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| From | Phil Hobbs <pcdhSpamMeSenseless@electrooptical.net> |
|---|---|
| Newsgroups | comp.dsp |
| Subject | Re: Variance of white noise |
| Date | 2020-10-10 13:24 -0400 |
| Organization | A noiseless patient Spider |
| Message-ID | <f99ceaf2-189d-4a33-ec82-8da166d44716@electrooptical.net> (permalink) |
| References | <8066780a-3f4a-4d4c-b145-bf0178230741n@googlegroups.com> <7d6a05fc-146c-5926-398e-302a7fb740f0@electrooptical.net> <rlr4mb$eha$1@dont-email.me> |
On 2020-10-09 22:01, Les Cargill wrote: > Phil Hobbs wrote: >> On 2020-09-27 23:38, Tom Killwhang wrote: >>> I was reading Box and Jenkins Time series analysis and noticed that >>> when they calculated power spectrum they had a factor 2 in the >>> numerator - see >>> http://www.ru.ac.bd/stat/wp-content/uploads/sites/25/2019/03/504_05_Box_Time-Series-Analysis-Forecasting-and-Control-2015.pdf >>> >>> >>> equation (3.1.12). >>> >>> I couldn't figure out where the 2 is coming from but then I wondered >>> if they define noise a different way in stats. Just like when we have >>> sine waves and take an FFT the magnitude is divided by 2 when we show >>> the two sided spectrum, is it fair to do the same with white-noise? I >>> think them may have multiplied it by 2 so that for the full spectrum >>> =pi to +pi it gets halved. We don't seem to do this in engineering do >>> we? >>> >> >> The analytic signal convention is used almost universally in test >> equipment and other areas. It allows one to use exp(i omega t) >> instead of sines and cosines, which saves half the algebra and >> therefore three quarters of the blunders. ;) >> >> You form the analytic signal from a real signal by adding +-i times >> its Hilbert transform (depending on your sign convention), which has >> the effect of : >> >> 1. doubling the positive frequency amplitudes >> 2. zeroing out the negative frequency ones >> 3. leaving DC alone. >> >> Normal people of course apply rules 1-3 instead of Hilbert >> transforming. ;) >> >> The analytic signal convention is responsible for many of those >> strange factors of 2 that show up in noise calculations, e.g. the 1-Hz >> shot noise density of a current I = e N is >> >> i_N = sqrt(2 e I) = e * sqrt(2N) >> >> rather than e * sqrt(N) >> >> The reason is that a 1-second boxcar has a bandwidth of 0.5 Hz on >> account of the negative frequencies being chopped off, so the sqrt(N) >> noise is compressed into half the bandwidth. >> >> Cheers >> >> Phil Hobbs >> > > Sp why do so many people treat the Hilbert transform as if it were > equivalent to the analytic signal? You get massive DC with the usual FFT > method of constructing a Hilbert transform. > > I will have to try your list, just for giggles. But anything that is > basically "cat signal | s/sin/cos/g " will not be pleasant with respect > to DC. Er, "what is cos(0)? :) > > -- > Les Cargill Well, you can't phase shift DC after all. (BTW remember to switch back to sines and cosines before doing anything very nonlinear such as computing the power. ) Cheers Phil Hobbs -- Dr Philip C D Hobbs Principal Consultant ElectroOptical Innovations LLC / Hobbs ElectroOptics Optics, Electro-optics, Photonics, Analog Electronics Briarcliff Manor NY 10510 http://electrooptical.net http://hobbs-eo.com
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Variance of white noise Tom Killwhang <gyansorova@gmail.com> - 2020-09-27 20:38 -0700
Re: Variance of white noise Phil Hobbs <pcdhSpamMeSenseless@electrooptical.net> - 2020-10-08 14:33 -0400
Re: Variance of white noise Les Cargill <lcargil99@gmail.com> - 2020-10-09 21:01 -0500
Re: Variance of white noise Phil Hobbs <pcdhSpamMeSenseless@electrooptical.net> - 2020-10-10 13:24 -0400
Re: Variance of white noise Phil Hobbs <pcdhSpamMeSenseless@electrooptical.net> - 2020-10-10 13:27 -0400
Re: Variance of white noise spope384@gmail.com (Steve Pope) - 2020-12-03 21:12 +0000
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