en>Materialscientist: Reverted edits by 218.248.0.2 (talk) to last revision by Yobot (HG)

2013-09-11T07:02:08Z

Reverted edits by 218.248.0.2 (talk) to last revision by Yobot (HG)

← Older revision		Revision as of 09:02, 11 September 2013
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	Hi there, ~~I am Andrew Berryhill~~. ~~Doing ballet~~ is ~~something she would never give up~~. ~~Office supervising~~ is ~~what she does~~ for a ~~living~~. ~~I've usually cherished residing~~ in ~~Kentucky~~ but ~~now I'm contemplating~~ other ~~choices~~.<br><br>~~Feel free~~ to ~~surf to my blog~~ ... ~~email psychic readings (~~[http://www.~~taehyuna~~.~~net~~/xe/~~?document_srl=78721~~ www.~~taehyuna~~.~~net~~])		A '''cyclostationary process''' is a [[signal (information theory)\|signal]] having statistical properties that vary cyclically with time.<ref name="gardner06">{{cite journal
			\| last = Gardner
			\| first = William A.
			\| coauthors = Antonio Napolitano, and Luigi Paura
			\| title = Cyclostationarity: Half a century of research
			\| journal = Signal Processing
			\| volume = 86
			\| issue = 4
			\| pages = 639–697
			\| publisher = Elsevier
			\| date = 2006
			\| doi = 10.1016/j.sigpro.2005.06.016 }}
			</ref>
			A cyclostationary process can be viewed as multiple interleaved [[stationary process]]es. For example, the maximum daily temperature in New York City can be modeled as a cyclostationary process: the maximum temperature on July 21 is statistically different from the temperature on December 20; however, it is a reasonable approximation that the temperature on December 20 of different years has identical statistics. Thus, we can view the random process composed of daily maximum temperatures as 365 interleaved stationary processes, each of which takes on a new value once per year.

			== Definition ==
			There are two differing approaches to the treatment of cyclostationary processes.<ref name="gardner91">{{cite journal
			\| last = Gardner
			\| first = William A.
			\| title = Two alternative philosophies for estimation of the parameters of time-series
			\| journal = IEEE Trans. Inf. Theory
			\| volume = 37
			\| issue = 1
			\| pages = 216–218
			\| date = 1991
			\| doi = 10.1109/18.61145 }}
			</ref>
			The probabilistic approach is to view measurements as an instance of a [[stochastic process]]. As an alternative, the deterministic approach is to view the measurements as a single [[time series]], from which a probability distribution can be defined as the fraction of time that events occurs over the lifetime of the time series. In both approaches, the process or time series is said to be cyclostationary if its associated probability distributions vary periodically with time. However, in the deterministic time-series approach, there is an alternative but equivalent definition: A time series that contains no additive finite-strength sine-wave components is said to exhibit cyclostationarity if there exists some nonlinear transformation of the signal that produces positive-strength additive sine wave components.

			== Wide-sense cyclostationarity ==
			An important special case of cyclostationary signals is one that exhibits cyclostationarity in second-order statistics (e.g., the [[autocorrelation]] function). These are called '''wide-sense cyclostationary''' signals, and are analogous to [[wide-sense stationary]] processes. The exact definition differs depending on whether the signal is treated as a stochastic process or as a deterministic time series.

			* For a stochastic process <math>x(t)</math>, we define the autocorrelation function as
			::<math>R_x(t;\tau) = E \{ x(t - \tau/2) x^*(t + \tau/2) \}.\,</math>
			:The signal <math>x(t)</math> is said to be wide-sense cyclostationary with period <math>T_0</math> if <math>R_x(t;\tau)</math> is cyclic in <math>t</math> with cycle <math>T_0,</math> i.e.,
			::<math>R_x(t;\tau) = R_x(t+T_0; \tau)\text{ for all }t, \tau.</math><ref name="gardner91"/>

			* For a deterministic time series <math>x(t)</math>, we define the cyclic autocorrelation function as
			::<math>\hat{R}_x^\alpha(\tau) = \lim_{T \rightarrow \infty} \frac{1}{T}
			\int_{-T/2}^{T/2} x(u+\tau/2) x^*(u-\tau/2) e^{-i 2\pi \alpha u}\, du.</math>
			:The time series <math>x(t)</math> is said to be wide-sense cyclostationary with period <math>T_0</math> if <math>\hat{R}_x^\alpha</math> is not identically zero for <math>\alpha = n T_0</math> for some integers <math>n</math>, but is identically zero for all other values of <math>\alpha</math>.<ref name="gardner91"/>
			:Equivalently, we may say that a time series having no finite-strength sine-wave components is wide-sense stationary if there exists a quadratic transformation of the time series that produces finite-strength sine-wave components.

			==Cyclostationary models==
			It is possible to generalise the class of [[autoregressive moving average model]]s to incorporate cyclostationary behaviour. For example, Troutman<ref>Troutman, B.M. (1979) "Some results in periodic autoregression." [[Biometrika]], 66 (2), 219–228</ref> treated [[autoregressive model\|autoregressions]] in which the autoregression coefficients and residual variance are no longer constant but vary cyclically with time. His work follows a number of other studies of cyclostationary processes within the field of [[time series analysis]].<ref>Jones, R.H., Brelsford, W.M. (1967) "Time series with periodic structure." [[Biometrika]], 54, 403–410</ref><ref>Pagano, M. (1978) "On periodic and multiple autoregreessions." Ann. Statist., 6, 1310–1317.</ref>
			== Notes ==
			{{reflist\|group=note}}

			== References ==
			<references/>

			== External links ==
			* Noise in mixers, oscillators, samplers, and logic: an introduction to cyclostationary noise [http://www.designers-guide.org/Theory/cyclo-paper.pdf manuscript] [http://www.designers-guide.org/Theory/cyclo-preso.pdf annotated presentation] [http://www.designers-guide.org/Theory/cyclo-pub.ppt presentation]

			[[Category:Stochastic processes]]
			[[Category:Signal processing]]
			[[Category:Time series analysis]]

en>Yobot: /* Thermodynamic reservoirs */WP:CHECKWIKI error fixes + general fixes using AWB (7536)

2011-01-11T00:43:31Z

Thermodynamic reservoirs: WP:CHECKWIKI error fixes + general fixes using AWB (7536)

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Hi there, I am Andrew Berryhill. Doing ballet is something she would never give up. Office supervising is what she does for a living. I've usually cherished residing in Kentucky but now I'm contemplating other choices.<br><br>Feel free to surf to my blog ... email psychic readings ([http://www.taehyuna.net/xe/?document_srl=78721 www.taehyuna.net])

Thermodynamic instruments - Revision history

en>Materialscientist: Reverted edits by 218.248.0.2 (talk) to last revision by Yobot (HG)

en>Yobot: /* Thermodynamic reservoirs */WP:CHECKWIKI error fixes + general fixes using AWB (7536)