Nonlinear and adaptive signal estimation

Kalman filter and its variations are commonly used to estimate signals observed under noise. The estimator is typically derived by minimizing the squared error between the observed and the estimated signals, and leads to a closed form optimal solution under Gaussian noise. However, such an approach leads to a solution where the Kalman gain is updated based on known noise statistics alone and not on the actual error. Hence the estimation could deviate from the actual if the noise statistics is different or time-varying, and could become unstable as well. In this paper we present a new nonlinear and adaptive signal estimator that introduces a closed loop between the error estimation and the gain adaptation and leads to an inherently robust estimator. Also, since nonlinear adaptive technique is utilized, complex error norms can as well be used. The design methodology basically involves making the combined estimator and the gain-update dynamics to resemble the dynamics of a nonlinear time varying (NLTV) electrical circuit having the required properties. This conceptually simple procedure leads to a new general class of complex nonlinear and adaptive estimation algorithms without the use of or the necessity for complex analytical tools, and hence can be understood and applied easily. We present below the basic methodology and simulation results to show the simplicity and the strength of the new approach.