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

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.