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The diffracted field due to a general 3-D Gaussian beam (GB) impinging close to an edge of a half-plane is expressed as a sum of diffracted GB's emerging from a discrete set of points and directions along the edge axis. The expansion utilizes an edge-fixed lattice of expansion beams, involving a phase-space beam expansion along the edge and an angular spectrum of beam around the edge. The excitation amplitudes of the diffracted beams, defined as the beam to beam (B2B) scattering matrix, are derived, interpreted, and validated via a comparison with an exact numerical solution. The representation is valid uniformly as a function of the incident beam direction and its distance from the edge. The asymptotic limits when the beam impinges far from the edge are interpreted via the geometrical theory of diffraction (GTD). The general procedure for calculating the B2B scattering coefficients may be applied for other wedge-type configurations. The new formulation enables the construction of self-consistent Gaussian beams summation (GBS) representations in complex configurations, in which the field is described as a sum of beam propagators, and the diffracted fields due to the propagators that hit near edges are also expanded in terms of beams. Applications to GBS modeling of urban propagation are discussed.