Theory of modulated filter banks and modulated wavelet tight frames

Perfect reconstruction (PR), finite-impulse-response (FIR), unitary modulated filter banks (MFBs) have been constructed by several authors. The forms of modulation used in these works are not the same. In the present work, the authors exhibit the most general form of modulation (which depends on an integral parameter, the modulation phase) and characterize PR for this general MFB problem (no FIR assumptions on the filters are made). An M-channel PR MFB reduces to J( approximately=M/2) two-channel PR FBs. Unitariness of an MFB is equivalent to the unitariness of each of the two-channel FBs. Type 1 and Type 2 MFBs are introduced. All FIR unitary MFBs are parametrized. Unitary MFBs are associated with modulated wavelet tight frames (MWTFs). All compactly supported MWTFs are parametrized. MFBs can be implemented efficiently using Type III and Type II discrete cosine transform (DCT) when the modulation phase is even and Type IV DCT when the modulation phase is odd.<>