Abstract
Many positioning and tracking applications use spatially distributed sensor stations, each equipped with coherent measurement channels. The coherent data set of each node is incoherently measured to the data sets of all other nodes, avoiding expensive synchronization procedures between the stations. Usually, the measurements are evaluated by mapping the incoherently measured complex valued data on real valued data like angle of arrival or received signal strength. After this preprocessing step, recursive filters fuse the real valued data to estimate a system state. Unfortunately, even though the original measurements are performed in additive white Gaussian noise environments, the preprocessing step can result in correlated, noise shaped errors, whose variance is system state dependent. Hence, the additive white Gaussian noise assumption, which is commonly drawn in Kalman filters, is violated. Therefore, this paper proposes an iterative extended Kalman filter, which estimates the state of a nonlinear real valued system via a complex valued measurement model that consists of incoherent sensor stations, each with several coherent measurement channels. Since the proposed iterative extended Kalman filter is well suited for distributed positioning systems with several stations, a transmitter is localized in a simulation via two sensor stations, each measuring the received signal's amplitude and phase at four channels. To illustrate the advantages of the proposed algorithm, the direct measurement evaluation using the proposed algorithm is compared to a Kalman filter that evaluates the received signal strengths and angle of arrivals at each sensor station. Finally, a reflecting wall is incorporated into the simulation scenario to demonstrate the flexibility of the proposed Kalman filter.
Highlights
In many applications, for example wireless positioning [1], [2], noise source and jammer detection [3], and imaging [4], receive signals are often evaluated using spatially distributed sensors, in which each sensor might be equipped with several coherent receive channels, each yielding complex valued measurement data
To address the general problem of incoherent measurements, this paper proposes a novel iterated EKF (IEKF) algorithm, which estimates the state of a real valued nonlinear system via the complex valued nonlinear incoherent measurements of sensor nodes, which possess several coherent channels
EVALUATION OF THE PROPOSED ALGORITHM As an example, the proposed algorithm is used to localize a transmitter in 2D using multiple antenna arrays and is compared to an extended Kalman filter (EKF) from [15], that evaluates the angle of arrival (AOA) and receive signal strength (RSS) at each array, which are estimated in a preprocessing step
Summary
For example wireless positioning [1], [2], noise source and jammer detection [3], and imaging [4], receive signals are often evaluated using spatially distributed sensors, in which each sensor might be equipped with several coherent receive channels, each yielding complex valued measurement data. Incoherent measurements at spatially distributed sensors, which are disturbed by normally distributed noise, are well suited for Kalman filter based state estimation. To address the general problem of incoherent measurements, this paper proposes a novel IEKF algorithm, which estimates the state of a real valued nonlinear system via the complex valued nonlinear incoherent measurements of sensor nodes, which possess several coherent channels. In comparison to other Kalman filter approaches, the proposed incoherent IEKF (ICIEKF) is estimating both the system state and the unknown phase shifts for each sensor station. The synchronization is not sufficient to assume constant phase relations between the sensor stations This is modeled by the random unknown phase shift αη,k at time instance k and yields complex valued measurements from a real valued system model as zk. The first-order Taylor series expansion of fk−1(xk−1, uk−1) is calculated at the current estimation xk−1|k−1 and the control input uk−1 as fk−1(xk−1, uk−1) ≈ fk−1(xk−1|k−1, uk−1)
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