Target audience

All who have some technical scientific interest and have read the chapter Background knowledge. Anyone who wants to understand things like "phase", "delay", time-corrected speakers, delay corrections with FIR filters in more detail.

The group delay (GD, symbol $\tau_{gr}$, measured in milliseconds) indicates how long it takes for a certain frequency to be reproduced after the signal has been applied to the input. In common usage, this often refers only to the frequency-dependent delay behavior of the transmission system. The constant component (the same for all frequencies) is often considered separately and referred to as signal delay. It is important, for example, to synchronize image and sound, but does not change the sound. The only decisive factor for the sound is whether different frequencies diverge in time. A "sluggish" woofer can be recognized, for example, by a longer group delay in the bass range. During playback, drums, for example, could appear "too little impulsive" as a result. At higher frequencies it becomes especially problematic if the delay time behavior of both stereo channels differs (e.g. due to an unbalanced listening room). Poor localization and sound coloration can be the result. Appropriate corrections, e.g. by using FIR filters, can lead to impressive improvements in my experience.

Initially somewhat misleading is the designation "Delay" for the corresponding control at the subwoofer output. Since only the frequency range of the subwoofer is affected here (i.e. the remaining speakers and thus their frequencies are not), it causes a frequency-dependent change of the group delay, so it definitely has an influence on the sound.

In principle, this time correctness is audible, otherwise a log sweep would sound just like a bang. What is disputed is a) from when and b) whether phase rotations are also audible. The effects are comparatively small. Without measures on the room acoustics, the optimization of the placement of the speakers and/or of the subwoofer(s) you probably do not need to deal with the GD, after that certainly in the high end area, otherwise maybe. Concrete facts that hopefully contribute a little to clarity:

Filters generate phase rotations ( minimum phase or stronger).
Not only crossovers, but also mechanical systems like bass reflex channels and loudspeaker chassis are filters.
This minimum phase can be calculated from the frequency response.
Everything beyond this in a real system is called excess phase.
These filters influence the GD. Only linear phase filters have a constant GD.
FIR filters can be linear-phase under certain conditions (search words: Parks-McClellan algorithm, Equiripple, Remez [Münker 2016]) - see also the example below.
For all other filters, the shape of a signal changes as it passes through the filter if it contains frequencies in the filter's operating range
Filters of the same type produce stronger phase rotations at stronger slopes (6 dB/oct., 12 dB/oct...)
A more pronounced change in the frequency response usually also causes a stronger phase rotation.
A 80 Hz Chebyshev low pass filter with 48 dB/oct has 40 to 60 ms delay depending on the parameterization.
Each zero / pole in the filter diagram generates a phase rotation by $\pi / 2$.
IIR and analog filters with RLC networks can be assumed to be minimum phase in a very good approximation, i.e. they have a lower, but frequency dependent GD for the same amplitude response.
Listening rooms cannot be assumed to be minimum-phase [Goertz 2001].
Therefore, the thesis "the excess phase comes from the listening room" could be a useful line of thought.
Therefore, pronounced, sharp room resonances could most likely cause strong delays.
These do not change when the frequency response is smoothed by equalizer.
Typical values for design-related delay in subwoofers are 10 to more than 30 ms.

At what point are runtime effects audible?

The threshold of hearing depends strongly on room, equipment and listener. Until today, [Blauert 1978] is often used as a reference:
"Frequency Threshold of Audibility
8 kHz 2 ms
4 kHz 1.5 ms
2 kHz 1 ms
1 kHz 2 ms
500 Hz 3.2 ms"
For frequencies in the range below 100 Hz, the hearing threshold is at much higher values, probably in the range of 10 to 20 ms [Neumann], an increase of the transit time from 10 to 40 ms leads to "relevant differences" [Goertz 2001].

The example measurements with the app "Subwoofer Optimizer" show irregularities in the group delay for the electrostatic speakers that are in this order of magnitude. The lower-cost dynamic speakers in a less damped listening room have significantly worse values. These differences are clearly audible. The less expensive loudspeakers provide more sound definition in a better damped reproduction room due to the reduced diffuse sound, but the accentuation of the electrostats cannot be achieved.

According to own experiments, the subjective difference depends on the sound material (impulsive, periodic). Even untrained listeners hear the differences effortlessly, as far as they have the inner peace to engage in the listening experience. Corrections of the GD by FIR filters improve the situation considerably, but do not close the gap between the different speakers.

According to the literature, monitoring volume and psychoacoustic discrepancies (small rooms that are supposed to sound like large concert halls) also play a role. The miking probably also influences the result: by having many microphones per instrument, time correctness could be affected, as one instrument radiates different parts of the sound spectrum in different directions. This would allow a microphone further away to pick up more low frequencies, for example, and a closer microphone (slightly earlier) to pick up more high frequencies.

The author of these lines lives near the Jesus-Christus-Kirche in Berlin Dahlem, which is internationally known for its excellent acoustics - despite "lousy" reverberation times measured against the specifications of DIN 18041 [Burkowitz, Fuchs 2009], and thus certainly corresponding GD. He was present at numerous sound recordings there and can confirm the sound quality.

If only a small frequency range is delayed, this is less audible. In [Goossens] this is investigated with artificially distorted hand claps. Obviously, this is due to the fact that then also only little energy is delivered late: Only when a clear second impulse (larger than the post-listening threshold) becomes visible in the energy curves due to changed group travel times does audibility begin.

To separate the effects from all these side effects by the group delay (only these can be corrected with the delay control), the following options are available:

Own listening test: The app "Subwoofer Optimizer" offers some short samples for the bass range.
There are plugins for media players that generate (via FIR filter) phase rotations and changes of GD, almost without changing other parameters. (Web search "perception group delay FIR").

Statements about the GD

On the one hand the GD belongs to the field of activity of component manufacturers and sound engineers with appropriate knowledge and years of experience, on the other hand today some components offer adjustment possibilities (filter characteristics like Bessel, Chebyshev, Butterworth or also linear phasing or minimum phasing with FIR filters), which basically require such knowledge. Whether a theoretical backgruond ultimately leads to better results, or whether the time spent (because of many side effects) is better invested in trial and error, is a complex question whose discussion usually ends without results.

Phase

You can rotate the phase of a subwoofer by 180° by reversing the polarity of the speaker. Especially with subwoofers placed on the side opposite to the speakers, this can improve the sound. Some sound processors also offer finer phase settings between 0 and 180°, which can further improve the sound.

Videos (German)

Use in YouTube ⚙ Subtitles for videos in German language.

Part 1 of 4: What is the phase control on the subwoofer for, what does delay mean? How can you adjust it with a centimeter ruler and smartphone?

Part 2 of 4: What exactly does the Speaker Management Unit Behringer Ultra Drive DCX 2496 do when you adjust phase and delay? Function generator and oscilloscope show details that cannot be seen with soundcard and software. By the way, an interesting peculiarity of the 2 kW Class D amplifier STA 2000D from IMG Stageline is found by chance.

Part 3 of 4: What do software products like REW or Acourate measure and show?

Part 4 of 4: : Interpretation of the results

Opportunities for improvement

In addition to the improvement possibilities described above, Hifi-Apps give further hints based on the impulse responses of the individual loudspeakers at the individual listening positions. The topic is constantly being developed, so only some general statements are made here: In particular, clear weaknesses in channel uniformity and also reverberation times that indicate very little damping can significantly degrade the sound. Both can be easily measured with Hifi-Apps and often just as easily corrected. Before any measurement, the general notes on speaker placement and subwoofer placement should be considered.

For technicians

By Fourier transforming the transfer function of a linear time-invariant system, a time shift (delay) $\tau_{d}$ becomes a frequency-proportional phase shift: $ \mathscr{F}\{ F(\omega)\} = f(t) \Rightarrow \mathscr{F}\{\exp(i \omega \tau_{d}) F(\omega)\} =f(t-\tau_{d})$ where $\omega$ is the angular frequency and $t$ is the time. In other words, in this simple case the transfer function can be taken as $H(\omega)=k\exp(i \omega \tau_{d}) $, i.e. $$ \begin{align} |H( \omega)| &= k \\ \angle H( \omega) &: = \varphi = -\omega \; \tau_{d}\\ \end{align} $$ Because of the linear relationship between $\omega$ and $\angle H(\omega)$, the delay can consequently be written both as a fraction and as a differential quotient: $$ \tau_{d} = - \frac{ \varphi( \omega)}{\omega} = - \frac{ \mathrm{d}\varphi( \omega)}{\mathrm{d}\omega} $$

Alternatively, one can assume a frequency dependent behavior of the transfer function. If one neglects the amount$\frac{\mathrm{d}}{\mathrm{d\omega}} |H(\omega)| = 0$, which is unimportant for the time behavior, then $\tau_{d}=-\varphi'$ describes the first element of the Taylor evolution of $H$ around $\omega_0$: $$ \begin{align} Y_k(\omega) &= H_k(\omega) X_k(\omega) \\ &= |H(\omega_0)| \exp\Big(i \varphi(\omega_0) + i (\omega-\omega_0) \varphi'(\omega_0) \Big) X_k(\omega) \\ &= \Bigg( |H(\omega_0)| \exp\Big(i \varphi(\omega_0) -i \omega_0 \varphi'(\omega_0) \Big) \Bigg) \exp\Big(i \varphi'(\omega_0)\omega\Big) X_k(\omega) \end{align} $$ The first bracket has a constant value, the second bracket characterizes the phase rotation by the delay.

Descriptively, both equations state that if one wave train of a certain frequency is needed to bridge a certain distance, WLOG two wave trains of twice the frequency are needed. In the real world, of course, this simple relationship is lost. But it remains useful to split the phase rotation into a trivial part $\tau_{pd}$ describing the time shift and the (mostly crucial) part determining the frequency dependence of the system due to resonances, filter effects etc. $$ \begin{align} \tau_{pd} &= -\frac{\mathrm{d} \varphi(\omega)}{\mathrm{d}\omega} \bigg \vert_{\omega = \omega_0} \\ \tau_{gr} &= -\frac{\mathrm{d} \varphi(\omega)}{\mathrm{d}\omega} \\ \end{align} $$ Software for displaying the group delay therefore offers an "unroll function", which automatically or manually allows the subtraction of a $\tau_{pd}$ portion. This gives the user the possibility to view the system behavior determined by resonances, filter effects etc. without disturbing phase rotations. In practice, this component plays in much slower time scales than the period of the signal, it rather changes the envelope. If one sets a signal with a carrier frequency $\omega$ modulated by an envelope, then $\tau_{gr}$ and $\tau_{pd}$ split accordingly in the transmission: $$ x(t) = \underbrace{ m(t)}_{\text{ Enveloping curve }} \underbrace{ \cos(\omega t)}_{\text{ Carrier freq }} \longrightarrow \underbrace{ m(t-\tau_{gr} )}_{\text{ Enveloping curve }} \; \underbrace{ \cos(\omega (t- \tau_{pd} ))}_{\text{ Carrier freq }} $$ A pure phase delay $\tau_{pd}$ can be accommodated in the cos term as described above, the crucial remainder shifts the envelope depending on which frequency it envelopes. The frequency-dependent change in amplitude was omitted for simplicity.

In discrete-time transmission systems, as represented by digital signal processing, the discrete group delay is related to the sampling interval $T$: $$ \frac{\tau_d(\Omega)}{T} = - \frac{\mathrm{d}\,\operatorname{arg}\{H(e^{i\Omega})\} }{\mathrm{d}\Omega} $$ with the angular frequency $\Omega$ normalized to the sampling frequency $f_s$: $$ \Omega = \frac{\omega}{f_\mathrm{s}} = \omega \cdot T $$ The advantage of the normalized form in discrete-time systems is the independence from concrete sampling frequencies.

Example

Let the transfer function of a discrete system be an averaging over the first 5 indices, i.e. $$ \begin{align} h[n] &= \frac{1}{5} (\delta(n) + \delta(n-1) + \delta(n-2) + \delta(n-3) + \delta(n-4)) \\ H(\Omega) &= \frac{1}{5} (e^{-i0} + e^{-i\Omega} + e^{-i2\Omega} + e^{-i3\Omega} + e^{-i4\Omega} ) \\ &= \frac{1}{5} ( e^{i2\Omega} + e^{i\Omega} + e^{0} + e^{-i\Omega} + e^{-i2\Omega} ) e^{-i2\Omega} \\ &= \frac{1}{5} ( 2 \cos(2 \Omega) + 2 \cos( \Omega) +1) e^{-i2\Omega} \\ \end{align} $$ The cos terms in the brackets (the amplitude response) are real, only the last multiplicand has influence on the phase. Consequently the group delay becomes $$ \tau_{\rm gr}(\Omega) = - \frac{\mathrm{d}\varphi(\Omega)}{\mathrm{d}\Omega} = - \frac{\mathrm{d} (-2\Omega)}{\mathrm{d}\Omega} = 2 $$ This can be understood if you imagine a step function as a signal, which jumps at $t=t_0=0$ from 0 to 1. When the signal reaches the system, at $t<0$ the output becomes 0, then at $t=0, 1, 2, 3, 4$ to $1/5$, $2/5$, $3/5$, $4/5$, $1$, i.e. after the group delay time the mean of the flank is reached.

By the way, the example is a linear phase filter: the phase includes only the $\arg \exp(-i2\Omega)$ term. Roughly speaking, this ultimately comes from the symmetrical structure of the 5 coefficients. While linear-phase filters usually have their maximum in the middle of the impulse response due to this symmetrical structure, the minimum-phase version of the same filter (with the same amplitude response) has the largest coefficients at the beginning of its impulse response. On [falstad.com] different filters can be simulated.

Some measuring devices can calculate (approximate values for) the group delay (directly) from two phase measurements at neighboring frequencies The app "Subwoofer Optimizer" determines the transfer function via logsweep, which is evaluated with Farina's algorithm. The group delay is determined (after smoothing) from the difference coefficient of the phase.

References

[Blauert 1978] Blauert, J. and Laws, P: "Group Delay Distortions in Electroacoustical Systems" Journal of the Acoustical Society of America Volume 63, Number 5, pp. 1478-1483 (May 1978)

[Burkowitz, Fuchs 2009] Peter K. Burkowitz, Helmut V. Fuchs "Das vernachlässigte Bass-Fundament" Vereinszeitschrift des Verbands Deutscher Tonmeister 2/2009 p. 35

[Goertz 2001] Goertz A, Wolff M (2001) "Neue Methoden zur Anpassung von Studiomonitoren an die Raumakustik mit Hilfe digitaler Filterkonzepte" Teil 1 von 2. Fortschritte der Akustik, DAGA 2002 http://www.ifaa-akustik.de/files/DAGA2002-Teil1.PDF http://www.ifaa-akustik.de/files/DAGA2002-Teil2.PDF

[Goossens] Sebastian Goossens "Wahrnehmbarkeit von Phasenverzerrungen" Institut für Rundfunktechnik, München https://forum2.magnetofon.de/bildupload/goosphase.pdf

[MSO] Multi Subwoofer Optimizer, Andy C https://www.avsforum.com/threads/optimizing-subwoofers-and-integration-with-mains-multi-sub-optimizer.2103074/

[Münker 2016] Christian Münker: "DSP auf FPGAs: Kap. 5-2 Do-It-Yourself FIR Filterentwurf" https://www.youtube.com/watch?v=y0PNXUI5x1U

[falstad.com] "...some educational applets I wrote to help visualize various concepts in math, physics, and engineering..."
http://www.falstad.com/mathphysics.html
http://falstad.com/dfilter/

[Welti Devantier] Todd Welti, Allan Devantier: Low-Frequency Optimization Using Multi Subwoofers. Harman International Industries Inc. Northbridge CA 91329 USA, Manuscript received 2006

[earl Geddes] mehlau.net/audio/multisub_geddes

[Welti Harman] Subwoofers: Optimum Number and Locationsby Todd Welti Research Acoustician, Harman International Industries, Inc.twelti@harman.com multsubs_0.pdf links folien rechts text Seite 4 "Multiple Subwoofers != Multiple Subwoofer Channels"

[Earl Geddes - YoutTube] Earl Geddes on Multiple Subwoofers in Small Rooms https://www.youtube.com/watch?v=SCWL-zusyqw

[MSO Software] https://www.andyc.diy-audio-engineering.org/mso/html/rev_hist.html

Forendiskussion. Aktuell (Okt 2020) 234 Seiten. https://www.diyaudio.com/forums/subwoofers/134568-multiple-subs-geddes-approach-149.html

Eine Art Review mit Raummoden, Welti, Geddes etc. Subwoofer / Low Frequency Optimization By Amir Majidimehr [Note: This article was published in the May/June 2012 issue of Widescreen Review Magazine]

Background knowledge: Group delay, Phase und Delay