Digital Filter Design Approach

The driving-point impedance $R(s)=ms$ of an ideal mass is an ideal differentiator (scaled by $m$):

$$R(j\omega) = m j \omega.$$

It is therefore natural to define the ideal digital differentiator as

$$H(e^{j\omega T}) = j\omega, \quad \omega T\in[-\pi,\pi)$$

• An exact match is not possible with a finite order digital filter (note frequency-response discontinuity at $z=-1$)
• In practice, we minimize $\left\Vert\,H(e^{j\omega T}) - {\hat H}(e^{j\omega T})\,\right\Vert$ where ${\hat H}$ is a digital filter frequency response
• We need some oversampling in order to have a
  guard band (e.g., from $20$ kHz to $22$ kHz)
• Desired response is unconstrained in the guard band