You haven't lost any information. There is nothing to "figure out" as the analog waveform will be exactly reproduced after the final filter stage.
Whether the lost information matters of course depends on the frequency range you can actually hear; in the case of CD-quality (44100Hz) audio you probably won't notice any issues unless you're a dog (or unless you're taking a higher-sample-rate recording and downsampling without any kind of anti-aliasing), but the article discusses a PCM chip that natively supports 32552Hz and is receiving a signal that was naïvely upsampled from 16276Hz. Especially in the latter case, you absolutely will hear the information loss.
D/A and A/D | Digital Show and Tell (Monty Montgomery @ xiph.org) 3/2/2013 https://youtube.com/watch?v=cIQ9IXSUzuM