### KDER – Derivation and filtering of the input signal Function Description
The KDER block is a Kalman-type filter of the norder-th order aimed at estimation of derivatives of locally polynomial signals corrupted by noise. The order of derivatives ranges from $0$ to $\mathtt{\text{norder}}-1$. The block can be used for derivation of almost arbitrary input signal $\mathtt{\text{u}}={u}_{0}\left(t\right)+v\left(t\right)$, assuming that the frequency spectrums of the signal and noise differ.

The block is configured by only two parameters pbeta and norder. The pbeta parameter depends on the sampling period ${T}_{S}$, frequency properties of the input signal u and also the noise to signal ratio. An approximate formula $\mathtt{\text{pbeta}}\approx {T}_{S}{\omega }_{0}$ can be used. The frequency spectrum of the input signal u should be located deep down below the cutoff frequency ${\omega }_{0}$. But at the same time, the frequency spectrum of the noise should be as far away from the cutoff frequency ${\omega }_{0}$ as possible. The cutoff frequency ${\omega }_{0}$ and thus also the pbeta parameter must be lowered for strengthening the noise rejection.

The other parameter norder must be chosen with respect to the order of the estimated derivations. In most cases the 2nd or 3rd order filter is sufficient. Higher orders of the filter produce better derivation estimates for non-polynomial signals at the cost of slower tracking and higher computational cost.

Input

 u Input signal to be filtered Double (F64)

Outputs

 y Filtered input signal Double (F64) dy Estimated 1st order derivative Double (F64) d2y Estimated 2nd order derivative Double (F64) d3y Estimated 3rd order derivative Double (F64) d4y Estimated 4th order derivative Double (F64) d5y Estimated 5th order derivative Double (F64)

Parameters

 norder Order of the derivative filter  $↓$2 $↑$10 $\odot$3 Long (I32) pbeta Bandwidth of the derivative filter  $↓$0.0 $\odot$0.1 Double (F64)

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