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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