Cramér–Rao lower bound of laser Doppler measurements at moving rough surfaces

Abstract

Laser Doppler techniques are widely used for measuring both fluid flows and moving solid surfaces. The measurement uncertainty of laser Doppler sensors is fundamentally limited by the uncertainty of the Doppler frequency estimation. Generally, the minimum achievable uncertainty of any unbiased estimator is given by the Cramér–Rao lower bound (CRLB). While the CRLB is well known for laser Doppler burst signals of single tracer particles used in flow research, no analytical expression for the CRLB has been known up to now for scattered light signals of rough solid surfaces where speckle effects occur. Therefore, the aim of this paper is to close this gap and to provide a simple analytical expression for the CRLB for the Doppler frequency estimation from scattered light signals of moving rough solid surfaces for the first time. A comparison with experimental data demonstrates the validity of the derived analytical CRLB formula, which is also proven to be consistent with previous works. The progress for science is that this analytical CRLB formula enables both an easy estimation of the minimum achievable uncertainty of laser Doppler measurements at moving rough surfaces and a direct analysis of the influences of certain system and signal parameters on the measurement uncertainty. This reveals specific measuring features and capabilities of different laser Doppler techniques. In addition, the CRLB is a valuable tool to evaluate the efficiency of applied signal processing techniques.