## Averaging ?

Gentlemen:

I hate to admit that my very strongly supported design idea of using a steep cut digital filter is doing nearly nothing to the overall performance of the radiometer. Allow me to rephrase this, the digital filter, plus the averaging renders a noise RMS almost equal to that of the average alone. In any case, our design should render a better NEDT (Noise Equivalent Delta Temperature) than the original radiometer, since that one had only a soft RC filtering with no averaging whatsoever.

The advantage of filtering stems from the fact that the useful temperature information in the signal lies mostly on part of the spectrum below 200 Hz (this is for 34 scans and 4 frames per second), so you may safely dispose of the spectral components above that frequency, thus reducing the total noise RMS. Averaging filtered white noise at 200 Hz, showed 33% improvement over the averaged unfiltered noise.

The effect of averaging is strong on white noise, unfortunately, the noise spectrum in our signal is reported to be of type 1/f or pink noise and this can make quite a difference.

Plain averaging of subsets of samples may also reduce the noise RMS. Say you divide the 2176 samples in a frame into 34 subsets of 64 samples, one subset for each vertical pixel. The variances of the averages, according to the central limit theorem, should be up to 64 times smaller than that of the parent distribution. The variance (noise power) reduction of 64 brings down the noise RMS in 8 or 87.5%. Of course, for this kind of variance reduction, the theorem requires total sample independence, in signal terms, this implies a white noise. For other distributions, like any of the 1/fn types, the values of the samples are not independent, but somewhat conditioned by the previous ones.

For brown noise (1/f2), which can be simulated by a random walker algorithm, this effect is so strong, that the variance (noise power) of the averages is only 6% smaller than that of the parent or a 3% in noise RMS. For pink noise the variance reduction must be somewhere between the 3% of the brown noise and the 87.5% of the white noise. After some numerical experimentation, we also found that:

• Averaging a filtered brown noise at 200 Hz, showed less than 3% improvement on noise RMS.
• Averaging an actual sample from our signal (already filtered), the averaging showed less than 3% improvement on noise RMS.

Though I can't get unfiltered samples of our signal for a direct prove, for all the above arguments, both the filtering and the averaging act upon the same upper part of the noise spectrum. So filtering + averaging is just an overkill..

Armando