Filter Group Delay
Group delay — the delay a filter imposes on the signal envelope — is set by how fast the phase rotates with frequency, which in turn is set by the filter's poles. This tool computes the band-center (DC) group delay of a Butterworth or Chebyshev lowpass prototype directly from its poles. Sharper responses (higher order, more ripple) delay more and ring longer, which matters for pulse and data waveforms.
Equations & Parameters ▸
\(\tau_g(\omega)=-\dfrac{d\phi}{d\omega}=\sum_k\dfrac{-\sigma_k}{\sigma_k^2+(\omega-\omega_k)^2}\)
\(\tau_g(0)=\sum_k\dfrac{-\sigma_k}{\sigma_k^2+\omega_k^2}\cdot\dfrac{1}{\omega_c},\quad \omega_c=2\pi f_c,\quad p_k=\sigma_k+j\omega_k\)
\(\tau_g(0)=\sum_k\dfrac{-\sigma_k}{\sigma_k^2+\omega_k^2}\cdot\dfrac{1}{\omega_c},\quad \omega_c=2\pi f_c,\quad p_k=\sigma_k+j\omega_k\)
| Response | Butterworth (maximally flat) or Chebyshev (equiripple). |
| n | Filter order (number of poles). |
| Rp | Passband ripple (dB) — Chebyshev only. |
| fc | Cutoff frequency (MHz) — the prototype's ripple/−3 dB edge. |
Reference: Standard lowpass-prototype pole locations; D. M. Pozar, Microwave Engineering, 4th ed. Delay computed at band center (DC) for the lowpass prototype.
Inputs
Prototype
Poles
dB
ChebyshevMHz
Band edgeResults
Group delay
Band-center delay τg(0)—
Normalized τ·ωc—
Delay in cycles at fc—
Diagram