Abstract: The formulation of a point target spectrum is a key step in deriving synthetic aperture radar focusing algorithms, which exploits the processing efficiency of the frequency domain. However, the existence of a double-square-root (DSR) in the bistatic range equation makes it difficult to find an exact analytical solution for the 2-D spectrum. In this paper, according to the idea of function optimal approach, we derive a new 2-D point target spectrum on the basis of Legendre polynomial expansion, which is more exact than the existing spectrums during the synthetic aperture time.