Digital Communication Systems

APPENDIX

D  Method of Lagrange Multipliers

D.1  Optimization Involving a Single Equality Constraint

Consider the minimization of a real-valued function ƒ(w) that is a quadratic function of a parameter vector w, subject to the constraint

where s is a prescribed vector and g is a complex constant; the superscript denotes Hermitian transposition. We may redefine the constraint by introducing a new function c(w) that is linear in w, as shown by

In general, the vectors w and s and the function c(w) are all complex. For example, in a beamforming application, the vector w represents a set of complex weights applied to the individual sensor outputs and s represents a steering vector whose elements are defined by a prescribed “look” direction; the function ƒ(w) to be minimized represents the mean-square value of the overall beamformer output. In a harmonic retrieval application, for another example, w represents the tap-weight vector of an FIR filter and s represents a sinusoidal vector whose elements are determined by the angular frequency of a complex sinusoid contained in the filter input; the function ƒ(w) represents the mean-square value of the filter output. In any event, assuming ...