# is there a second order approximation to perceived loudness

Quoted from this

As a first order approximation it takes ten times the energy to double the perceived loudness

So is there a 2nd order approximation. If so then how much change in RMS value doubles perceived loudness? Is it governed by any system of equations?

Thanks

• When they say "first-order approximation" they just mean "first approximation". They're not implying a first-order polynomial or anything like that, so "2nd-order approximation" doesn't really make sense. There are more and more complicated models that give you better results. – endolith Mar 14 '14 at 0:00
• @endolith, So by first order approximationit doesn't imply applying nth order calculus, rather first approximation(of loudness) would be nice to accept, isn't it? – nmxprime Mar 14 '14 at 4:04
• yeah it just means "a simple approximation". There are more complex ways to more accurately approximate it, but none would be called "2nd order" – endolith Mar 14 '14 at 16:39

This depends highly on the signal and it's content. For narrow band signal, the loudness can fairly well be estimated through the equal loudness curves as published in ISO 226 (see for example) http://en.wikipedia.org/wiki/Equal-loudness_contour

For wide band signal, things are more complicated. If the signals are stationary, you apply frequency weighting curves for the appropriate overall loudness (such as "A" or "C" weighting), which gives in some cases a reasonable estimate but can also be quite off by a fair amount.

Next step up in complexity would be a loudness calculation based on on a detailed perceptual model. So you split the signal into "critical bands", apply forward, backward, simultaneously masking thresholds, apply frequency weightings and than integrate this all up. The resulting loudness is often measure in "Sone" which intended to be a "ratio scale", meaning it has an absolute zero value (threshold of audibility) and 6 Sones is twice as loud as 3 Sone.

So overall it's possible to get very accurate estimates of perceived loudness even for non stationary signals, but it requires deploying very complicated perceptual models. These models basically the same ones as used in perceptual codecs such as MP3, AAC, Dolby Digital etc.

• Hilmar, is perceived loudness and perceptual loudness same? As of my understanding, i can't get right way understanding both! Any explanation or better links other than wiki? – nmxprime Mar 14 '14 at 3:58

No, by "first order approximation" they probably mean "rule of thumb". Loudness is a subjective measure related to sound intensity, but there are no exact relationships.

• That feels like on overly simplistic statement to me. It's entirely possible to get very good estimates for percieved loudness, it's just a lot of work – Hilmar Mar 13 '14 at 13:51
• @Hilmar I think we do not disagree. By "no exact relationships" I mean that all possible estimates are based on some sort of model which cannot be exact. I tried to clear up what was (probably) meant by "first order approximation", and that there is no straightforward generalization to a "second-order approximation". – Matt L. Mar 13 '14 at 13:57
• @Hilmar, you gave great explanation on possibility of estimating non stationary signal's loudness.That's great and thanks alot . Also i wanted to know what and how 1st order and 2nd order estimates would be – nmxprime Mar 14 '14 at 4:00