Filter Design Approach

• Ideal Digital Differentiator:
  $$H(e^{j\omega T})\approx j\omega, \quad \omega\in[-\pi/T,\pi/T]$$

• Ideal Digital Integrator
  $$H(e^{j\omega T})\approx \frac{1}{j\omega}, \quad \omega\in[-\pi/T,\pi/T]$$

• Exact match is not possible in finite order
• Minimize $\left\Vert\,H(e^{j\omega T}) - {\hat H}(e^{j\omega T})\,\right\Vert$ where ${\hat H}$ is the digital filter frequency response