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In this paper, we investigate a relative positioning technique using the vehicle-to-vehicle communication signal. Specifically, we use the delay and Doppler information provided by the channel impulse response. Since the geometry of the vehicles has a direct impact on the delay and Doppler frequency, it is possible to deduce the relative positions of the vehicles. This particular relationship can be exploited by using prolate spheroidal coordinates. By doing an algebraic analysis in this coordinate system closed form solutions for delay and Doppler frequency can be derived. The delay and Doppler information is then used to determine the relative position similar to using the bearing and distance to a lighthouse in marine navigation or distance measuring equipment (DME) and VHF omnidirectional radio range (VOR) in aeronautical navigation. INTRODUCTION The estimation of the relative positions and velocities of transmitter and receiver is important in several wireless applications, like resource management, adaptive modulation and coding, channel assignment, channel estimation, and localization. According to [1] the velocity estimation techniques can be divided into four categories: crossing-based, covariance-based; power spectrum-based; and maximum likelihood-based methods. The absolute value of the velocity with only the transmitter moving is presented in [2]. The authors use a covariance-based algorithm to estimate the speed of the mobile receiver. Another covariance-based method in estimating the maximum Doppler and thus the absolute value of the velocity is shown in [3]. A power spectrum based method is presented in [4], where the absolute value of the velocity of a mobile receiver is estimated. The first publication estimating the transmitter and receiver velocity jointly with the relative direction of movement is given in [1] based on the crossing method for the velocity estimation and correlation based for the direction estimation, i.e., usage of stochastic operators. In [5] the absolute values of transmitter and receiver velocity are estimated based on the maximum likelihood approach. Our novel approach takes into account the information of the instantaneous delay-dependent Doppler power spectrum close to the line-of-sight and far away from it. Since the limiting frequencies are dependent on the relative movement and position, i.e., the geometry, between the vehicles, the velocity components relative to the line-of-sight can be deduced. With this knowledge the angle between the velocity vector of the vehicle and the line-of-sight (LOS) is determined. If the line-of-sight is available, the relative positioning is done by angle (using the Doppler spectrum) and distance (using range or pseudo-ranging information) estimation. This approach is similar to estimating the bearing and distance to a lighthouse in marine navigation or DME and VOR in aeronautical navigation. The difference is that both transmitter (Tx) and receiver (Rx) are moving during the velocity and position estimation process. In contrast to the stochastic methods mentioned above, we use the time-variant, distance-dependent relationship between the vehicle geometry and Doppler spectrum to deduce the velocity components and the relative position. Specifically, the Doppler spectrum close to the line of sight (LOS) enables us to estimate the orientation of the velocity vector components relative to the LOS instead of the magnitude of the velocity vectors only. Furthermore, the method does not depend on the particular stochastic distribution of the scatterers as long as scattering from all directions can be observed. Our approach, thus, ensures a time-variant estimation of the velocity vector components and the relative positions between two vehicles. Usually, a time-invariant, narrow-band model is used when estimating the Doppler frequency from the Doppler spectrum. In this approach, however, we use the time-variant, distance-dependent Doppler spectrum, since the limiting frequencies are dependent on the distance for vehicle-to-vehicle (V2V) channels. Closed-form solutions exist for delays close to the LOS and very large delays. A co-moving coordinate system with prolate spheroidal coordinates [6] allows for an adequate description of the V2V communication scenario. In this particular coordinate system the delay coordinate refers to a constant delay between transmitter, scatterer, and receiver. We use the Doppler spectrum at the LOS and in direct vicinity of the LOS. Additionally, it might be necessary to use the Doppler spectrum for large delays of the scatterers to estimate the orthogonal component relative to the LOS of the other vehicle’s velocity vector. For the relative positioning, however, the Doppler spectrum very close to the line-of-sight is sufficient, which is a very important fact, since most of the scattering in V2V communication occurs close to the vehicles. PROLATE SPHEROIDAL COORDINATE SYSTEM The ability to derive closed-form solution for the Doppler frequencies depends mainly on the adequate selection of an appropriate coordinate system. In the case of V2V communication, the prolate spheroidal coordinate system (PSCS) represents a suitable coordinate system for two center problems as shown in [7]. The PSCS is a three-dimensional, curvilinear, orthogonal coordinate system as shown in Fig. 1. The origin of the coordinate system is by definition always in the middle between Tx an Rx and co-moves with the movement of both. The relationship between Cartesian and prolate spheroidal coordinates is given by [8] x = l √ (ξ2 − 1)(1− η2) cos(θ) y = l √ (ξ2 − 1)(1− η2) sin(θ) z = lξη , (1) with l being the focus distance, i.e. the distance from Tx and Rx to the origin, ξ ∈ [1,∞), and η ∈ [−1, 1]. ξ = τ τlos represents a normalized delay. The absolute delay is calculated by τ = τlosξ = 2lξ c with c being the speed of light. Isosurfaces of ξ represent prolate spheroidal ellipsoids, whereas values of constant η are displayed as hyperboloid surfaces. Geometrically, a fixed ξ-coordinate represents locations where the sum of the delays to the foci, i.e., Tx and Rx, is constant. A fixed η-coordinate describes locations where the difference of the delays to the Tx and Rx is constant. Note that the third orthogonal coordinate in the PSCS constitutes a half-plane as iso-surface. This particular coordinate transform effectively leads to polynomial expressions in the ξand η-coordinates in the Doppler frequency. The big advantage of the PSCS is that by fixing the ξ-coordinate, we automatically obtain a fixed delay. Thus, the delaydependent Doppler probability density function (pdf) becomes only dependent on two coordinates in the PSCS; in a planar geometry this dependency reduces even to a single PSCS coordinate. In contrast, fixing the delay in a Cartesian coordinate system still implies the dependency on all three spatial coordinates x and y and z. This makes the resulting expressions more difficult. The description in the PSCS allows us to express both delay and Doppler frequency in a more compact notation in comparison to Cartesian coordinates. The delay from Tx to an arbitrary scatter dt and the delay from this scatterer to the Rx dr, can be defined as dt = (ξ + η) l , (2) dr = (ξ − η) l , (3) dsc = dt + dr = 2ξl , (4) where the total distance dsc, or the total delay τsc = dsc c , depend only on the ξ-coordinate. As has been derived in [9], the Doppler frequency is calculated as spatial gradient of the Tx and Tx distances dt and dr projected onto their velocity vectors. Thus, the Doppler frequency is given by fd(x) = ( v T t ∇dt(x) + v r ∇dr(x) ) fc c . (5) As mentioned above a delay-dependent description of the Doppler frequency in Cartesian coordinates depends on the x-, y-, and z-coordinate. Thus, the calculation of the Doppler pdf becomes very cumbersome. In order to simplify the calculation, we transform the mathematical analysis in the PSCS and express the gradient in (5) in prolate spheroidal coordinates. Consider an arbitrary scalar function Ψ(ξ, η, θ) represented in the PSCS. Its gradient ∇Ψ is computed according to [10] by ∇Ψ(ξ, η, θ) = 1 hξ ∂Ψ ∂ξ eξ + 1 hη ∂Ψ ∂η eη + 1 hθ ∂Ψ ∂θ eθ (6) where eξ, eη, and eθ are the orthonormal basis vectors of the PSCS. They are obtained by transforming the Cartesian basis vectors ex, ey and ez using the following equations eα = ∂x ∂α ex hα + ∂y ∂α ey hα + ∂z ∂α ez hα , (7) where the subscript α ∈ {ξ, η, θ}. In (6) and (7) the scalars hξ, hη, and hθ are called scale factors. They ensure the normalization of the basis vectors in the transformed coordinate system. For orthogonal coordinate systems the scale factors are the roots of the three non-zero elements hi = √ gii of the metric tensor g, see [10] for details. In the PSCS, the scale factors hα are calculated as hξ = l √ (ξ2 − η2) (ξ2 − 1) , hη = l √ (ξ2 − η2) (1− η2) , hθ = l √ (ξ2 − 1)(1− η2) . (8) By transforming (5) into prolate spheroidal coordinates, the Doppler frequency is given by fd(t; ξ, η, θ) = fc c { √ (ξ2 − 1) (1− η2) ξ + η (v t cosθ+ v y t sinθ) + ξη + 1 ξ + η v t + √ (ξ2 − 1) (1− η2) ξ − η (v x r cosθ+ v y r sinθ) + ξη − 1 ξ − η v z r } . (9) with vt = [v x t , v z t ] T being the velocity vector of the Tx and vr = [v x t , v z t ] T being the velocity vector of the Rx, respectively. Note that due to the implicit time-variance of the velocity vectors, the Doppler frequency becomes time-variant which is taken into account by the variable t. Time-Variant, Delay-Dependent Doppler PDF In the subsequent algebraic analysis, we reduce the general 3D mobile-to-mobile (M2M) description to a 2D V2V description, which corresponds to (ξ, η) coordinates in the PSCS and (x, z) in the Cartesian coordinate system, see Fig.1. The condition for the coordinate system becomes y = l √ (ξ2 − 1)(1− η2) sin(θ) = 0 , (10) which is true for θ = {0, π}. The mathematical derivation relies on the geometric-stochastic description of the vehicular environment to obta

proceedings of the satellite division's international technical meeting (online)/proceedings of the satellite division's international technical meeting (cd-rom)

Relative Positioning and Velocity Estimation Using V2V Delay and Doppler Information