SOME RESULTS IN THE THEORY MODULATED FILTER BANKS AND MODULATED WAVELET TIGHT FRAMES

Perfect reconstruction (PR) FIR filter banks, obtained by modulation o f a linear-phase, lowpass, prototype filter and of length 2Mm are well known. Recently PR modulated filter banks (MFBs) with the analysis an d synthesis banks obtained from different prototypes have been reporte d. This paper describes a general form of modulation that includes mod ulations used in the literature. This modulation depends on an integer parameter, the modulation phase. The PR property is characterized for MFBs with finite and infinite impulse response filters. The MFB PR pr oblem reduces to roughly M/2 two-channel PR problems. A natural dichot omy in the PR conditions leads us to the concepts of Type 1 and Type 2 MFBs. Unitary MFBs are characterized by the M/2 two-channel PR filter banks also being unitary (for FIR filters of length N = 2Mm, these re sults are given in (Malvar, Electr. Lett. 26, June 1990, 906-907; Koil pillai and Vaidyanathan, IEEE Trans. SP 40, No. 4, Apr. 1992, 770-783) ). We also give a necessary and sufficient condition for a large class (including FIR) unitary MFB prototypes to have symmetric (even or odd ) prototype filters, and exhibit unitary MFBs without symmetric protot ypes. A parameterization of all FIR unitary MFBs is also given. An eff icient design procedure for FIR unitary MFBs is developed. it turns ou t that MFBs can be implemented efficiently using Type III and Type IV DCTs. Compactly supported modulated wavelet tight frames are shown to exist and completely parameterized. K-regular modulated WTFs are desig ned numerically and analytically by solving a set of non-linear equati ons over the parameters. Design of optimal modulated WTFs for the repr esentation of any given signal is described with examples, and this is used to design smooth modulated WTFs. (C) 1995 Academic Press. Inc.