Discrete Lagrangian methods for optimizing the design of multiplierless QMF banks

In this paper, we present a new discrete Lagrangian method for designing multiplierless quadrature mirror filter banks. The filter coefficients in these filter banks are in powers-of-two, where numbers are represented as sums or differences of powers of two (also called canonical signed digit representation), and multiplications are carried out as additions, subtractions, and shifts. We formulate the design problem as a nonlinear discrete constrained optimization problem, using reconstruction error as the objective, and stopband and passband energies, stopband and passband ripples, and transition bandwidth as constraints. Using the performance of the best existing designs as constraints, we search for designs that improve over the best existing designs with respect to all the performance metrics. We propose a new discrete Lagrangian method for finding good designs and study methods to improve the convergence speed of Lagrangian methods without affecting their solution quality. This is done by adjusting dynamically the relative weights between the objective and the Lagrangian part. We show that our method can find designs that improve over Johnston's benchmark designs using a maximum of three to six ONE bits in each filter coefficient instead of using floating-point representations. Our approach is general and is applicable to the design of other types of multiplierless filter banks.