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In [[mathematics]], the '''Bussgang theorem''' is a [[theorem]] of [[Stochastic process|stochastic analysis]]. The theorem states that the crosscorrelation of a Gaussian signal before and after it has passed through a nonlinear operation are equal up to a constant. It was first published by Julian J. Bussgang in 1952 while he was at the [[Massachusetts Institute of Technology]].<ref>J.J. Bussgang,"Cross-correlation function of amplitude-distorted Gaussian signals", Res. Lab. Elec., Mas. Inst. Technol., Cambridge MA, Tech. Rep. 216, March 1952.</ref> | |||
==Statement of the theorem== | |||
Let <math> \left\{X(t)\right\} </math> be a zero-mean stationary [[Gaussian process|Gaussian random process]] and <math> \left \{ Y(t) \right\} = g(X(t)) </math> where <math> g(\cdot) </math> is a nonlinear amplitude distortion. | |||
If <math> R_X(\tau) </math> is the [[autocorrelation function]] of <math> \left\{ X(t) \right\}</math>, then the [[cross-correlation function]] of <math> \left\{ X(t) \right\}</math> and <math> \left\{ Y(t) \right\}</math> is | |||
:<math> R_{XY}(\tau) = CR_X(\tau), </math> | |||
where <math>C</math> is a constant that depends only on <math> g(\cdot) </math>. | |||
It can be further shown that | |||
: <math> C = \frac{1}{\sigma^3\sqrt{2\pi}}\int_{-\infty}^\infty ug(u)e^{-\frac{u^2}{2\sigma^2}} \, du. </math> | |||
==Application== | |||
This theorem implies that a simplified correlator can be designed.{{clarify|reason=compared to what?|date=December 2010}} Instead of having to multiply two signals, the cross-correlation problem reduces to the gating{{clarify|reason=undefined|date=December 2010}} of one signal with another.{{citation needed|date=December 2010}} | |||
==References== | |||
{{reflist}} | |||
==Further reading== | |||
* E.W. Bai; V. Cerone; D. Regruto (2007) [http://diegoregruto.com/P14.pdf "Separable inputs for the identification of block-oriented nonlinear systems"], ''Proceedings of the 2007 American Control Conference'' (New York City, July 11–13, 2007) 1548–1553 | |||
{{DEFAULTSORT:Bussgang Theorem}} | |||
[[Category:Probability theorems]] | |||
[[Category:Stochastic processes]] | |||
Revision as of 03:45, 10 April 2013
In mathematics, the Bussgang theorem is a theorem of stochastic analysis. The theorem states that the crosscorrelation of a Gaussian signal before and after it has passed through a nonlinear operation are equal up to a constant. It was first published by Julian J. Bussgang in 1952 while he was at the Massachusetts Institute of Technology.[1]
Statement of the theorem
Let be a zero-mean stationary Gaussian random process and where is a nonlinear amplitude distortion.
If is the autocorrelation function of , then the cross-correlation function of and is
where is a constant that depends only on .
It can be further shown that
Application
This theorem implies that a simplified correlator can be designed.Template:Clarify Instead of having to multiply two signals, the cross-correlation problem reduces to the gatingTemplate:Clarify of one signal with another.Potter or Ceramic Artist Truman Bedell from Rexton, has interests which include ceramics, best property developers in singapore developers in singapore and scrabble. Was especially enthused after visiting Alejandro de Humboldt National Park.
References
43 year old Petroleum Engineer Harry from Deep River, usually spends time with hobbies and interests like renting movies, property developers in singapore new condominium and vehicle racing. Constantly enjoys going to destinations like Camino Real de Tierra Adentro.
Further reading
- E.W. Bai; V. Cerone; D. Regruto (2007) "Separable inputs for the identification of block-oriented nonlinear systems", Proceedings of the 2007 American Control Conference (New York City, July 11–13, 2007) 1548–1553
- ↑ J.J. Bussgang,"Cross-correlation function of amplitude-distorted Gaussian signals", Res. Lab. Elec., Mas. Inst. Technol., Cambridge MA, Tech. Rep. 216, March 1952.