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I am going through the basic tutorials for GNURadio, and I have a question related to the sample rate tutorial. In it they demonstrate the degradation effects of sampling at too low a sampling rate.

I understand why sampling below the Nyquist rate will result in degradation due to aliasing, but in one of their examples they get significant signal degradation already when sampling a 15 kHz sinusoid at a rate of 32 kHz. That is above the Nyquist rate, so why is the signal degraded?

When they go on to sample an 18 kHz sinusoid at a rate of 32 kHz, i.e. below the Nyquist rate, the frequency plot shows the expected aliasing, producing a spike at 14 kHz. But the time domain plot still shows something unexpected, because it does not show a 14 kHz sinusoid (as I would expect after reconstruction) but some other significantly degraded waveform.

So there seems to be some source of degradation other than the sampling rate. Could someone clarify? Is it related to the fact that a DFT (specifically an FFT) is used rather than an ideal DTFT?

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  • $\begingroup$ I am not sure what you mean by “Ideal DTFT” as if the DFT is some sort of approximation. The DFT samples are exactly the same as what the DTFT provides at those sampling locations in frequency associated with each DFT bin. $\endgroup$ Jun 17 '21 at 18:35
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That is above the Nyquist rate, so why is the signal degraded?

It's not degraded in any way form or shape. The perceived degradation is purely cosmetic but not functional. See for example: How is sampling affecting this sine wave?

But the time domain plot still shows something unexpected

No it doesn't. It looks exactly as it should. If that's unexpected, the expectations are are wrong.

The effect is purely visual and caused by sloppy plotting. You need to be clear about what exactly your are plotting: a discrete sequence (which ideally should be a stem plot) or the representative continuous waveform. The latter needs proper interpolation between the sampled points. At high frequencies the basic "connect the dots" method of interpolation gives visually poor results.

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  • $\begingroup$ Thank you, I remember this now. Your answer led me to review some old notes on sampling and reconstruction from when I studied signal processing a few years ago, and I see now where I got confused. The essential thing with sampling above the Nyquist rate is that we can reconstruct the continuous signal with sinc PAM of the correct scaling (so no information has been lost). But the sampled signal may still "look wonky" (especially when drawn with linear interpolation) if there are just a few samples per cycle. $\endgroup$
    – ummg
    Jun 17 '21 at 21:01
  • $\begingroup$ It certainly didn't help that they called it "significant degradation of the waveform" in the tutorial. (Even for the example with sampling above the Nyquist rate, that is.) $\endgroup$
    – ummg
    Jun 17 '21 at 21:10
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    $\begingroup$ I agree. "Degradation" is IMO inappropriate especially for a tutorial $\endgroup$
    – Hilmar
    Jun 18 '21 at 23:20
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Part of me wants to put on my Mr Innocent face and say "what degradation?".

The signal isn't -- in my opinion -- degraded. All of the information of the original unsampled signal is still there. It's just that in its unfiltered version it doesn't look pretty.

What those "degraded" signals are is the sum of two sampling products: $\cos \left(2 \pi 15000 t\right) + \cos \left(2 \pi 17000 t\right)$ in the first case, and $\cos \left(2 \pi 14000 t\right) + \cos \left(2 \pi 18000 t\right)$. No information has been lost, and the "bad" factor could be filtered out if you desired (even in the "undersampled" case).

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