# 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.