Nuts & Volts - May 2004

In The Trenches

usually these numbers are created by someone who has an agenda. They fail to consider outside factors (a falling piece of foam) and the fact that many events are not random (especially the behavior of people). Is it a machine failure if the design causes the operators to make poor decisions? Remember, according to Mark Twain, there are three types of lies: lies, damn lies, and statistics.

improve the S/N ratio. Gee, statistics is becoming more interesting.

You can use the same technique to improve the resolution of A/D converters. Suppose you want to measure a DC voltage to 0.1%, but you only have an eight-bit A/D on your microcomputer (µC). Eight bits gives you 1/256 — 0.4% resolution. Assuming that the DC voltage varies little when compared to the sampling speed of the A/D, you can take 16 eight bit measurements and convert them to a single 10-bit measurement or take 256 measurements and get 12 bits of resolution.

There is a special requirement for this. There must be enough noise in the system to cause variations in measurement. Here's why. Suppose your real value is 4. 9 volts and you can measure only in full volts. You'll get either " 4" or " 5" volt readings. If there is some noise, the reading will vary between 4 and 5 volts. Since the real value is so close to 5 volts and, with a normal noise distribution (Gaussian noise), the 5 value will occur more times than the 4 reading.

Knowing the distribution of the noise (or measuring it) allows you to calculate the actual reading of 4. 9 volts. Now you can see why a perfect

system that only reads a single value — either 4 or 5 — cannot be improved upon with this method. In fact, some special A/D systems actually add noise for better performance!

Conclusion

Signals and Noise

Earlier, I mentioned using statistics to improve an S/N ratio or increase the resolution of an A/D converter. Here's how you do that. If you take random noise and add different random noise to it, there is an increase in noise. However, some noise peaks are partially canceled out with noise troughs. The result is that, instead of doubling, the noise increases by the square root of two. If you add a signal to an identical signal, there is twice the signal — or an increase by a factor of two. So, if you can superimpose many signals, their sum increases faster than the noise and your S/N ratio increases. The amount of increase is equal to the square root of the number of summed measurements.

If you take four measurements, you can increase your S/N by a factor of two — or 6 dB. If you take 100 measurements, it increases by a factor of 10 — or 20 dB. There is no theoretical limit to this. This approach has been used very successfully in deep space communication, sensitive radar systems, geophysical measurements, medical imaging, and many other areas.

The key to this is that you must have your signals properly aligned so that they sum. Normally, this is accomplished with some form of triggering that you control, for example the pulse from your radar. If you can't do that, but you do have a powerful computer, you could sum lots of pieces of noise and then use autocorrelation to identify similar signals and then add those signals to further

MAY 2004

Being familiar with statistics is an important part of being a complete engineer. It's a powerful tool that can allow you to do things that you thought were impossible. If you understand the fundamental aspects of statistics, it becomes more intuitive and much less intimidating. Next month, we'll continue with probabilities. NV

For Your Info ...

Book Recommendation. The most readable book on statistics I have found is Statistical Analysis in Psychology and Education by George A. Ferguson, published by McGraw-Hill. My copy is very old —a third edition from 1971. Newer editions are available (used) for a few dollars. The latest in-print edition also features co-author Yoshio Takane. This edition is quite expensive; I have yet to review it personally.

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