Discrete Lagrangian Method for Optimizing the Design of Multiplierless QMF Filter Banks

In this paper, we present a new discrete Lagrangian optimization method for designing multiplier-less QMF (quadrature mirror lter) lter banks. In multiplierless QMF lter banks, lter coeecients are powers-of-two (PO2) where numbers are represented as sums or diierences of powers of two (also called Canonical Signed Digit{CSD{representation), and multiplications can be carried out as additions , subtractions and shifting. We formulate the design problem as a nonlinear discrete constrained optimization problem, using the reconstruction error as the objective, and other performance metrics as constraints. One of the major advantages of this formulation is that it allows us to search for designs that improve over the best existing designs with respect to all performance metrics, rather than nding designs that trade one performance metric for another. We show that our design method can nd designs that improve over Johnston's benchmark designs using a maximum of three to six ONE bits in each lter coeecient. Digital lter banks have been applied in many engineering elds. Their design objectives consist of their overall metrics and the metrics of each individual lter. Figure 1 summarizes the various design objectives for measuring design quality. In general, lter bank-design problems are multi-objective, continuous, nonlinear optimization problems. Algorithms for designing lter banks are either optimization-based or non-optimization based. In optimization-based methods, a design problem is formulated as a multi-objective nonlinear optimization problem whose form may be application-and lter-dependent. The problem is then converted into a single-objective optimization problem and solved by existing optimization methods, such as gradient-descent, Lagrange-multiplier, quasi-Newton, simulated-annealing, and genetics-based methods. On the other hand, lter-bank design problems have been solved by non-optimization-based algorithms, which include spectral factorization and heuristic methods. These methods generally do not continue to nd better designs once a suboptimal design has been found. In this paper, we study global optimization methods to design multiplierless QMF lter banks. In a two-band QMF lter bank, the reconstructed signal is 3]: