A linear programming approach to bounding spectral power

The mapping from a finite set of correlation samples to a power density spectrum is not unique. Furthermore, power density spectra exist that take on arbitrary values at a particular frequency and yet are consistent with the correlation samples. Thus values of the spectral density function at a particular frequency cannot be determined without further prior information. In a recent paper, Cybenko comments that tight upper and lower bounds on linear functionals of the spectral density can be obtained as solutions of semi-infinite linear programming problems. In this paper, the primal linear programming problem is interpreted as a search for extremal spectra and the dual linear programming problem is interpreted as a data-adaptive window design procedure. The effect of discretization on both the primal and dual problems is noted. Finally, it is shown how the dual window design problem can be used to design fixed classical type windows for the computation of suboptimal bounds.