We consider the 2D Helmholtz equation with a complex wavenumber in the exterior of a convex polygonal obstacle, for a general family of Robin type boundary conditions. Using the principle of the Half-Space Matching method, the problem is formulated as a system of coupled Fourier-integral equations: the unknowns are the Robin traces on the infinite straight lines supported by the edges of the polygon. We prove that this system is a Fredholm equation of the second-kind, in an L 2 functional framework. The corresponding numerical method is also analyzed, including the effect of truncation of the Fourier integrals and of the finite element approximation. The theoretical results are supported by various numerical experiments.