Editing Group delay and phase delay (section)

== 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.