First- and second-order characterization of direction dispersion and space selectivity in the radio channel

We present analytical results relating to first- and second-moment characterization of direction dispersion and spatial selectivity in the radio channel as well as to the duality between these two effects. Dispersion in direction can be characterized either by the direction power spectrum or by a family of spatial Doppler power spectra at the reception site. Two measures called the direction spread and the maximum spatial Doppler spread are introduced which describe the extent of channel dispersion in direction and in spatial Doppler frequency, respectively. Both measures are analogous to the delay and Doppler spreads, which are commonly employed to describe the extent of dispersion in delay and Doppler frequency, respectively. The relationships between the two approaches for characterizing spatial dispersion and especially between the direction and maximum spatial Doppler spreads are analytically established. The coherence distance at a certain level summarizes certain features of space selectivity in the radio channel which impact on the performance of communication systems. Two uncertainty relations between the direction spread and the coherence distance as well as between the maximum spatial Doppler spread and the coherence distance express the duality between direction dispersion and space selectivity. These relations are analogous to those established previously between the delay spread and the coherence bandwidth and between the Doppler spread and the coherence time. Examples relevant to mobile communications in the case where the waves propagate only horizontally illustrate the theoretical results. An application of these results to the design of uniform linear antenna arrays is also discussed.