Theoretical Limits of Parameter Estimation Based on Quantized Data

Abstract

Parameter estimation of band-limited periodic signals (sine and multisine waves) is a very common task in the field of measurement technology and control engineering. In the overwhelming majority of data acquisition and control systems the analog signals of the real world are sampled an quantized using analog-to-digital converters (ADCs). To estimate the parameters of the analog signal and the parameters of the quantizer from the same measurement record is an obvious need in these cases. The parameters of the recorded signal can be used to calculate the response of our system (e.g. signals of the actuators) while the parameters of the quantizer can be used to identify the transfer characteristic of the measurement channel. Maximum likelihood (ML) estimation of the quantizer and analog signal parameters has been developed to perform this task and to provide asymptotically unbiased and efficient estimators for the quantizer and signal parameters. This paper investigates the theoretical limits of this kind of estimation: provides the Cramér-Rao Lower Bound (CRLB) for the covariance of the achieved estimators and compares them to CRLB values obtained using less complex signal and channel models. This article also provides a comparison of the empirical covariance of estimator populations achieved different ways to the CRLB of estimation. The major tendencies are drawn and explanation for them is provided as well.