World Library  
Flag as Inappropriate
Email this Article

Group delay and phase delay

Article Id: WHEBN0000041222
Reproduction Date:

Title: Group delay and phase delay  
Author: World Heritage Encyclopedia
Language: English
Subject: Bessel filter, True time delay, Latency (audio), Outline of television broadcasting, Signal processing
Collection: Optics, Signal Processing, Waves
Publisher: World Heritage Encyclopedia
Publication
Date:
 

Group delay and phase delay

In signal processing, group delay is a measure of the time delay of the amplitude envelopes of the various sinusoidal components of a signal through a device under test, and is a function of frequency for each component. Phase delay is similar, however it is a measure of the time delay of the phase as opposed to the time delay of the amplitude envelope.

All frequency components of a signal are delayed when passed through a device such as an amplifier, a loudspeaker, or propagating through space or a medium, such as air. This signal delay will be different for the various frequencies unless the device has the property of being linear phase. (Linear phase and minimum phase are often confused. They are quite different.) The delay variation means that signals consisting of multiple frequency components will suffer distortion because these components are not delayed by the same amount of time at the output of the device. This changes the shape of the signal in addition to any constant delay or scale change. A sufficiently large delay variation can cause problems such as poor fidelity in audio or intersymbol interference (ISI) in the demodulation of digital information from an analog carrier signal. High speed modems use adaptive equalizers to compensate for non-constant group delay.

Contents

  • Introduction 1
  • Group delay in optics 2
  • Group delay in audio 3
  • See also 4
  • References 5
  • External links 6

Introduction

Group delay is a useful measure of time distortion, and is calculated by differentiating, with respect to frequency, the phase response versus frequency of the device under test (DUT): the group delay is a measure of the slope of the phase response at any given frequency. Variations in group delay cause signal distortion, just as deviations from linear phase cause distortion.

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

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

Or, in the frequency domain,

Y(s) = H(s) X(s) \,

where

X(s) = \mathcal{L}\left \{ x(t) \right \} \ \stackrel{\mathrm{def}}{=}\ \int_{-\infty}^{\infty} x(t) e^{-st}\, dt
Y(s) = \mathcal{L}\left \{ y(t) \right \} \ \stackrel{\mathrm{def}}{=}\ \int_{-\infty}^{\infty} y(t) e^{-st}\, dt

and

H(s) = \mathcal{L}\left \{ h(t) \right \} \ \stackrel{\mathrm{def}}{=}\ \int_{-\infty}^{\infty} h(t) e^{-st}\, dt .

Here \displaystyle h(t) is the time-domain impulse response of the LTI system and \displaystyle X(s), \displaystyle Y(s), \displaystyle H(s), are the Laplace transforms of the input \displaystyle x(t), output \displaystyle y(t), and impulse response \displaystyle h(t), respectively. \displaystyle H(s) is called the transfer function of the LTI system and, as does the impulse response, \displaystyle h(t), 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 \displaystyle a(t), relative to the rate of change of phase, \displaystyle \omega, of the sinusoid),

x(t) = a(t) \cos(\omega t + \theta) \

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

y(t) = |H(i \omega)| \ a(t - \tau_g) \cos \left( \omega (t - \tau_\phi) + \theta \right) \

if

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

and \displaystyle \tau_g and \displaystyle \tau_\phi, the group delay and phase delay respectively, are equal to the expressions below (and potentially are functions of angular frequency \displaystyle \omega). In a linear phase system (with non-inverting gain), both \displaystyle \tau_g and \displaystyle \tau_\phi are constant and equal to the same overall delay of the system and the unwrapped phase shift of the system is negative with magnitude increasing linearly with frequency \displaystyle \omega.

It can be shown that for an LTI system with transfer function \displaystyle H(s) driven by a complex sinusoid of unit amplitude,

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

the output is

\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} \

where the phase shift \displaystyle \phi is

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

Additionally, it can be shown that the group delay, \displaystyle \tau_g, and phase delay, \displaystyle \tau_\phi, are related to the phase shift \displaystyle \phi as

\tau_g(\omega) = - \frac{d \phi(\omega)}{d \omega} \
\tau_\phi(\omega) = - \frac{\phi(\omega)}{\omega} \ .

Group delay in optics

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,
\tau_g = -\frac{d\phi}{d\omega}
through a device or transmission medium, where \phi \ is the total phase shift in radians, and \omega \ is the angular frequency in radians per unit time, equal to 2 \pi f \ , where 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 \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., \phi(\omega) = \phi(0) - \tau_g \omega \ where the group delay \tau_g \ is a constant). The degree of nonlinearity of the phase indicates the deviation of the group delay from a constant.

Group delay in audio

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).

Frequency Threshold
500 Hz 3.2 ms
1 kHz 2 ms
2 kHz 1 ms
4 kHz 1.5 ms
8 kHz 2 ms

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]

See also

References

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

  1. ^ Flanagan; Moore, Brian C. J.; Stone, Michael A. (2005), "Discrimination of Group Delay in Clicklike Signals Presented via Headphones and Loudspeakers", Journal of the Audio Engineering Society 53 (7/8): 593–611 
  2. ^ http://www.rp-photonics.com/group_delay.html
  3. ^ http://www.rp-photonics.com/spectral_phase.html
  • Blauert, J.; Laws, P. (May 1978), "Group Delay Distortions in Electroacoustical Systems", Journal of the Acoustical Society of America 63 (5): 1478–1483,  

External links

  • Discussion of Group Delay in Loudspeakers
  • Group Delay Explanations and Applications
This article was sourced from Creative Commons Attribution-ShareAlike License; additional terms may apply. World Heritage Encyclopedia content is assembled from numerous content providers, Open Access Publishing, and in compliance with The Fair Access to Science and Technology Research Act (FASTR), Wikimedia Foundation, Inc., Public Library of Science, The Encyclopedia of Life, Open Book Publishers (OBP), PubMed, U.S. National Library of Medicine, National Center for Biotechnology Information, U.S. National Library of Medicine, National Institutes of Health (NIH), U.S. Department of Health & Human Services, and USA.gov, which sources content from all federal, state, local, tribal, and territorial government publication portals (.gov, .mil, .edu). Funding for USA.gov and content contributors is made possible from the U.S. Congress, E-Government Act of 2002.
 
Crowd sourced content that is contributed to World Heritage Encyclopedia is peer reviewed and edited by our editorial staff to ensure quality scholarly research articles.
 
By using this site, you agree to the Terms of Use and Privacy Policy. World Heritage Encyclopedia™ is a registered trademark of the World Public Library Association, a non-profit organization.
 


Copyright © World Library Foundation. All rights reserved. eBooks from Project Gutenberg are sponsored by the World Library Foundation,
a 501c(4) Member's Support Non-Profit Organization, and is NOT affiliated with any governmental agency or department.