Measurement-driven quality assessment of nonlinear systems by exponential replacement

Abstract

Abstract-We discuss the problem how to determine the quality of a nonlinear system with respect to a measurement task. Due to amplification, filtering, quantization and internal noise sources physical measurement equipment in general exhibits a nonlinear and random input-to-output behaviour. This usually makes it impossible to accurately describe the underlying statistical system model. When the individual operations are all known and deterministic, one can resort to approximations of the input-to-output function. The problem becomes challenging when the processing chain is not exactly known or contains nonlinear random effects. Then one has to approximate the output distribution in an empirical way. Here we show that by measuring the first two sample moments of an arbitrary set of output transformations in a calibrated setup, the output distribution of the actual system can be approximated by an equivalent exponential family distribution. This method has the property that the resulting approximation of the statistical system model is guaranteed to be pessimistic in an estimation theoretic sense. We show this by proving that an equivalent exponential family distribution in general exhibits a lower Fisher information measure than the original system model. With various examples and a model matching step we demonstrate how this estimation theoretic aspect can be exploited in practice in order to obtain a conservative measurement-driven quality assessment method for nonlinear measurement systems.Index Terms-nonlinear systems, Fisher information, Cramér-Rao lower bound, exponential family, Saleh model, Rician model, regression, Wiener system, measurement uncertainty I. MOTIVATION The characterization of nonlinear systems and the development of appropriate processing algorithms forms a problem for engineering tasks like signal processing or system identification while the increasing demand on low-cost, energy-efficient and fast measurement devices makes it inevitable to operate such systems outside their linear regime. In order to provide high processing accuracy under these circumstances, it is important to investigate nonlinear models and to develop generic approaches and methods for such kind of problems. The key challenge for efficient solutions to nonlinear measurement problems lies in the fact that access to an appropriate model p(z; θ) is required. In theory, this probabilistic description establishes the statistical relationship of the system parameter θ ∈ R and the output measurement z ∈ R. This allows to formulate efficient signal processing algorithms and to describe the achievable performance by analytical tools. However, in practice the system model p(z; θ) is rarely known exactly and therefore has to be established by numerical calculations or approximated by physical measurements.If the analytical form of the system model p(z; θ) is known but to complicated to work with, one can resort to an direct approximation of the distribution function like performed in [1] for the Rician model. When the nonlinear...