Ideal FIlters

Ideal Filter has gain of unity in the pass band |T| = 1 or α=0 db and has gain of zero or infinite attenuation (α = infinity) in stop band. Ideal filters are theoretical (mathematical) filters that have no transition region or gradual roll-off between the passband and stopband, perfectly passing some frequencies while blocking others. Despite not being physically achievable, they serve as a conceptual model in signal processing because they capture the "perfect" filter behavior.

fig: Ideal Vs Real Filter

Important Features of the Ideal Filters

• Sharp cutoff: At the cutoff frequency, the filter's frequency response abruptly shifts from pass (magnitude = 1) to stop (magnitude = 0).
• No distortion: All frequencies are passed through the filter with equal gain (often magnitude 1) within the passband.
• Zero response: All frequencies (magnitude 0) outside the passband are completely eliminated. An ideal filter in the time domain necessitates a filter with an infinite impulse response that extends both non-causal and infinite impulse response.
• Non-causal and infinite impulse response: It is impossible to implement an ideal filter in practice because it necessitates a filter with an infinite impulse response that extends both forward and backward in time.

Ideal Low-Pass Filter:

\[
|H(\omega)| = 
\begin{cases}
1 & |\omega| \leq \omega_c \\
0 & |\omega| > \omega_c
\end{cases}
\]

Ideal High-Pass Filter:

\[
|H(\omega)| = 
\begin{cases}
0 & |\omega| \leq \omega_c \\
1 & |\omega| > \omega_c
\end{cases}
\]

Ideal Band-Pass Filter:

\[
|H(\omega)| = 
\begin{cases}
1 & \omega_1 \leq |\omega| \leq \omega_2 \\
0 & \text{otherwise}
\end{cases}
\]

Ideal Band-Stop Filter:

\[
|H(\omega)| = 
\begin{cases}
0 & \omega_1 \leq |\omega| \leq \omega_2 \\
1 & \text{otherwise}
\end{cases}
\]