Group delay and phase delay

In linear time-invariant (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, that is 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 ${\displaystyle \displaystyle \tau _{g}}$ and ${\displaystyle \displaystyle \tau _{\phi }}$, the group delay and phase delay respectively, are given by the expressions below (and potentially are functions of the angular frequency ${\displaystyle \displaystyle \omega }$). The sinusoid, as indicated by the zero crossings, is delayed in time by 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, and they can be computed from the properly unwrapped phase shift ${\displaystyle \displaystyle \phi }$ by

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

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

In physics, and in particular in optics, the term group delay has the following meanings:

1. The rate of change of the total phase shift with respect to angular frequency,

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

through a device or transmission medium, where ${\displaystyle \phi \ }$ is the total phase shift in radians, and ${\displaystyle \omega \ }$ is the angular frequency in radians per unit time, equal to ${\displaystyle 2\pi f\ }$, where ${\displaystyle f\ }$ is the frequency (hertz if group delay is measured in seconds).

2. In an optical fiber, the transit time required for optical power, traveling at a given mode's group velocity, to travel a given distance.

Note: 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 ${\displaystyle \tau _{g}(\omega )=-{\frac {d\phi }{d\omega }}}$, as defined in (1), 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.

Group delay has some importance in the audio field and especially in the sound reproduction field. Many components of an audio reproduction chain, notably loudspeakers and multiway loudspeaker crossover networks, introduce group delay in the audio signal. It is therefore important to know the threshold of audibility of group delay with respect to frequency, especially if the audio chain is supposed to provide high fidelity reproduction. The best thresholds of audibility table has been provided by Blauert & Laws (1978).

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.[1]

A transmitting apparatus is said to have true time delay (TTD) if the time delay is independent of the frequency of the electrical signal.[2][3] 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."[4]
• Spectral phase — "The group delay can be defined as the derivative of the spectral phase with respect to angular frequency."[5]

 This article incorporates public domain material from the General Services Administration document: "Federal Standard 1037C".

• Blauert, J.; Laws, P. (May 1978), "Group Delay Distortions in Electroacoustical Systems", Journal of the Acoustical Society of America, 63 (5): 1478–1483, Bibcode:1978ASAJ...63.1478B, doi:10.1121/1.381841

• Discussion of Group Delay in Loudspeakers
• Group Delay Explanations and Applications
• Blauert, J.; Laws, P. (May 1978), "Group Delay Distortions in Electroacoustical Systems", Journal of the Acoustical Society of America 63 (5): 1478–1483
• "Introduction to Digital Filters with Audio Applications", Julius O. Smith III, (September 2007 Edition).