Thursday, June 18, 2009

Error Analysis

I recently had a need to do some error analysis. I had forgotten the some rules, or so I decided to brush up. Really the two main ones I did remember were that PERCENTWISE when you multiply(divide) two numbers the percent error adds, and when you add(subtract) the percents stay the same. On the bright side these basic non-calculus rules can be quite helpful most of the time.

I don't do a lot of analog circuits, or statistics any more. Typical problems involve selecting circuit components to guarantee the parameters which are desired for a circuit. As you start getting into capacitors, inductors, and transistors, the linearity goes away. Also if there's some sort of statiscical distribution your function things can also be quite complicated. In these situations you must use calulus which starts to get VERY complicated once you start getting to deal with statistical distrubutions. Error anlysis can also be extended to computational algorithms, estimation functions, and a whole lot of different concepts. See wikipedia for some complex examples. Still even you can get pretty far doing simple integrals.

The additive property seems quite obvious. The multplicative (or really any rule) is pretty clear if you approach it like a circuit problem (or at least when I do) and think of the biggest variety doing a sweep in (the old days when I used to do) simulations. Fortunately On the bright side quite often you can reasonably simplify a problem to a set of linear equations. For example, despite the complexity of the circuitry inside the op-amps which were the cornerstone of the instrumentation amplifier I was investigating you can see from wikipedia the it's just the basic math operations because of the confiuration used. so extracting the net error is simple following the basic rules above. If you wanted a more exact number you might consider nonlinearity in your opamps, but using basic design rules you learn in EE 101 the high gain of an omp amp can be used to idealize a non linear circuit to something linear.

This is often what you want for something like an instrumentation amp where you're trying to amplify a signal. Still these concepts can be applied to more complicated scenarios. Quite often for analog circuits that have a transfer function and are applied to AC signals you can use a fourier (continuous signals) or laplace transfor (unit step or other functions with instantanous changes) to put them into a linear set of equations.

Error anlysis is pretty pervasive. A good open course from Columbia is a good refresher. Enjoy.

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