Constrained quadratic minimizations for signal processing and communications

Constrained minimization problems considered here arise in the design of multi-dimensional subspace beamformers for radar, sonar, seismology, and wireless communications, and in the design of precoders and equalizers for digital communications. The problem is to minimize a quadratic form, under a set of linear or quadratic constraints. We derive the solutions to these problems and establish connections between them. We show that the quadratically-constrained problem can be solved by solving a set of linearly-constrained problems and then using a majorization argument and Poincare's separation theorem to determine which linearly-constrained problem solves the quadratically-constrained one. In addition, we present illuminating circuit diagrams for our solutions, called generalized sidelobe canceller (GSC) diagrams, which allow us to tie our constrained minimizations to linear minimum mean-squared error (LA/IMSE) estimations.