Return stroke current pulses can propagate at speeds approaching the speed of light c. Such a fast-moving pulse is expected to radiate differently than conventional dipole emitters. In this study, we revisit the theoretical analysis for the high-speed effect on the radiation beam pattern. Instead of starting with specific return stroke models, as has been done before by other investigators, we start the analysis with a general moving current pulse. Through a simple differential transformation between the retarded time and stationary time/space, the so-called F factor (1 - v cos $theta$/c)-1 can be readily obtained. This factor is found to be fundamental and is explicitly associated with the radiation beam pattern but is not limited only to the lossless transmission line (TL) return stroke model. It is demonstrated that different beam pattern factors could be derived from this fundamental factor under different return stroke model assumptions.