# Is phase and amplitude information necessarily lost when undersampling?

Is phase and amplitude information necessarily lost when undersampling if you have a constant periodic single frequency sinusoidal?

My second question is: How can one determine the undersampling frequency for such a case? And how to retrieve the amplitude and phase.

I want to know if this is possible for a sampling frequency much lower than the Nyquist rate.

• What exactly do you mean by undersmapling? – Phonon May 20 '14 at 0:51
• Do you have any other interfering signals (across the entire bandwidth visible to the sampler, even beyond its Nyquist rate), or are you observing only a single sinusoid and nothing else? If you don't, then undersampling should work just fine. Phase measurements are somewhat tricky in this case; you need to know the frequency a priori in order to reference its phase to anything meaningful. – Jason R May 20 '14 at 0:56
• @Phonon: Sampling below the Nyquist rate. – iQt May 20 '14 at 0:57
• @JasonR: Only one single sinusoid (ideally). There will be some noise ofcourse, but assume it is noise free. – iQt May 20 '14 at 0:58

## 2 Answers

From an RF perspective... since you specified just the one sine wave then yes, Nyquist holds because you are operating at baseband. Any sampling rate less than 2B will result in aliasing, because you are undersampling. In this case your bandwidth is the frequency of the sine wave, since it is the highest frequency you want to recover.

However if you have a low frequency signal modulated onto a high frequency carrier signal, you can get away with undersampling because the carrier has no information. Can use spectral folding to recover the baseband signal and discard the carrier.

Now about phase recovery, there are multiple methods to accomplish it. One is to use complex signals, where you create an I/Q signal by complex mixing. With an I/Q signal you can recover instantaneous phase because you have a reference, the other copy of the signal.

If you sample a constant periodic signal of frequency f with a frequency less than twice the frequency (below Nyquist rate or Undersampling) you will lost the data due to aliasing. Lets say $f=100 Hz$ then by Nyquist rate we should sample the signal with a sampling rate higher than 200 Hz. If we under sample the signal (<200) by a sampling frequency say $f_s=180 Hz$. Then due to aliasing the reconstructed signal will be a periodic signal with a frequency less than the original signal. Here the reconstructed signal will be a 80 Hz signal instead of 100 HZ.

So under sampling will replace a signal with a low frequency signal.

• That's not strictly true; there isn't necessarily information loss if you sample at less than the Nyquist rate. The technique is called bandpass sampling. – Jason R May 20 '14 at 19:49