On precision limits for derivatives numerically calculated from noisy data

This paper has three purposes. (1) To verify an error formula from which the maximal precision in derivatives obtained from noisy measurement data can be calculated. The formula is verified by comparison with the resulting noise in derivatives obtained by local least squares polynomial fitting. It is also verified that differentiation by Fourier series expansion gives noise exceeding the minimum value according to the error formula. (2) An index called the "Relative Noise Amplification" (RNA) is introduced. For an arbitrary differentiating filter it is defined as the noise transmission of the filter divided by the minimal noise transmission according to the above mentioned error formula. When the filter produces unbiased estimates the value of the RNA always exceeds one. The RNA can be used as a quality index for differentiating filters. The filter with the smallest value on the RNA also has the smallest amount of noise superimposed on the calculated derivatives provided the input noise is white and additive. (3) The bandwidth and the Relative Noise Amplification of differentiating filters obtained by local least squares polynomial fitting are presented. The 0th, 1st and 2nd order derivatives have been investigated for polynomials up to 8th order. These results can be used for the determination of suitable filter parameters in various applications.