Critically sampled discrete wavelet transforms with rational dilation factor of 3/2

An approach for designing a critically sampled FIR rational rate filter bank satisfying conditions such as perfect reconstruction, regularity orders and good frequency selectivity is proposed. This approach is used to design a rational discrete wavelet transform with dilation factor of 3/2 and satisfies a shift invariant property approximately. The rational filter bank design problem is transformed to an equivalent uniform M-channel filter bank design. The perfect reconstruction (PR) condition gives rise to a set of non-linear conditions between the synthesis and analysis filter coeficients while regularity gives rise to a set of linear constraints. The design problem is solved using a constrained nonlinear optimisation program in which the PR and regularity are the constraints to an objective function defined as sum of the squared of the difference between the magnitude responses of the designed analysis/synthesis filters and the ideal brick-wall responses. The optimisation program used for the design is a simple linear-search built-in program in Matlab. The obtained filter bank designs show excellent frequency selectivity and their iterations are approximately shift invariant.