Theorem of Bertini - Revision history

en>Yobot: WP:CHECKWIKI error fixes using AWB (10093)

2014-05-05T11:35:01Z

WP:CHECKWIKI error fixes using AWB (10093)

← Older revision		Revision as of 13:35, 5 May 2014
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	The concept of a '''linewidth''' is borrowed from [[laser spectroscopy]]. The linewidth of a laser is a measure of its [[phase noise]]. The spectrogram of a laser is produced by passing its light through a prism. The spectrogram of the output of a pure noise-free laser will consist of a single infinitely thin line. If the laser exhibits phase noise, the line will have non-zero width. The greater the phase noise, the wider the line. The same will be true with oscillators. The spectrum of the output of a noise-free oscillator has energy at each of the harmonics of the output signal, but the bandwidth of each harmonic will be zero. If the oscillator exhibits phase noise, the harmonics will not have zero bandwidth. The more phase noise the oscillator exhibits, the wider the bandwidth of each harmonic.

	~~<div style="float:right;width:245px;">~~
	<!-- Image with unknown copyright status removed: [[Image:Linewidth-Sphi.gif\|frame\|right\|Increasing levels of phase noise (shown on a log-log scale).\|{{deletable image-caption\|1=Monday, 9 March 2009}}]] -->
	<!-- Image with unknown copyright status removed: [[Image:Linewidth-Sv.gif\|frame\|right\|Effect of increasing levels of phase noise on the linewidth of an oscillator (shown on a log-log scale).\|{{deletable image-caption\|1=Monday, 9 March 2009}}]] -->
	~~</div>~~
	~~[[Phase noise]] is a noise in the phase of the signal. Consider the following noise free signal:~~
	~~:''v''(''t'') = ''A''cos(2π''f''<sub>0</sub>''t'').~~
	~~Phase noise is added to this signal by adding a [[Stochastic Process\|stochastic process]] represented by φ to the signal as follows:~~
	~~:''v''(''t'') = ''A''cos(2π''f''<sub>0</sub>''t'' + φ(''t'')).~~
	If the phase noise in an oscillator stems from [[white noise]] sources, then the power spectral density (PSD) of the phase noise produced by an oscillator will be ''S''<sub>φ</sub>(''f'') = ''n''/''f''<sup> 2</sup>, where ''n'' specifies the amount of noise (as shown in the top figure on the right). The PSD of the output signal would then be

	~~:<math>S_v(f) = \frac{A^2}{2}\frac{cf_0^2}{c^2 \pi^2 f_0^4 + (f - f_0)^2}~~,~~</math>~~		They're always ready to help, and they're always making changes to the site to make sure you won't have troubles in the first place. The next step is to visit your Word - Press blog dashboard. If you have any concerns regarding where and how to use [http://2am.eu/wordpressbackupplugin546867 wordpress dropbox backup], you can speak to us at our web site. I thought about what would happen by placing a text widget in the sidebar beneath my banner ad, and so it went. If you are using videos on your site then this is the plugin to use. Also our developers are well convergent with the latest technologies and bitty-gritty of wordpress website design and promises to deliver you the best solution that you can ever have. <br><br>Choosing what kind of links you'll be using is a ctitical aspect of any linkwheel strategy, especially since there are several different types of links that are assessed by search engines. WPTouch is among the more well known Word - Press smartphone plugins which is currently in use by thousands of users. Well Managed Administration The Word - Press can easily absorb the high numbers of traffic by controlling the server load to make sure that the site works properly. E-commerce websites are meant to be buzzed with fresh contents, graphical enhancements, and functionalities. By using Word - Press, you can develop very rich, user-friendly and full-functional website. <br><br>It is also popular because willing surrogates,as well as egg and sperm donors,are plentiful. s cutthroat competition prevailing in the online space won. We can active Akismet from wp-admin > Plugins > Installed Plugins. Every single Theme might be unique, providing several alternatives for webpage owners to reap the benefits of in an effort to instantaneously adjust their web page appear. Converting HTML to Word - Press theme for your website can allow you to enjoy the varied Word - Press features that aid in consistent growth your online business. <br><br>Additionally Word - Press add a default theme named Twenty Fourteen. And, that is all the opposition events with nationalistic agenda in favor of the individuals of Pakistan marching collectively in the battle in opposition to radicalism. This allows for keeping the content editing toolbar in place at all times no matter how far down the page is scrolled. Giant business organizations can bank on enterprise solutions to incorporate latest web technologies such as content management system etc, yet some are looking for economical solutions. This includes enriching the content with proper key words, tactfully defining the tags and URL. <br><br>Website security has become a major concern among individuals all over the world. Being a Plugin Developer, it is important for you to know that development of Word - Press driven website should be done only when you enable debugging. As a result, it is really crucial to just take aid of some experience when searching for superior quality totally free Word - Press themes, Word - Press Premium Themes for your web site. Page speed is an important factor in ranking, especially with Google. I have never seen a plugin with such a massive array of features, this does everything that platinum SEO and All In One SEO, also throws in the functionality found within SEO Smart Links and a number of other plugins it is essentially the swiss army knife of Word - Press plugins.

	~~where~~ ''~~n'' = 2''cf''<sub>0<~~/~~sub><sup>2<~~/~~sup>~~. ~~This is shown~~ in the ~~lower figure~~ on the ~~right~~.
	~~Define~~ the ~~corner frequency ''f''~~<~~sub~~>Δ<~~/sub~~> = '~~'c''π ''f''<sub>0</sub><sup>2</sup> as~~ the ~~'''linewidth'''~~ of the ~~oscillator~~. ~~Then~~

	:<~~math~~>~~S_v(f_0 + \Delta f) = \frac{A^2}{2\pi}\frac{f_\Delta}{f_\Delta^2 + \Delta f^2}.~~<~~/math~~>

	It is ~~more common to report oscillator phase noise~~ as ~~''L''~~, the ~~ratio of the~~ single~~-sideband~~
	~~(SSB) phase noise power~~ to the ~~power~~ in the ~~fundamental (~~in ~~dBc/Hz)~~. ~~In this case~~

	:<~~math~~>~~L(\Delta f) = \frac{1}{\pi}\frac{f_\Delta}{f_\Delta^2 + \Delta f^2}.~~<~~/math~~>

	~~Adding phase noise neither increases nor decreases the power of the signal~~. ~~It simply redistributes the power by increasing the bandwidth over which the signal~~ is ~~present while decreasing~~ the ~~amplitude~~ of the ~~signal that occurs at~~ the ~~nominal oscillation frequency~~. ~~The total noise power, as found by integrating~~ the ~~power spectral density over~~ all ~~frequencies, remains constant regardless of~~ the ~~amount of phase noise. This~~ is ~~illustrated in the figures on the right~~. It can ~~be proven by integrating ''L'' over all frequencies~~ to ~~compute~~ the ~~total power of~~ the ~~signal~~.

	:<~~math~~>~~\int_{~~-~~\infty}^{\infty}L(\Delta f)d\Delta f = \frac{f_\Delta}{\pi}\int_{-\infty}^{\infty}\frac{d\Delta f}{f_\Delta^2 + \Delta f^2} = \left~~.~~\frac{1}{\pi}\tan^{~~-~~1}(\frac{\Delta f}{f_\Delta})\right\|_{~~-~~\infty}^{\infty}= 1</math>~~

	~~==See also==~~
	*[[Laser linewidth]]
	*[[Spectral linewidth]]
	*[http://www.~~designers-guide~~.~~org/Analysis/rf~~-~~sim~~.~~pdf ''Introduction to RF Simulation and its Application'' by Ken Kundert]~~

	~~[[Category:Oscillators]]~~
	~~[[Category:Stochastic processes]]~~
	~~[[Category:Time series models]]~~

en>Suslindisambiguator at 14:15, 6 November 2013

2013-11-06T14:15:14Z

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en>Jakob.scholbach: /* Remarks */ Kleiman's Bertini theorem

2012-06-19T12:06:07Z

Remarks: Kleiman's Bertini theorem

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