Precise sinusoidal signal extraction from noisy waveform in vibration calibration

Precise extraction of sinusoidal vibration parameters is essential for the dynamic calibration of vibration sensors, such as accelerometers. However, several standard methods have not yet been optimized for large background noise. In this work, signal processing methods to extract small vibration signals from noisy data in the case of accelerometer calibration are discussed. The results show that spectral leakage degrades calibration accuracy. Three methods based on the use of a filter, window function, and numerical differentiation are investigated to reduce the contribution of the calibration system noise. We demonstrate the effectiveness of these methods with theoretical calculations, simulations, and experiments. The uncertainty of microvibration calibration in the National Metrology Institute of Japan is reduced by two orders of magnitudes using the proposed methods. We recommend to use a combination of numerical differentiation and either a time-domain window function or a bandpass filter for most accelerometer microvibration sensitivity calibrations. For suppressing the effects of line noise, adjusting the data set length is also effective.

💡 Research Summary

The paper addresses a critical problem in the dynamic calibration of vibration sensors—accurately extracting the amplitude and phase of a very small sinusoidal vibration when the measurement is contaminated by large background noise. The authors focus on accelerometer micro‑vibration calibration, where the target vibration amplitude can be as low as 10⁻³ m/s², and the conventional method prescribed by ISO 16063‑11, the sine‑approximation method (SAM), is shown to be sub‑optimal under such noisy conditions.

SAM works by fitting a sinusoid to the sampled data using a rectangular time window, which is mathematically equivalent to taking a discrete Fourier transform (DFT) of the data. The authors derive the SAM estimator in closed form and demonstrate that, because the rectangular window has a sinc‑shaped frequency response, energy from frequencies other than the excitation frequency f v leaks into the estimator (spectral leakage). This leakage makes the estimated complex amplitude ˆx_est a convolution of the true spectrum X(f) with the window spectrum W̃(f − f v). Consequently, even if the noise power spectral density (PSD) is low at f v, noise present at other frequencies can degrade the signal‑to‑noise ratio (SNR) of the calibration.

The paper categorises noise sources into four groups: (i) independent random noise of the sensor (n_s) and reference (n_r), (ii) common random vibration noise (n_x) that appears in both signals, (iii) independent line‑frequency or harmonic noise (l_s, l_r), and (iv) common line‑frequency or harmonic noise (l_x). By modelling random noise as a stationary process with one‑sided PSD G(f), the authors obtain an analytical expression for the variance of the amplitude estimator (Eq. 12‑13). In the limit of long measurement time T, the variance reduces to G(f v)/T, confirming the familiar √T improvement, but the presence of the window’s side‑lobes prevents this ideal behaviour in practice.

To mitigate spectral leakage and improve estimator robustness, three complementary signal‑processing strategies are investigated:

1. Band‑pass filtering – A digital FIR/IIR filter centred on f v removes out‑of‑band noise before applying SAM. The filter must be designed with linear phase (e.g., symmetric FIR) to avoid introducing additional phase error.

2. Window‑function modification – Replacing the rectangular window with a tapered window (Hamming, Hann, Blackman, etc.) dramatically reduces side‑lobe levels of |W̃(f)|², thereby suppressing leakage from distant frequencies. The window length remains equal to the total acquisition time, preserving frequency resolution.

3. Numerical differentiation – Because the accelerometer output is an acceleration (second derivative of displacement) while the reference measures displacement, applying a numerical differentiation (central difference) to both signals before SAM accentuates high‑frequency noise in the acceleration channel but simultaneously attenuates common low‑frequency disturbances (e.g., power‑line noise). Moreover, the differentiation converts the common vibration noise n_x into a term that is proportional to (2πf v)² n_x, making its contribution to the estimator more predictable and easier to suppress.

The authors validate these concepts through Monte‑Carlo simulations. For a 1 Hz excitation with 1 m/s² amplitude, they generate synthetic data with either flat PSD noise or a coloured PSD that is larger away from f v. The simulated standard deviations of the amplitude estimator match the analytical predictions from Eq. 13, confirming the model’s accuracy. When a Hamming window and numerical differentiation are combined, the estimator’s standard deviation drops by roughly an order of magnitude compared with plain SAM; adding a modest‑order band‑pass filter yields a further reduction, achieving a total uncertainty improvement of about two orders of magnitude.

Experimental verification is performed on the National Metrology Institute of Japan (NMIJ) calibration setup. The original SAM‑based calibration yielded an amplitude uncertainty of ≈0.1 % (the target for primary calibration). By applying the proposed workflow—numerical differentiation, Hamming window, and a 0.5 Hz‑wide FIR band‑pass filter—the authors reduce the uncertainty to ≈0.001 %, a 100‑fold improvement. They also demonstrate that slight adjustments of the data‑set length (making it a non‑integer multiple of the vibration period) can further suppress line‑frequency leakage, an effect that is especially useful when mains hum (50/60 Hz) contaminates the measurement.

In conclusion, the paper provides a rigorous theoretical framework for understanding how window‑induced leakage and various noise sources affect sinusoidal parameter estimation in vibration calibration. It shows that a practical, low‑complexity processing chain—numerical differentiation plus a tapered window (or equivalently a band‑pass filter) —offers the best trade‑off between accuracy, computational cost, and ease of implementation. The authors recommend this combination for most micro‑vibration sensitivity calibrations of accelerometers and suggest that the same principles can be transferred to other dynamic calibration domains, such as MEMS accelerometers, broadband seismometers, and optical displacement sensors.