# Sampling below Nyqust Frequency

My understanding of Nyquist's theorem is that you need to sample at a rate that's twice the bandwidth of a signal to fully recover it.

What exactly does the bandwidth mean in this context and what happens if you sample below this frequency?

I am trying to measure a physical signal that is filtered at 100 kHz, but which only has content I care about below 1 Hz. Since I only care about the low frequency components, can my bandwidth be considered 1 Hz for this purpose? Is there any reason to sample the signal at 200 kHz?

• Depends, do you think the noise can be significant between 1 Hz and 100 kHz? – Ben Nov 29 '17 at 16:40
• There could be. What implications would that have? – ishmandoo Nov 29 '17 at 18:35
• If there is negligible noise between 1 Hz and 100 kHz, then yes... you could sample at 2 Hz even though your anti-aliasing filter only filters frequencies from 100 khz and up... However, that is a big assumption.. – Ben Nov 29 '17 at 20:14

## 2 Answers

One does need to (not a necessary condition)! But if you do, you can recover (at least theoretically) the continuous signal from the discrete samples (sufficient condition).

The bandwidth most usually denotes the span between the zeroth and the maximum frequency (in the analog signal). Sometimes, for a band-pass signal, it denotes the span between the minimum (positive) and the maximum frequency, and there are similar theorems: twice the bandwidth (sometimes a little more depending on the location of the minimum or the maximum) is enough.

If you sample below, aliasing might occur, but not every-time, this depends on the structure of the spectrum.

You care only about 1 Hz, it is dangerous to simply resample: it is advised to filter a little above 1 Hz (say 1.5) and then resample consistently with 1.5 and the precision of your filter. Could be 4 Hz (above $2*1.5$ Hz). Those figures should not be taken for granted, just an idea of the whole method.

Why what it set to 100 kHz initially? Several possibilities, for instance:

• people did not know initially in which part of the spectrum the information was located.
• the acquisition scheme was designed in a "the more data you have/the more information" mood
• My understanding is that the response of the instrument I'm using falls off above 100 kHz, so it was filtered only in that sense. I'm interested in behavior at about 1 Hz and below. I sampled at 2 Hz, thinking it would be enough. – ishmandoo Nov 29 '17 at 18:36
• Your guess might just be right. Many persons and processing schemes confuse "data" and "information". However, don't forget that some information may exist between 1 and 100 Hz. A mere down-sampling may work, but risks a shuffling of unneeded information and the useful one – Laurent Duval Nov 30 '17 at 20:38

For baseband sampling, if there is any spectrum content above half the sample rate, it will appear aliased (mixed in) with the spectrum below half the sample rate, potentially contaminating any lower frequency data that you care about.

If the baseband bandwidth is below half the sample rate, that means there is no (significant) high frequency spectrum content that could potentially mess up your low frequency sample data.

In your case, if you care about spectrum data around 1 Hz, and you sample at only 10 Hz, then any spectrum with a frequency around 9 Hz or 11 Hz (etc.) will get mixed up with your 1 Hz data and potentially mess up the data you care about. If you sample at 200 kHz, then only stuff above 100 kHz can potentially mess up your data. If you have a filter that removes anything at or above 100 kHz, then it won't be there to contaminate the 1 Hz low frequency data that you care about.