Analytical and numerical methods for efficient calculation of edge diffraction by an arbitrary incident signal

In the present study, diffraction of plane and spherically spreading signals by half-planes is considered. An existing analytical impulse response is investigated, which is exact for plane and approximate for spherical incident signals. It is shown that all its primitive functions with respect to time exist and have an explicit form involving elementary functions. The primitive functions are employed to (i) prove that the convolution of the impulse response with any bounded signal is also bounded for all times, (ii) obtain analytically the diffraction response as a combination of elementary functions for any incident signal approximated piecewise by fitting polynomials, (iii) improve the performance of the numerical convolution by orders of magnitude, and (iv) handle the convolution of very coarsely sampled incident signals. An impulse response is presented for finite-length edges, which, unlike the traditional integration formulas along the edge, is an explicit form of time. Because it is based on the impulse response for infinite edges, it inherits all aforementioned benefits associated with its primitive functions. Furthermore, it offers a substantial computational benefit compared to traditional integration formulas along the edge. Finally, the requirements for the direct application of the presented results to other impulse responses are discussed.