LF: Optimum weighting in the presence of variable noise

[Markus Vester <markusvester@aol.com>]

When combining a number of measurements with uncorrelated noise, the in-phase signal contributions add up linearly while the noise adds up quadratically. If the individual measurements are scaled by weighting factors w, the combined SNR is

 (S/N)combined = (w1 S1 + w2 S2 + ...) / sqrt(|w1 N1|^2 + |w2 N2|^2 + ...)

Combined SNR can be maximized if the weight factors are chosen proportional to the expected signal voltages divided by the noise powers:

 w ~ S*/|N|^2,

with the asterisk denoting the complex conjugate which is used to align the expected signal phases. The rule is easy to derive using the well-known differentiation formula (u/v)' = (vu'-uv')/v^2.

The optimum weighting rule is rather generic and has been applied in the frequency domain (Wiener matched filter), spatial domain (maximum ratio combined array antennas), or time domain (EbNaut day/night stacking). It could conceptually be decomposed into two steps:

 w ~ (1/|N|) (S*/|N|),

where we first normalize to constant noise level, and then weight the measurements according to their individual SNR.

For example, let's assume that a VLF signal is 6 dB stronger at night, but the noise increases by 10 dB. Thus the nighttime measurement should be down-weighted by (+6-20) dB= -14 dB, which is more redunction than noise normalization.

The optimum weighting concept could also be applied to spherics blanking. The classic "noise blanker" basically works by nulling all samples above a pre-set threshold. This can be viewed as a crude binary approximation of optimum inverse-noise-power weighting, but requires an empirical  selection of a threshold depending on QRN statistics (sparse local lightning crashes versus many distant spherics). An optimum weighting "super-AGC" would continuously monitor the noise level, and reduce the gain proportional to it's square with a short time constant ("The Strong will become Weak").

One caveat is that a statistically significant measurement of "instantaneous" noise power requires a large number of samples (e.g. 100) to be incoherently added. If the noise measurements are taken only from within the signal channel itself, gain adaptation needs to be slow compared to the symbol rate. For the purpose of spherics blanking, one would want to evaluate the noise in a relatively large bandwidth (preferably several kHz).

Best 73,
Markus (DF6NM)