I want to differentiate a noisy signal with many, randomly located, missing data values. Which smoothing techniques should be applied before differentiating the signal?

I have a velocity signal at 200 Hz, but due to wireless transmission about half the data values are missing (NaN), up to 8 in a row (40 ms between data points). I would like to get the derivative signal (acceleration), but without smoothening, it is too noisy. I'm mostly interested in a rough (!) estimation of maximum acceleration. What is the best techniques to handle the missing data? Linear interpolation between known data points?

  • $\begingroup$ A Kalman Filter can be formulated on non uniformly spaced samples. $\endgroup$
    – user28715
    Mar 28, 2018 at 3:54
  • $\begingroup$ Good idea! Thanks! A bit more effort then I was looking for but this seems like a good solution. $\endgroup$ Mar 28, 2018 at 8:44

1 Answer 1


The best tool for this job is normalized convolution. It can deal with missing samples as well as uncertainty.

The paper describing the method is "Normalized and Differential Convolution -- Methods for Interpolation and Filtering of Incomplete and Uncertain Data" by Hans Knutsson Carl-Fredrik Westin. There is a PDF on Semantic Scholar.

Normalized convolution basically takes a convolution with a Gaussian of the data, with 0 for the missing samples, and divides sample-wise with the result of the same convolution on a weight signal, which is 1 for known samples and 0 for the samples you set to zero. More formally:

$$ g = (( f m ) * h ) / (m * h) $$

(With $h$ the convolution kernel and $m$ the weight signal). You can derive the analytic derivative of that equation, leading to a derivative operator that is robust to missing samples. See the paper for more details.


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