In this paper, a new numerical method is proposed to solve the exponential interpolation problem via solving a special polynomial system. Exploiting the special structure of the polynomial system, it is proved that: for generic data, all its solutions belong to one equivalence class. Furthermore, an efficient polynomial homotopy continuation method, in which only one path is needed to be traced, is proposed. The proposed method does not require the assumption of equally spaced sampling or positive weighted coefficients as the existing algorithms like Prony method and Newton homotopy method, and it is globally convergent for generically chosen parameters in the homotopy. Preliminary numerical tests
show that the new method is promising.