Resonance parameter estimation from spectral data: Cramér–Rao lower bound and stable algorithms with application to liquid sensors

Abstract

A recently introduced method for robust determination of the parameters of strongly damped resonances is evaluated in terms of achievable accuracy. The method extracts and analyzes the locus of the resonant subsystem of noisy recorded complex spectra, such that the interfering influences of many environmental factors are eliminated. Estimator performance is compared to the absolute lower limit determining the Cramér–Rao lower bound (CRLB) for the variance of the estimated parameters. A generic model that is suitable for representation of a large class of sensors is used and analyzed. It is shown that the proposed robust method converges to the CRLB for low measurement noise.

Modeling-free evaluation of resonant liquid sensors for measuring viscosity and density

“…In this context a signal-to-noise ratio (SNR) can be defined as ratio of the diameter of the Nyquist circleỸ and the standard deviation σ of the noise ε. The given variances are equal to the Cramér-Rao lower bound for Q and very close to it for f r [20]. The lowest possible value for q is 1 and is only achieved for a uniform distribution of sampling points around the resonance circle in the Nyquist plot.…”

“…These effects are estimated and compensated, such that f r and Q of the motional part can be extracted. For given complex noise ε on the recorded spectra (zero mean AWGN, with variance var {ε} = σ 2 and var {Im {ε}} = var {Re {ε}} = σ 2 /2), the parameter noise on the estimates (estimated values are marked with a hat symbol) of f r and Q is governed by [19], [20] var…”

Monitoring of Monosodium Urate Crystallization for the Detection of Crystal Arthropathies in Human Joints

“…The fluid viscosity ηf introduces a damping factor ∆c associated with a hydrodynamic function Γη (see Equation (1b)) . According to [9], these changes follow the functional dependency (2) where the constants mρ, mρη, cη and cρη can be determined by model adjust measurements using a set of test liquids [4,9]. The angular resonance frequency ωr and the Q-factor are given by (3) where m and c are the mass and the damping factor of the unloaded sensor, respectively.…”

“…This is mainly attributed to the low penetration depth of the shear waves of approximately 200 nm and the missing mass effect [2] (i.e., the crystals are not rigidly attached to the sensor but form a soft layer). We therefore employed a low frequency piezoelectric quartz tuning fork (TF) sensor setup with a penetration depth of 3.3 µm in water, using the MFA200K44 impedance analyzer from Microresonant with the robust resonance base fitting algorithm [3,4]. Contrary to TSM, using the tuning fork also yields the possibility to separate density and viscosity of the liquid under test [5].…”

Application of resonant steel tuning forks with circular and rectangular cross sections for precise mass density and viscosity measurements

“…(5) and (6)) depend on density and viscosity. For proper measurements of the resonance characteristics processed with the estimation procedure from [18] it was shown in [40] that there is a relation between the relative standard deviations of f r and Q which is determined by the signal-to-noise ratio 1 (SNR) of the acquired frequency spectra and the number of frequency points M:…”