A concept which seems to exist, is that certain standard Fourier Transforms do not produce desired results, and that therefore, They must be modified for use with compressed sound.
What I have noticed is that often, when we modify a Fourier Transform, it only produces a special case of an existing standard Transform.
For example, we may start with a Type 4 Discrete Cosine Transform, that has a sampling interval of 576 elements, but want it to overlap 50%, therefore wanting to double the length of samples taken in, without doubling the number of Frequency-Domain samples output. One way to accomplish that is to adhere to the standard Math, but just to extend the array of input samples, and to allow the reference-waves to continue into the extension of the sampling interval, at unchanged frequencies.
Because the Type 4 applies a half-sample shift to its output elements as well as to its input elements, this is really equivalent to what we would obtain, if we were to compute a Type 2 Discrete Cosine Transform over a sampling interval of 1152 elements, but if we were only to keep the odd-numbered coefficients. All the output elements would count as odd-numbered ones then, after their index is doubled.
The only new information I really have on Frequency-Based sound-compression, is that there is an advantage gained, in storing the sign of each coefficient, notwithstanding.
(Edit 08/07/2017 : )
But, there is a weakness in what I have just suggested.
The assumption could be, that when encoding, the 1152-sample interval in the time-domain is not in fact applied with a rectangular windowing function, but in fact computed with a sinusoidal – aka Hanning Window.
At that point, the modified transform would no longer work. However, in the first blog-entry I linked to, at the top of this blog-entry, I have updated my account of how Frequency-Domain-Compressed sound works, and to explain why it ultimately does.