Abstract: The radar signature of complex target is a multi-dimensional function with many variables. The measurement from a special radar only obtains a slice or a projection of the multi-dimensional function in a lower dimensional space. Fully exploring the target signature from the limited radar measurement is helpful in target detection and recognition. Based on the three dimensional scattering distribution function(SDF), the Fourier transform relation between the radar image and the integral projection of the SDF is deduced; the implicit simplification in the SDF is analyzed based on the complex scattering phenomenon of real targets; and the formalized multi-dimensional scattering function is derived. The low dimensional representation of this multi-dimensional scattering function is analyzed from the aspects of scattering center model and radar image, and several examples are presented, such as the phase reversal of edge scatterers, the polarization scattering matrix of multiple scattering, the scattering dispersion of cavities in spatial targets, and the image defocus of flexible structures, and so on. The multi-dimensional scattering model and its lower dimensional representation can be implemented in analyzing the scattering signature of complex radar targets.