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:

y(t) = h(t) * 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 \displaystyle x(t), \displaystyle y(t), and \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 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

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

and \displaystyle\tau_g and \displaystyle\tau_\phi, 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 \displaystyle\tau_g and \displaystyle\tau_\phi are equal to the same constant delay of the system and the 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,

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 = - \frac{d \phi(\omega)}{d \omega} \

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

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

- 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).

- 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.

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 \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.