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)
