| dc.description.abstract | Perfect reconstruction filter banks (PRFBs) are widely used for signal decomposition, subband coding, subband adaptive filtering, etc. Both IIR and FIR filter banks are used in the applications mentioned above. Causal stable IIR PRFBs are popular as they have good responses and preprocessing of input is not necessary. For the 2?channel case, efficient design methods have been developed in the literature. For the M?channel case, existing design procedures are complicated. General design methods for causal stable IIR PRFBs have received less interest. FIR PRFBs are popular as they are easy to implement. For a class of PRFBs, namely paraunitary PRFBs, complete characterizations and efficient design methods have been developed. However, the problem of complete characterization of general FIR PRFBs has received less attention. Recently, complete characterizations of general FIR PRFBs for which the analysis polyphase is of order one have been developed. However, the design of a large class of general FIR PRFBs with analysis polyphase order greater than one has not been provided. In this work, the problem of designing wider classes of IIR causal stable PRFBs and general FIR PRFBs, with the order of the analysis polyphase matrix greater than one, is addressed.
First, some design approaches for M?channel IIR causal stable PRFBs are developed. The analysis polyphase matrix is realized in state?space form, and minimal characterizations are used to avoid pole?zero cancellation. The inverse system is explicitly given if the analysis polyphase matrix is invertible at z=?z = \inftyz=?. All the design methods are based on forcing the poles of the analysis polyphase matrix and its inverse system inside the unit circle. A design method based on the concept of a function of a matrix is proposed. A design method based on similarity transformation is also developed. Further, a factorization?based approach is developed where the analysis polyphase matrix is factorized into degree?1 terms using theorems on factorization of rational matrix functions. Several design examples are provided that compare favorably with existing IIR PRFB designs.
The second part of the work deals with the design of a very wide class of general M?channel FIR PRFBs with the order of the analysis polyphase matrix greater than one (say lll). The analysis polyphase matrix is treated as a regular matrix polynomial so that its inverse exists. The design problem is treated as the inverse problem of constructing a regular matrix polynomial given the spectral data (degree of the determinant of the analysis polyphase matrix) of the regular matrix polynomial. An explicit formula for the inverse given the spectral data is provided. The proposed design allows restrictions on reconstruction delay and the order of the inverse polynomial (synthesis polyphase). Further, near?linear?phase filter banks for which the length of all filters is equal to (l+1)M(l + 1)M(l+1)M are developed. Here also, the design method allows restrictions on reconstruction delay and the order of the inverse polynomial. Lastly, low reconstruction delay is achieved with unimodular matrix polynomials, and a design of low?delay filter banks is also developed. All the proposed designs are illustrated with a few simulation examples. | |
