On the study of lossless time-varying filter banks

We study the fundamentals of lossless time-varying filter banks (TVFB). We derive the necessary and sufficient condition of the impulse response of a lossless linear time-varying (LTV) system. We show that some properties of LTI lossless systems continue to hold while some properties fail in the LTV case. For example, all lossless LTV systems are invertible. Moreover the inverse is the FIR if the original lossless system is the FIR. Therefore if the analysis bank of a TVFB is the FIR and lossless, the synthesis bank is also the FIR. Explicit construction of the inverses is given. However, unlike in the LTI case, we show that the inverse system is not necessarily unique or invertible. In fact, the inverse of a lossless LTV system is not necessarily lossless. Thus in a general perfect reconstruction (PR) TVFB, the losslessness of the analysis bank does not imply that of the synthesis bank. Moreover we show that replacing the delay z/sup -1/ in an implementation of a lossless LTV system with z/sup -L/ for integer L in general will result in a non-lossless system. Finally, a PR lossless TVFB in general will not generate a discrete-time orthonormal basis for l/sub 2/.