Time-Frequency Localization of Linear Signal Spaces

Abstract

Several basic properties of the WD of a linear signal space have been discussed in the previous chapter. In this chapter, we consider properties that are related to the geometry of the WD and to the TF localization of a space. For example, we study the relation between the orthogonality of two spaces and the TF disjointness of their WDs. We introduce a distinction between “simple” and “sophisticated” spaces that has a direct bearing on the shape of the WD and the TF shifts caused by the space’s projection operator. Quantitative measures of a space’s TF concentration are introduced, bounds for these concentration measures (generalizing conventional uncertainty relations) are formulated, and the “maximally concentrated” spaces attaining these bounds are explicitly characterized.