Kirchhoff's complex spatial convolution formula is developed as a new algorithm for radiation from a aperture. It is based on hte idea of analytic ectention that the original aperture field is extended to a complex plane.and then the kirchhoff's complex spatial-convolution formula makes up an indirect and rigorous representation of radiation from a plane aperture.Numerical tests confirm its correctness and efficiency.

The wave equation initial value problems are solved numerically on the grid of hexagonal sample by fourier method. A numerical test by this approach shows the availability and the efficiency over the rectangular sample.