Maximally flat differentiators through WLS Taylor decomposition

Instantaneous derivative estimates of a signal are obtained using the wighted least square (WLS) approximation of a Taylor (WLST) signal model, using classical windows as weighting factors. The WLST approximation in time corresponds to a Taylor approximation at the origin of the windowed signal spectrum. And the successive application of the WLST approximation leads to a filter bank whose frequency responses approach the set of ideal differentiator gains on the baseband, providing maximally flat differentiators on that band. Examples of these differentiator banks are designed with the Rectangular, Kaiser and Hamming windows, and their frequency and impulse responses are illustrated. Due to the strong symmetry of the signal model, this method achieves linear phase filter banks with equal delay for all the derivative estimates, which are very useful in applications where synchronized derivative of a bandlimited signal are desired.