Hehe...
I guess I'll have to defer to you on this one as I read your reply like 3 times and am still having trouble with the 'math.'
Ok...
Say you're a digital system encoding a bitstream...
If you have an amplitude value of 14,280 in quantization interval A, in binary this is represented as:
11011111001000
If your value at the next quantization interval is an amplitude value of 14,283, this is represented as:
11011111001011
In both intervals, 14-bit words are required which comprises a total of 28 bits of data.
Now, imagine you're an ADPCM (Adaptive Delta PCM) encoder...
Instead of encoding the two 14-bit values above, the value recorded at quantization interval B = B - A. Therefore, 14283 - 14280 = a change (or "delta") value of 3.
So in interval B your value of 3 is represented in binary as:
11
Since the system calculates the amplitude value of every quantization interval as a difference from the preceding value, the exact same waveform can be reconstructed with much less data. Only the first quantization interval needs to be represented in absolute form, every successive interval's value can be calculated by adding or subtracting the difference from the preceding value.
Now, another thing about ADPCM is that the quantization intervals are throttled such that if an interval needs only two bits that's what it's going to use. In Linear PCM, 14 bits are still used where in this last example of a value of 3, recorded as 11, would consist of, first, ten zeros ending with two 1's.
So in this one interval you've just conserved 12 bits of data to quantify the exact same amplitude. Granted, the ADPCM decoder needs to have the basic math to understand what its reading, but that's what coding schemas do...
ADPCM is a lossless coding schema because of a couple things:
1. The sound itself is not filtered in any way to reduce or cap the bitstream requirements.
2. More importantly, the encoding used is capable of reproducing the original analogue waveform in a manner that is absolutely indistinguishable from the original.
Granted, AAC is not technically a lossless format but it uses the same principles of more efficient coding to reduce the required bandwidth to reconstruct the same analogue waveform. At certain bitstream thresholds, perceptual transparency relative to PCM is entirely possible.
But it is worth noting that the Audio Engineering Society has stated that 128Kbps AAC is perceptually indistinguishable from 16-bit Linear PCM.