How to convert multi-band Energy Time Curve (ETC) to Impulse Response (IR)?
You can't really get the impulse response from the ETC. This is because ETC represents the energy of the signal, while the impulse response contains time information too.
One may argue that ETC also has some time information but it's not the same. You need phase to be able to reconstruct the impulse response and this is simply not present in the ETC.
What you could possibly try to do is to sum the energy of the frequency bands to acquire the total energy (broadband), calculate the square root of the result to acquire the amplitude and add random phase, or even impose the phase characteristics you want. Keep in mind that this may lead to unstable results or complex values in your time series (IR).
In order to impose your desired phase response, you could convolve your signal (square root of total energy) with the IFFT of a &quot;manufactured&quot; signal that will have constant amplitude spectrum and the desired phase.
Once more, please keep in mind that arbitrary phase curves may lead to results that &quot;do not make sense&quot;. No matter what, the resulting sound may, or may not sound natural/real. Most probably (haven't tested though) parts of it, such as the late reverberation tale may sound natural (it exhibits random-like phase characteristics) while the fist parts of the IR may not sound convincing.
The main issue with the manufactured phase is that real spaces exhibit mixed-phase characteristics. Some parts of the IR (early reflections) have minimum phase (most often) while late reverberation shows maximum phase characteristics.