Return-Path: X-Spam-Checker-Version: SpamAssassin 3.4.0 (2014-02-07) on lipkowski.org X-Spam-Level: X-Spam-Status: No, score=-2.3 required=5.0 tests=FREEMAIL_FROM,HTML_MESSAGE, RCVD_IN_DNSWL_MED,SPF_PASS,T_DKIM_INVALID,T_FREEMAIL_FORGED_FROMDOMAIN, T_HEADER_FROM_DIFFERENT_DOMAINS autolearn=unavailable autolearn_force=no version=3.4.0 X-Spam-DCC: : mailn 1480; Body=2 Fuz1=2 Fuz2=2 Received: from post.thorcom.com (post.thorcom.com [195.171.43.25]) by mailn.lipkowski.org (8.14.4/8.14.4/Debian-8+deb8u1) with ESMTP id uATCDiEn002720 for ; Tue, 29 Nov 2016 13:13:45 +0100 Received: from majordom by post.thorcom.com with local (Exim 4.14) id 1cBhAa-0005iq-89 for rs_out_1@blacksheep.org; Tue, 29 Nov 2016 12:05:56 +0000 Received: from [195.171.43.32] (helo=relay1.thorcom.net) by post.thorcom.com with esmtp (Exim 4.14) id 1cBhAP-0005id-Mc for rsgb_lf_group@blacksheep.org; Tue, 29 Nov 2016 12:05:45 +0000 Received: from omr-a018e.mx.aol.com ([204.29.186.64]) by relay1.thorcom.net with esmtps (TLSv1:DHE-RSA-AES256-SHA:256) (Exim 4.87) (envelope-from ) id 1cBhAI-0005r5-CF for rsgb_lf_group@blacksheep.org; Tue, 29 Nov 2016 12:05:44 +0000 Received: from mtaomg-aaj02.mx.aol.com (mtaomg-aaj02.mx.aol.com [172.27.3.208]) by omr-a018e.mx.aol.com (Outbound Mail Relay) with ESMTP id DDED8380006E for ; Tue, 29 Nov 2016 07:05:35 -0500 (EST) Received: from core-acd01h.mail.aol.com (core-acd01.mail.aol.com [172.27.22.11]) by mtaomg-aaj02.mx.aol.com (OMAG/Core Interface) with ESMTP id 5ED7038000087 for ; Tue, 29 Nov 2016 07:05:34 -0500 (EST) Received: from 80.146.228.69 by webprd-a60.mail.aol.com (10.72.5.229) with HTTP (WebMailUI); Tue, 29 Nov 2016 07:05:34 -0500 Date: Tue, 29 Nov 2016 07:05:34 -0500 From: Markus Vester To: rsgb_lf_group@blacksheep.org Message-Id: <158aff9cf67-1fc0-11313@webprd-a60.mail.aol.com> In-Reply-To: <583C799E.4090608@posteo.de> MIME-Version: 1.0 X-MB-Message-Source: WebUI X-MB-Message-Type: User X-Mailer: JAS STD X-Originating-IP: [80.146.228.69] x-aol-global-disposition: G DKIM-Signature: v=1; a=rsa-sha256; c=relaxed/relaxed; d=mx.aol.com; s=20150623; t=1480421135; bh=6+Cu7ToLf0izIsjwP5e7cTktyNxe4kQ1zZmVNvND7Vo=; h=From:To:Subject:Message-Id:Date:MIME-Version:Content-Type; b=htZ9thXGTTluPEOhzeoRm5rVPlzEnaJoYWuVq/FhkuFg80nF7ZY35y697vFc3trYM /6jphuTMCyXLPvtWqrlr2nRbvo5Al3+nevjTaKS9r1+f+DOzl68SqxMnU3QUhu8SoS Ja+QLGMkxbAsoI+bc82SlvF1wVuwK2IaxFsVyc3g= x-aol-sid: 3039ac1b03d0583d6f0e5d6e X-Scan-Signature: 1908b49407fc556987226b536cc4bad0 Subject: LF: Optimum weighting in the presence of variable noise Content-Type: multipart/alternative; boundary="----=_Part_89940_390303429.1480421134182" X-SA-Exim-Scanned: Yes Sender: owner-rsgb_lf_group@blacksheep.org Precedence: bulk Reply-To: rsgb_lf_group@blacksheep.org X-Listname: rsgb_lf_group X-SA-Exim-Rcpt-To: rs_out_1@blacksheep.org X-SA-Exim-Scanned: No; SAEximRunCond expanded to false X-Scanned-By: MIMEDefang 2.75 Status: O X-Status: X-Keywords: X-UID: 9614 ------=_Part_89940_390303429.1480421134182 Content-Type: text/plain; charset=utf-8 Content-Transfer-Encoding: 7bit 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) ------=_Part_89940_390303429.1480421134182 Content-Type: text/html; charset=utf-8 Content-Transfer-Encoding: quoted-printable
When combining a number of measurements with uncorrelate= d noise, the in-phase signal contributions add up linearly while the noise = adds up quadratically. If the individual measurements are scaled by weighti= ng factors w, the combined SNR is

 (S/N)combined = =3D (w1 S1 + w2 S2 + ...) / sqrt(|w1 N1|^2 + |w2 N2|^2 + ...)

Combined SNR can be maximized if the weight factors are chosen prop= ortional 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 p= hases. The rule is easy to derive using the well-known differentiation form= ula (u/v)' =3D (vu'-uv')/v^2.

The optimum weighting r= ule 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 decompos= ed into two steps:

 w ~ (1/|N|) (S*/|N|),
<= div style=3D"color: black; font-family: arial,helvetica,sans-serif; font-si= ze: 10pt;">
where we first normalize to constant noise level, and then w= eight the measurements according to their individual SNR.

The optimum weighting concept could a= lso 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, b= ut requires an empirical  selection of a threshold depending on QRN st= atistics (sparse local lightning crashes versus many distant spherics). An = optimum weighting "super-AGC" would continuously monitor the noise level, a= nd reduce the gain proportional to it's square with a short time constant (= "The Strong will become Weak").

One caveat is that a s= tatistically significant measurement of "instantaneous" noise power require= s a large number of samples (e.g. 100) to be incoherently added. If the noi= se 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 la= rge bandwidth (preferably several kHz).

Best 73,
Ma= rkus (DF6NM)

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