= Discrete Fourier transform of a real signal
See sections: "Example 1 - N even", "Example 2 - N odd" and "Representation in terms of sines and cosines" of https://www.statlect.com/matrix-algebra/discrete-Fourier-transform-of-a-real-signal
The transform still has complex numbers.
Summary:
* $X_0$ is real
* $X_1 = \conj{X_{N-1}}$
* $X_2 = \conj{X_{N-2}}$
* $X_k = \conj{X_{N-k}}$
Therefore, we only need about half of $X_k$ to represent the signal, as the other half can be derived by conjugation.
"Representation in terms of sines and cosines" from https://www.statlect.com/matrix-algebra/discrete-Fourier-transform-of-a-real-signal then gives explicit formulas in terms of $X_k$.
<NumPy> for example has "Real FFTs" for this: https://numpy.org/doc/1.24/reference/routines.fft.html\#real-ffts