Wei, LiyingKennedy, RodneyLamahewa, Tharaka2015-12-101053-587Xhttp://hdl.handle.net/1885/63528This paper introduces a quadratic variational framework for solving a broad class of signal design problems on the 2-sphere. The functional, to be extremized, combines energy concentration measures using a weighting function in the spatial domain, multiplicative weights in the spectral domain, and a total energy constraint. This leads to two formulations of the signal design problem on the 2-sphere, one a Fredholm integral equation in the spatial domain and the other an infinite matrix equation in the spectral domain. The framework is illustrated by deriving the key equations for the two classical spatio-spectral concentration problems on the 2-sphere, and for an isotropic filter design that maximizes the filtered energy. In addition, using the proposed framework, we formulate a joint 3-D beamforming application which achieves optimal directivity and spatial resolution simultaneously.Keywords: 2-sphere; Directional derivative; Directivity; Energy concentration; Filter designs; Fredholm integral equations; isotropic convolution; Key equation; Matrix equations; Signal design; Spatial domains; Spatial resolution; Spectral domains; Total energy; un Directional derivative; isotropic convolution; uncertainty principle; unit sphere201110.1109/TSP.2011.21625062016-02-24