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A Graphical Explanation involving Sampling

The questions about modulation of sine waves near the Nyquist limit seem to be common and repeated. The image URL here has a plot that, I think, explains how the "beating" comes about due to adding in frequencies that are solely above the Nyquist limit, and thus how filtering them out removes any "beating" one observes in raw, unfiltered data from a DAC (before the anti-imaging filter).

Start at the BOTTOM. That's the waveform you see at the DAC output, before the anti-imaging (reconstruction) filter. In this example, the beating is quite obvious.

Now, going from top down, let us see how this staircase wave was constructed. In the TOP graph, we see the original sine wave (cyan) and the first two images. (magenta and yellow). The black waveform is the SUM of the three. You notice how already, at this point, we have a lot of "beating" happening, and that also, we've very carefully and quite specifically added ONLY sine waves ABOVE the anti-imaging filter cutoff.

The middle three rows add more of the images, adding 2 sets (4 total sine waves) each time. (c,m,y,g in order, lowest to highest image)

The final row is the sum of the first 1000 images. I didn't bother to show the other 900+ very high frequency sine waves.

So, very clearly, shown in simple pictures, the result is clear, the waveform that shows the modulations at the output of the DAC arises from OUT OF BAND signals.

The graph, btw, is Copyright J. Johnston, 1999, 2000.

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Topic - A Graphical Explanation involving Sampling - jj 09:25:41 02/17/00 (43)

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