Re: LF: Optimum weighting in the presence of variable noise

[IZ7SLZ <iz7slz.domenico@gmail.com>]

-------- Forwarded Message --------

Subject:	Re: LF: Optimum weighting in the presence of variable noise
Date:	Wed, 30 Nov 2016 10:19:53 -0500
From:	Markus Vester <markusvester@aol.com>
To:	iz7slz.domenico@gmail.com
CC:	selberdenken@posteo.de, riccardozoli@libero.it

Hi Domenico,

re 1) Yes it should be applicable whenever the noise level is changing over time. Note that normalization (ie. dividing by the noise voltage to make the noise constant) does only half the job, for the full benefit you'll need to divide by the square of the noise (-2 dB/dB).

Here's a simple numerical example: Assume two periods with same signal (10 uV), but noise 1 uV during the first segment and 2 uV during the second.

SNR for first segment alone: 10 / 1 = 20 dB
SNR for second segment alone: 10 / 2 = 13.98 dB
Combined SNR with equal weighting: (10+10)/sqrt(1+4) = 8.94 = 19.03 dB
Noise normalisation: Second segment reduced by 6 dB: (10+5)/sqrt(1+1) = 10.61 = 20.51 dB
Optimally combined: second segment reduced by 12 dB: (10+2.5)/sqrt(1+0.25) = 11.18 = 20.97 dB

Maximum available SNR = sum of squares: sqrt(10^2 + 5^2) = 11.18 = 20.97 dB
 
re 2) Yes why not. But it might be much simpler to do a single transmission with three times slower symbols, or alternatively split the message text into 3 shorter ones. The advantage of the stacking approach is only that you can get the decode earlier if you have more SNR than you need. Note that LF propagation is normally phase-stable over many hours during the day but not at dusk, night and dawn.

Best 73,
Markus
 

-----Ursprüngliche Mitteilung-----
Von: IZ7SLZ <iz7slz.domenico@gmail.com>
An: Markus Vester <markusvester@aol.com>
Cc: DK7FC <selberdenken@posteo.de>; riccardozoli <riccardozoli@libero.it>
Verschickt: Di, 29 Nov 2016 7:04 pm
Betreff: Re: LF: Optimum weighting in the presence of variable noise

Thanks Stefan for sharing this message that gives me the opportunity to ask Markus some questions that are a little bit off-VLF-thread (sorry !):

1) in Stefan's VLF experiments, Paul Nicholson has found that normalizing the signal with its mean value may gives some improvements in decoding the single transmission; can this be applied/tested also in LF ?
2) is it possibile increase  Eb/N0 by adding 2 or three short LF transmissions  ? (of course, not of differents days like in VLF, but consecutive i.e. differentiate of 15 or 30 minutes).

Thanks Markus in advance.

Cheers
Domenico, iz7slz


On 11/29/2016 1:53 PM, DK7FC wrote:

Hi Markus,

Thanks, i've been waiting for that message.
It seems i need a firmware update to decode that message, also my RAM is close to the limit. It's about M copy only :-)

73, Stefan

Am 29.11.2016 13:05, schrieb Markus Vester:

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)