What Is Group Delay?

A phase response gives how far each frequency has advanced through its cycle by the time it leaves the system. If the phase falls faster as frequency rises, the higher frequencies have been held back for longer. Group delay is that rate of fall expressed as a time. In words, it is minus the change in phase divided by the change in frequency, with phase in radians and frequency in radians per second, which gives seconds.

Group delay

Minus the slope of phase against angular frequency, in seconds. It gives how late a narrow band of frequencies around that point arrives.

A pure delay of 3 ms shifts phase by an amount that grows in a straight line with frequency. The slope of a straight line is constant, so the group delay is 3 ms at every frequency. That case is constant group delay, and it is the same thing as latency. A filter whose phase curves has a slope that changes, so its group delay changes with frequency. Varying group delay is the case with a measured threshold of audibility.

The ideal 4th-order Linkwitz-Riley crossover is a worked example. At the lowest frequencies its group delay is 2 times the square root of 2, divided by 2π times the crossover frequency. At 80 Hz that is 5.6 ms. At 300 Hz it is 1.5 ms, and at 2 kHz it is 0.23 ms. The delay is inversely proportional to the corner, so a crossover set ten times lower carries ten times the delay.

Sound takes 2.9 ms to travel one metre, so a microphone 3 m from a speaker records a group delay near 8.7 ms before the speaker has done anything to the signal. That flat offset is arrival time. Measurement software estimates it and subtracts it. What remains is the excess group delay, the part that makes some frequencies late relative to others.

Latency is the delay of the whole signal through a device, the same at every frequency. It adds to the arrival time and nothing else. A converter with 1 ms of latency moves the entire group delay trace up by 1 ms and leaves its shape alone. Only the shape of the trace, the excess, changes how the timing between frequencies sounds.

Latency matters for other reasons. Anything you play or sing through the device arrives late by that amount. Between two speakers that share a band, a difference in latency is a difference in arrival time, which is covered in the article on polarity, delay and phase.

Every filter that changes level also changes phase, and so adds group delay near its corner. In a speaker there are two places this happens. The first is the crossover between drivers. The second is the low-frequency roll-off of the box, which is a high-pass filter whether or not anyone designed it as one.

An ideal LR4 crossover adds 5.6 ms at 80 Hz, 1.5 ms at 300 Hz and about 0.2 ms at 2 kHz. A sealed box rolling off at 50 Hz adds about 5 ms, and a ported two-way at 35 Hz 15 to 20 ms. These are textbook models, not measurements, and the full source table is in the article linked below.

Group delay follows the corner frequency. A lower corner means a longer delay, so the large figures are all in the bass. The ported box is the largest case here because the model gives it a 4th-order high-pass at a low frequency. The study's models put every driver at the same depth. Real drivers sit at different depths, which the study estimates adds a few tenths of a millisecond of excess at high frequencies (about 0.1 ms for every 3.4 cm of offset), still under the 1 to 2 ms thresholds there.

Blauert and Laws measured in 1978 how much variation in group delay listeners could detect. The thresholds are 3.2 ms at 500 Hz, 2 ms at 1 kHz, 1 ms at 2 kHz, 1.5 ms at 4 kHz and 2 ms at 8 kHz. A 2 kHz crossover adds about 0.2 ms against a threshold of about 1 ms. Below 500 Hz the data stops, and later work, including Liski et al. (2018), finds larger thresholds there. Scaling the threshold with the period below 500 Hz is the study's modelling assumption, not a measured value.

Read a group delay plot from the definition. It is a slope, so each point gives how fast phase is changing at that frequency. A flat line at any height means the phase is a straight line there, and a straight line is a pure delay. The height of that flat line is arrival time. Changes in height are the excess.

Spikes at deep nulls follow from the same definition. Near a null the level has dropped almost to nothing and the phase swings quickly across it, so its slope becomes very large. There is almost no sound at that frequency to be late. In the model above the spike points downward. In a real measurement it can point either way, and its height reflects the depth of the null, not timing.

A slope also magnifies small wiggles in the phase, so noise in a measurement shows up in the group delay trace as roughness. How much windowing or smoothing a given piece of software applies before drawing group delay changes how the trace looks, and the amount varies between products.

Compare any group delay figure with the threshold at its own frequency before acting on it. A modelled two-way from 200 Hz up sits at 0.10 times the threshold before correction, and linearising its 2 kHz crossover costs 0.8 ms. The large figures are in the bass, where the box roll-off and the sub crossover sit, and where the threshold is disputed.

Removing group delay costs latency. A filter that flattens group delay has to delay everything to match the latest frequency, and in the bass that delay is 19 ms for an 80 Hz crossover and 82 ms for a ported box. That trade is the subject of Where the Latency in Linear-Phase Room Correction Comes From.