A question jumped into my head while processing data from MEMS sensors and Vicon: Is there any better ways of estimating first-order derivative of a function using samples at a fixed time interval, i.e. a "table"?
To be precise, suppose we have a function which is smooth enough, and a series of samples where .
The most intuitive way would be:
By looking into the Taylor Series, we found that the estimation error is the same order of :
But if we examine another Taylor Series,
it is obvious that we can eliminate the term by subtracting from :
Thus obtaining an estimation with error which is the same order of :
seems to be a better estimation. But it depends on the smoothness of as it is possible that is much larger than .