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Analog Roosters Taking Digital Drinks

It’s tough not being a domain specialist at times. You get hit by scary-sounding jargon that you mostly hope you can evade without having to parse and understand it. And none sounds more imposing than the thing that came up in the wind sensor article, the delta-sigma modulator

That sounds like one of two things: some top-secret military weapon that will do things to you that you really don’t want to think about or the guy pouring beer at a frat

Well, it’s neither.

It’s a way of adding precision to data in a noisy environment. The goal is to encode data as a pulse train. The more frequent the pulse (or the higher the “pulse density”), the higher the “amplitude” of the signal. 

Think of some poor dude at an old telegraph office providing an ongoing report of the local temperature. Let’s say that, if the temperature is 0, he sends one pulse per hour, and, if the temperature is 60, he sends one per minute. For any other temperature, he clicks at the appropriate proportional rate. Well, he’s pulse-coded the temperature data; the guy at the other end can then figure out the temperature by counting the frequency of clicks.

The idea is that the information being transmitted is in the very low frequency band (as compared to the frequencies involved in generating the pulses). The noisy stuff is all in the high frequencies. So whoever receives this pulse train can apply a low-pass filter to retrieve the data and discard the noise. 

Not sure if that lonely guy listening to the telegraph clicks will appreciate being called a low-pass filter.

The concept of taking noise in a measurement and, effectively, moving it into a higher frequency band than the underlying data has the awesome-sounding name “noise shaping.”

Given that concept, do you encode the data? If you look at some of the descriptions of ΔΣ modulators, you’d be forgiven for immediately throwing up your hands: some people immediately revert to arcane math in an attempt either to be more precise than necessary, look smart, or scare away anyone trying to invade their turf. Or all of the above. (OK, precision is good, but it’s best after you know the rough concepts.)

And, for this, there’s a really simple analog. Have you ever seen those garden water features with a rooster or some other bird that periodically dips its head in the water? That’s a ΔΣ modulator. Of sorts.

Those things generally work on some variation of the following: a flow of water gradually fills some kind of vessel, and, when the vessel reaches a certain point, it tips, and the bird takes a drink. After the water empties out (due to the tipping), it starts over. 

Usually this just goes on at a constant rate. But you can imagine that, if the flow of water were increasing and decreasing, it would take the vessel a different amount of time to fill, and so the bird would tip more frequently with faster flow and less frequently with slower flow.

(It occurs to me that we could conjoin this analogy with the earlier fraternity suspicion by doing things like replacing the vessel with a beer bong, watching the frat guy eventually tip over, etc. But I’ll leave that for you to work out.)

And that’s how a ΔΣ modulator works. Except that, instead of water and a vessel and a bird, you have a voltage driving an integrator (which slowly fills), which drives a comparator, which, when a threshold is reached, sends out a pulse. The higher the voltage, the faster the integrator “fills up,” and the more frequent the pulses. 

The pulse is also fed back via a one-bit DAC to be subtracted out of the input to the integrator – this is the step that effectively drains the vessel so it can start over.

You can get fancy by adding another integrator; this is a 2nd-order modulator, which has less noise. (Almost seems like double-buffering to ward off metastability.) And, in fact, you can keep adding integrators to get an nth-order modulator (except that, apparently, if you just do that willy-nilly, you’re going to have some instability issues beyond 3rd-order).

And yes: for precision, you can go to the math. There’s lots of it.

I sort of prefer the pastoral garden image.

I guess that’s why I’m not actually designing these things.

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