Geometric Analysis of the Doppler Frequency for General Non-Stationary 3D Mobile-to-Mobile Channels Based on Prolate Spheroidal Coordinates

Abstract

Mobile-to-mobile channels often exhibit timevariant Doppler frequency shifts due to the movement of transmitter and receiver. An accurate description of the Doppler frequency turns out to be very difficult in Cartesian coordinates, and any subsequent algebraic analysis of the Doppler frequency is intractable. In contrast to other approaches, we base our investigation on a geometric description of the Doppler frequency with the following three mathematical pillars: prolate spheroidal coordinate system, algebraic curve theory, and differential forms. The prolate spheroidal coordinate system is more appropriate to algebraically investigate the problem. After the transformation into the new coordinate system, the theory of algebraic curves is needed to resolve the ambiguities. Finally, the differential forms are required to derive the joint delay Doppler probability density function. This function is normalized by the equivalent ellipsoidal area of the scattering plane bounded by the delay ellipsoid. The results generalize in a natural way our previous model to a complete 3D description. Our solutions enable insight into the geometry of the Doppler frequency and we were able to derive a Doppler frequency that is dependent on the delay and the scattering plane. The presented theory allows describing any time-variant, single-bounce, mobile-to-mobile scattering channel.

Application of Path-Based Multipath Component Tracking on Air-Ground Channel Measurement Data

“…This is not a coincidence, but can be explained by the fact, that in the air-ground channel, most of the reflectors causing the MPCs are located on the ground comparatively close to the GS. At the same time, each reflector is located on the surface of a prolate spheroid having the transmitter and receiver in its focal points [20]. The size of the spheroid a given reflector is located on is determined by the corresponding MPC's delay.…”

Empirical Fading Model and Bayesian Calibration for Multipath-Enhanced Device-Free Localization

“…Besides these unique expressions for the locations of VT and VR, the propagation delay of an MPC caused by SBR allows to constrain the possible locations of the corresponding RP. Specifically, this means that the possible locations of an RP coincide with the locations on the surface of an ellipsoid, each of which equally causes the propagation delay τ l,n [26], [27]. For reflections from surfaces orthogonal to the axis between the transmitting and receiving nodes, e.g., vertical walls of buildings, the possible locations are reduced to the circumference of an ellipse, as shown in Fig.…”

“…far from the propagation path of an MPC for modeling the variance. Specifically, we use the definition of the excess path length describing an ellipse with foci at (virtual) transmitting and receiving nodes, see (26) and (27). By introducing a threshold ξ th for the excess path length, we therefore achieve an inherent geometric separation, i.e., between locations near the propagation path with excess path lengths below the threshold and locations far from the propagation path with excess path lengths above it.…”

Non-Stationary 3D M2M Channel Modeling and Verification With Aircraft-to-Aircraft, Drone-to-Drone, Vehicle-to-Vehicle, and Ship-to-Ship Measurements

“…The authors in [18] address the non-stationarity of the channel by accounting, among others, for the probabilistic presence and movement of clusters of scatterers. Based on their work in [19] and [20] for V2V and A2A channels, respectively, Walter et al proposed in [21] a general way of analytically deriving the delay-dependent and joint delay Doppler probability density functions (pdfs) of M2M channels. The approach is based on defining an infinite arbitrarily-oriented scattering plane where the effective scatterers are located.…”

“…This, together with the demonstration that the time-variant squared delay/Dopplerspread function remains proportional to the time-variant joint delay Doppler pdf [22], allows for an analytical description of the time-variant channel in any 3D M2M scenario where the scatterers can be considered to be uniformly distributed over a plane. The potential of such an approach has already been shown with measurement data in V2V [23]- [25] and A2A [26], [27] scenarios. However, the technique proposed in [21] has two key constraints: the scatterers can only be assumed to be distributed over a single plane, which must also be infinite.…”