Frequency-Domain Modeling of Correlated Gaussian Noise in Kalman Filtering

This article derives an automated method to obtain models for time-correlated noise that are guaranteed to produce an upper bound on the Kalman filter (KF) estimate error covariance matrix. The noise is assumed to be zero-mean Gaussian and stationary over the filtering duration, but otherwise has no known structure. We first show that the covariance matrix predicted by the KF upper bounds the true error covariance matrix when the noise model's power spectral density (PSD) function exceeds the true PSD at every frequency. An approach is then developed to automatically obtain autoregressive models up to second order that satisfy this criterion. The method is evaluated using covariance analysis for an example application in GPS-based relative positioning.