The Most General Time-Varying Filter Bank and Time-Varying Lapped Transforms

Abstract

In this paper, a model-based method is used to analyze the most general time-varying filter bank, where all parameters in the filter bank change with time, including filter length, decimator factor, and the number of channel. Through modeling the time-varying filter bank at a fixed time point as a time-invariant filter bank, we change the study of the most general time-varying filter bank to study of the established model. Based on the model, the most general time-varying filter bank is analyzed in both polyphase domain and modulation domain. The general perfect reconstruction (PR) conditions of the most general time-varying filter bank are given using time-varying polyphase matrices in the polyphase domain. In the modulation domain, the time-varying distortion-matrix is presented, and the distortion-free condition of the most general time-varying filter bank is given. Based on the established theory for the most general time-varying filter bank, in the second part of this paper, the time-varying cosine-modulated lapped transform (TV-MLT) as an example is analyzed in detail. An interesting design approach called the window-switching technique, used to design the time-varying prototype windows for a perfectly reconstructed TV-MLT, is introduced. The window-switching technique provides us with the possibility to design a perfectly reconstructed TV-MLT without complex mathematical optimization computation, and simply by graphically building the prototype windows in the transition period based on the given diagrams of the prototype windows before and after the transform changing.