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.
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.
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
500 Hz
1 kHz
2 kHz
4 kHz
8 kHz
Delay Threshold
3.2 msec
2 msec
1 msec
1.5 msec
2 msec
3.2 msec
2 msec
1 msec
1.5 msec
2 msec
Cycles
1.6
2
2
6
16
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.
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)
The formula being: GD = -(phase at f2 * phase at f1) / (f2 * f1)