Updated: Wednesday, 11 November 2015

Selected Posts on Audio Topics
by John L. Murphy

Subject:  Equalizers and Group Delay in Loudspeakers

(posted 23Feb01  to Bass List)

I wrote previously:
> "Well actually, If you do the EQ correctly, the group delay response of the
> net system (speaker + EQ) will reflect the new overall response.  The EQ can
> (and does) correct the group delay by correcting the phase response."
S wrote:
> I have my doubts about this.
> _Conceptually_:
> It is impossible to "catch up" phase lag, unless you are capable of time travel,
> i.e., by implementing a sort of "phase precendence" before the actual event. I
> concede that it is possible to impose a phase lag on the upper frequencies in
> order to flatten phase response, but then if it is not applied to the whole
> system, you'll still have "late bass". It will just not be confined to the
> lowest frequencies. In an active sub application this kind of phase EQ can be
> problematic.
> That's just my thoughts on the matter...
When I first approached the subject the notion of time delay that somehow could be undone bugged me too.  But it does appear to be true.  It is all a matter of how you define your delays.
We can resolve this question of whether the EQ corrects the group delay by simulating the two different circuits in a SPICE type circuit simulator. Let System A be a closed box with F(3) = 20 Hz.  Let System B be a closed box with F(3) = 100 Hz in combination with an EQ that corrects the response to 20 Hz).
System A can be modeled in SPICE as: 
A second order high-pass filter with Q = .707 and F(3) = 20 Hz
(representing a closed box with F(3) = 20 Hz, the particular implementation is not relevant)
System B can be modeled as: 
A second order high-pass filter with Q = .707 and F(3) = 100 Hz
followed by a Linkwitz Transform circuit designed to transform the response to
Q = .707 and F(3) = 20 Hz.
(representing a closed box with F(3) = 100 Hz EQ'd to 20 Hz)
The SPICE simulation will show that the outputs of these two systems will have the same frequency response, phase response and group delay response. As a fine point note that the Linkwitz Transform circuit will invert the polarity (and throw in a fixed 180 deg phase shift) but you could include an additional inverter op amp stage to flip the polarity back to match the input polarity.  Either way the group delay (as calculated from the net phase response of the two circuits) will be the same for System A and System B.
The same behavior holds true when most types of EQ correction are applied to our playback systems.  Any frequency response correction tends to restore the frequency, phase and group delay responses.  Even the shape of the waveform tends to be "corrected". 
The reason the phase and group delay responses are corrected when the frequency response is corrected is that the systems involved are generally accepted to be "minimum phase" systems.  The phase of a minimum phase system is directly related to the frequency response and can even be calculated from the frequency response.  Change one and you change the other.  Correct one and you correct the other.  There is some ongoing discussion about the minimum phase nature of loudspeakers, but the largest consensus seems to be that, on the whole, they behave as minimum phase systems.
It is important to note that we a talking about a very particular type of "delay" here.  This discussion has been about, so called, "group delay".  As opposed to, say, phase delay or the everyday notion of "time delay".  Group delay is usually defined as the delay of a signal envelope.  While group delay is similar to our notion of "time delay", strictly speaking, it is not the same.  One excellent paper on the subject was written by Dr. Leach of Ga. Tech. Here's the reference:
"The Differential Time-Delay Distortion and Differential Phase-Shift Distortion as Measures of
Phase Linearity", W.Marshall Leach, Jr., J. Audio Eng.Soc., Vol 37, No. 9, 1989 September
I would highly recommend this particular paper to anyone who is curious about the various ways in which audio signals can be delayed or "phase shifted" by our amplifiers, crossovers and loudspeakers.  Leach comes to the conclusion that what is most relevant is the difference between the system's group delay and its phase delay, or the "differential delay distortion" (as he defines it) .  I personally think Leach has the best explanation of the problem I have ever read. In order to minimize differential delay we need to avoid any type of delay that would allow the waveform to get out of alignment with the envelope of the waveform packet.  Evenly delaying both the waveform (phase delay) and the packet envelope (group delay) is benign and constitutes what we normally think of as simple "time delay".  Leach shows some representative differential delay responses for various high-pass filters. 
Anyone who wants to learn more about the Linkwitz Transform is invited to visit True Audio and read my Tech Topic on the subject:
You can find more bass list discussion of group delay here:
The topic of "delay distortion" or "phase distortion" in audio systems is certainly not trivial and has been a source of considerable mystery in the audio enthusiasts (and engineers) over the years. I think we can achieve a good understanding of the topic only by starting with traditional electrical engineering circuit analysis and then studying of the best technical papers on the subject...such as the Leach paper cited above.
John L. Murphy
Physicist/Audio Engineer
True Audio
Check out my recent book "Introduction to Loudspeaker Design" at Amazon.com

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