The Nyquist Sampling Theorem

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Sampled Waveforms

For demonstrations and explanations, we'll look at three closely related waveforms:

y = sin(2 π t), a 1 cycle per second or 1 Hertz sine wave.

y = sin(2 π t * 0.9), a 0.9 Hertz sine wave.

y = sin(2 π t * (1+ 0.02t)), a "chirped pulse," where the frequency continuously increases with time. If you want to see the original Excel 2007 file, click here.

We can write these down as continous functions, but any digital device will measure the signal only at discrete, specified times (typically, a signal is sampled, a digital number corresponding to the signal computed with an Analog to Digital Converter, and then another sample is taken. Each conversion takes a measureable amount of time). Here we show two of the three functions plotted as if they were continuous (they're actually computed in Microsoft Excel at discrete, 0.01 s intervals), and the third function showing each point discretely. Click on the picture to get a larger image.

The same waveforms, viewed over 20 s instead of 2 s look like this:

Notice how the waveforms drift in and out of phase with each other. At 10 s and 20 s, all the waveforms pass through zero.