Subwoofer tools.
SYSTEM PLANNING / THE TOOLKIT

Group delay.

Group delay is the negative slope of unwrapped phase. Inspect an ideal sealed system, or paste measured phase data with optional response magnitudes to flag unreliable nulls.

01 / THE INPUTS
Response model
Sealed system
Hz
Hz
Propagation

Keep at zero for excess group delay. Do not add distance twice to uncompensated measured data.

Air & conventions
m/s
kg/m3
02 / THE RESULTS
Group delay at 50 Hz
4.501ms
Delay at system resonance
4.501ms
Added propagation delay
0.000ms

Ideal sealed-system group delay

LOG FREQUENCY
Ideal sealed-system group delay: Group delay (ms) against frequency in hertz.01.172.353.524.695.875102050100200Group delay (ms)Frequency (Hz)

Read this result with its assumptions.

Exact analytical sealed-system group delay; no smoothing. Propagation is added only by the entered distance.

Units & assumptions

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03 / UNDER THE HOOD

The math, in plain sight.

Understand the timing behind the response.

Differentiate unwrapped phase

Phase is converted to radians and unwrapped before differentiation. Interior samples use central differences and endpoints use one-sided differences. The selected moving-average smoothing, if any, is applied to unwrapped phase before differentiation. It changes resolution.

τg(ω)=−dϕ(ω)dω=−12πdϕ(f)df\tau_g(\omega)=-\frac{d\phi(\omega)}{d\omega}=-\frac1{2\pi}\frac{d\phi(f)}{df}
τg(fi)≈−ϕ(fi+1)−ϕ(fi−1)2π(fi+1−fi−1)\tau_g(f_i)\approx-\frac{\phi(f_{i+1})-\phi(f_{i-1})}{2\pi(f_{i+1}-f_{i-1})}

Exact ideal sealed-system result

For fc = 50 Hz and Qtc = 0.707, the excess group delay at resonance is about 4.5 ms. Uncompensated propagation adds approximately r/c.

H(s)=s2s2+(ωc/Qtc)s+ωc2H(s)=\frac{s^2}{s^2+(\omega_c/Q_{tc})s+\omega_c^2}
τg(ω)=(ωc/Qtc)(ωc2+ω2)(ωc2−ω2)2+(ωωc/Qtc)2\tau_g(\omega)=\frac{(\omega_c/Q_{tc})(\omega_c^2+\omega^2)}{(\omega_c^2-\omega^2)^2+(\omega\omega_c/Q_{tc})^2}
τg(ωc)=2Qtcωc=Qtcπfc\tau_g(\omega_c)=\frac{2Q_{tc}}{\omega_c}=\frac{Q_{tc}}{\pi f_c}

Interpret with care

At a cancellation or response zero, phase becomes highly noise-sensitive. Supplied magnitudes more than 40 dB below the largest sample are flagged as unreliable, and neighboring numerical derivatives are excluded. This 40 dB flag is an explicit screening rule, not a universal measurement criterion. Without magnitude data, nulls cannot be identified. One large delay number does not by itself establish audibility or integration quality.