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 03/07/2017 : )