Side-by-side comparison
| Parameter | Ideal | Practical Low Pass Filter |
|---|---|---|
| Passband gain | Exactly 1 (0 dB) for all |f| ≤ fc | Approximately 1; ripple possible (Chebyshev: ±0.5 dB) |
| Stopband attenuation | Exactly 0 (−∞ dB) for |f| > fc | Finite attenuation; Butterworth: −20n dB/decade |
| Transition band | Zero width — brick-wall cutoff at fc | Non-zero; Butterworth 2nd order: 1 octave past fc is −12 dB |
| Impulse response | sinc(2fc·t) — infinite in time, non-causal | Decaying exponential-like; causal and finite settling |
| Phase response | Linear phase — constant group delay τg = 1/2fc | Non-linear phase for Butterworth/Chebyshev IIR filters |
| Group delay | Constant at all frequencies | Varies; peaks near fc — causes waveform distortion |
| Causality | Non-causal — cannot be built | Causal — physically realisable |
| Real implementation | Approximated by ideal brickwall in MATLAB using fir1() with large N | MFB Butterworth LPF using TL072, fc = 1 kHz, Q = 0.707 |
| Gibbs phenomenon | Overshoot at fc when FIR approximation is truncated | No Gibbs; smooth roll-off avoids ringing |
| Filter order tradeoff | Not applicable — ideal is infinite order | Higher order → sharper rolloff but more phase distortion and cost |
Key differences
The ideal LPF has a rectangular frequency response: unity gain for |f| ≤ fc, zero elsewhere. Its time-domain equivalent is sinc(2fct), which exists for all time including t < 0 — making it non-causal and unrealisable. Practical filters approximate this: a 2nd-order Butterworth rolls off at −12 dB/octave past fc; a 4th-order drops at −24 dB/octave. But higher order also means more phase distortion — the group delay of a Chebyshev filter peaks sharply near fc, distorting pulse edges. Bessel filters sacrifice rolloff sharpness to get maximally flat group delay.
When to use Ideal
Use the ideal LPF as a theoretical benchmark — for example, in a MATLAB simulation using fir1(512, 0.4) with a 512-tap FIR to approximate a brick-wall filter at 0.4·fs/2 for an offline ECG analysis script.
When to use Practical Low Pass Filter
Use a practical LPF when hardware is involved — for example, a 2nd-order Sallen-Key Butterworth filter using a TL072 with R = 15.9 kΩ and C = 10 nF giving fc ≈ 1 kHz for anti-aliasing before a 8 kHz ADC.
Recommendation
For exams, choose the ideal LPF for theoretical derivations involving convolution with a sinc function. Choose the practical Butterworth or Chebyshev filter for any design problem that involves component values, order selection, or −3 dB frequency. In placements, name the filter type (Butterworth for maximally flat, Chebyshev for sharper rolloff, Bessel for linear phase) — generic answers will cost you.
Exam tip: Examiners ask you to sketch the impulse and frequency response of an ideal LPF and then state why it is non-causal — show h(t) = 2fc·sinc(2fct), note it is non-zero for t < 0, and conclude it violates causality.
Interview tip: An interviewer at an analog design company will ask you to choose between Butterworth and Chebyshev for an anti-aliasing filter — say Butterworth for maximally flat passband with no ripple, Chebyshev Type I if you need steeper rolloff and can tolerate ±0.5 dB passband ripple.