Group delay and phase delay

The group delay and phase delay properties of a linear time-invariant (LTI) system are functions of frequency, giving the time from when a frequency component of a time varying physical quantity—for example a voltage signal—appears at the LTI system input, to the time when a copy of that same frequency component—perhaps of a different physical phenomenon—appears at the LTI system output.

A varying phase response as a function of frequency, from which group delay and phase delay can be calculated, typically occurs in devices such as microphones, amplifiers, loudspeakers, magnetic recorders, headphones, coaxial cables, and antialiasing filters.[2] All frequency components of a signal are delayed when passed through such devices, or when propagating through space or a medium, such as air or water.

Group delay

Figure 1: Outer and Inner LTI Devices

The group delay is a convenient measure of the linearity of the phase with respect to frequency in a modulation system.[4][5]

In LTI system theory, control theory, and in digital or analog signal processing, the relationship between the input signal, ${\displaystyle \displaystyle x(t)}$ and the output signal, ${\displaystyle \displaystyle y(t)}$, of an LTI system is governed by a convolution operation:

${\displaystyle y(t)=(h*x)(t)\ {\stackrel {\mathrm {def} }{=}}\ \int _{-\infty }^{\infty }x(u)h(t-u)\,du}$

Or, in the frequency domain,

${\displaystyle Y(s)=H(s)X(s)\,}$

where

${\displaystyle X(s)={\mathcal {L}}{\Big \{}x(t){\Big \}}\ {\stackrel {\mathrm {def} }{=}}\ \int _{-\infty }^{\infty }x(t)e^{-st}\,dt}$

${\displaystyle Y(s)={\mathcal {L}}{\Big \{}y(t){\Big \}}\ {\stackrel {\mathrm {def} }{=}}\ \int _{-\infty }^{\infty }y(t)e^{-st}\,dt}$

and

${\displaystyle H(s)={\mathcal {L}}{\Big \{}h(t){\Big \}}\ {\stackrel {\mathrm {def} }{=}}\ \int _{-\infty }^{\infty }h(t)e^{-st}\,dt}$.

Here ${\displaystyle \displaystyle h(t)}$ is the time-domain impulse response of the LTI system and ${\displaystyle \displaystyle X(s)}$, ${\displaystyle \displaystyle Y(s)}$, ${\displaystyle \displaystyle H(s)}$, are the Laplace transforms of the input ${\displaystyle \displaystyle x(t)}$, output ${\displaystyle \displaystyle y(t)}$, and impulse response ${\displaystyle \displaystyle h(t)}$, respectively. ${\displaystyle \displaystyle H(s)}$ is called the transfer function of the LTI system and, like the impulse response ${\displaystyle \displaystyle h(t)}$, fully defines the input-output characteristics of the LTI system.

Suppose that such a system is driven by a quasi-sinusoidal signal, such as a sinusoid having an amplitude envelope ${\displaystyle \displaystyle a(t)>0}$ that is slowly changing relative to the frequency ${\displaystyle \displaystyle \omega }$ of the sinusoid. Mathematically, this means that the quasi-sinusoidal driving signal has the form

${\displaystyle x(t)=a(t)\cos(\omega t+\theta )\ }$

and the slowly changing amplitude envelope ${\displaystyle \displaystyle a(t)}$ means that

${\displaystyle \left|{\frac {d}{dt}}\log {\big (}a(t){\big )}\right|\ll \omega \ .}$

Then the output of such an LTI system is very well approximated as

${\displaystyle y(t)={\big |}H(i\omega ){\big |}\ a(t-\tau _{g})\cos {\big (}\omega (t-\tau _{\phi })+\theta {\big )}\;.}$

Here ${\displaystyle \displaystyle \tau _{g}}$ is the group delay and ${\displaystyle \displaystyle \tau _{\phi }}$ is the phase delay, and they are given by the expressions below (and potentially are functions of the angular frequency ${\displaystyle \displaystyle \omega }$). The phase of the sinusoid, as indicated by the positions of the zero crossings, is delayed in time by an amount equal to the phase delay, ${\displaystyle \displaystyle \tau _{\phi }}$. The envelope of the sinusoid is delayed in time by the group delay, ${\displaystyle \displaystyle \tau _{g}}$.

In a linear phase system (with non-inverting gain), both ${\displaystyle \displaystyle \tau _{g}}$ and ${\displaystyle \displaystyle \tau _{\phi }}$ are constant (i.e., independent of ${\displaystyle \displaystyle \omega }$) and equal, and their common value equals the overall delay of the system; and the unwrapped phase shift of the system (namely ${\displaystyle \displaystyle -\omega \tau _{\phi }}$) is negative, with magnitude increasing linearly with frequency ${\displaystyle \displaystyle \omega }$.

More generally, it can be shown that for an LTI system with transfer function ${\displaystyle \displaystyle H(s)}$ driven by a complex sinusoid of unit amplitude,

${\displaystyle x(t)=e^{i\omega t}\ }$

the output is

${\displaystyle {\begin{aligned}y(t)&=H(i\omega )\ e^{i\omega t}\ \\&=\left({\big |}H(i\omega ){\big |}e^{i\phi (\omega )}\right)\ e^{i\omega t}\ \\&={\big |}H(i\omega ){\big |}\ e^{i\left(\omega t+\phi (\omega )\right)}\ \\\end{aligned}}\ }$

where the phase shift ${\displaystyle \displaystyle \phi }$ is

${\displaystyle \phi (\omega )\ {\stackrel {\mathrm {def} }{=}}\ \arg \left\{H(i\omega )\right\}\;.}$

Additionally, it can be shown that the group delay, ${\displaystyle \displaystyle \tau _{g}}$, and phase delay, ${\displaystyle \displaystyle \tau _{\phi }}$, are frequency-dependent.[6] They can be computed from the unwrapped phase shift ${\displaystyle \displaystyle \phi }$ by

${\displaystyle \tau _{g}(\omega )=-{\frac {d\phi (\omega )}{d\omega }}\ }$

${\displaystyle \tau _{\phi }(\omega )=-{\frac {\phi (\omega )}{\omega }}\ }$ .

That is, the group delay at each frequency equals the negative of the slope of the phase at that frequency[7] (compare to instantaneous frequency).

Group delay has some importance in the audio field and especially in the sound reproduction field.[10][11] Many components of an audio reproduction chain, notably loudspeakers and multiway loudspeaker crossover networks, introduce group delay in the audio signal.[2][11] It is therefore important to know the threshold of audibility of group delay with respect to frequency,[12][13][14] especially if the audio chain is supposed to provide high fidelity reproduction. The best thresholds of audibility table has been provided by Blauert and Laws.[15]

Flanagan, Moore and Stone conclude that at 1, 2 and 4 kHz, a group delay of about 1.6 ms is audible with headphones in a non-reverberant condition.[16] Other experimental results suggest that when the group delay in the frequency range from 300 Hz to 1 kHz is below 1.0 ms, it is inaudible.[13]

The waveform of an audio signal can be reproduced exactly by a system that has a flat frequency response over the bandwidth of the signal and a phase delay that is equal to the group delay. Leach[17] introduced the concept of differential time-delay distortion, defined as the difference between the phase delay and the group delay, which is given by:

${\displaystyle \Delta \tau =\tau _{\phi }-\tau _{g}}$.

An ideal system should exhibit zero or negligible differential time-delay distortion.[17]

It is possible to use digital signal processing techniques to correct the group delay distortion that arises due to the use of crossover networks in multi-way loudspeaker systems.[18] This involves considerable computational modeling of loudspeaker systems in order to successfully apply delay equalization,[19] using the Parks-McClellan FIR equiripple filter design algorithm.[1][5][20][21]

Group delay is important in physics, and in particular in optics.

In an optical fiber, group delay is the transit time required for optical power, traveling at a given mode's group velocity, to travel a given distance. For optical fiber dispersion measurement purposes, the quantity of interest is group delay per unit length, which is the reciprocal of the group velocity of a particular mode. The measured group delay of a signal through an optical fiber exhibits a wavelength dependence due to the various dispersion mechanisms present in the fiber.

It is often desirable for the group delay to be constant across all frequencies; otherwise there is temporal smearing of the signal. Because group delay is ${\textstyle \tau _{g}(\omega )=-{\frac {d\phi }{d\omega }}}$, it therefore follows that a constant group delay can be achieved if the transfer function of the device or medium has a linear phase response (i.e., ${\displaystyle \phi (\omega )=\phi (0)-\tau _{g}\omega }$ where the group delay ${\displaystyle \tau _{g}}$ is a constant). The degree of nonlinearity of the phase indicates the deviation of the group delay from a constant value.

A transmitting apparatus is said to have true time delay (TTD) if the time delay is independent of the frequency of the electrical signal.[22][23] TTD is an important characteristic of lossless and low-loss, dispersion free, transmission lines. TTD allows for a wide instantaneous signal bandwidth with virtually no signal distortion such as pulse broadening during pulsed operation.

• Audio system measurements
• Bessel filter
• Eye pattern
• Group velocity — "The group velocity of light in a medium is the inverse of the group delay per unit length."[24]

 This article incorporates public domain material from Federal Standard 1037C. General Services Administration. Archived from the original on 2022-01-22.

• 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).