Editing Group delay and phase delay (section)

== 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" />