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 ambiguit y functions are derived. Based on a similar formulation for the projections of the auto and cross Wigner distributions and the well known two-dimensio nal (2-D) Fourier transformation relationship between the ambiguity and Wig ner domains, closed-form expressions are obtained for the slices of both th e Wigner distribution and the ambiguity function, By using discretization o f the obtained analytical expressions, efficient algorithms are proposed to compute uniformly spaced samples of the Wigner distribution and the ambigu ity function located on arbitrary line segments. With repeated use of the p roposed algorithms, samples in the Wigner or ambiguity domains can be compu ted on non-Cartesian sampling grids, such as polar grids.