Group delay and phase delay

In signal processing, group delay and phase delay are two related ways of describing how a signal's frequency components are delayed in time when passing through a linear time-invariant (LTI) system (such as a microphone, coaxial cable, amplifier, loudspeaker, telecommunications system, ethernet cable, digital filter, or analog filter). Phase delay describes the time shift of a sinusoidal component (a sine wave in steady state). Group delay describes the time shift of the envelope of a wave packet, a "pack" or "group" of oscillations centered around one frequency that travel together, formed for instance by multiplying (amplitude modulation) a sine wave by an envelope (such as a tapering function).

These delays are usually frequency dependent,^[1] which means that different frequency components experience different delays. As a result, the signal's waveform experiences distortion as it passes through the system. This distortion can cause problems such as poor fidelity in analog video and analog audio, or a high bit-error rate in a digital bit stream. However, for the ideal case of a constant group delay across the entire frequency range of a bandlimited signal and flat frequency response, the waveform will experience no distortion.

Figure 2: Negative group delay filter circuit

Circuit with negative group delay of $\displaystyle \tau _{g}$ = −RC = −1 ms for frequencies much lower than 1⁄RC = 1 kHz.

LTspice AC simulation of $\displaystyle \tau _{g}$ from 1 Hz ( $\displaystyle \tau _{g}$ ≅ −1 ms) to 10 kHz ( $\displaystyle \tau _{g}$ ≅ 0 ms).

Transient simulation of an input (green) wave whose output (red) is ahead by 1 ms, but with instability when the input turns on and off.

True time delay[edit]

A transmitting apparatus is said to have true time delay (TTD) if the time delay is independent of the frequency of the electrical signal.^[25]^[26] TTD allows for a wide instantaneous signal bandwidth with virtually no signal distortion such as pulse broadening during pulsed operation.

TTD is an important characteristic of lossless and low-loss, dispersion free, transmission lines. Telegrapher's equations § Lossless transmission reveals that signals propagate through them at a speed of $1/{\sqrt {LC}}$ for a distributed inductance $L$ and capacitance $C$ . Hence, any signal's propagation delay through the line simply equals the length of the line divided by this speed.

Audio system measurements

Linear phase

— low pass filter with maximally-flat group delay

Bessel filter

— from the example section

Legendre filter

Eye pattern

— "The group velocity of light in a medium is the inverse of the group delay per unit length."^[28]

Group delay and phase delay

True time delay[edit]

Audio system measurements

Linear phase

Bessel filter

Legendre filter

Eye pattern

Group velocity

Phase velocity

Wave packet

Discussion of Group Delay in Loudspeakers

Group Delay Explanations and Applications

"Introduction to Digital Filters with Audio Applications", Julius O. Smith III, (September 2007 Edition).