Part of Respawned Fluff's answer to this recent question on headphones got me thinking about low-pass filters:
It seems they actually invert the transfer function of the dummy head/ears via software because they say right before that that "Theoretically, this graph should be a flat line at 0dB."… but I'm not entirely sure what they do… because after that they say "A “natural sounding” headphone should be slightly higher in the bass (about 3 or 4 dB) between 40Hz and 500Hz." and "Headphones also need to be rolled-off in the highs to compensate for the drivers being so close to the ear; a gently sloping flat line from 1kHz to about 8-10dB down at 20kHz is about right." Which doesn't quite compile for me in relation to their previous statement about inverting/removing the HRTF.
This is talking about headphones, not circuits, but it made me wonder if it's possible to create such a transfer function with an analog circuit. First-order filters have a slope of -20 dB/decade. Is there anything weaker? I suppose the transfer function would be something like this:
$$H(s) = \frac 1 {1 + \sqrt{s / \omega_c}}$$
Best Answer
Yes it is but it's more complex because you have to use breakpoints formed by a multiple array of resistors and capacitors:
The above is a piecemeal 3dB per octave (10 dB per decade) filter. It was designed to convert white noise to pink noise. See this link.
Here's another white to pink noise filter using an op-amp with a few more breakpoints:
You could convert it to 2 dB per octave or 4 dB per octave but the accuracy comes from the number of breakpoints and therefore the number of RCR stages.
Note that pink noise rolls off at 3 dB per octave and here's the final "circuit" and graph: