A description of periodically time-varying systems was developed which completely characterizes the relationship between the spectra of the input and output signals of these systems. The description is the bi-frequency map. This description completely separates the time-invariant and time-varying portions of the system. It can be used to analyze each of these portions of the system. When this description was applied to a number of periodically time-varying systems, the analysis often provided new insights and facts about the characteristics of the systems which led to new and improved designs. For example, it was shown that the bi-frequency map of any discrete time-varying system could be approximated with a periodically time-varying system and an optimal design procedure was developed. New criteria for designing the filters with decimator-interpolator structures were derived. With these new design criteria more efficient filters were designed. When these techniques were applied to filters with highly quantized coefficients that dither or vary periodically with time, it was shown that under general circumstances the optimal structure is time-invariant (i.e., non-dithering). In addition to analyzing previously developed filters, a new class of filters, called operator filters, was found using the techniques developed in this thesis.