On algebraic differentiators
============================

Algebraic differentiators have been derived and discussed in the systems
and control theory community. The initial works based on
differential-algebraic methods have been developed by Mboup, Join, and
Fliess in `[2] <#2>`__. These numerical, non-asymptotic approximation
approaches for higher-order derivatives of noisy signals are well suited
for real-time embedded systems. A historical overview and a detailed
discussion of these differentiators and their time-domain and
frequency-domain properties are given in the survey `[1] <#1>`__.

The approximation-theoretic derivation recalled in the survey
`[1] <#1>`__ permits the interpretation of the estimation process by the
following three steps illustrated in the figure below stemming from
`[1] <#1>`__:

1. Projection: At time :math:`t`, the sough :math:`n`-th order time derivative :math:`y^{(n)}` over the
   interval :math:`I_T(t)` is projected onto the space of polynomials of degree :math:`N`. This
   yields the polynomial :math:`p_N` depicted in the left and middle part of Figure
   2.
2. Evaluation: The polynomial :math:`p_N` is evaluated at :math:`t-\delta_t`, which gives an estimate :math:`\hat{y}^{(n)}(t)=p_N(t-\delta_t)`
   for the derivative :math:`y^{(n)}` as depicted in the central part of Figure 2.
   Choosing the delay to be the largest root of a special Jacobi
   polynomial increases the approximation order by 1 with a minimal
   delay. Alternatively, a delay-free estimation or even a prediction of
   the future derivative might be selected, at the cost of a reduced
   accuracy.
3. Repetition: The first two steps are repeated at each discrete time
   instant while keeping the parameters of the differentiator constant.
   This yields the estimate :math:`\hat{y}^{(n)}` depicted in the right part of the Figure 2.

.. figure:: interpretationDifferentiators.png
   :scale: 80 %
   :alt: Three-step process of the estimation

   Figure 2. Three-step process of the estimation of the :math:`n`-th order derivative :math:`y^{(n)}\mapsto y^{(n)}(t)` of a signal :math:`y\mapsto y(t)` using algebraic differentiators (figure from `[1] <#1>`__)
   
Algebraic differentiators can be interpreted as linear time-invariant
filters with a finite-duration impulse response. Figure 3 visualizes the
online estimation process of the first derivative of a noisy signal. The
filter window, the buffered signal, and the filter kernel can be clearly
seen.


.. figure:: animationEstimation.gif
   :scale: 50 %
   :alt: Visualization of the online estimation

   Figure 3. Visualization of the online estimation of the first derivative a noisy signal using an algebraic differentiator. 
   
These filters can be approximated as lowpass filters with a known cutoff
frequency and a stopband slope. Figure 4 presents the amplitude and
phase spectra of two exemplary filters. The lowpass approximation is
also shown.


.. figure:: filterSpectrum.png
   :scale: 50 %
   :alt: Visualization of the online estimation

   Figure 4. Amplitude and phase spectra of two different filters and the corresponding lowpass approximation of the amplitude spectrum. 
   
   
See `[1] <#1>`__, `[3] <#3>`__, `[4] <#4>`__, and `[5] <#5>`__ for more
details on the parametrization of these differentiators.