Misunderstanding of Nyquist sampling theorem and minimum sampling rate

Question

I sample my time-domain (TD) signal using a distance between time-samples of $\delta = (t_{max} - t_{min}) / N_t$, where $N_t$ is the number of samples taken. The sampling rate is $1 / \delta$. I have read 10's of lectures about DFT, but it seems I still don't understand the basics.

I know the sampling theorem: if the signal is bandwidth limited to $f_c$, that is, its FT is 0 for any $f$ NOT in the interval $[- f_c, ..., f_c]$, the signal can be perfectly reconstructed from its discrete samples taken at a rate $\delta = 1 / (2 * f_c)$, or higher.

I also know the Nyquist criterion: I need to sample my time domain signal at a rate which is at least twice the value of the maximum frequency present in the signal. That would be $1/\delta > 2 * f_{max}$.

According to my understanding of the accepted answer from here: Discrete inverse Fourier transform, it seems I need to sample at a rate $ 1/\delta > f_{max}$, where $f_{max}$ is the highest frequency present in the signal.

What I am missing, the above two facts seem different for me, by a factor of 2.

Starting from page 2 of these notes, it seems that the condition from the DSP answer above is in agreement with both TD->FD sampling theorem and also with FD->TD sampling theorem. https://ocw.mit.edu/courses/electrical-engineering-and-computer-science/6-341-discrete-time-signal-processing-fall-2005/lecture-notes/lec15.pdf. The second picture of the diagram shown on page 2 makes sense to me visually: in order to avoid folding back to the 1st Nyquist zone, given that the length of the base of any of the triangles is $\Omega_0$, then $\Omega_0 / 2$ (RHS of the center triangle) + $\Omega_0 / 2$ (LHS of the 1st triangle to the right of the center triangle) < $ 2\pi / (\Delta T)$ (the center of the 1st triangle to the right of the center triangle).

My question is probably more general:

When designing the input 1D vector of length $N_t$ for a FFT algorithm for computing a direct discrete FT, what shall I look for, if we consider that I have infinite computational power (hypothetically)? Do I just need to make sure that the sampling rate is twice the bandwidth of the signal, where the bandwidth is $f_{max} - f_{min}$? I.e. only to respect the Nyquist criterion?

Answer 1

If the signal is complex, then $1/\delta > f_{max}$ with consideration to the bins of the Discrete Fourier Transform (DFT) extending from $k=0 \ldots N-1$ since the Discrete Fourier Transform will be unique in that case from bin $0$ to bin $N-1$. If the signal is real, then the DFT will be Hermitian symmetric, so we may only consider bins $0$ to $(N-1)/2$ since bins $(N-1)/2$ to $N-1$ will be the complex conjugate of bins$1$ to $(N-1/2)$. For example, with $N$ odd and a real signal, $k_1 = k^*_{N-1}$, $k_2 = k^*_{N-2}$, etc. This may lead one to state that $f_{max}$ is less than $(N-1)/2$ for that case.

In general, the Nyquist sampling theorem is to sample complex baseband signals extending from $-B/2$ to $+B/2$ at a sampling rate greater than $B$. When you consider the DFT bins can equally represent signals from DC to (nearly) the sampling rate $f_s$ (since for $N$ bins for $k$ from $0$ to $N-1$, bin $N$ would represent the sampling rate $f_s$) or equivalently (nearly) $-f_s/2$ to $+f_s/2$ the above is not in conflict with the Nyquist sampling theorem.

Answer 2

Note that your statement of the Nyquist sampling theorem only works for infinite length signals. If N is finite, then you need to sample above twice the highest spectrum frequency present by some amount.

That's because any finite length window has infinite support in the frequency domain. Same with any finite length reconstruction or anti-aliasing filter. Thus, the windowing cause by a finite length of sampling will spread your signal's spectrum. Beyond the edges of your supposed band-limit.

So you have to sample at a rate at or above 2X the frequency where the window's stopband energy goes below your desired noise floor. Which will never be zero even with "perfect" reconstruction for a finite length of samples (unless assumed to be perfectly periodically extended to beyond the lifetime of the known universe.)