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The sampling theorem requires that your function is bandlimited to < samplerate/2. If it is, then there is no ambiguity in what continuous time function produced your set of samples.

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To visualize the frequencies of discrete time signals beyond the sampling rate, simply insert $M-1$ zeros in between each sample and scale the signal by $M$. This will extend the frequency axis by $M$ where $M$ is any positive integer. What you will see is the periodicity in the frequency domain as given for discrete time signals.

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There is aliasing at 3 kHz and 4.5 kHz. The 2 kHz does not, but since 2 kHz and 3 kHz are equally far apart from the 2.5 kHz Nyquist frequency, they look similar. Same with why 0.5 kHz and 4.5 kHz look similar.

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Zigbee is primarily designed to be cheap, not fast or efficient. Maximum data rate is 250kb/s so 2 MHz sample rate is more than enough to capture any data signal. The "2MHz bandwidth" number mainly refers to the RF channel layout. Zigbee channels are 2 MHz wide and spaced 5 MHz apart. This primarily determines what RF filtering needs to happen to ...

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One note before going into this. If we talk on discrete data, then having more samples within the same time interval means higher frequency sampling. Now, if you remember how to remember how is discrete noise is derived as samples from continuous white noise (See my answer to How to Simulate AWGN (Additive White Gaussian Noise) in Communication Systems for ...

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Take your sine wave at $f = 5\mathrm{kHz}$; $x(t) = \sin 2 \pi f\, t$. If you sample it at $F_s$ then you get a result $y[k] = \sin \frac{2 \pi f}{F_s} k$*. Because of the trigonometric identity $\sin \theta = \sin (2\pi + \theta)$, you can't tell the difference between sine waves at $f$ and $f \pm nF_s$. Because of the trigonometric identity \$\sin \theta ...

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I just ran into this problem of unevenly sampled data and couldn't find a simple solution available online. So I wrote a quick (naive) one that uses RBF kernel to do the interpolation. Shared it here https://github.com/juliusbierk/gautocorr if anyone else needs it.

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