Why group delay measurements are usually useless

I used to wonder why the measurements and simulation plots for group delay usually look different.

A simulation plot will start at a few milliseconds at DC. It will rise to a peak near the lower rolloff frequency, then fall gradually down to zero at infinity Hz with maybe one or two more peaks on the way. You rarely see a negative GD.

Most plots I have seen based on measurements look like the Himalayas. Up and down far too much for the order of the system, and frequently entering the negative, even for nominally simple cabinets.

This presents a dilemma: which do you believe? And how do you choose your delay settings for your LMS?

Now, both simulations and measurements have lots of extraneous issues that can make them invalid. One that affects measurements is reflections. You can do measurement without reflections if you have the kit, and I suspect such measurments would be OK. But most do involve reflections and for GD purposes they are totally up the kazoo.

If reflections ruin GD measurments, why do they not also ruin amplitude response measurements or directivity measurements? Well, actually they can, but if you make sure the reflections are from distant objects that are not too big, then their magnitude will be small and they will just add a "little wiggle" to the measured amplitude - easily smoothed away using a computer or a discerning eye. These slightly wiggly measuremnts are still OK to use.

The massive fluctuations in most GD plots I see resembles the same sort of wiggle but on a much bigger scale. Enough to force the GD number into the negative for cabinets that mathematically can never have negative GD.

Why would GD be so much more easily "disrupted" by reflections? The answer is in the maths, and it's pretty simple once you get it. Imagine you are measuring a speaker. Speaker is perfect (yeah, I know, run with it) and mic is perfect, and mic is too small to generate any reflections itself. You are floating in the air with no reflective boundaries at all (ignore ground and top of atmosphere). Speaker measures flat, GD measures as distance from mic to speaker, say 3 feet.

Now add a single reflector, a 1 foot by 1 foot tile. Set it up nearby, so it reflects a second path from speaker to mic. Second path might be 6 feet, say. Second path comes in 6dB down (twice the distance) and 3ms after the first path. Adding under relative delay causes cacellation and coupling at different frequencies, so you get a rippple. But since reflected signal is lower in amplitude, ripple is limited to +3 to -6dB.

Now move reflector away, so reflected path is 12feet, then 24 then 48 etc. Each time, the reflected signal is smaller and so the ripple is lower. But also, since the difference in group delays is getting bigger, the ripples will get denser in frequency domain - I mean closer together on the graph. At 3ft relative delay, peaks are about 330Hz apart. But with 45 feet the peaks are only 22Hz apart. So amplitude measurment shows lower, denser fluctuations and the graph looks better and better.

The same thing will apply to a phase graph. As in, using measurments, plot phase angle against frequency. Again, as you move the reflector away, the ripples get lower and denser. This suggests a graph you can trust, and you can, if you are interested in reading off phase angle at some frequency.

But GD is calculated as the rate of change of phase angle with respect frequency. It's a differential, and that means (basically) you get the group delay by measuring the gradient of the phase graph at each freuqency. Now, the fact that the fluctuations get lower would naturally tend to make the fluctuations in the gradient lower (differentiation is linear). So GD groups should be manageable just like amplitude and phase right?

Well, no. When a graph is squashed up in the frequency direction (going from one cycle of fluctuation every 330Hz to one cycle every 22Hz would be an example) this makes the differential bigger, by the same proportion as the squashing-up (15 in this example). Mathematically, this is the chain rule with a scaling on the parameter. Intuitively, if you have to fit more mountains of the same height into the same range, their slopes will have to be steeper.

The differentiation process enlarges the effect of the dense phase fluctuations, making the fluctuations in the GD graph much larger, big enough to completely obliterate the original curve. Distant reflections can be very quiet, maybe 30dB down, but the long delay makes for dense fluctuations when added back to the direct path, which grow in size during the differentiation process used to get a GD graph.

For a known simple situation with a single reflection, you might be able to mathematically correct the graph. But for real wold situations you are in trouble. I dont think smoothing is enough - multiple distant reflections cause irregular fluctuations and you could confuse a real GD peak with a fluctuation because the fuctuations are as large if not larger then the direct-path GD you are trying to measure.

One possibility might be to perform the measurement for less time than than any reflective path - literally turning off the mic before the first reflection of the test sound reaches it. This is only OK if you can live with reduced precision - you might not get the true maximum value of a GD peak for example, though I suspect that due to the Haas effect, you'll get something that corresponds much more closely with what you actually hear.

Any thoughts?