A non redundant complex source expansion derived by using spherical modes

In this paper we have derived an analytical expression of the coefficients of a beam expansion from a spherical wave representation by direct complex sampling of the currents. Although many authors have treated beam and CSP expansions, the formulation presented here provides a direct analytical relationship with the spherical harmonics, and clarifies the way to limit the redundancy of the representation with respect the minimum possible number of wave objects, given by the degrees of freedom.