Nyquist's theories (and Shannon's sampling theorem based on it) only concerned themselves with the time-based discretization of analog signals. As Shannon showed, to accurately reconstruct a signal of bandwidth x, you need to take samples at a constant frequency of 2x.
But when speaking of digital signals, discretization is only half the story. The other part of the story is quantization, i.e. how many bits to use to represent each sample. And that amplitude quantization is a lossy operation, regardless of Shannon's theorem about discretization in the time domain. You can't simply say that an analog signal of 40kHz therefore has a bitrate of 80kbps -- using just one bit to represent each sample will not accurately reproduce the original analog waveform, unless the original waveform was a square wave (and you would still need that bit of information to accurately reconstruct it).
So what, then, is the bitrate of a discretized, analog signal? It is still infinite, because the amplitude of every sample can be any element of R! Only once you decide on a quantization resolution, i.e. once you determine your acceptable level of signal loss, can you speak of "the bitrate of an analog signal".
But that bitrate is only an approximation of the original signal, Shannon notwithstanding.
Nyquist's theories (and Shannon's sampling theorem based on it) only concerned themselves with the time-based discretization of analog signals. As Shannon showed, to accurately reconstruct a signal of bandwidth x, you need to take samples at a constant frequency of 2x.
But when speaking of digital signals, discretization is only half the story. The other part of the story is quantization, i.e. how many bits to use to represent each sample. And that amplitude quantization is a lossy operation, regardless of Shannon's theorem about discretization in the time domain. You can't simply say that an analog signal of 40kHz therefore has a bitrate of 80kbps -- using just one bit to represent each sample will not accurately reproduce the original analog waveform, unless the original waveform was a square wave (and you would still need that bit of information to accurately reconstruct it).
So what, then, is the bitrate of a discretized, analog signal? It is still infinite, because the amplitude of every sample can be any element of R! Only once you decide on a quantization resolution, i.e. once you determine your acceptable level of signal loss, can you speak of "the bitrate of an analog signal".
But that bitrate is only an approximation of the original signal, Shannon notwithstanding.