Some observations on Group Delay
...and how to minimize its effects in a vented design
There seems to be no
clear consensus
concerning the
audibility of group
delay as applied to
speaker design. This is
due in part to the
psychoacoustics of
hearing rather than
some mathematical
constraint. The
prevalent thinking
appears to be that
some amount of group
delay is inaudible
except under special
conditions, and that at
lower or higher
frequencies more delay
can be tolerated before
its perception. How
much is too much?
Blauert and Laws, (1)
suggest the  thresholds
of audibility listed in the
table below: For
comparison, I added
the number of cycles of
phase rotation
represented for each
threshold.
Unfortunately, I know of
no similar study that
explored the
frequencies below 500
Hz to establish
thresholds of audibility.
Other studies I
reviewed however,
tended to indicate the
audibility of group
delay, and phase
distortion in general,
roughly followed the
Fletcher Munson curve.
Another interesting fact to ponder is that group delay
accumulates throughout the entire analog recording chain, due
to the limited bandwidth of each mic, preamp, amp, recording
medium, etc.
Frequency
500 Hz
1 kHz
2 kHz
4 kHz
8 kHz
Delay Threshold
3.2 msec
2 msec  
1 msec  
1.5 msec
2 msec  
Cycles
1.6
2
2
6
16
Without delving into calculus, I'll offer this layman's definition: Group delay (GD) can be thought of as
related to the time elapsed between a signal of a specific frequency applied to the driver and the cone's
attempt to recreate that stimulus, as compared to the next adjacent frequency. (And the next -ad infinum.)
This delay is a function of the phase of the system at those frequencies. For a constant group delay, and
freedom from waveform distortion, the system phase has to change linearly with the frequency response.

The plots above show the group delay and response plots of a typical driver in a sealed enclosure with Qtc
ranging from .5 to 1.2. With a Q of 1.2 the group delay peak occurs at a higher frequency, compared to
lower Q's. It also has the lowest peak delay of the group. While the onset of GD occurs lower in frequency
with a Q of 0.5, it ultimately has the greatest GD of 7.5 milliseconds at 20 Hz. This Qtc, generally referred to
as  'critically damped', and 'transient perfect' would indicate that higher values of GD has little effect on the
perceived transient response at lower frequencies.

Group delay is not a function of the response transfer function, but rather the changes in phase that
accompanies those changes in response amplitude. This might seem a trivial point, but important
nonetheless, as response changes, perhaps even below the frequency band of interest, will affect the
relative phase response over a significant frequency band. It can also be demonstrated that the higher
order transfer functions exhibit more relative phase change per octave than lower order transfer functions,
therefore GD increases with higher order transfer functions. The response transfer function is affected by
the driver characteristics, i.e. the low-end roll off, the type and compliance of the enclosure, and the
electrical characteristics of any associated crossover. To narrow the scope of this discourse, the article is
confined to the GD of an enclosed woofer at its low frequency roll off.

Another perspective to consider is that wavelengths are longer at lower frequencies.  A 1 kHz sine wave
requires 1 millisecond to complete a cycle while a 20 Hz sine wave takes 50 milliseconds to complete a
cycle. In essence: If the GD was expressed in degrees of rotation instead of milliseconds of delay, 50 msec
of delay at 20 Hz is the same amount of group delay as 1 msec at 1 kHz. Since a 4th order acoustic transfer
function as commonly used crossover design also results in a relative delay of one cycle, I hypothesize by
extension that at least 1 cycle of GD at lower frequencies will also be relatively inaudible.
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Group delay is defined as the negative derivative of the
phase slope. The formula being:
GD = -(phase at f2 * phase at f1) / (f2 * f1)