4
$\begingroup$

I do understand that for a signal to be properly sampled, it has to be done on the Nyquist sampling rate. What I do not understand is, what happens at sampling rates lower rate than the Nyquist rate, and at rates higher than the Nyquist rate, for that matter.

$\endgroup$
  • 1
    $\begingroup$ thanks @AliceD, that means the higher the sampling rate the better a signal is been reconstructed. $\endgroup$ – wokoro douye samuel Oct 7 '17 at 19:27
  • $\begingroup$ Yes exactly! I have made that more explicit, thanks. By the way - if you think the answer is useful a 'thanks' is certainly appreciated, but the convention is to let others know it is helpful by upvoting the answer (use the up-arrow). If the answer answers your question properly, you can also accept it by hitting the check mark. $\endgroup$ – AliceD Oct 7 '17 at 19:33
  • 1
    $\begingroup$ i would love to upvote, but i don't have enough reputation to do that @AliceD. $\endgroup$ – wokoro douye samuel Oct 7 '17 at 20:26
  • $\begingroup$ That's interesting... Anyway - glad to help :-) $\endgroup$ – AliceD Oct 7 '17 at 21:13
  • $\begingroup$ 'Aliasing' is the keyword for what happens... en.wikipedia.org/wiki/Aliasing $\endgroup$ – Memming Oct 9 '17 at 1:36
3
$\begingroup$

Short practical answer

for a signal to be properly sampled, it has to be done [at] the Nyquist sampling rate

Up front: sampling at the Nyquist frequency is the bare minimum rate to reproduce the frequency of the input signal. Dependent on your demands, I would advise to go at least 2 times that, if not more, to reproduce amplitude and shape of the signal. In other words, higher sampling rates yield better reconstructed signals.

Background
The problem is that in this digital age, analogue signals are sampled digitally, i.e. at fixed sampling rates. The Nyquist criterion states that the sampling rate should be at least twice the target frequency of the signal.

Suppose the target signal is a simple sinusoid with a frequency f (Fig. 1). And let us start with the worst-case scenario, namely a digital sampling rate (SR) equaling f. In this scenario, we end up with a straight line in the digitally recorded signal (upper panel).

In case we slightly increase the SR to 4/3 the result is not much better (lower panel).

In fact, it is not until we we go to doubling the SR to 2 times f, that we obtain a saw-tooth with a frequency equaling the input signal. The shape is, however, unlike the input signal, but at least we have the target input frequency right. However, this does not mean we can faithfully reproduce the signal. For one thing, the amplitude of the input signal will depend on the phase shift between SR and the signal. From here on, you can imagine that going to 4 f will substantially improve the reconstructed signal in terms of shape and amplitude (see, e.g. this web page of Cardif Universiy).

Nyquist
Fig. 1. Discrete sampling of an analogue target signal. source: National Instruments

$\endgroup$
  • $\begingroup$ as a digital signal processing person: uuuh, these graphs are just wrong! The linear interpolation is simply not what you do when you've got a sampled signal. Never. It's never bandwidth-limited, so you know that this interpolation must be wrong. The statement is still true – a sampling rate too low can't unambiguously allow you to recreate your signal – but you can't use linear interpolation as an argument. $\endgroup$ – Marcus Müller Jul 29 at 10:29
  • $\begingroup$ @MarcusMüller - thanks for the comment. What do you mean with interpolation? It's not mentioned anywhere. In the image the points are simply connected, not interpolated in any way. It's just an illustrative graph to visualize undersampling graphically. $\endgroup$ – AliceD Jul 29 at 11:02
  • $\begingroup$ "Connecting with a line" means "interpolating the path that the signal takes between the original points", i.e. "connected" = "interpolated". $\endgroup$ – Marcus Müller Jul 29 at 21:48
2
$\begingroup$

Here is the 10 KHz signal (the maximum frequency of the signal is 10 KHz):

enter image description here

Now, if we sample the signal at 5 KHz and 10 KHz, the signal will look like as shown below (the brown points):

enter image description here

It is clear that we can't get any useful information from the sampled signal. Now, to get some information about the signal, Nyquist said that we must sample the signal at least 2*max signal frequency. Now, here are the signals sampled at the Nyquist frequency and more than it:

enter image description here

It can be seen that by sampling at the Nyquist rate, we can get the frequency information about the signal. However, to faithfully reconstruct the signal, we have to increase the sampling rate even more.

For more details please visit https://www.gaussianwaves.com/tag/sampling-theorem/

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.