The previous posting looked at what happens with a square-law (power) detector in the situation where signal is added to a masking pseudo-noise field. Power is additive there. And it gets added incoherently.
That is, if two signal sources are uncorrelated, then the power of their additive combination is the sum of their powers. But if two sources are correlated, then we get a coherent sum, and the resulting power would be substantially larger, if added in phase, or substantially smaller if phase cancellation occurs. Incoherent signal summing ignores signal phase.
And we could quibble about thresholds of audibility, but a square law detector will exhibit a transition from seeing 1/3 false events, to 2/3 real events, under the circumstances discussed in the previous posting.
But our hearing is not square-law detection. We know that our hearing, at normal environmental sound levels, is cube-root compressive. Power detection is convenient because we can do incoherent direct summing to obtain the resulting power in signal combinations. Cube root compression throws a monkey wrench into the works. We can’t simply add the cube-root compressed signals together to obtain their composite effect.
[Side note: people with normal hearing really are power detectors at threshold levels. And so the analysis of the previous post might be correct for them. But then, they don’t ever really need any hearing tests anyway…]
A digression on hearing damage…
Common folklore has that hearing damage is caused by dead hair cells in the cochlea. That may be true to an extent, but probably not quite in the way you imagine.
If we assume that impaired hearing is the result of dead, non-responding, hair cells, then we end up with some major contradictions with reality. Dead, non-responding, hair cells cannot produce recruited hearing, they would only cause attenuation (signals would sound fainter than they are), as only the few remaining live hair cells contribute to sound sensation.
As a result, an absurd contradiction arises in that someone with, say 60 dB threshold elevation, could stand next to a running jet engine at 120 dBSPL and hear only something about as loud as an adult conversation at 1 meter separation. A mere 60 dB threshold elevation would also call for nearly 99.96% of hair cells to be dead.
So, instead of treating the dead hair cells as inert objects, recall the last time you burned a finger and you had searing pain for several minutes. That pain gradually subsided, but only partly because of body repair mechanisms setting in.
Another reason that pain subsided is that it was a constant stimulation, and our brains gradually ignore constancy. We cannot see, unless our eyeballs jitter about to cause minute scene changes on the retina. We cannot feel after some period of time with unceasing pinching. We even gradually ignore constant sounds.
So, suppose instead that we postulate that the “dead hair cells” are screaming at the top of their lungs, at 120 dB equivalent signal level. (the actual level doesn’t much matter)
In that case, a 60 dB threshold elevation could be caused by having a mere 1% of hair cells being impaired. A person with 60 dB of threshold elevation, standing next to a running jet engine at 120 dB would hear a sound as loud as 119.9 dB – essentially the full sound of the jet engine.
This last view corresponds to reality, and produces elevated thresholds with recruited hearing. Faint sounds need a lot of help to overcome the elevated threshold. But loud sounds seem about as loud as they really are.
Back to Hearing Tests…
When we view hearing damage in light of the previous interpretation, we see that we aren’t performing additive signal and masker noise power summing. Instead, we have signal competing against a separate channel to the brain from the screaming dead hair cells. It is a contest against a threshold of attention. We don’t get noise-assisted boosts in threshold level signal sound power.
We really do get false detections, but that may simply be the result of conjured sonic memories playing out, in anticipation of faint sounds from a signal.
So at this point, we are at a loss to explain the mechanism of attention redirection. It isn’t as simple as square law detection using a 2/3 event threshold. It may well be that as soon as the incoming signal is strong enough to excite the majority live hair cells to form a cube-root compressive response as strong as the screaming signal from the few dead cells, then we detect the sound. Or there may be some additional dB strength needed from the live signal to the brain before it switches its attention. This part is beyond gedanken physics analysis.
But I feel pretty confident in stating that loudness response, as discussed here, depends on the amplitude of vibration in the basilar membrane, as picked up by the hair cells – and not directly on the dB strength of the incoming signal. By that I mean that we know cube root compressive behavior sets in. We are operating in Sones space, not signal dB space. (The EarSpring equation)
Hence a 60 dB threshold elevation corresponds to only a small few hair cells being damaged out of hundreds for the critical band, not from a damaged majority of millions of hair cells. There aren’t millions of hair cells corresponding to one critical band of hearing. And an overly simplified description of damage cannot produce recruitment hearing.