Optimal Two-Sided Diagonal Scaling for Filter Bank Frames

Abstract

Two-sided diagonal scaling for transfer matrices is formulated. An efficient algorithm is proposed to compute globally optimal solutions using the iterated bi-section and linear matrix inequality (LMI) solver. It is shown that the two-sided scaling of filter bank (FB) frames can be implemented by the adjustment of channel gains and the periodic precoding of source signal, and that the frame-bound-ratio of FB frames can be effectively improved by such scaling. Explicit formulas are established for both uniform and nonuniform FB frames, including detail formulas for discrete Weyl-Heisenberg frames and tree-structured FBs (discrete wavelets). Different examples show the effectiveness of the obtained results.Index Terms-Condition number, diagonal scaling, discrete wavelets, filter banks, frame bound ratio.