Instantaneous Erasures in Oversampled Filter Banks: Conditions for Output Perfect Reconstruction

In this paper, we aim at finding the conditions that an oversampled filter bank (OFB) should satisfy, in order to maintain its perfect reconstruction property when erasures happen in the subband domain. This problem has been addressed before mainly from frame-theoretic point of view and only for the case of what we call in this paper classic erasure. In the frame-theoretic approach, the stable filter banks are associated with frames in ℓ²(BBZ) and a subband erasure is defined as the deletion of the frame expansion coefficients corresponding to the frame vectors resulting from each filter and all its translated versions. This is equivalent to assuming that all the samples of a subband have been completely lost (classic erasure) and this is why in this approach it is always assumed that each channel is either working perfectly or not at all. In this paper, we extend this notion of erasure to a situation where subband channels are allowed to be on or off arbitrarily in each time instance and we define a new type of erasure called instantaneous erasure. Using an approach based on the time-domain analysis of perfect reconstruction property, we introduce general conditions for perfect reconstruction of the output and also the sufficient conditions for two classes of filter banks: Causal OFBs with causal inverse and OFBs with maximum robustness against classic erasure.