Poles and Zeros in Filter S-Parameters
A filter's frequency response is determined by the location of poles and zeros in the complex s-plane. S21 transmission zeros are frequencies where the filter blocks all signal (S21 → 0, infinite attenuation). Poles determine the passband and rolloff characteristics.
Identifying Transmission Zeros
| Filter Type | Transmission Zeros | S21 Signature |
|---|---|---|
| Butterworth | None (at infinity) | Smooth monotonic rolloff |
| Chebyshev | None (at infinity) | Equiripple passband, then monotonic |
| Elliptic (Cauer) | N/2 finite zeros in stopband | Notches visible in stopband of S21 |
| SAW/BAW filter | Near-band zeros | Sharp stopband notch near passband edge |
Reading Transmission Zeros from RF View
Load elliptic filter .s2p → S21 dB view Transmission zeros appear as deep notches in stopband Single marker at each notch → read exact zero frequency Example LTE Band 3 duplexer RX filter (elliptic): TX rejection zero at 1750 MHz: S21 = −72 dB (notch!) Without elliptic design, S21 would only be −50 dB at same offset
Why Elliptic Filters Have Finite Zeros
Elliptic (Cauer) filters place transmission zeros at specific stopband frequencies to achieve the steepest possible rolloff. This "steals" attenuation bandwidth to create narrow but very deep notches exactly where interference is strongest — e.g., at the TX band frequency, directly across the duplexer from the RX passband.
RF View: Load elliptic filter .s2p → S21 plot shows clear notches at transmission zeros. Single marker reads exact notch frequency and depth. Compare with Butterworth .s2p to see the selectivity difference. Free on Android.