# Electronic – Puzzled by Nyquist frequency

analysis

Say I have a 1kHz sine, so no higher harmonics, then I need to sample it at least at 2kHz to be able to reconstruct it.
But if I sample at 2kHz, but all my samples are on the zero-crossing, then my sampled signal doesn't show a sine at all, rather the ECG of a deceased patient. How can that be explained?

This can be expanded to higher sampling frequencies too. If I sample a more complex waveform at 10kHz, I should at least get the first 5 harmonics, but if the waveform is such that the samples are each time zero, then again we get nothing. This isn't far-fetched, it's perfectly possible for a rectangle wave with a duty cycle < 10%.

So why is it that the Nyquist-Shannon criterion seems to be invalid here?

You actually need just over 2 kHz sampling rate to sample 1 kHz sine waves properly. It's $$f_N < f_S / 2$$ not $$f_N \le f_S / 2$$

P.S. If you took your signal into complex space, where a sinusoid is of the form $$v(t) = Ae^{j(2 \pi f t - \theta)} = A(\cos(2 \pi f t - \theta) + j \sin(2 \pi f t - \theta))$$ where t is time, A is amplitude, f is frequency, and θ is phase offset, $$f_N = f_S / 2$$ is the point where the frequency "folds over", i.e. you cannot distinguish f from -f. Further increases in frequency will appear, after sampling, to have the sampling frequency subtracted from them, in the case of a pure sinusoid.

# Non-Sinusoids

For the case of a square wave at 1 kHz with a duty cycle less than or equal to 10% which is sampled at 10 kHz, you are misunderstanding the input.

First you would need to decompose your waveform into a Fourier series to figure out what the amplitudes of the component harmonics are. You will probably be surprised that the harmonics for this signal are quite large past 5 kHz! (The rule of thumb of third harmonic being 1/3 as strong as the fundamental, and 5th being 1/5 of fundamental, only applies to 50% duty cycle square waves.)

The rule of thumb for a communications signal is that your complex bandwidth is the same as the inverse of the time of your smallest pulse, so in this case you're looking at a 10 kHz bandwidth minimum (-5 kHz to 5 kHz) for a 10% duty cycle with the fundamental at 1 kHz (i.e. 10 kbps).

So what will ruin you is that these strong higher-order harmonics will fold over and interfere (constructively or destructively) with your in-band harmonics, so it's perfectly expected that you might not get a good sampling because so much information is outside the Nyquist band.