Effect of multiplicative noise on least-squares parameter estimation with applications to the atomic force microscope

Area of Science:

• Physics
• Measurement Science
• Nanotechnology

Background:

• Power spectral density analysis of Brownian fluctuations is key for micro/nano-device characterization.
• Measurement noise, particularly multiplicative noise, can distort parameter estimation.
• Least-squares analysis is commonly used but susceptible to noise interference.

Purpose of the Study:

• To theoretically analyze the impact of multiplicative noise on least-squares fitting of spectral density data.
• To develop a general framework for quantifying noise-induced uncertainty in fit parameters.
• To provide a method for a priori determination of measurement uncertainty effects.

Main Methods:

• Theoretical analysis of multiplicative noise in least-squares fitting.
• Derivation of closed-form expressions for expected value and variance of fit parameters.
• Validation via Monte Carlo simulations and atomic force microscope cantilever measurements.

Main Results:

• Multiplicative noise introduces significant uncertainty in extracted fit parameters.
• Analytical formulas for uncertainty in resonant frequency and quality factor were derived.
• Individual spectral density measurements can yield highly deviated fit parameters.

Conclusions:

• Precise parameter measurements are inherently challenging in the presence of noise.
• Understanding multiplicative noise effects is critical for reliable micro/nano-device analysis.
• This work offers a rigorous framework for assessing measurement uncertainty in spectral density analysis.