Chebyshev Response
|H(jω)|² = 1 / [1 + ε²·Tₙ²(ω/ω_c)] Tₙ = Chebyshev polynomial of order N ε = √(10^(Rp/10) − 1) Rp = passband ripple (dB) Passband: 0 ≤ ω ≤ ω_c, response oscillates between 1/(1+ε²) and 1 Above cutoff: |H|² falls faster than Butterworth (better selectivity)
Chebyshev vs Butterworth Rejection
| Order | Atten at 2ω_c (Butterworth) | Atten at 2ω_c (Cheby 0.5 dB) |
|---|---|---|
| 3 | −36 dB | −47 dB |
| 4 | −48 dB | −63 dB |
| 5 | −60 dB | −80 dB |
| 7 | −84 dB | −112 dB |
Prototype Element Values (0.5 dB ripple)
| N | g₁ | g₂ | g₃ | g₄ |
|---|---|---|---|---|
| 2 | 1.4029 | 0.7071 | — | — |
| 3 | 1.5963 | 1.0967 | 1.5963 | — |
| 4 | 1.6703 | 1.1926 | 2.3661 | 0.8419 |
| 5 | 1.7058 | 1.2296 | 2.5408 | 1.2296 |
Chebyshev Group Delay
Unlike Butterworth, Chebyshev filters have significant group delay peaking near the band edge. For 0.5 dB ripple, N=5 Chebyshev BPF: group delay peaks at ≈3× the passband average near cutoff. For LTE data (OFDM): group delay ripple must be <5 ns, often requiring BAW filters rather than LC Chebyshev.
RF View: Plot both S21 and Group Delay from a measured Chebyshev filter .s2p file. BW Marker reads 3 dB bandwidth; Group Delay view shows peaking near band edges — the characteristic Chebyshev signature.