Fast computation of the ambiguity function and the Wigner distribution on arbitrary line segments

By using the fractional Fourier transformation of the time-domain signals, closed-form expressions for the projections of their auto or cross ambiguity functions are derived. Based on a similar formulation for the projections of the auto and cross Wigner distributions and the well known two-dimensional (2-D) Fourier transformation relationship between the ambiguity and Wigner domains, closed-form expressions are obtained for the slices of both the Wigner distribution and the ambiguity function, By using discretization of the obtained analytical expressions, efficient algorithms are proposed to compute uniformly spaced samples of the Wigner distribution and the ambiguity function located on arbitrary line segments. With repeated use of the proposed algorithms, samples in the Wigner or ambiguity domains can be computed on non-Cartesian sampling grids, such as polar grids.