Abstract
Through a simple angle substitution, we can obtain the corresponding relationship between frequency shift and phase difference. Based on this relationship, the Doppler shift can be obtained by simple phase difference measurement. However, the reason why the function relation between phase shift and frequency shift can produce the expansion efficiency is also due to the relationship, that is obtained by the first order change of distance, between the rate of change of phase shift and the Doppler frequency shift. It's just a simple mathematical generalization of what's already known. Further, based on the two basic modes of angle permutation and firstorder change, various relations between phase shift and frequency shift, as well as their first-order change, can be determined. This mathematical description extends to physical applications, enabling many motion parameters and observations that would otherwise be difficult to detect to be obtained by simple phase shift/difference detection.