Theory and design of two-parallelogram filter banks

It is well known that the analysis and synthesis filters of orthonormal I)FT filter banks tan not have good frequency selectivity. The reason for this is that each of the analysis and synthesis filters have only one passhand. Such frequency stacking (or configuration) in general does not allow alias cancelation when the individual filters have good stopband attenuation. A frequency stacking of this nature is called nonpcrmissible and should be avoided if good filters are desired. In a usual Mchanncl filter bank with real-coefficient filters, the analysis and synthesis filters have two passhands. It can he shown that the configuration is permissible in this case. Many designs proposed in the past demonstrate that filter banks with such configurations can have perfect reconstruction and good niters at the same time. In this paper, we develop the two-parallelogram filter banks, which is the class of 2-D filter banks in which the supports of the analysis and synthesis filters consist of two parallelograms. The two-parallelogram filter banks are analyzed from a pictorial viewpoint by exploiting the concept of permissibility. Based on this analysis, we construct and design a special type of twoparallelogram filter banks, namely, cosine-modulated filter banks (CMFB). In two-parallelogram CMFB, the analysis and synthesis filters are cosine-modulated versions of a prototype that has a parallelogram support. Necessary and sufficient conditions for perfect reconstruction of two-parallelogram CMFB will be derived in the paper.