A reader recently asked:
T, from my mechanical engineering world we have strict rules on sampling rates vs. signal frequency rates. Ie you cannot reliably measure a 60hz ac sine wave with a 5hz analog sampling device. The result ends up being strange results that don’t show spikes well and also might not show averages well either. Can you help me understand how 120 year sampling proxies can resolve relatively high frequency temperature spikes?
This objection comes up so often from those who are accustomed to data which are evenly sampled in the time domain, and the misconception is so firmly imprinted on so many people, that it’s worth illustrating how uneven time sampling overcomes such limitations.
Let’s start with a signal consisting of a 60 Hz sine wave of amplitude 1. We’ll even add noise with standard deviation equal to 1. We’ll measure it at a sampling rate well below the signal frequency — but we won’t use even time sampling. Instead, I’ll let the time differences between samples equal 0.2 plus uniform random noise. Hence the maximum possible sampling rate will be 1/0.2 = 5 Hz, and in fact for my sample the maximum sample rate is 4.8 Hz and the mean sampling rate is a paltry 1.42 Hz. By conventional wisdom among engineers, this is nowhere near enough properly to characterize a signal at 60 Hz.
Here’s the data:
Here’s the discrete Fourier transform of the data:
The signal frequency (at 60 Hz) is plainly evident. Its signal power doesn’t significantly leak to other frequencies. There’s really no ambiguity about it, despite the fact that its frequency is much higher than the maximum sampling rate.
In fact a correct analysis reveals that the signal has frequency 60.00006 +/- 0.00075 Hz, amplitude 1.09 +/- 0.21.
This is not due to the use of a uniform distribution for the sampling time intervals. It’s a much more general property of uneven time sampling — just about any time sampling other than regular will reveal the signal as well.
Astronomy is plagued by sampling problems. We just don’t get to observe when we would like to. Most targets can’t be seen during the day, or when the sun or moon is too close, or when the weather doesn’t cooperate. And for professionals rather than amateurs, competition for telescope time at major observatories is fierce (which is a distinct advantage for amateur astronomers).
That’s why astronomers are used to irregular time sampling. It has some disadvantages, but it also has advantages such as the ability to search frequency space far far beyond the “Nyquist frequency” or, frankly, any frequency limit you care to define. In my opinion, in all but very specific circumstances the advantages of uneven time sampling outweight the disadvantages, by a lot. Uneven time sampling is sometimes thought to be a bane for data analysis, but it’s far more likely to be a boon.