Measurement uncertainty - Revision history

en>Fgnievinski: /* Further reading */

2014-12-18T19:41:23Z

Further reading

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	The '''raised-cosine filter''' is a [[filter (signal processing)\|filter]] frequently used for [[pulse-shaping]] in digital [[modulation]] due to its ability to minimise [[intersymbol interference]] (ISI). Its name stems from the fact that the non-zero portion of the [[frequency spectrum]] of its simplest form (<math>\beta = 1</math>) is a [[cosine]] function, 'raised' up to sit above the <math>f</math> (horizontal) axis.		Greetings. Let me begin by telling you the author's name - Phebe. Years in the past we moved to North Dakota. To gather coins is what his family and him appreciate. Managing individuals has been his day job for a whilst.<br><br>Feel free to surf to my website - [http://www.animecontent.com/blog/124234 http://www.animecontent.com]

	~~==Mathematical description==~~
	~~[[Image:Raised-cosine filter.svg\|thumb\|right\|300px\|Frequency response of raised-cosine filter with various roll-off factors]]~~
	~~[[Image:Raised-cosine-impulse.svg\|thumb\|300px\|right\|Impulse response of raised-cosine filter with various roll-off factors]]~~
	The raised-cosine filter is an implementation of a low-pass [[Nyquist ISI criterion\|Nyquist filter]], i.e., one that has the property of vestigial symmetry. This means that its spectrum exhibits odd [[symmetry]] about <math>\frac{1}{2T}</math>, where <math>T</math> is the symbol-period of the communications system.

	~~Its frequency-domain description is a [[piecewise]] [[function (mathematics)\|function]], given by:~~

	~~:<math>H(f) = \~~begin~~{cases}~~
	T,
	~~& \|f\| \leq \frac{1 - \beta}{2T} \\~~
	~~\frac{T}{2}\left[1 + \cos\left(\frac{\pi T}{\beta}\left[\|f\| - \frac{1 - \beta}{2T}\right]\right)\right],~~
	~~& \frac{1 - \beta}{2T} < \|f\| \leq \frac{1 + \beta}{2T} \\~~
	0,
	~~& \mbox{otherwise}~~
	~~\end{cases}</math>~~
	~~:<math>0 \leq \beta \leq 1</math>~~

	~~and characterised~~ by ~~two values; <math>\beta</math>,~~ the '~~'roll-off factor'', and <math>T</math>, the reciprocal of the symbol~~-~~rate.~~

	~~The [[impulse response]] of such a filter <ref>[http://www.commsys.isy.liu.se/TSKS04/lectures/3/MichaelZoltowski_SquareRootRaisedCosine~~.~~pdf Michael Zoltowski - Equations for the Raised Cosine and Square-Root Raised Cosine Shapes]</ref> is given by:~~

	~~:<math>h(t) = \mathrm{sinc}\left(\frac{t}{T}\right)\frac{\cos\left(\frac{\pi\beta t}{T}\right)}{1 - \frac{4\beta^2 t^2}{T^2}}</math>,~~ in ~~terms of~~ the ~~normalised [[sinc function]].~~

	~~===Roll-off factor===~~
	~~The [[roll-off]] factor, <math>\beta</math>, is a measure of the ''excess bandwidth'' of the filter, i.e. the bandwidth occupied beyond the Nyquist bandwidth of <math>\frac{1}{2T}</math>. If~~ we ~~denote the excess bandwidth as <math>\Delta f</math>, then:~~

	~~:<math>\beta = \frac{\Delta f}{\left(\frac{1}{2T}\right)} = \frac{\Delta f}{R_S/2} = 2T\Delta f</math>~~

	~~where <math>R_S = \frac{1}{T}</math> is the symbol-rate~~.

	~~The graph shows the amplitude response as <math>\beta</math>~~ is ~~varied between 0~~ and ~~1, and the corresponding effect on the [[impulse response]]~~. As can be seen, the time-domain ripple level increases as <math>\beta</math> decreases. This shows that the excess bandwidth of the filter can be reduced, but only at the expense of an elongated impulse response.

	~~====<math>\beta = 0</math>====~~
	~~As <math>\beta</math> approaches 0, the roll-off zone becomes infinitesimally narrow, hence:~~

	~~:<math>\lim_{\beta \rightarrow 0}H(f) = \mathrm{rect}(fT)</math>~~

	~~where <math>\mathrm{rect}(~~.)<~~/math~~> ~~is the [[rectangular function]], so the impulse response approaches~~ <~~math~~>~~\mathrm{sinc}\left(\frac{t}{T}\right)</math>. Hence, it converges~~ to ~~an ideal or [[brick-wall filter]] in this case.~~

	~~====<math>\beta = 1</math>====~~
	~~When <math>\beta = 1</math>, the non-zero portion of the spectrum is a pure raised cosine, leading to the simplification:~~

	~~:<math>H(f)\|_{\beta=1} = \left \{ \begin{matrix}~~
	~~\frac{T}{2}\left[1 + \cos\left(\pi fT\right)\right],~~
	~~& \|f\| \leq \frac{1}{T} \\~~
	0,
	~~& \mbox{otherwise}~~
	~~\end{matrix} \right.</math>~~

	~~===Bandwidth===~~
	~~The bandwidth of a raised cosine filter is most commonly defined as the width of the non-zero portion of its spectrum, i.e.:~~

	~~:<math>BW = \frac{1}{2}R_S(\beta+1)</math>(0<T<1)~~

	~~===Auto-correlation function===~~

	~~The [[auto-correlation]] function of raised cosine function is as follows:~~

	<math>R\left(\tau\right) = T \left[\mathrm{sinc}\left( \frac{\tau}{T} \right) \frac{\cos\left( \beta \frac{\pi \tau}{T} \right)}{1 - \left( \frac{2 \beta \tau}{T} \right)^2} - \frac{\beta}{4} \mathrm{sinc}\left(\beta \frac{\tau}{T} \right) \frac{\cos\left( \frac{\pi \tau}{T} \right)}{1 - \left( \frac{\beta \tau}{T} \right)^2} \right]</math>

	~~The auto-correlation result can be used~~ to ~~analyze various sampling offset results when analyzed with auto~~-~~correlation.~~

	~~==Application==~~
	[~~[Image~~:~~Raised-cosine-ISI.png\|thumb\|500px\|Consecutive raised-cosine impulses, demonstrating zero-ISI property]]~~

	~~When used to filter a symbol stream, a Nyquist filter has the property of eliminating ISI, as its impulse response is zero at all <math>nT<~~/~~math> (where <math>n<~~/~~math> is an integer), except <math>n = 0</math>~~.

	~~Therefore, if the transmitted waveform is correctly sampled at the receiver, the original symbol values can be recovered completely~~.

	~~However, in many practical communications systems, a [[matched filter]] is used in the receiver, due to the effects of [[white noise]]. For zero ISI, it is the <u>net<~~/~~u> response of the transmit and receive filters that must equal <math>H(f)<~~/~~math>''':'''~~

	~~:<math>H_R(f)\cdot H_T(f) = H(f)</math>~~

	~~And therefore''':'''~~

	~~:<math>\|H_R(f)\| = \|H_T(f)\| = \sqrt{\|H(f)\|}</math>~~

	~~These filters are called [[root-raised-cosine filter\|root-raised-cosine]] filters.~~

	~~==References==~~

	~~{{Reflist}}~~

	* Glover, I.; Grant, P. (2004). ''Digital Communications'' (2nd ed.). Pearson Education Ltd. ISBN 0-13-089399-4.
	* Proakis, J. (1995). ''Digital Communications'' (3rd ed.). McGraw-Hill Inc. ISBN 0-07-113814-5.
	* Tavares, L.M.; Tavares G.N. (1998) ''Comments on "Performance of Asynchronous Band-Limited DS/SSMA Systems" ''. IEICE Trans. Commun., Vol. E81-B, No. 9
	~~==External links==~~
	*[http://www.~~nonstopsystems~~.com~~/radio/article-raised-cosine.pdf Technical article entitled "The care and feeding of digital, pulse-shaping filters"] originally published in RF Design, written by Ken Gentile.~~

	~~[[Category:Linear filters]]~~
	~~[[Category:Telecommunication theory]~~]

en>Az628: Undid revision 589626915 by 67.107.43.228 (talk). Typical measurand for scales is mass (in e.g. kg), rather than weight (e.g. N). Calibration takes into account gravitational acceleration

2014-01-15T15:02:12Z

Undid revision 589626915 by 67.107.43.228 (talk). Typical measurand for scales is mass (in e.g. kg), rather than weight (e.g. N). Calibration takes into account gravitational acceleration

← Older revision		Revision as of 17:02, 15 January 2014
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	~~Hello~~ and ~~welcome~~. ~~My title~~ is ~~Irwin~~ and ~~I totally dig~~ that ~~title~~. ~~Bookkeeping~~ is ~~my occupation~~. ~~To play baseball~~ is the ~~hobby he will never stop performing~~. ~~California~~ is ~~our birth location~~.<br><br>~~My website~~ ... [http://~~immbooks~~.com/~~blogs~~/~~post/44599 home std test~~]		The '''raised-cosine filter''' is a [[filter (signal processing)\|filter]] frequently used for [[pulse-shaping]] in digital [[modulation]] due to its ability to minimise [[intersymbol interference]] (ISI). Its name stems from the fact that the non-zero portion of the [[frequency spectrum]] of its simplest form (<math>\beta = 1</math>) is a [[cosine]] function, 'raised' up to sit above the <math>f</math> (horizontal) axis.

			==Mathematical description==
			[[Image:Raised-cosine filter.svg\|thumb\|right\|300px\|Frequency response of raised-cosine filter with various roll-off factors]]
			[[Image:Raised-cosine-impulse.svg\|thumb\|300px\|right\|Impulse response of raised-cosine filter with various roll-off factors]]
			The raised-cosine filter is an implementation of a low-pass [[Nyquist ISI criterion\|Nyquist filter]], i.e., one that has the property of vestigial symmetry. This means that its spectrum exhibits odd [[symmetry]] about <math>\frac{1}{2T}</math>, where <math>T</math> is the symbol-period of the communications system.

			Its frequency-domain description is a [[piecewise]] [[function (mathematics)\|function]], given by:

			:<math>H(f) = \begin{cases}
			T,
			& \|f\| \leq \frac{1 - \beta}{2T} \\
			\frac{T}{2}\left[1 + \cos\left(\frac{\pi T}{\beta}\left[\|f\| - \frac{1 - \beta}{2T}\right]\right)\right],
			& \frac{1 - \beta}{2T} < \|f\| \leq \frac{1 + \beta}{2T} \\
			0,
			& \mbox{otherwise}
			\end{cases}</math>
			:<math>0 \leq \beta \leq 1</math>

			and characterised by two values; <math>\beta</math>, the ''roll-off factor'', and <math>T</math>, the reciprocal of the symbol-rate.

			The [[impulse response]] of such a filter <ref>[http://www.commsys.isy.liu.se/TSKS04/lectures/3/MichaelZoltowski_SquareRootRaisedCosine.pdf Michael Zoltowski - Equations for the Raised Cosine and Square-Root Raised Cosine Shapes]</ref> is given by:

			:<math>h(t) = \mathrm{sinc}\left(\frac{t}{T}\right)\frac{\cos\left(\frac{\pi\beta t}{T}\right)}{1 - \frac{4\beta^2 t^2}{T^2}}</math>, in terms of the normalised [[sinc function]].

			===Roll-off factor===
			The [[roll-off]] factor, <math>\beta</math>, is a measure of the ''excess bandwidth'' of the filter, i.e. the bandwidth occupied beyond the Nyquist bandwidth of <math>\frac{1}{2T}</math>. If we denote the excess bandwidth as <math>\Delta f</math>, then:

			:<math>\beta = \frac{\Delta f}{\left(\frac{1}{2T}\right)} = \frac{\Delta f}{R_S/2} = 2T\Delta f</math>

			where <math>R_S = \frac{1}{T}</math> is the symbol-rate.

			The graph shows the amplitude response as <math>\beta</math> is varied between 0 and 1, and the corresponding effect on the [[impulse response]]. As can be seen, the time-domain ripple level increases as <math>\beta</math> decreases. This shows that the excess bandwidth of the filter can be reduced, but only at the expense of an elongated impulse response.

			====<math>\beta = 0</math>====
			As <math>\beta</math> approaches 0, the roll-off zone becomes infinitesimally narrow, hence:

			:<math>\lim_{\beta \rightarrow 0}H(f) = \mathrm{rect}(fT)</math>

			where <math>\mathrm{rect}(.)</math> is the [[rectangular function]], so the impulse response approaches <math>\mathrm{sinc}\left(\frac{t}{T}\right)</math>. Hence, it converges to an ideal or [[brick-wall filter]] in this case.

			====<math>\beta = 1</math>====
			When <math>\beta = 1</math>, the non-zero portion of the spectrum is a pure raised cosine, leading to the simplification:

			:<math>H(f)\|_{\beta=1} = \left \{ \begin{matrix}
			\frac{T}{2}\left[1 + \cos\left(\pi fT\right)\right],
			& \|f\| \leq \frac{1}{T} \\
			0,
			& \mbox{otherwise}
			\end{matrix} \right.</math>

			===Bandwidth===
			The bandwidth of a raised cosine filter is most commonly defined as the width of the non-zero portion of its spectrum, i.e.:

			:<math>BW = \frac{1}{2}R_S(\beta+1)</math>(0<T<1)

			===Auto-correlation function===

			The [[auto-correlation]] function of raised cosine function is as follows:

			<math>R\left(\tau\right) = T \left[\mathrm{sinc}\left( \frac{\tau}{T} \right) \frac{\cos\left( \beta \frac{\pi \tau}{T} \right)}{1 - \left( \frac{2 \beta \tau}{T} \right)^2} - \frac{\beta}{4} \mathrm{sinc}\left(\beta \frac{\tau}{T} \right) \frac{\cos\left( \frac{\pi \tau}{T} \right)}{1 - \left( \frac{\beta \tau}{T} \right)^2} \right]</math>

			The auto-correlation result can be used to analyze various sampling offset results when analyzed with auto-correlation.

			==Application==
			[[Image:Raised-cosine-ISI.png\|thumb\|500px\|Consecutive raised-cosine impulses, demonstrating zero-ISI property]]

			When used to filter a symbol stream, a Nyquist filter has the property of eliminating ISI, as its impulse response is zero at all <math>nT</math> (where <math>n</math> is an integer), except <math>n = 0</math>.

			Therefore, if the transmitted waveform is correctly sampled at the receiver, the original symbol values can be recovered completely.

			However, in many practical communications systems, a [[matched filter]] is used in the receiver, due to the effects of [[white noise]]. For zero ISI, it is the <u>net</u> response of the transmit and receive filters that must equal <math>H(f)</math>''':'''

			:<math>H_R(f)\cdot H_T(f) = H(f)</math>

			And therefore''':'''

			:<math>\|H_R(f)\| = \|H_T(f)\| = \sqrt{\|H(f)\|}</math>

			These filters are called [[root-raised-cosine filter\|root-raised-cosine]] filters.

			==References==

			{{Reflist}}

			* Glover, I.; Grant, P. (2004). ''Digital Communications'' (2nd ed.). Pearson Education Ltd. ISBN 0-13-089399-4.
			* Proakis, J. (1995). ''Digital Communications'' (3rd ed.). McGraw-Hill Inc. ISBN 0-07-113814-5.
			* Tavares, L.M.; Tavares G.N. (1998) ''Comments on "Performance of Asynchronous Band-Limited DS/SSMA Systems" ''. IEICE Trans. Commun., Vol. E81-B, No. 9
			==External links==
			*[http://www.nonstopsystems.com/radio/article-raised-cosine.pdf Technical article entitled "The care and feeding of digital, pulse-shaping filters"] originally published in RF Design, written by Ken Gentile.

			[[Category:Linear filters]]
			[[Category:Telecommunication theory]]

en>WikHead: Cleared the cite-error

2012-08-21T17:43:44Z

Cleared the cite-error

New page

Hello and welcome. My title is Irwin and I totally dig that title. Bookkeeping is my occupation. To play baseball is the hobby he will never stop performing. California is our birth location.<br><br>My website ... [http://immbooks.com/blogs/post/44599 home std test]