It has been reported, and verified by us, that hearing damage can be modeled as a masking pseudo-noise in the critical band. In fact, we can induce recruited hearing in people with normal hearing by presenting ambient noise, and that was the subject of one of our patents for cell phones. (Don’t worry, their hearing returns to normal when the noise quits).
For those of us with permanent recruited hearing, we can’t hear any noise. But that may either be due to the fact that (A) such noise does not actually exist and the brain just responds as though it does, or (B) it actually exists as activations of neurons in the hearing system, but since it is constant we become inured to its presence.
(Given the strong similarity between permanent recruitment hearing, and that induced with people having normal hearing due to ambient noise fields, I am inclined to believe that the noise actually does exist in our heads.)
Either way, we can model the condition of hearing loss against a pseudo-noise field to derive the relationship between measured threshold elevations and the power level in that pseudo-noise.
When you take an audiology exam, you are placed in a sound isolation booth (very quiet) and then asked to click a button when you hear a sound coming through the headphones. The usual criteria call for sound recognition in at least 2 of 3 attempts. The level of sound at which you can achieve 2 out of 3 recognitions is considered the elevated threshold for your hearing at that frequency.
So let’s consider a Gaussian zero-mean noise signal, whose variance is equal to the noise power of the pseudo-noise field at that frequency. Critical bands are narrow enough that we can consider the noise as spectrally white across the critical band (equal power in every subband of equal frequency length).
Any random process has a finite, if vanishingly small, probability of presenting itself at any amplitude. When we consider a zero-mean Gaussian signal, and look at the expectation for its power we get a curve like this:
The curve shows the probability density times the power value for power arising from a zero-mean Gaussian noise signal. The peak of that curve coincides with its expected value (= 1 sigma^2) which is also the variance (sigma squared, i.e., power) of the Gaussian signal.
This is a graph of:
Now consider hearing thresholds: we know that signal amplitude of a Gaussian process will exceed 1 standard deviation about 1/3 of the time. (actually 31.7%) It will exceed smaller values more often. So we probably don’t want to set our threshold much below the 1 sigma^2 level.
A possibly more meaningful graph is shown here, where we plot the cumulative probability of excedence versus dB power relative to the mean noise field power. For any chosen power level, relative to the mean noise field power, the curve shows the probability of noise randomly exceeding that power level. The curve is just a plot of the Erfc function with argument Sqrt[p/2], against a dB axis for power, p, in units of variance.
If we set our threshold equal to the mean noise field power, then how much signal should be added to reach a noise-assisted perception of sound about 2/3 of the time?
During a hearing test, there are 4 possible situations that can arise:
Signal absent – listener falsely percieves a signal as present, or doesn’t hear a signal at all.
Signal present – listener hears the signal, or doesn’t hear it.
A valid test result is for the listener to respond 2/3 times when the signal is present, and substantially less often than 2/3 times when the signal is absent. The signal level must be increased until this can be achieved. When that finally happens, the level of the signal is considered to be the threshold elevation at that frequency.
So by adding signal power to the noise field, equal to the difference between the noise power where sound would be falsely heard 2/3 of the time, and the 1-sigma^2 level, then we should be hearing something real, even if assisted by random noise, and it will occur about 2/3, or more, of the time.
The level at which the Gaussian noise exceeds a 2/3 threshold on its own occurs at a power level about -7.4 dB below its mean power level (18% of mean power).
So we need to add a signal with power 82% of the mean power level, or about -0.9 dB below the mean power level. But since 1 dB is a just noticeable difference under ideal conditions (i.e., not with hearing impairment at threshold levels…) we might as well just add a signal with 100% of mean power. That way we are almost certain to hear it about 2/3+ times.
In other words, when we measure a threshold elevation using the 2/3 rule, we can expect that the pseudo-noise power is just about equal to that measured threshold, to within experimental errors.
Experimental conditions seem to limit the accuracy of audiology to about +/- 5 dB. So our figures of 0.9 dB are “in the noise”, so to speak.
Now, I have been talking as though we have some freedom to choose thresholds and power levels for the pseudo-noise field. In fact we have none. All we have are measured signal levels at which the listener correctly identifies the presence of a signal at least 2/3 time. And they do not identify a signal when it is absent more than about 1/3 time. It is from these boundaries that we can identify a mean power level for the pseudo-noise field.
What we have shown is that when using the 2/3 rule, the mean power level of the noise field must be very near (within 1 dB of) the signal level needed to achieve recognition.
PS: An interesting question arises here… For normal threshold level hearing, we know that our ears respond as linear transducers – discerned power is additive. But at normal environmental sound levels, above 30 dBSPL, our hearing becomes cube-root compressive.
So, for a hearing impaired person with a signifiant threshold elevation, more than 20 dB of elevation, we are operating in the region of compressive sound levels. The hearing physics is a mechanical phenomenon, and does not depend on perceptions. What does this do, if anything, to the 2/3 rule for measurement?
I don’t think the conversion to Sones makes any difference here. Hearing something is simply a detection event. We aren’t trying to measure sound levels, just a binary hear it / don’t hear it.