Editing Group delay and phase delay (section)

== 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/>