On the information content of scattered waves

A basic trade-off between spatial and frequency diversities available for communication with multiple scattered waves is presented. A rigorous notion of the total number of space-frequency degrees of freedom of the field is introduced based on Slepian´s theory of spectral concentration of L² functions. This definition agrees with the “rule of thumb” of multiplying the degrees of freedom available in space and frequency. However, while the mathematical definition requires both frequency and spatial bandwidths to scale simultaneously, a physical argument precludes such a scaling to occur in practice. Hence, since the nature of wireless communication prevents unbounded and simultaneous spatial and frequency diversity gains, rule of thumb definitions of degrees of freedom should be taken with care as they are inherently imprecise.