Linear estimation with transformed measurement for nonlinear estimation

The linear minimum mean-square error (LMMSE) estimation plays an important role in nonlinear estimation. It is the best of all estimators that are linear in the measurement. However, it may not perform well for a highly nonlinear problem. A generalized linear estimation framework, namely, linear in transform (LIT) estimation, is proposed in this work. It employs a measurement transform function (MTF) and finds the best estimator among all estimators that are linear in MTF, rather than in the measurement itself. The performance of LIT estimation with a proper MTF can be superior to LMMSE estimation due to the benefit of a more appropriate or larger candidate set of estimators determined by the MTF (compared to the set of all linear estimators in LMMSE estimation). Since systematic procedures for constructing an MTF that guarantees enhanced performance for the general case are difficult to produce, we provide several design guidelines, which are illustrated by a numerical example. Further, similar to LMMSE estimation, moments involved in LIT estimation are difficult to compute analytically in general. Fortunately, many numerical approximations for LMMSE estimation are also applicable to LIT estimation. Approximation of LIT estimation based on the Gauss-Hermite quadrature is presented. Our LIT estimation is demonstrated by applications to target tracking. Its performance is compared with LMMSE estimation by Monte Carlo simulations.