**Originally posted Apr 1, 2013**

**Negative infinity may be just a femtowatt away**

As RF engineers we deal with log-scaled data and displays so often that they feel like the only natural way of doing things, and we take this type of scaling for granted. We even tend to convert measures such as EVM that were originally linear (percent) into log (dB) and rename them as modulation error ratio (MER) or relative constellation error (RCE) etc. The log scale often enhances visualization of different measurements to provide insight.

The log scaling (of the vertical axis) of a power spectrum display is a good example. In RF measurements we often deal with signals whose amplitude varies really widely. R E A L L Y widely. Just try switching your spectrum/signal analyzer from log to linear sometime, and see how difficult it is to understand the relationships of signals or signal components such as spurs, harmonics, and intermodulation.

The log scale improves the dynamic range of displays, generally compressing the power axis to accurately represent large differences in amplitude at the expense of small (in relative terms) ones.

However as is so often the case, noise and measuring near it makes things tricky. Experienced RF engineers are always on the alert when dwelling in the domain of the really small. Consider the examples of noise floor measurements below:

The difference between adding or subtracting a small amount of power from a measurement can be very large on a log scale

For a realistic figure and a round number, I assume the average noise power is -150 dBm/Hz. This is a power spectral density (PSD) of 1 femtowatt/kHz. Or an attowatt/Hz!

The situation gets more interesting when you’re taking individual samples of this noise, or small signals with it. As it does so often, the random noise complicates things and the two example pairs above show how. In the first one successive measurements of the same signal vary by equal amounts of power, ±0.75 fW. The dB variations are quite different, however, at +2.4 and -6 dB. The log and linear power averages are different as well, with the average of the dB readings in error by 1.8 dB from the true -150 dBm figure.

Things get curiouser and curiouser when you increase the sample-to-sample variation by just a little bit, to ±1.0 fW. The positive dB variation increases modestly to +3 dB but the negative dB variation goes to negative infinity. That will foul up your averages!

Despite all this drama on the negative side of the log scale the linear power averaging that is so reliable for time-varying signals produces an accurate result whether that result is expressed in (linear) watts or (log) dBm. That’s one good lesson from this mathematical thought experiment. For another see my earlier post **The Average of the Log is not the Log of the Average**

So the log scaling that is so useful most of the time can be a reality distortion field, with very high numerical sensitivities to very small power changes. Fortunately Agilent R&D engineers work hard to keep you out of this kind of trouble, choosing and linking averaging scales and detectors and programming measurement applications to yield correct answers unless you forcibly interfere.