Group delay and phase delay

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The phase delay property of a linear time invariant (LTI) system or device such as an amplifier, filter, or telecommunications system, gives the time delay of the various frequency components of a signal to pass through from input to output. In some cases this time delay, as revealed by the phase delay property, will be different for the various frequency components, in which case the signal comprising these signal components will suffer distortion because these components are not delayed by the same amount of time at the output of the device. A sufficiently large time delay variation can cause signal problems such as poor fidelity in video or audio, for example.

In a system consisting of multiple devices where the output of one device feeds the next device, the group delay over a straight-line portion (aperture) of the device's phase response, where the device is passing a modulated signal adds directly to the phase delay of the entire system.

Discussed in this article is some background theory on a device’s phase response, from which phase delay and group delay can be calculated exactly. There is also background theory on fourier series to aid in the understanding of a device’s phase response. The heart of this article is the theory of group delay and phase delay in the context of a device’s phase response.

Introduction

The phase delay of each block is directly additive to the phase delay of the entire system.

The phase delay property of a linear time invariant (LTI) system or device such as a filter or amplifier, is a function of frequency that gives the time that it takes for the various frequency components of a signal to pass through the device from input to output. Sufficient variations of phase delay over the range of the signal’s frequency components indicate that the time delay of those frequency components will be a contributing factor of signal distortion at the output. Averaging phase delay over that same frequency range when the phase delay variations are sufficiently small gives a direct measure of the signal time delay.

If the (red block) device phase response is a straight line over the range of frequencies contained in the signal, then, directly additive to the total system phase delay are: phase delay of the devices in white, and group delay of the device in red.

In a system that cascades individual devices or stages in a chain, connecting the output of one to the input of the next, the phase delay of each individual device is directly additive to the total system phase delay, with the notable exception of a device in a system chain positioned after a modulator and before a demodulator. In that case, if the device phase response is a straight line over the frequencies contained in the signal, then that device's group delay property is a function of frequency that is directly additive to the total system phase delay.

Group delay and phase delay are calculated exactly from the phase response property of an LTI device or system.

Background

Frequency components of a signal

For a periodic signal, a frequency component is a sinusoid with properties that include time-based frequency and phase.

Generating a basic sinusoid

The sinusoid, with or without a time based frequency property, is generated by a circle as shown in the figure. In this example, the sinusoid is a sine wave that is traced out using the sin trig function.

Tracing a sinusoid from a circle: y=sin(x). In this example, the sin trig function is used. For both the sinusoid and the unit circle, the dependent output variable y is on the vertical axis. For the sinusoid only, the angle in degrees is the independent input variable x on the horizontal axis. For the unit circle only, the angle in degrees is the independent input value x, represented as the actual angle in the diagram made between the horizontal axis and the red vector, currently at zero degrees in the image, but can be at any angle

Tracing two cycles of cosine and sine functions from unit circle. (Fast animation.)

Tracing two cycles of cosine and sine functions from unit circle. (Slow animation.)

When an increasing angle x makes a complete CCW rotation around the circle, one cycle of the function’s pattern is generated. Further increasing the angle beyond 360 degrees simply rotates around the circle again, completing another cycle, where each succeeding cycle repeats the same pattern, making the function periodic. (See slow animation.) The angle value has no limit, and so the number of times the pattern repeats itself also has no limit. Because of this, a sinusoid has no beginning and no end. A sinusoid function is based on either or both of the trig functions sin(x) and cos(x).

Theory

In linear time-invariant (LTI) system theory, control theory, and in digital or analog signal processing, the relationship between the input signal, ${\displaystyle \displaystyle x(t)}$ , to 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

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

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,
${\displaystyle \tau _{g}=-{\frac {d\phi }{d\omega }}}$
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 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 Periods (Cycles)
500 Hz 3.2 ms 1.6
1 kHz 2 ms 2
2 kHz 1 ms 2
4 kHz 1.5 ms 6
8 kHz 2 ms 16

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]

True time delay

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.