What is a Relevant Signal-to-Noise Ratio for Numerical Differentiation?

TL;DR

This work identifies the inadequacy of traditional RMS-based SNR for numerical differentiation of sampled sensor data and derives derivative-based SNR definitions for two noise models. For harmonic noise, it yields ${\rm RMSE}_{\rm sd}=\frac{A_n \omega_n}{\sqrt{2}}$, ${\rm RMSE}_{\rm dd}=\frac{A_n \omega_n^2}{\sqrt{2}}$ with ${\rm SNR}_{\rm sd}=\frac{1}{A_n \omega_n}$ and ${\rm SNR}_{\rm dd}=\frac{1}{A_n \omega_n^2}$, while for white noise it obtains ${\rm RMSE}_{\rm sd}=\frac{\sqrt{2}\sigma}{T_s}$, ${\rm RMSE}_{\rm dd}=\frac{\sqrt{6}\sigma}{T_s^2}$ with ${\rm SNR}_{\rm sd}=\frac{T_s}{\sigma}$ and ${\rm SNR}_{\rm dd}=\frac{T_s^2}{\sigma}$. These metrics better predict derivative accuracy than conventional SNR, and the paper demonstrates this using backward-difference and adaptive input and state estimation, highlighting implications for Kalman-filter-based differentiation and future research directions.

Abstract

In applications that involve sensor data, a useful measure of signal-to-noise ratio (SNR) is the ratio of the root-mean-squared (RMS) signal to the RMS sensor noise. The present paper shows that, for numerical differentiation, the traditional SNR is ineffective. In particular, it is shown that, for a harmonic signal with harmonic sensor noise, a natural and relevant SNR is given by the ratio of the RMS of the derivative of the signal to the RMS of the derivative of the sensor noise. For a harmonic signal with white sensor noise, an effective SNR is derived. Implications of these observations for signal processing are discussed.

Figures (7)

• Figure 1: Example \ref{['eg:harmonic_motivation']}: Numerical differentiation with harmonic sensor noise using BD. The left axis label shows the RMSE of the estimate of a single derivative of $y_{\rm m}$ versus $\omega_{{\rm n}}$. The right axis label shows ${\rm SNR}_{\rm amp}$ and ${\rm SNR}_{\rm eng}$. Note that neither SNR correlates with the accuracy of the estimates.
• Figure 2: Example \ref{['eg:harmonic_motivation']}: Numerical differentiation with harmonic sensor noise using BD. The left axis label shows the RMSE of the estimate of a double derivative of $y_{\rm m}$ versus $\omega_{{\rm n}}$. The right axis label shows ${\rm SNR}_{\rm amp}$ and ${\rm SNR}_{\rm eng}$. Note that neither SNR correlates with the accuracy of the estimates.
• Figure 3: Example \ref{['eg:nd_harmonic_noise']}: Numerical differentiation with harmonic sensor noise. The left axis label shows the ${\rm RMSE}_{\rm sd}$ of the estimates of $\dot y_{\rm m}$ versus $\omega_{\rm n}$. The derivative is computed using BD, AISE, and exact \ref{['sd_rmse_harmonic']}. The right axis label shows ${\rm SNR}_{\rm sd}$. Note that SNR$_{\rm sd}$ is highly correlated with the accuracy of the estimates.
• Figure 4: Example \ref{['eg:nd_harmonic_noise']}: Numerical differentiation with harmonic sensor noise. The left axis label shows the ${\rm RMSE}_{\rm dd}$ of the estimates of $\ddot y_{\rm m}$ versus $\omega_{\rm n}$. The derivative is computed using BD, AISE, and exact \ref{['dd_rmse_harmonic']}. The right axis label shows ${\rm SNR}_{\rm dd}$. Note that SNR$_{\rm dd}$ is highly correlated with the accuracy of the estimates.
• Figure 5: Example \ref{['eg:nd_white_noise']}: Numerical differentiation with white sensor noise. The left axis label shows ${\rm RMSE}_{\rm sd}$ of $\dot y_{\rm m}$ versus $\sigma^2$. The derivatives are computed using BD, AISE, and exact \ref{['sd_rmse_white']}. The right axis label shows ${\rm SNR}_{\rm sd}$\ref{['prop_snr_white_sd']}. Note that SNR$_{\rm sd}$ is highly correlated with the accuracy of the estimates.

Theorems & Definitions (3)

• Example II.1
• Example III.1
• Example IV.1