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I need to calculate the PSD of a discrete signal and want to compare it to other processes. By Nyquist theorem, I only can account half of the frequencies.

Assume I have a signal of length $N=100$, with sampling distance $\Delta n = 1$, and we have a periodically sampled signal. The (non-aliasing) frequencies a common FFT algorithm accounts are then

$$ f_i = [0, 1, ..., 49] \cdot \frac{1}{N\Delta n}\,.$$

Hence, the smallest non-zero and highest frequencies are $f_1 = \frac{1}{N\Delta n}$ and $f_{49} = \frac{49}{N\Delta n}$. In real-space, the corresponding signal spacing is $x = 1/f_1 = N\Delta n$ and $x = 1/f_{49} \approx 2\Delta n$.

My questions are now:

How can we have information about the correlation length $x = N\Delta n$, if the largest distance in our periodic signal is $x_\mathrm{max} = N\Delta n/2$?

Why is the resolution at maximum $2\Delta n$, whereas in the real space correlation function it is $\Delta n$?

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  • $\begingroup$ Nyquist says there's a maximum frequency you can recover. Where did you see "half the frequencies" ? Half of what? In addition, please provide reference for the "real space" value presented. $\endgroup$
    – Carl Witthoft
    Commented Mar 23, 2022 at 12:32
  • $\begingroup$ Exactly, this is what I mean. As far as I understood, the frequencies are $f_i = i/(N \Delta n)$ with $i=1,...,N$, since we have a signal of N points, we also have N points in Fourier space. The Nyquist frequency is $\frac{1}{2 \Delta n}$, right? Thus, only half of the mathematically accessible frequencies are significant. The real space values I present are obtained from the relation $x_i = 1/f_i$, that typically is used as an estimation to relate the correlations in Fourier space back to real space correlations. $\endgroup$
    – Lew Dawidowitsch
    Commented Mar 23, 2022 at 13:04
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    $\begingroup$ Typically the output of the DFT includes both positive and negative frequencies. Is that the factor of 2 you are looking for? $\endgroup$
    – Carl Witthoft
    Commented Mar 23, 2022 at 17:37

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