In this paper we consider a deterministic worst-case framework for perfect reconstruction of discrete data transmission through a dispersive communication channel. More specifically, we extend our previous work to capture time-varying transmission dynamics that also include the case of linear time-varying preprocessing of the data without increasing the power of the transmitted signal. We present necessary and sufficient conditions for perfect reconstructability and formulate a framework for the synthesis of an optimal preprocessor. In the case of periodic preprocessing followed by (periodic) DFE at the receiving end, we provide a design procedure based on ℓ1-optimization. The application of this procedure to representative communication channels suggests that careful choice of linear time-varying preprocessing can significantly enhance our ability for perfect reconstruction without increasing the power of the transmitted signal.