In grating-based x-ray phase contrast imaging, Fourier component analysis (FCA) is usually recognized as a gold standard to retrieve the contrasts including attenuation, phase and dark-field, since it is well-established on wave optics and is of high computational efficiency. Meanwhile, an alternative approach basing on the particle scattering theory is being developed and can provide similar contrasts with FCA by calculating multi-order moments of deconvolved small-angle x-ray scattering, so called as multi-order moment analysis (MMA). Although originated from quite different physics theories, the high consistency between the contrasts retrieved by FCA and MMA implies us that there may be some intrinsic connections between them, which has not been fully revealed to the best of our knowledge. In this work, we present a Fourier-based interpretation of MMA and conclude that the contrasts retrieved by MMA are actually the weighted compositions of Fourier coefficients, which means MMA delivers similar physical information as FCA. Based on the recognized cosine model, we also provide a truncated analytic MMA method, and its computational efficiency can be hundreds of times faster than the original deconvolution-based MMA method. Moreover, a noise analysis for our proposed truncated method is also conducted to further evaluate its performances. The results of numerical simulation and physical experiments support our analyses and conclusions.