Uniform sampling of a periodic signal

According to the Nyquist sampling theorem, a sampling frequency for a periodic signal $x(t)$ with a period $T$ and bandlimited by $f_m$ Hz is given by

$$f_s > 2f_m.$$

Is it possible to relax this constraint for periodic signals?

• so what does being periodic have to do with being bandlimited? why should the periodic nature of $x(t)$ change what the sampling theorem requires for it to be adequately sampled? – robert bristow-johnson Mar 24 '18 at 11:21

Depends a bit on your application

First of all the Nyquist/Shannon Criteria that you quoted is somewhat over simplified. It's actually sufficient to have two samples for every Hz of bandwidth. This makes a big difference for signals that are by nature "bandpass signals" as compared to "lowpass signals".

See https://en.wikipedia.org/wiki/Nyquist%E2%80%93Shannon_sampling_theorem, section for "non-baseband signals".

Back to your question: If your signal is strictly periodic, it can be represented as a Fourier Series. For most application, this would be a much more efficient representation than the time domain sampling. If you still need time domain sampling, you can determine the bandwidth by looking at the difference of the first and last non-zero coefficient of the Fourier Series

Examples for a signal with 10 Hz peridocity

1. The signal has the components: 0, 10, 20, 50, 90, 120 Hz. Bandwidth is 120 Hz and you need to sample at, at least, 260 Hz
2. Signal has the components: 1000, 1010, 1020, 1040 Hz. Bandwidth is 40 Hz, so you need to sample at, at least, 80 Hz

There a bunch of technical wrinkles in there (base-band sampling needs proper accounting of the carrier frequency, critical sampling doesn't work in practice, no real world signals are truly periodic, negative&positive frequencies, etc.) but these are still the high level rules.

You can, in a sense, if you can also sample your signal at an incommensurate sampling rate for the same period.

Any periodic signal can be decomposed into its constituent tones with a DFT and a proper algorithm. (I have an algorithm I developed about 10 years ago that I have in the pipeline for my blog that I recently found out that McLeod described about 20 years ago.)

So, reduce your question down to a single pure tone. The frequency for a pure tone, or an alias tone, can be found exactly (see my blog if you doubt this) for any frequency.

So, reduce your question down to a pure tone with a whole number of cycles in your sample frame.

The limitation with a DFT due to the discrete sampling is the aliasing effect. If a tone is higher than the Nyquist frequency it will appear in the DFT as a frequency less that the Nyquist limit. So, if a tone appears in bin $k$, where $k < N/2$, the will be a conjugate value in bin $N-k$, which is over the Nyquist limit. In actuality, you can't tell with the frequency is $k$ cycles per frame, or $N-k$. The frequency could also be any multiple of N over those.

The possible frequency can be: $k$, $N-k$, $k+N$, $2N-k$ ...

By sampling at two different truly incommensurate rates, the sets of possible frequencies from each of the DFTs will only have one value in common, and that will be the actual frequency. If the rates are nearly incommensurate, the next possible common value will be well out of range.

Here is a concrete example:

60 Hz Sampling  N = 60

49 Hz Sampling  N = 49

49                   60

5 Hz   5, 44, 54, 93 ...    5, 55, 65, 115 ...

47 Hz   2, 47, 51, 96 ...   13, 47, 63, 107 ...

97 Hz   1, 48, 50, 97 ...   23, 37, 63, 97  ...

Now, suppose you thought you could be clever and use every other point of your 60Hz sampling, and every third point as your two different sampling rates.

One second sample frame

30 Hz Sampling  N = 30

20 Hz Sampling  N = 20

Possible Frequencies:

20                                30

5 Hz   5, 15, 25, 35, 45, 55, 65 ...   5, 25, 35, 55, 65 ....

47 Hz   7, 13, 27, 43, 47, 63, 67 ...  13, 17, 43, 47, 73 ....

You can't tell whether the 5 Hz is 5, 25, 35, 55, etc.

You can't tell whether the 47 Hz is 13, 43, 47, etc.

So that doesn't work so well.

Hope this helps.

Ced

Followup:

This technique assumes you have "ideal" instantaneous sampling, so it is more theoretical than practical.