Design of symmetric bi-orthogonal double density wavelet filter banks

We look at the design of a class of oversampled filter banks and the resulting framelets. The oversampled property is achieved via an extra subband resulting in double density filter banks (DDFB´s). We design a class of such filters with linear phase property. We look at a special class of framelets from a filter bank perspective, in that we design double density filter banks (DDFB´s). We define type 1 polyphase representation as X (z) = Σ_k=0¹ z^-kX_k(z²) and type 2 polyphase representation as X (z) = Σ_k=0¹ z^kX_k(z²). Polyphase matrices are given in the article, where H˜(z) is the type 1 analysis polyphase matrix, and H(z) is the type 2 synthesis polyphase matrix, we can write the perfect reconstruction condition as [H(z)]^T H˜(z) = I.