To illustrate the process let’s consider case when . In the same way we can obtain expressions for any . Red dashed line is the magnitude derivative questions and solutions pdf of an ideal differentiator .
In practice there is no need for ideal differentiators because usually signals contain noise at high frequencies which should be suppresed. From the plot we can see that central differences don’t resemble such behavior, all they care about is to get as closer as possible to the response of ideal differentiator, without supression of noisy high frequencies. As a consequence they perform well only on exact values, which contain no noise. Different technique is needed for robust derivative estimation of noisy signals. Second order central difference is simple to derive.
We use the same interpolating polynomial and assume that . I just wanted to say how much i enjoyed finding this resource as i am taking my first course in numerical differential equations. I am having some confusion based on the definitions for the central difference operator that i am given and the one you are using. Also I have used least-squares instead of interpolation. 5 grid has only 25 degrees of freedom. I can cite the reference in a paper I’m writing.
Well, I’ve derived this formula by myself. I have no idea is it published somewhere or not. I think there is no problem. You can tell me more about your task, maybe I can derive more suitable filters for your conditions. Congratulations for your good and well organised work. I’m not mathematic and I’m writing an algorithm to derivate a discrete function. My set of points are not equidistant, so they have x values that are no constant.
I read your advanced work about derivatives for noisy functions and probably I will use it in a future. For the moment, I will derivate with central differences method. I wonder if it’s easy to extend the method when points are not equidistant. Is it ok to divide by the x increment? Let me know about the results please.
Thank you very much for your fast and good answer. I need to smooth the function for display purpose and I need also to derivate the function in order to find inflexion points, local maxima and other features that I use to do measurements of the object that I’ve enhanced the profile. At this moment I’m smoothing the function using Savitzky-Golay filter and I apply central differences to the smoothed function. I read your work and I realized that smooth noise-robust differentiators would be really useful to my work, but I don’t know how to use it to smooth the function for display purposes. I think your application could benefit from using SNRD filters. Just let me know what polynomial degree and number of points you are using now in Savitzky-Golay filter. Knowing that info I can build SNRD filters with better properties for your task.