Spatial Configurations

In this chapter we consider the information associated with the different spatial configurations of a waveform.We consider propagation of sinusoidal waves in arbitrary multiple scattering environments, and we compute the spatial bandwidth of the field measured in the space. By “spatial bandwidth” we mean the measure of the support set of the signal in the wavenumber domain – a more appropriate term would be “wavenumber bandwidth.” A minor disquieting fact of life is that the name spatial bandwidth is standard, and we shall use the two interchangeably. We obtain a simple formula for the number of spatial degrees of freedom of the received signal, showing that this number is limited by the wavelength-normalized size of the cut through which the information must flow.

A heuristic principle of communication is that in the presence of multiple scattering the number of possible spatial configurations of the field, and thus the amount of information it can carry over space, is increased. Loosely speaking, the superposition in space of many waveforms from many different multiple scattered paths “creates” bandwidth and provides independent parallel channels in the spatial–wavenumber domain. In the jargon of communication theory, the term “rich scattering” is used to denote an environment capable of providing an unlimited number of parallel spatial channels between transmitters and receivers. The intuition used to explain this phenomenon is that if the received field is the superposition of many waveforms coming from many different multiple scattered paths to the receivers, and each path can carry an independent stream of information, then many communications can occur in parallel by using a pair of antennas for communication along each scattered path. If the environment provides a sufficiently large number of independent paths, then this spatial multiplexing capability can grow proportionally to the number of antennas.

For any arbitrary scattering environment, however, the wavenumber bandwidth is not an unlimited resource.