Sample a signal too slowly and the samples themselves can't tell the difference between the real signal and a completely different, lower frequency one — that ambiguity, not measurement error, is what aliasing actually is.
Digital signal processing (DSP) works with signals that have been sampled — converted from a continuous waveform into a discrete sequence of numbers. The Nyquist-Shannon sampling theorem is the foundational rule governing that conversion: to uniquely capture a signal containing frequencies up to some maximum, you must sample at a rate strictly more than twice that maximum frequency (the Nyquist rate), or the sampled data becomes fundamentally ambiguous.
When a signal is sampled below its Nyquist rate, the resulting samples are mathematically indistinguishable from samples of a different, lower-frequency signal — as shown above, both the true 10 Hz wave and a much lower 'alias' frequency pass through exactly the same sample points. This isn't a measurement error to be corrected; it's an inherent ambiguity in the sampled data itself, which is why it must be prevented before sampling, not fixed afterward.
Because aliasing can't be undone after the fact (the information distinguishing the true signal from its alias is simply gone from the samples), practical systems use an analog anti-aliasing filter before the analog-to-digital converter, removing any signal content above the Nyquist frequency before sampling occurs — trading away information you couldn't have captured correctly anyway, rather than letting it corrupt the frequencies you can capture.
Audio CDs sample at 44.1 kHz specifically because human hearing tops out around 20 kHz, and 44.1 kHz safely exceeds twice that. Any measurement or control system sampling a physical process — vibration, temperature, pressure — has to be designed with its sample rate chosen relative to the highest frequency content actually present in the signal, or risk misreading real high-frequency behavior as false, misleading low-frequency behavior.
The Nyquist rate is twice the highest frequency present in a signal. Sampling at or above this rate (strictly greater, per the sampling theorem) allows the original continuous signal to be perfectly reconstructed from its samples; sampling below it makes reconstruction fundamentally ambiguous — aliasing.
No — once aliasing has occurred, the information needed to distinguish the true signal from its alias is genuinely gone from the sampled data; no amount of digital processing afterward can recover it. This is exactly why anti-aliasing filtering must happen in the analog domain, before sampling.
Human hearing is generally considered to top out around 20 kHz. The Nyquist rate for 20 kHz is 40 kHz; 44.1 kHz provides a practical margin above that to allow a physically realizable anti-aliasing filter (which can't have a perfectly sharp cutoff) some transition room.
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