Group delay and phase delay
41222
198029765
2008-03-13T19:44:30Z
Oli Filth
547762
rm stuff which is about practice, not theory
{{context}}
'''Group delay''' is a measure of the transit time of a signal through a [[device under test]] (DUT), versus frequency. Group delay is a useful measure of phase distortion, and is calculated by differentiating the insertion [[phase response]] of the DUT versus frequency. Another way to say this is that group delay is a measure of the slope of the transmission [[phase response]]. The linear portion of the [[phase response]] is converted to a constant value (representing the average signal-transit time) and deviations from linear phase are transformed into deviations from constant group delay. The variations in group delay cause signal distortion, just as deviations from linear phase cause distortion. Group delay is just another way to look at linear phase distortion.
In [[LTI system theory]], [[control theory]], and in [[digital signal processing|digital]] or [[analog signal processing|analog]] [[signal processing]], the relationship between the input signal, <math>\displaystyle x(t)</math>, to output signal, <math>\displaystyle y(t)</math>, of an LTI system is governed by:
: <math> y(t) = h(t) * x(t) \ \stackrel{\mathrm{def}}{=}\ \int_{-\infty}^{\infty} x(u) h(t-u) \, du </math>
Or, in the [[frequency domain]],
: <math> Y(s) = H(s) X(s) \, </math>
where
: <math> X(s) = \mathcal{L}\left \{ x(t) \right \} \ \stackrel{\mathrm{def}}{=}\ \int_{-\infty}^{\infty} x(t) e^{-st}\, dt </math>
: <math> Y(s) = \mathcal{L}\left \{ y(t) \right \} \ \stackrel{\mathrm{def}}{=}\ \int_{-\infty}^{\infty} y(t) e^{-st}\, dt </math>
and
: <math> H(s) = \mathcal{L}\left \{ h(t) \right \} \ \stackrel{\mathrm{def}}{=}\ \int_{-\infty}^{\infty} h(t) e^{-st}\, dt </math>.
Here <math>\displaystyle h(t)</math> is the time domain [[impulse response]] of the LTI system and <math>\displaystyle X(s)</math>, <math>\displaystyle Y(s)</math>, <math>\displaystyle H(s)</math>, are the [[Laplace transform]]s of <math>\displaystyle x(t)</math>, <math>\displaystyle y(t)</math>, and <math>\displaystyle h(t)</math>, respectively. <math>\displaystyle H(s)</math> is called the [[transfer function]] of the LTI system and, as does the impulse response, <math>\displaystyle h(t)</math>, ''fully'' defines the input-output characteristics of the LTI system.
When such a system is driven by a quasi-sinusoidal signal, (a [[sinusoid]] with a slowly changing amplitude envelope <math>\displaystyle A(t)</math>, relative to the change of phase, <math>\displaystyle\omega</math>, of the sinusoid),
: <math> x(t) = A(t) \cos(\omega t + \theta) \ </math>
the output of such an LTI system is very well approximated as
: <math> y(t) = |H(i \omega)| A(t-\tau_g) \cos\left(\omega (t-\tau_{\phi}) + \theta\right) \ </math>
if
: <math> \frac{d \log \left( A(t) \right)}{dt} \ll \omega \ </math>
and <math>\displaystyle\tau_g</math> and <math>\displaystyle\tau_\phi</math>, the '''group delay''' and '''phase delay''' respectively, are as shown below and potentially functions of ω. In a [[linear phase]] system (with non-inverting gain), both <math>\displaystyle\tau_g</math> and <math>\displaystyle\tau_\phi</math> are equal to the same constant delay of the system and the [[Phase (waves)|phase shift]] of the system increases linearly with frequency ω.
It can be shown that for an LTI system with transfer function ''H''(''s'') that if such is driven by a complex sinusoid of unit amplitude,
: <math> x(t) = e^{i \omega t} \ </math>
the output is
: <math> \begin{align}
y(t) & = H(i \omega) e^{i \omega t} \ \\
& = \left( |H(i \omega)| e^{i \phi(\omega)} \right) e^{i \omega t} \ \\
& = |H(i \omega)| e^{i \left(\omega t + \phi(\omega) \right)} \ \\
\end{align} \ </math>
where the phase shift <math>\displaystyle\phi</math> is
: <math> \phi(\omega) \ \stackrel{\mathrm{def}}{=}\ \arg \left\{ H(i \omega) \right\} \ </math>
Additionally, it can be shown that the group delay, <math>\displaystyle\tau_g</math>, and phase delay, <math>\displaystyle\tau_\phi</math>, are related to the phase shift <math>\displaystyle\phi</math> as
: <math> \tau_g = - \frac{d \phi(\omega)}{d \omega} \ </math>
: <math> \tau_{\phi} = - \frac{\phi(\omega)}{\omega} \ </math>.
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]],
::<math> \tau_g = -\frac{d\phi}{d\omega}</math>
:through a device or [[transmission medium]], where <math> \phi \ </math> is the total phase shift in [[radians]], and <math>\omega \ </math> is the [[angular frequency]] in [[radian]]s per unit time, equal to <math>2 \pi f \ </math>, where <math> f \ </math> is the [[frequency]] ([[hertz]] if group delay is measured in seconds).
:'''2.''' In an [[optical fiber]], the transit [[time]] required for optical [[Power (physics)|power]], traveling at a given [[Transverse mode|mode]]'s [[group velocity]], to travel a given distance.
:''Note:'' For optical fiber [[dispersion (optics)|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 [[Signalling (telecommunication)|signal]] through an optical fiber exhibits a [[wavelength]] dependence due to the various [[dispersion (optics)|dispersion]] mechanisms present in the fiber.
:Source: from [[Federal Standard 1037C]]
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 <math> \tau_g(\omega) = -\frac{d\phi}{d\omega}</math>, 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., <math>\phi(\omega) = \phi(0) - \tau_g \omega \ </math> where the group delay <math>\tau_g \ </math> is a constant).
The degree of nonlinearity of the phase indicates the deviation of the group delay from a constant.
== Group delay in the audio field ==
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 loudspeakers [[Audio crossover|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 a [[high fidelity]] reproduction. At the time of writing no extensive data is available, and the concept is often treated by "rule of thumb" or based on hunches and received wisdom. The best thresholds of audibility table has been provided by [[Blauert]] and Laws:
{| class="wikitable"
! Frequency !! Threshold
|-
| 500 [[hertz|Hz]] || 3.2 [[millisecond|ms]]
|-
| 1 [[kilohertz|kHz]] || 2 ms
|-
| 2 kHz || 1 ms
|-
| 4 kHz || 1.5 ms
|-
| 8 kHz || 2 ms
|}
The table above has been published into the following article:
Blauert, J. and Laws, P "Group Delay Distortions in Electroacoustical Systems", Journal of the Acoustical Society of America, Volume 63, Number 5, pp. 1478–1483 (May 1978)
== See also ==
* [[Audio system measurements]]
* [[Bessel filter]]
== External links ==
* [http://www.trueaudio.com/post_010.htm Discussion of Group Delay in Loudspeakers]
* [http://www.radiolab.com.au/DesignFile/DN004.pdf Group Delay Explanations and Applications]
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[[Category:Optics]]
[[Category:Waves]]
[[Category:Signal processing]]