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If the OP's reference is showing a unique spectrum after the Nyquist frequency corresponding to $f_s/2$ where $f_s$ is the sampling rate (and perhaps this was the motivation for the question?), then this would imply the time domain waveform is complex as in that case the entire spectrum spanning over the width of the sampling rate (in any frequency interval) ...

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It appears the OP is missing a critical point about the effects of sampling a continuous time signal, so I provide some slides I have below that may be helpful demonstrating the periodic frequency spectrum that results due to sampling. The spectrum is only unique from $-f_s/2$ to $+f_s/2$ where $f_s$ is the sampling rate and for real waveforms that spectrum ...

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Is it possible to perform a dft looking after the Nyquist frequency. Of course. Every DFT will do this. Here is why: Applying a DFT requires the signal to be time discrete, which means that it is periodic in the frequency domain with the sample rate being the period. Let's say you are sample rate is 40 kHz. Then the value of the DFT at 1kHz will be the same ...

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At Nyquist the signal goes $[1, -1, 1, -1,...]$ - it's the fastest possible discrete variation for any input length. Zero padding won't help: it'll only lower the lowest possible (non-zero) frequency. Going beyond Nyquist thus necessarily implies increasing the physical sampling rate, or "rate of information", such that the same discrete variation $... 0 In a digitized signal, the minimum time gap between two samples would be 1/(sampling frequency). Hence the maximum frequency component can't exceed sampling frequency (which can be Nyquist rate or greater than that) For a real signal, the maximum frequency component with unique information will not exceed$\frac{sampling frequency}{2} $( because of ... 0 Reconstruction is possible so long as NOLA is obeyed - which is an easier criterion (on synthesis information) to meet than what you seek (analysis information). To discriminate temporal variations finer than$T$, the window's temporal width must be$\leq T$. You can use ssqueezepy's window_resolution with appropriate unit conversion (mult by$f_s\$) to ...

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Is there any general thumb rule as to what should be the optimal sampling rate (4x,5x...10x)? This depends a lot on your application, your specific requirements and the typical spectrum of the signals. 9kHz sounds like audio, where something like 2.5x-3x would be a good starting point. A non trivial aspect of sample rate selection is "what can the HW ...

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