= Aliasing
{title2=$e^{2\pi i(\nu+mf_s)n/f_s}=e^{2\pi i\nu n/f_s},\quad m,n\in\mathbb Z$}
<Aliasing> is the inability to distinguish different continuous <frequencies> from sampled values. The identity above says that <frequencies> separated by an integer multiple of the sampling rate $f_s$ give identical samples of a complex <Fourier mode>. For real <cosines>, changing the sign of a <frequency> also gives the same waveform, explaining the $2$-versus-$8$ example at rate $10$. The <Nyquist–Shannon sampling theorem> prevents this ambiguity within its specified class of <bandlimited functions>. Before sampling, unwanted higher <frequencies> must be suppressed if they would violate that assumption; once two waveforms give the same samples, those samples cannot identify which was present.
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