= Aliasing of discrete Fourier modes
{title2=$e^{2\pi i(m+qN)n/N}=e^{2\pi imn/N}$}
For <integers> $m,q,n$, adding $qN$ to a <frequency> index changes the <complex exponential> by $e^{2\pi iqn}=1$. Thus different continuous <Fourier modes> become identical <discrete Fourier modes> on the sampling grid. This is <aliasing>, caused by sampling before any <discrete Fourier transform> is computed.
For $N=8$, compare the continuous <functions> $\cos(2\pi mt)$ and $\sin(2\pi mt)$ with $m=1$ and $m=9$, sampled at $t=n/8$, $n=0,\ldots,7$. With $a=\sqrt2/2$, both <cosines> give exactly the first row and both <sines> give exactly the second:
$$
\begin{array}{c|rrrrrrrr}
n&0&1&2&3&4&5&6&7\\\hline
\cos(2\pi n/8)=\cos(18\pi n/8)&1&a&0&-a&-1&-a&0&a\\
\sin(2\pi n/8)=\sin(18\pi n/8)&0&a&1&a&0&-a&-1&-a
\end{array}
$$
The nine-cycle <sine> agrees in sign as well as magnitude with the one-cycle <sine>: the <frequency> indices differ by $8$, rather than by a sign reversal. Consequently, either <cosine> produces the same <discrete Fourier transform>, $X_1=X_7=4$ with all other <coefficients> zero; either <sine> produces $X_1=-4i$, $X_7=4i$, with all other <coefficients> zero.
\Image[/discrete-sampling-frequency-aliasing.png]
{title=One-cycle and nine-cycle cosine and sine waves have exactly the same eight samples}
An observed low <frequency> therefore does not establish that physical high <frequencies> are absent. To exclude such alternatives, the <Nyquist–Shannon sampling theorem> requires a specified class of <bandlimited functions>. For arbitrary pure <sinusoids> with <frequencies> at most $B$, choose sampling rate $f_s>2B$; equality needs care because $\sin(2\pi(f_s/2)t)$ vanishes at every $t=n/f_s$. The finite-energy <L2 space> version of the <Nyquist–Shannon sampling theorem> permits equality under its own hypotheses. In either case, exact continuous reconstruction generally requires the entire infinite sample <sequence>; eight observed samples alone do not supply it. Demanet derives the grid <aliasing> condition as a difference by an <integer> multiple of the sampling <angular frequency>.https://math.mit.edu/icg/resources/teaching/18.330/18.330-Mar21.pdf[Laurent Demanet, MIT 18.330 notes, §6.2, Definition 11, p. 57: aliases on a grid, 2012]{ref}
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