10; 11; 12; 13; 14]. However, most studies are focused on the PCBs whose DOCs are of Gaussian function because it is a hard task to devise a physically realizable CSD function with arbitrary DOC. Recently, [PERSON] and co-workers proposed a sufficient condition for mathematically devising a genuine CSD function [15; 16]. Since then, researchers have turned their attention to study PCBs with special DOCs (non-Gaussian function) [17; 18; 19; 20; 21]. Owing to their special DOCs, these beams exhibit peculiar propagation features even in free space. For example, a non-uniformly correlated (NUC) beam displays \"self-focusing\" and lateral shift of intensity maxima during free-space propagation [17]. The multi-Gaussian-correlated beam and Laguerre-Gaussian (LG)-correlated beam will produce a flat-topped beam profile and ring-shaped beam profile in the far field, respectively [18; 19]. In addition, the DOCs may carry the information of the beam itself or of the object with which the PCBs interact [22]. For example, by measuring the far-field DOC [23], the azimuthal and radial mode orders of a partially coherent LG\({}_{pl}\) beam can be determined. In [14], the authors point out that, when a PCB is obstructed by an opaque obstacle, its DOC can self-reconstruct. In addition, several methods have been proposed to experimentally generate the prescribed DOC of a PCB, for example by using a spatial light modulator, intra-cavity modulation and a deformable mirror, etc. [24; 25; 26; 27]. However, the aforementioned studies are confined to PCBs whose initial intensity distributions have a Gaussian function. On the other hand, PCBs with both initial non-Gaussian beam profiles and DOCs have also received considerable attention in recent years [28; 29; 30; 31; 32; 33; 34]. In the following text, we use the term general PCBs (GPCBs) to include a class of PCBs with non-Gaussian amplitudes and DOCs. Through controlling the DOCs and beam profiles in the source plane simultaneously, GPCBs can generate a far-field beam array with controllable forms [32; 33], and the profile of each beamlet in the array is also adjustable, making them ideal sources for particle trapping and manipulation. Nevertheless, in these studies, the evolution properties of the DOC during propagation were neglected. Though other studies [35; 36] presented the reciprocity relation between the initial beam profile and the DOC in the far filed, this is only applicable for special kinds of GPCBs, known as quasi-homogeneous beams. In such beams, the variation of the DOCs with the spatial point r is much faster than that of intensity distribution with r. Up to now, only few papers have dealt with the evolution of the DOCs during GPCBs passage through free space or paraxial optical systems, especially for those beams whose DOCs or initial intensity distributions are impossible to obtain using an analytical expression. In this paper, we propose a numerical approach for calculating the DOC of a GPCB in any transverse plane perpendicular to the propagation axis during propagation in free space or paraxial optical systems. In this approach, we introduce white noises [37; 38; 39] into the convolution approach [40] to generate speckle patterns in any transverse plane of propagation.