“Are they still dithering about?”
One of my technical articles seemed to have fallen into a black hole. The magazine that had agreed to publish it apparently couldn’t figure out where or when to place it.
But as soon as I wrote that line above to my hard-pressed editor (yes—hard though it may be to believe, I actually let a grown-up comment and even edit my work), I thought “Ah, now there’s a Front End piece just waiting to be written.”
Dither, you see, is a term that the practicing system designer will encounter quite frequently these days. And, whatever we might conclude that it does mean, one thing it does not imply is the inability of a system or circuit to make a single, particular decision.
Both statistical economics and game theory distinguish between two transactional modes: the single encounter and the repeated encounter. The expected value of a business transaction equals the actual value of the transaction if it occurs, times the probability that it will occur.
The same applies to lottery tickets. You might be able to win a million pounds by purchasing a one pound ticket in the UK’s National Lottery, but the expected value of that transaction, allowing for the cost of your ticket, is about −0.6 of a pound, i.e. a loss. We all have some level of intuitive understanding of this, though the prevalence of lottery participants indicates some natural variability.
So your economic strategy, whether as a lottarian (I made that up, I admit it) or a businessperson, depends on whether you are making one bet, or executing a large number of transactions whose average value is more important than the results of one particular “move.”
Our everyday interpretation of dithering involves failure to decide on a one-off action. Even when we say of a person that “that one’s a ditherer” because they often exhibit this behavior, we’re focusing on uncorrelated single events and not on the aggregate “value” of their actions.
But in the world of the system designer, dither is in fact a rather splendid way of ensuring that—on average—the quality or outcome of your decisions is a close approximation to the “right value,” or desired outcome. That’s because dither inherently works its magic on a sequence of events. To get the true value out of dither, you must keep dithering! To dither a system is to give it, over time, a richer “search volume” in which it can find the ‘answers’ to the ‘decisions’ that need to be made.
Let’s look at a really simple story example. In your organization, the HR and catering departments are at loggerheads with each other. HR wants you to have free coffee, but the careering department doesn’t want its budget to get hit, so they want you to pay for it in the lovely new canteen.
So, after many meetings, they come to a compromise. This compromise was strongly influenced by the catering department, staffed as it is with wannabe economists with a spare-time game theory obsession. The compromise is this: HR will give you a token, valid for one day, for $1, to buy coffee. And the catering department will set the price of the coffee at $2. Sounds like it might be hard for you to get your caffeine fix, seeing as you have no other source of funds…
Then I come along, and offer you a deal. I will randomly give you $1, or take $1 from you, with a probability P=0.5 for each. So the net monetary value of that deal in the long term—a repeated encounter, in the game theory sense—is zero. But look what it has enabled. Now, with a probability of P=0.5, you have enough money for a coffee that day. You’re able to enjoy, on average, half a cup of coffee per day. Which is exactly what $1 per day would buy you if it wasn’t for this pesky quantization of coffee states. My zero net value deal has enabled you to get your coffee—it has dithered your coffee consumption.
Now some of you may say “yes, but that’s just obvious, a child could see that this is how to solve the problem.” And I think that’s part of the point that I’d like to make. Dither isn’t some arcane technical concept; it’s something else that we seem to have a strong intuitive understanding of.
I mentioned quantization, and of course this is where as a system designer you’re most likely to encounter the use of dither. The obstacle to getting your coffee—the quantization of coffee delivery in $2 steps—is akin to the stepwise transfer function of a signal quantizer, which turns a continuous signal (usually, but not always, a physical quantity such as a voltage), into a countable, discrete quantity. That’s usually—but not always—a digital signal. It’s no coincidence that the word “digital” derives from the Latin for finger, and it was upon fingers that Man used to count, before we invented Excel. And the lineage of the word “quantization” can be traced back to the Latin for “how many?”
Dither, then, is just the zero net value deal that your system offers to a quantizer so that the mean value of the results you get from it are a better reflection of the variability of the underlying signal. The application of dither doesn’t improve the “accuracy” of a single conversion result. But when properly applied, it can significantly improve the resolution of a system in the long term. Because in the long term, all the noise to do with quantization can be averaged away, to reveal how good the system is at capturing very small changes in the underlying process being monitored.
And what of that technical article I mentioned? Well, as I write, they do still seem to be dithering. I just hope they aren’t getting jittery about publishing my work. Wait, jitter…