The Sampled Sine Wave Theorem
Given a sampling rate of SR hertz,
and an integer k,
A sine wave at a frequency of F is indistinguishable from a sine wave at a frequency of F + (k × SR) after sampling.
If we’re sampling at a rate of 6 Hz , this theorem tells us that a sine wave with a frequency of 1 Hz is indistinguishable from sine waves at 7 Hz, 13 Hz, 19 Hz and so on after the sampling process.
Figure 1 shows an animation which demonstrates this phenomenon for 8 potential aliases of a 1 Hz sine wave which was sampled at a rate of 6 Hz. I’ve decided to loop the animation after showing 8 aliases, but I assure you that it’s possible to continue playing this game for an infinite number of aliases to the left and right of the original (blue) sine wave. While we wont really deal with negative frequencies in this essay, it’s worth noting that they are a useful construct that you might encounter in the literature.
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Figure 1. All Sampled Sine Waves Have Aliases Aliases of a 1 Hz sine wave sampled at 6 Hz |
This theorem implies a quite impressive conclusion: Every sampled sine wave has an infinite number of aliases. You can easily identify these aliases by adding multiples of the sampling frequency to the frequency of the original sine wave. This theorem tells us that - without context - any sine wave is indistinguishable from an infinite number of other sine waves after sampling.
I’m not introducing this theorem to dishearten you, but simply to reinforce the notion that sampling isn’t a trivial thing. We’ve seen how sampling can go wrong, and we’ve introduced a theorem which allows us to ensure that our sampling is sound. For now, we leave sampling behind and dive into the Discrete Fourier Transform.