> Are there any compressors that can figure out that they are dealing with something that already has undergone the "throw out imperceptible parts to make it more compressible" step done and just skip to the next stage, so they produce a C' that is a lossless representation of S'?
The compressor will try throwing away exactly the information that was thrown away the first time you compressed it. So basically it will leave the content as is, because there is nothing extra to throw away.
You can easily see this with an MP3 file at 128 kbps - the first time you compress it most of the very high frequency content will be thrown away - you can see this on a spectogram of the uncompressed file compared with one of the compressed file. But the if you compress it again, the second compression spectogram will look very similar to the first compression one, because there is not anything else that you can throw away.
But there is a complication - the audio file is typically stored in the time domain (PCM), but the compressor operates in the frequency domain (FFT), and there will be a conversion between these two domains that you can't avoid. This conversion unfortunately will lose a bit of information and degrade quality a bit.
The compressor will try throwing away exactly the information that was thrown away the first time you compressed it. So basically it will leave the content as is, because there is nothing extra to throw away.
You can easily see this with an MP3 file at 128 kbps - the first time you compress it most of the very high frequency content will be thrown away - you can see this on a spectogram of the uncompressed file compared with one of the compressed file. But the if you compress it again, the second compression spectogram will look very similar to the first compression one, because there is not anything else that you can throw away.
But there is a complication - the audio file is typically stored in the time domain (PCM), but the compressor operates in the frequency domain (FFT), and there will be a conversion between these two domains that you can't avoid. This conversion unfortunately will lose a bit of information and degrade quality a bit.