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)
