Efficient design of perfect-reconstruction biorthogonal cosine-modulated filter banks using convex Lagrangian relaxation and alternating null-space projections

In essence, designing a perfect-reconstruction (PR) biorthogonal cosine-modulated filter bank (BCM) is a non-convex constrained optimization problem that can be solved in principle using general optimization solvers. However, when the number of channels is large and the order of the prototype filter (PF) is high, numerical difficulties in using those optimization solvers often occur, and the computational efficiency also becomes a concern. This paper proposes an algorithm that carries out the design in two stages. In the first stage, a convex Lagrangian relaxation technique is used to obtain a near PR (NPR) filter bank and, in the second stage, the coefficient vector of the PF obtained is alternately projected onto the null-spaces that are associated with the PR constraints, which turns the NPR filter bank into a PR filter bank. Simulation results are included to demonstrate the robustness of the proposed algorithm for designing BCM filter banks with a large number of channels and high-order PF as well as satisfactory design efficiency.