Three-dimensional multipath shape factors for spatial modeling of wireless channels

This paper develops a generalized theory of multipath shape factors used to study the small-scale fading statistics in three-dimensional non-omnidirectional multipath channels. Previous results for horizontal propagation channels are special cases of the present framework when the multipath angular power density (APD) and the travel direction of the mobile unit are confined to a plane. The three-dimensional multipath shape factors are derived in terms of the first three degrees of unnormalized spherical harmonics of the APD. Based on the so derived shape factors, analytical expressions for the level crossing rate, average fade duration and approximate envelope correlation are derived for a Rayleigh fading three-dimensional channel.