Use integrator/differentiator with vanishing group delay

I discovered issues with the current implementation of the integrator that's used to calculate the local_timer_deviations. It is done by computing the cumulative sum (numpy.cumsum). The equivalent IIR filter reads y[n] = y[n-1] + x[n] \Delta t. The problem with is the group delay of negative half a sample. This is not a problem if the differentiation is performed by the standard operation y[n] = \frac{x[n] - x[n-1]}{\Delta t} since it will compensate perfectly the group delay to zero again. The problem becomes noticeable if the sampling rate is changed between integration and differentiation since a half sample delay will be different in terms of absolute time before and after decimation.

I would suggest to use integration and differentiation operations that have zero group delay:

• integration: y[n] = y[n-1] + \frac{x[n-1] + x[n]}{2}\Delta t
• differentiation: y[n] = \frac{x[n+1] - x[n-1]}{2\Delta t}