Spatial Bandwidth: a Heuristic Calculation with Applications

About 2WT real numbers are needed to fully characterize a signal s(t) of effective duration T and bandwidth W ; we then say that the approximate dimension of the signal space is 2WT [1]. In a more general case, and in addition to time t, s(r, t) also depends on the position in space r. Such a model appears, for instance, in the electromagnetic analysis of waveguides or optical fibres [2]. The boundary conditions for s(r, t) are stated in terms of feasible regions A and directions Ω: the field in a waveguide must be confined to its interior, and there is a limited range of directions (angles) that remain guided and therefore do not radiate. We then speak of spatial modes, which are a set of orthonormal functions onto which the signal can be uniquely decomposed. Past tentative applications of this spatial analysis to communication theory were made by Gabor [3] and Levitin and Lebedev [4]. Recent work by Poon, Brodersen, Tse [5] on multiple-antenna channels combines electromagnetism and antenna theory to extend the 2WT -theorem and include the spatial dimensions. In the present work we analyze the dimension of the signal space with fundamental methods based on the uncertainty principle, thus providing an alternative to the counting method in [5]. As application, we also estimate the dimension of the signal space for optical transmission and storage and for satellite links.