Impulse Response of Ideal LPF

$$\begin{eqnarray*} h_{\mbox{ideal}}(n) &\mathrel{\stackrel{\Delta}{=}}& \mbox{DTFT}^{-1}_n\left(H_{\mbox{ideal}}\right) \\ [10pt] &\mathrel{\stackrel{\Delta}{=}}& \frac{1}{2\pi}\int_{-\omega_c}^{\omega_c} e^{j\omega n }d\omega \\ [10pt] &=& \frac{\sin(\omega_c n)}{\pi n} \\ [10pt] &=& 2f_c \mbox{sinc}(2f_c n ), \hspace{2cm} n \in \mathbb{Z} \end{eqnarray*}$$

Problems:

• Infinitely long
• Non-causal
• Cannot be shifted to make it causal
• Cannot be implemented in practice

Conclusion:

• We must accept some compromise(s) in the design of any practical lowpass filter.
• Managing such trade-offs is the topic of digital filter design.