Filter Classification by Frequency Response
The four classical filter response types offer different tradeoffs between passband flatness, transition band selectivity, group delay linearity, and implementation complexity. All derive from a normalized lowpass prototype with unit cutoff frequency ω_c = 1 rad/s that is frequency-transformed to the desired passband.
Butterworth (Maximally Flat Magnitude)
|S21(jω)|² = 1 / [1 + (ω/ω_c)^(2N)] N = filter order No passband ripple — flattest possible passband −3 dB exactly at ω_c for all orders Roll-off: −20N dB/decade beyond cutoff Group delay: monotonically increases at band edge (not flat) Moderate selectivity — between Bessel and Chebyshev
Chebyshev Type I (Equiripple Passband)
|S21(jω)|² = 1 / [1 + ε² · TN²(ω/ω_c)] TN = Chebyshev polynomial of order N ε = √(10^(Rp/10) − 1), Rp = passband ripple in dB Passband ripple: ε is set to match Rp (0.1 dB, 0.5 dB, 1 dB common) Steeper rolloff than Butterworth for same N and passband ripple Worse group delay (larger variation at band edges) More sensitive to component tolerances than Butterworth
Bessel (Maximally Flat Group Delay)
H(s) = d_N / B_N(s) [Bessel polynomial in denominator] Maximally flat group delay at ω = 0 → extends across passband Slowest rolloff of all four types for same N Best for digital signal transmission (OFDM, QAM): minimal ISI Used in oscilloscope bandwidth stages, baseband pulse shaping
Elliptic (Cauer) Filter
Equiripple in BOTH passband and stopband Fastest rolloff for given N, passband ripple, and stopband attenuation Sharpest shape factor: BW_60dB / BW_3dB can be <1.2 Worst group delay (large ripple near cutoff) Finite transmission zeros in stopband Used when sharp cutoff is critical (duplexer receive band protection)
Filter Comparison Table (N=5, 0.5 dB ripple)
| Characteristic | Butterworth | Chebyshev | Bessel | Elliptic |
|---|---|---|---|---|
| Passband ripple | 0 dB (flat) | 0.5 dB | 0 dB (flat) | 0.5 dB |
| Attn at 2ωc | −60 dB | −84 dB | −34 dB | >90 dB |
| Group delay flatness | Moderate | Poor | Excellent | Very poor |
| Phase linearity | Moderate | Poor | Excellent | Very poor |
| Component sensitivity | Low | Moderate | Low | High |
| Best use | General RF | Diplexer, BPF | Data signals | Duplexer Rx |
Filter Classification by Frequency Band
- Low-Pass Filter (LPF): passes DC to f_c, attenuates above
- High-Pass Filter (HPF): passes above f_c, blocks below
- Bandpass Filter (BPF): passes f_low to f_high
- Band-Stop / Notch: attenuates f_low to f_high, passes outside
RF View: Load filter S2P files into RF View to measure actual passband IL, stopband rejection, 3 dB bandwidth, and group delay. Compare multiple filter vendors on a single chart to identify best-in-class. The Circuit Simulator can add ideal Butterworth/Chebyshev filter blocks to your simulation chain.