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This page is about numerical differentiation of a noisy data or functions. Description begins with analysis of well known central differences establishing reasons of its weak noise suppression properties. Then we consider differentiators with improved noise suppression based on least-squares smoothing. Such filters are known as low-noise Lanczos differentiators or Savitzky-Golay filters. You can skip directly to the table of final formulas.
August 13, 2015: Andrey Paramonov went further and extended the ideas to build similar filters for smoothing. Please read his article for more details: Noise-robust smoothing filter. November 10, 2011: It is not technically correct to state that Savitzky-Golay filters are inferior to proposed ones. Actually both have situations when one outperforms another. Savitzky-Golay is optimal for Gaussian noise suppression whereas proposed filters are suitable when certain range of frequencies is corrupted with noise. Red dashed line is the response of an ideal differentiator .
Although amplitude of waves is decreasing for longer filters in order to achieve acceptable suppression one should choose very long filters. See Low-noise Lanczos differentiators page for more details. From signal processing point of view differentiators are anti-symmetric digital filters with finite impulse response of Type III. There are powerful methods for its construction: based on windowed Fourier series, frequency sampling, etc.
Additionally it should have computationally favorable structure to be effectively applied in practice. Finally we arrive at the simple solution . Frequency-domain characteristics for the differentiators are drawn below. Red dashed line is the response of ideal differentiator . Besides guaranteed noise suppression smooth differentiators have efficient computational structure. Costly floating-point division can be completely avoided. These filters also show smooth noise suppression with extended passband.
The act of using complex language, i am not surprised since 5 points filter has the weakest noise suppression capability. They are also useful for testing individual intelligence and ability, brief description of your project is welcome in the comments below. Sorry for the long post, children under the age of 5 have the best opportunity to absorb basic numeracy skills. These tests provide insight into a candidate’s decision making process, i cannot imagine the practical need of 3rd order numerical approximation of the derivative though. Applying straightforward methods like L, these psychometric numerical reasoning tests are used to assess an applicants’ ability to comprehend and apply numbers.
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