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Why do I see amplitude modulation for frequencies near the half sample rate?

This has more to due with the algorithm for Kyma's OscilloscopeDisplay than it does with the analog to digital converters. You won't see the same effect if you look at the output of the Capybara on an external analog oscilloscope.

Here's why. Imagine a sine wave whose frequency is exactly half the sample rate. In other words, you will take two samples of this sine wave on each cycle. What if you happen to sample it on the two zero crossings that a sine wave has on each cycle? That is why the Nyquist Theorem says that the highest frequency you can represent in a digital sampled system is less than half of the sample rate.

OK, so if you can't get exactly half the sample rate, let's take a sine wave that is a little bit less than half the sample rate: (SR/2 - 1) hz. On the first sample, you might get unlucky again and hit right on the zero crossing. But on the next sample is going to come a little bit sooner in the cycle than the zero crossing. And on subsequent cycles, your sample points will slowly drift with respect to the waveform. After one second, you have drifted all the way through the waveform and end up back at the initial zero again.

What does the result of this sampling look like? It looks like a square wave that repeats (SR/2-1) times per second and has an amplitude that grows and shrinks at a rate of once per second. NOT a sine wave a full amplitude. So does this mean that Nyquist was wrong?!

What Nyquist was saying was not that you would see a sine wave directly after sampling—just that you would have enough information to determine which sine wave it must have been. If you eliminate all the frequencies greater than or equal to SR/2, then there is exactly one sine wave that could have passed through the points that you sampled.

The last step of conversion is to take the sampled points and pass them through a perfect low pass filter whose cutoff is set at SR/2. And that is where the problem lies. Kyma's OscilloscopeDisplay doesn't do any low pass filtering on the sample (the stairsteps are visible, especially when you zoom way in). In a sense, the OscilloscopeDisplay is showing you an intermediate stage in the conversion process—before the pulses or spikes have been perfectly interpolated to reveal the one and only sine wave they could have come from. The Capybara's output converters have very good (though not "perfect") low pass filters (in part because they use oversampling which is another trick we haven't talked about yet). So you would not see the same "amplitude modulation" effect on the resynthesized sine wave coming out of the Capybara D/A converters.

-- CarlaScaletti - 11 Oct 2004

Klar wie Kloßbrühe, thank you! (Clear like meatball broth ;-) ) I realized my 22k5 sine wave CD-player generator now with a square wave of 22k5. It works splendidly. I guess my CD player would never have thought that he will have such a responsible job ever in his life.

-- MathisNitschke - 11 Oct 2004

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