Editing Group delay and phase delay

{{Short description|Delays experienced through a linear time-invariant system}}

In [[signal processing]], '''group delay''' and '''phase delay''' are functions that describe in different ways the delay times experienced by a signal’s various sinusoidal frequency components as they pass through a [[linear time-invariant system|linear time-invariant (LTI) system]] (such as a [[microphone]], [[coaxial cable]], [[amplifier]], [[loudspeaker]], [[communications system]], [[ethernet cable]], [[digital filter]], or [[analog filter]]).

Unfortunately, these delays are sometimes [[frequency]] dependent,<ref name="RabinerGold1975" /> which means that different sinusoid frequency components experience different time delays. As a result, the signal's [[waveform]] experiences [[distortion]] as it passes through the system. This distortion can cause problems such as poor [[High fidelity|fidelity]] in [[analog video]] and [[analog audio]], or a high [[bit-error rate]] in a digital bit stream.

== Background ==
=== Frequency components of a signal ===
{{Main|Sine wave|Fourier analysis|Linear time-invariant system}}
[[Fourier analysis]] reveals how [[signals]] in time can alternatively be expressed as the sum of [[sinusoidal]] [[frequency component]]s, each based on the trigonometric function <math>\sin(x)</math> with a fixed amplitude and phase and no beginning and no end.

[[Linear time-invariant system]]s process each sinusoidal component independently; the property of [[linearity]] means they satisfy the [[superposition principle]].

== Introduction ==
The group delay and phase delay properties of a linear time-invariant (LTI) system are functions of frequency, giving the time from when a [[#Frequency components of a signal|frequency component]] of a time varying physical quantity&mdash;for example a voltage signal&mdash;appears at the LTI system input, to the time when a copy of that same frequency component&mdash;perhaps of a different physical phenomenon&mdash;appears at the LTI system output.

A varying [[phase response]] as a function of frequency, from which group delay and phase delay can be calculated, typically occurs in devices such as microphones, amplifiers, loudspeakers, magnetic recorders, headphones, coaxial cables, and antialiasing filters.<ref name="Preis1982" /> All frequency components of a signal are delayed when passed through such devices, or when propagating through space or a medium, such as air or water.

While a phase response describes [[phase shift]] in [[angular units]] (such as [[Degree (angle)|degrees]] or [[radians]]), the phase delay is in [[units of time]] and equals the negative of the phase shift at each frequency divided by the value of that frequency. Group delay is the negative [[derivative]] of phase shift with respect to frequency.

=== Phase delay ===
A linear time-invariant system or device has a [[phase response]] property and a phase delay property, where one can be calculated exactly from the other. Phase delay directly measures the device or system time delay of individual ''sinusoidal'' frequency components. If the phase delay function at any given frequency&mdash;within a frequency range of interest&mdash;has the same constant of proportionality between the phase at a selected frequency and the selected frequency itself, the system/device will have the ideal of a flat phase delay property, a.k.a. [[linear phase]].<ref name="RabinerGold1975" /> Since phase delay is a function of frequency giving time delay, a departure from the flatness of its function graph can reveal time delay differences among the signal’s various sinusoidal [[#Frequency components of a signal|frequency components]], in which case those differences will contribute to signal distortion, which is manifested as the output signal waveform shape being different from that of the input signal.

The phase delay property in general does not give useful information if the device input is a [[Modulation|modulated]] signal. For that, group delay must be used.

=== Group delay ===
The group delay is a convenient measure of the linearity of the phase with respect to frequency in a modulation system.<ref name="OppenheimSchaferBuck1999" /><ref name="OppenheimSchafer2014" /> For a modulation signal (passband signal), the information carried by the signal is carried exclusively in the [[wave envelope]]. Group delay therefore operates only with the frequency components derived from the envelope.

==== Basic modulation system ====

[[File:Outer and Inner LTI Device.png|Figure 1: Outer and Inner LTI Devices|frame]]

A device's group delay can be exactly calculated from the device's phase response, but not the other way around.  The simplest use case for group delay is illustrated in Figure 1 which shows a conceptual [[modulation]] system, which is itself an LTI system with a baseband output that is ideally an accurate copy of the baseband signal input. This system as a whole is referred to here as the outer LTI system/device, which contains an inner (red block) LTI system/device.  As is often the case for a radio system, the inner red LTI system in Fig 1 can represent two LTI systems in cascade, for example an amplifier driving a transmitting antenna at the sending end and the other an antenna and amplifier at the receiving end.

==== Amplitude Modulation ====
[[Amplitude modulation]] creates the passband signal by shifting the baseband frequency components to a much higher frequency range.  Although the frequencies are different, the passband signal carries the same information as the baseband signal. The demodulator does the inverse, shifting the passband frequencies back down to the original baseband frequency range. Ideally, the output (baseband) signal is a time delayed version of the input (baseband) signal where the waveform shape of the output is identical to that of the input.

In Figure 1, the outer system phase delay is the meaningful performance metric. ''For amplitude modulation, the inner red LTI device group delay becomes the outer LTI device phase delay''. If the inner red device group delay is completely flat in the frequency range of interest, the outer device will have the ideal of a phase delay that is also completely flat, where the contribution of distortion due to the outer LTI device's phase response&mdash;determined entirely by the inner device's possibly different phase response&mdash;is eliminated. In that case, the group delay of the inner red device and the phase delay of the outer device give the same time delay figure for the signal as a whole, from the baseband input to the baseband output. It is significant to note that it is possible for the inner (red) device to have a very non-flat phase delay (but flat group delay), while the outer device has the ideal of a perfectly flat phase delay.  This is fortunate because in LTI device design, a flat group delay is easier to achieve than a flat phase delay.

==== Angle Modulation ====
In an angle-modulation system&mdash;such as with frequency modulation (FM) or phase modulation (PM)&mdash;the (FM or PM) passband signal applied to an LTI system input can be analyzed as two separate passband signals, an in-phase (I) amplitude modulation AM passband signal and a quadrature-phase (Q) amplitude modulation AM passband signal, where their sum exactly reconstructs the original angle-modulation (FM or PM) passband signal. While the (FM/PM) passband signal is not amplitude modulation, and therefore has no apparent outer envelope, the I and Q passband signals do indeed have separate amplitude modulation envelopes.  (However, unlike with regular amplitude modulation, the I and Q envelopes do not resemble the wave shape of the baseband signals, even though 100 percent of the baseband signal is represented in a complex manner by their envelopes.)  So, for each of the I and Q passband signals, a flat group delay ensures that neither the I pass band envelope nor the Q passband envelope will have wave shape distortion, so when the I passband signal and the Q passband signal are added back together, the sum is the original FM/PM passband signal, which will also be unaltered.

== Theory ==
According to [[LTI system theory]] (used in [[control theory]] and [[digital signal processing|digital]] or [[analog signal processing]]), the output signal <math>\displaystyle y(t)</math> of an LTI system can be determined by [[convolution|convolving]] the time-domain [[impulse response]] <math>\displaystyle h(t)</math> of the LTI system with the input signal <math>\displaystyle x(t)</math>. {{Slink|Linear time-invariant system|Fourier and Laplace transforms}} expresses this relationship as:

: <math> y(t) = (h*x)(t) \mathrel{\stackrel{\text{def}}{=}} \int_{-\infty}^\infty h(t - \tau) \, x(\tau) \, \mathrm{d} \tau \mathrel{\stackrel{\text{def}}{=}} \mathcal{L}^{-1}\{H(s) \, X(s)\} \, , </math>

where <math>*</math> denotes the convolution operation, <math>\displaystyle X(s)</math> and <math>\displaystyle H(s)</math> are the [[Laplace transform]]s of the input <math>\displaystyle x(t)</math> and impulse response <math>\displaystyle h(t)</math>, respectively, {{Mvar|s}} is the [[Laplace transform#Formal definition|complex frequency]], and <math>\mathcal{L}^{-1}</math> is the inverse Laplace transform. <math>\displaystyle H(s)</math> is called the [[transfer function]] of the LTI system and, like the impulse response <math>\displaystyle h(t)</math>, ''fully'' defines the input-output characteristics of the LTI system. This convolution can be evaluated by using the integral expression in the [[time domain]], or (according to the rightmost expression) by using multiplication in the [[Laplace domain]] and then applying the inverse transform to return to time domain.

=== LTI system response to wave packet ===
Suppose that such a system is driven by a wave packet formed by a [[Sine wave|sinusoid]] multiplied by an amplitude envelope <math>\displaystyle A_\text{env}(t)>0</math>, so the input <math>\displaystyle x(t)</math> can be expressed in the following form:

: <math> x(t) = A_\text{env}(t) \cos(\omega t + \theta) \, . </math>

Also suppose that the envelope <math>\displaystyle A_\text{env}(t)</math> is slowly changing relative to the sinusoid's frequency <math>\displaystyle \omega</math>. This condition can be expressed mathematically as:

: <math> \left| \frac{d}{dt} \log \big( A_\text{env}(t) \big) \right| \ll \omega \ .</math>

Applying the earlier convolution equation would reveal that the output of such an LTI system is very well approximated{{Clarification needed|reason=Some of the missing math steps (maybe as a footnote) would be nice here to show how the earlier condition allows for this to be "very well approximated".|date=June 2023}} as:

: <math> y(t) = \big| H(i \omega) \big| \ A_\text{env}(t - \tau_g) \cos \big( \omega (t - \tau_\phi) + \theta \big) \; .</math>

Here <math>\displaystyle \tau_g</math> is the group delay and <math>\displaystyle \tau_\phi</math> is the phase delay, and they are given by the expressions below (and potentially are functions of the [[angular frequency]] <math>\displaystyle \omega</math>). The phase of the sinusoid, as indicated by the positions of the zero crossings, is delayed in time by an amount equal to the phase delay, <math>\displaystyle \tau_\phi</math>. The envelope of the sinusoid is delayed in time by the group delay, <math>\displaystyle \tau_g</math>.

=== Mathematical definition of group delay and phase delay ===
The '''group delay''', <math>\displaystyle \tau_g</math>, and '''phase delay''', <math>\displaystyle \tau_\phi</math>, are (potentially) frequency-dependent<ref name="Ambardar1999" /> and can be computed from the [[phase unwrapping|unwrapped]] phase shift <math>\displaystyle \phi( \omega )</math>. The '''phase delay''' at each frequency equals the negative of the phase shift at that frequency divided by the value of that frequency:

: <math> \tau_\phi(\omega) = - \frac{\phi(\omega)}{\omega} \, . </math>
The '''group delay''' at each frequency equals the negative of the ''slope'' (i.e. the [[derivative]] with respect to frequency) of the phase at that frequency:<ref name="OppenheimWillskyNawab1997" />
: <math> \tau_g(\omega) = - \frac{d \phi(\omega)}{d \omega} \, . </math>

In a [[linear phase]] system (with non-inverting gain), both <math>\displaystyle \tau_g</math> and <math>\displaystyle \tau_\phi</math> are constant (i.e., independent of <math>\displaystyle \omega</math>) and equal, and their common value equals the overall delay of the system; and the unwrapped [[Phase (waves)|phase shift]] of the system (namely <math>\displaystyle -\omega \tau_\phi</math>) is negative, with magnitude increasing linearly with frequency <math>\displaystyle \omega</math>.

=== LTI system response to complex sinusoid ===
More generally, it can be shown that for an LTI system with transfer function <math>\displaystyle H(s)</math> driven by a [[phasor|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( \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{align} \ </math>

where the phase shift <math>\displaystyle \phi</math> is

: <math> \phi(\omega) \ \stackrel{\mathrm{def}}{=}\ \arg \left\{ H(i \omega) \right\}  \;. </math>

=== 1st order low- or high-pass RC filter example ===
The phase of a 1st-order [[low-pass filter]] formed by a [[RC circuit]] with [[cutoff frequency]] <math> \omega_o {=} \frac{1}{RC} </math> is:<ref>https://www.tedpavlic.com/teaching/osu/ece209/lab3_opamp_FO/lab3_opamp_FO_phase_shift.pdf {{Bare URL PDF|date=August 2024}}</ref>

<math display="block"> \phi(\omega) = -\arctan(\frac{\omega}{\omega_o}) \, . </math>

Similarly, the phase for a 1st-order RC [[high-pass filter]] is:

<math display="block"> \phi(\omega) = \frac{\pi}{2} -\arctan(\frac{\omega}{\omega_o}) \, . </math>

Taking the negative derivative with respect to <math> \omega </math> for either this low-pass or high-pass filter yields the same group delay of:<ref name="aolson"/>

<math display="block"> \begin{align}
\tau_g(\omega) &= \frac{\omega_o}{\omega^2 + \omega_o^2} \, . \\
\end{align} </math>

For frequencies significantly lower than the cutoff frequency, the phase response is approximately linear (arctan for small inputs can be approximated as a line), so the group delay simplifies to a constant value of:

<math display="block"> \begin{align}
\tau_g(\omega \ll \omega_o) &\approx \frac{1}{\omega_o} = RC \, . \\
\end{align} </math>

Similarly, right at the cutoff frequency, <math> \tau_g(\omega {=} \omega_o) = \frac{1}{2 \omega_o} = \frac{RC}{2} \, . </math>

As frequencies get even larger, the group delay decreases with the inverse square of the frequency and approaches zero as frequency approaches infinity.

=== Negative group delay ===
<gallery mode="packed" perrow="1" caption="Figure 2: Negative group delay filter circuit">
File:Ltspice-negative-1ms-group-delay.png|[[Electronic circuit|Circuit]] with ''negative'' group delay of <math>\displaystyle \tau_g</math> = {{Nowrap|−RC}} = {{Nowrap|1=−1 ms}} for frequencies much lower than {{Fraction|1|RC}} = {{Nowrap|1 kHz}}.
File:Negative-1ms-group-delay.png|[[LTspice]] [[Alternating current|AC]] simulation of <math>\displaystyle \tau_g</math> from {{Nowrap|1 Hz}} {{Nowrap|(<math>\displaystyle \tau_g</math> ≅ −1 ms}}) to {{Nowrap|10 kHz}} (<math>\displaystyle \tau_g</math> ≅ {{Nowrap|0 ms}}).
File:100Hz-negative-group-delay-wave-1Ghz-bandwidth-opamp.png|[[Transient response|Transient]] simulation of an input (green) wave whose output (red) is ahead by {{Nowrap|1 ms}}, but with instability when the input turns on and off.
</gallery>
Filters will have ''negative'' group delay over frequency ranges where its phase response is positively-sloped. If a signal is [[band-limited]] within some maximum frequency B, then it is predictable to a small degree (within time periods smaller than {{Fraction|1|B}}). A filter whose group delay is negative over that signal's entire frequency range is able to use the signal's predictability to provide an illusion of a non-causal time advance. However, if the signal contains an unpredictable event (such as an abrupt change which makes the signal's spectrum exceed its band-limit), then the illusion breaks down.<ref name="Bariska" /> Circuits with negative group delay (e.g., Figure 2) are possible, though [[causality]] is not violated.<ref name="NakanishiSugiyamaKitan2002" />

Negative group delay filters can be made in both digital and analog domains. Applications include compensating for the inherent delay of low-pass filters, to create ''zero phase'' filters, which can be used to quickly detect changes in the trends of sensor data or stock prices.<ref name="CastorPerry" />

== Group delay in audio ==
Group delay has some importance in the audio field and especially in the sound reproduction field.<ref name=PlompSteeneken1969/><ref name=Ashley1980/> Many components of an audio reproduction chain, notably [[loudspeakers]] and multiway loudspeaker [[Audio crossover|crossover networks]], introduce group delay in the audio signal.<ref name=Preis1982/><ref name=Ashley1980/> It is therefore important to know the threshold of audibility of group delay with respect to frequency,<ref name=Moller1975/><ref name=Liski2018/><ref name=Liski2021/> especially if the audio chain is supposed to provide [[high fidelity]] reproduction. The best thresholds of audibility table has been provided by Blauert and Laws.<ref name="BlauertLaws1978"/>

{| class="wikitable" style=text-align:center
! Frequency<br />(kHz)
! Threshold<br />(ms)
! Periods<br />(Cycles)
|-
| 0.5 || 3.2 || 1.6
|-
| 1 || 2 || 2
|-
| 2 || 1 || 2
|-
| 4 || 1.5 || 6
|-
| 8 || 2 || 16 
|}

Flanagan, Moore and Stone conclude that at 1, 2 and 4&nbsp;kHz, a group delay of about 1.6&nbsp;ms is audible with headphones in a non-reverberant condition.<ref name="FlanaganMooreStone2005"/> Other experimental results suggest that when the group delay in the frequency range from 300&nbsp;Hz to 1&nbsp;kHz is below 1.0&nbsp;ms, it is inaudible.<ref name="Liski2018"/>

The waveform of any signal can be reproduced exactly by a system that has a flat frequency response and group delay over the bandwidth of the signal. Leach<ref name=Leach1989/> introduced the concept of differential time-delay distortion, defined as the difference between the phase delay and the group delay:

: <math> \Delta\tau = \tau_\phi - \tau_g </math>.

An ideal system should exhibit zero or negligible differential time-delay distortion.<ref name=Leach1989/>

It is possible to use digital signal processing techniques to correct the group delay distortion that arises due to the use of crossover networks in multi-way loudspeaker systems.<ref name=Adam2007/> This involves considerable computational modeling of loudspeaker systems in order to successfully apply delay equalization,<ref name=Makivirta2018/> using the [[Parks–McClellan filter design algorithm|Parks-McClellan FIR equiripple filter design algorithm]].<ref name=RabinerGold1975/><ref name=OppenheimSchafer2014/><ref name=McClellanParksRabiner1973/><ref name=OppenheimSchafer2010/>

== Group delay in optics ==
Group delay is important in [[physics]], and in particular in [[optics]].

In an [[optical fiber]], group delay is the transit [[time]] required for optical [[Power (physics)|power]], traveling at a given [[Transverse mode|mode]]'s [[group velocity]], to travel a given distance. For optical fiber [[dispersion (optics)|dispersion]] measurement purposes, the quantity of interest is group [[Propagation delay|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 (optics)|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 <math display="inline"> \tau_g(\omega) = -\frac{d\phi}{d\omega}</math>, 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 value.

{{anchor|Differential}}The '''differential group delay''' is the [[Difference (mathematics)|difference]] in [[phase velocity|propagation time]] between the two [[eigenmode]]s ''X'' and ''Y'' [[Polarization (waves)|polarizations]].  Consider two [[eigenmodes]] that are the 0° and 90° [[Linearity|linear]] [[polarized light|polarization]] states.  If the state of polarization of the input signal is the linear state at 45° between the two eigenmodes, the input signal is divided equally into the two eigenmodes.  The power of the [[Transmitter|transmitted signal]] ''E''<sub>''T'',total</sub> is the combination of the transmitted signals of both ''x'' and ''y'' modes.

: <math>E_T  =  (E_{i,x} \cdot t_x)^2 + (E_{i,y} \cdot t_y)^2 \, </math>

The differential group delay ''D''<sub>''t''</sub> is defined as the difference in propagation time between the eigenmodes: ''D''<sub>''t''</sub>&nbsp;=&nbsp;|''t''<sub>''t'',''x''</sub>&nbsp;&minus;&nbsp;''t''<sub>''t'',''y''</sub>|.

==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.<ref name="TrueTimeDelay"/><ref name="Smith"/> TTD allows for a wide instantaneous signal [[Bandwidth (signal processing)|bandwidth]] with virtually no signal distortion such as pulse broadening during pulsed operation.

TTD is an important characteristic of lossless and low-loss, dispersion free, [[transmission lines]]. {{Slink|Telegrapher's equations|Lossless transmission}} reveals that signals propagate through them at a speed of <math>1 / \sqrt{LC}</math> for a distributed inductance {{Mvar|L}} and capacitance {{Mvar|C}}. Hence, any signal's propagation delay through the line simply equals the length of the line divided by this speed.

== Group delay from transfer function polynomials ==
If a transfer function or Sij of a [[Scattering parameters|scattering parameter]], is in a polynomial [[Laplace transform]] form, then the [[#Mathematical definition of group delay and phase delay|mathematical definition for group delay]] above may be solved analytically in closed form.  A [[polynomial]] [[transfer function]] <math>P(S)</math> may be taken along the <math>j\omega</math> axis and defined as <math>P(j\omega)</math>.  <math>\phi(\omega)</math> may be determined from <math>P(j\omega)</math>, and then the group delay may be determined by solving for <math>-d\phi(\omega)|/d\omega</math>.

to determine <math>\phi(\omega)</math> from <math>P(j\omega)</math>, use the definition of <math>\phi(\omega) = tan^{-1}(P(j\omega)_{imag}/P(j\omega)_{real})</math>.  Given that <math>j^{2N}</math> is always real, and <math>j^{2N+1}</math> is always imaginary, <math>\phi(\omega)</math> may be redefined as <math>\phi(\omega) = tan^{-1}(-jP(j\omega)_{odd}/P(j\omega)_{even})</math> where ''even'' and ''odd'' refer to the polynomials that contain only the even or odd order coefficients respectively.  The <math>-j</math> in the numerator merely converts the imaginary <math>P(j\omega)_{odd}</math> numerator to a real value, since <math>P(j\omega)_{odd}</math> by itself is purely imaginary.

<math>\begin{align}
&\frac{dtan^{-1}(f(x))}{dx} = \frac{df(x)/dx}{1+f(x)^2} \\
&f(x) = \frac{-jP(j\omega)_{odd}}{P(j\omega)_{even}} \\
&\frac{df(x)}{dx} = \frac{P(j\omega)_{even} \frac{d(P(j\omega)_{odd})}{dx} - 
-jP(j\omega)_{odd} \frac{d(-jP(j\omega)_{even})}{dx}}
{P(j\omega)_{even}^2}
\end{align}</math>

The above expressions contain four terms to calculate:

<math>\begin{array}{lcl}
Se = P(j\omega)_{even} &=& \sum_{k=0}^{N/2}P_{2k}(j\omega)^{2k} &=& \sum_{k=0}^{N/2}P_{2k}(-1)^{k}(\omega)^{2k}\\
So = P(j\omega)_{odd} &=& -j\sum_{k=1}^{(N+1)/2}P_{2k-1}(j\omega)^{2k-1} &=& \sum_{k=1}^{(N+1)/2}P_{2k-1}(-1)^{k-1}(\omega)^{2k-1}\\
De = \frac{d(P(j\omega)_{even})}{dx} &=& -j\sum_{k=1}^{N/2}2kP_{2k}(j\omega)^{2k-1} &=& \sum_{k=1}^{N/2}2kP_{2k}(-1)^{k-1}(\omega)^{2k-1} \\
Do = \frac{d(P(j\omega)_{odd})}{dx} &=& \sum_{k=1}^{(N+1)/2}{(2k-1)}P_{2k-1}(j\omega)^{2k-2} &=& \sum_{k=1}^{(N+1)/2}{(2k-1)}P_{2k-1}(-1)^{k-1}(\omega)^{2k-2}\\
\\
\frac{df(x)}{dx} &=& \frac{SeDo - SoDe}{Se^2}
\end{array}</math>

The equations above may be used to determine the group delay of polynomial <math>P(S)</math>  in closed form, shown below after the equations have been reduced to a simplified form.

<math>\text{Group Delay} =gd(P(j\omega))= -\frac{d\phi(\omega)}{d\omega} = -\frac
{(So*De + Se*Do)}
{(Se^2 + So^2)}
\text{ sec}</math>

=== Polynomial ratio ===
A polynomial ratio of the form <math>P2(S) = P_{num}(S)/P_{den}(S)</math>, such as that typically found in the definition of [[filter design]]s, may have the group delay determined by taking advantage of the phase relation, <math>\phi(P1/P2) = \phi(P1) - \phi(P2)</math>.

<math>\text{Group Delay} = gd(P2) = gd(P2_{num}) - gd(P2_{den}) sec</math>

=== Simple filter example ===
A four pole Legendre filter transfer function used in the [[Optimum "L" filter#Example: 4th order transfer function|Legendre filter example]] is shown below.

<math>T_4(j\omega) = \frac{1}{2.4494897(j\omega)^4 + 3.8282201(j\omega)^3 + 4.6244874(j\omega)^2 + 3.0412127(j\omega) + 1}</math>

The numerator group delay by inspection is zero, so only the denominator group delay need be determined.

<math>\begin{align} 
&Pe_{den} = 2.4494897\omega^4  - 4.6244874\omega^2 + 1  \\
&Po_{den} = -3.8282201\omega^3  + 3.0412127\omega       \\
&De_{den} = 4(2.4494897)\omega^3 - 2(4.6244874)\omega  \\
&Do_{den} = 3(-3.8282201)\omega^2 + 3.0412127
\end{align}</math>

Evaluating at <math>\omega</math> = 1 rad/sec:

<math>\begin{align} 
&Pe_{den} = -1.1749977  \\
&Po_{den} = -0.7870074       \\
&De_{den} = -0.548984 \\
&Do_{den} = -8.4434476
\end{align}</math>

<math>\begin{align}
&\text{Group Delay} =gd(T_4(j\omega))= -\frac{d\phi(\omega)}{d\omega} \\
&= \bigg[0--\frac
{((-0.7870074*-0.548984)  + (-1.1749977*-8.4434476))}
{((-1.1749977)^2 + (-0.7870074)^2)}\bigg]  \\
&= 5.1765430\text{ sec} \\
&\text{at }\omega = 1\text{ rad/sec}
\end{align}</math>

The group delay calculation procedure and results may be confirmed to be correct by comparing them to the results derived from the digital [[derivative]] of the phase angle, <math>\phi(\omega)</math>, using a small delta <math>\Delta\omega</math> of +/-1.e-04 rad/sec.

<math>\begin{align}
&\text{Group Delay} =gd(T_4(j\omega))= -\frac{d\phi(\omega)}{d\omega} \\
&= - (\phi(1+1e-04)-\phi(1.-1e04))/2e-04\\

&= 5.1765432\text{ sec} \\
&\text{at }\omega = 1\text{ rad/sec}
\end{align}</math>

Since the group delay calculated by the digital derivative using a small delta is within 7 digits of accuracy when compared to the precise analytical calculation, the group delay calculation procedure and results are confirmed to be correct.

== Deviation from Linear Phase ==
Deviation from [[Linear phase|Linear Phase]], <math>\phi_{DLP}(\omega)</math>, sometimes referred to as just, "phase deviation", is the difference between the phase response, <math>\phi(\omega)</math>, and the [[Linearity|linear]] portion of the phase response <math>\phi_L(\omega)</math>,<ref name="Keysight"/> and is a useful measurement to determine the linearity of <math>\phi(\omega)</math>.

A convenient means to measure <math>\phi_{DLP}(\omega)</math> is to take the [[simple linear regression]] of <math>\phi(\omega)</math> sampled over a frequency range of interest, and subtract it from the actual <math>\phi(\omega)</math>. The <math>\phi_{DLP}(\omega)</math> of an ideal linear phase response would be expected to have a value of 0 across the frequency range of interest (such as the pass band of a filter), while the <math>\phi_{DLP}(\omega)</math> of a real-world approximately linear phase response may deviate from 0 by a small finite amount across the frequency range of interest.

=== Advantage over group delay ===
An advantage of measuring or calculating <math>\phi_{DLP}(\omega)</math> over measuring or calculating group delay, <math>gd(\omega)</math>, is <math>\phi_{DLP}(\omega)</math> always converges to 0 as the phase becomes linear, whereas <math>gd(\omega)</math> converges on a finite quantity that may not be known ahead of time. Given this, a linear phase optimizing function may more easily be executed with a <math>\phi_{DLP}(\omega){=}0</math> goal than with a <math>gd(\omega){=}constant</math> goal when the value for <math>constant</math> is not necessarily already known.

== See also ==
* [[Audio system measurements]]
* [[Linear phase]]
* [[Bessel filter]] — low pass filter with maximally-flat group delay
* [[Legendre filter]] — from the example section
* [[Eye pattern]]
* [[Group velocity]] — "The group velocity of light in a medium is the inverse of the group delay per unit length."<ref>{{cite web|url=https://www.rp-photonics.com/group_delay.html|title=Group Delay|author=<!--Not stated-->}}</ref>
* [[Phase velocity]]
* [[Wave packet]]

==References==
{{FS1037C}}

{{reflist|refs=
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<ref name="FlanaganMooreStone2005">{{cite journal |last1=Flanagan |first1=Sheila |last2=Moore |first2=Brian C. J. |last3=Stone |first3=Michael A. |title=Discrimination of Group Delay in Clicklike Signals Presented via Headphones and Loudspeakers |journal=Journal of the Audio Engineering Society |year=2005 |volume=53 |issue=7/8 |pages=593–611 |url=http://www.aes.org/e-lib/browse.cfm?elib=13428}}</ref>
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<ref name="PlompSteeneken1969">{{cite journal |last1=Plomp |first1=R. |last2=Steeneken |first2=H. J. M. |title=Effect of Phase on the Timbre of Complex Tones |journal=The Journal of the Acoustical Society of America |volume=46 |number=2B |pages=409–421 |year=1969 |doi=10.1121/1.1911705 |pmid=5804112 |bibcode=1969ASAJ...46..409P}}</ref>
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<ref name="CastorPerry">{{cite web |last=Castor-Perry |first=Kendall |date=2020-03-18 |title=Five things to know about prediction and negative-delay filters |url=https://www.planetanalog.com/five-things-to-know-about-prediction-and-negative-delay-filters/ |url-status=live |archive-url=https://web.archive.org/web/20220628090824/https://www.planetanalog.com/five-things-to-know-about-prediction-and-negative-delay-filters/ |archive-date=2022-06-28 |access-date=2023-06-13 |website=planetanalog.com}}</ref>
<ref name="aolson">{{cite web |title=EELE503: Modern filter design |author=<!--Not stated--> |year=2011 |url=https://www.montana.edu/aolson/ee503/EELE503_filters.pdf |page=11}}</ref>
<ref name="Keysight">{{cite web |title=Deviation from Linear Phase |author=Keysight Technologies, Inc. |url=https://helpfiles.keysight.com/csg/N52xxB/Tutorials/Phase_Devi.htm |access-date=2024-09-09}}</ref>
}}

== External links ==
* [http://www.trueaudio.com/post_010.htm Discussion of Group Delay in Loudspeakers]
* [http://www.radio-labs.com/DesignFile/DN004.pdf Group Delay Explanations and Applications]
* [https://ccrma.stanford.edu/~jos/filters/Phase_Group_Delay.html "Introduction to Digital Filters with Audio Applications", Julius O. Smith III, (September 2007 Edition).]

{{Authority control}}

[[Category:Optics]]
[[Category:Waves]]
[[Category:Signal processing]]
[[Category:Electrical engineering]]