noise_notes_set4

noise_notes_set4 - ECE594I notes, M. Rodwell, copyrighted...

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ECE594I notes, M. Rodwell, copyrighted ECE594I Notes set 4: More Math: Expectations of 1-2 R.V.'s Mark Rodwell University of California, Santa Barbara [email protected] 805-893-3244, 805-893-3262 fax
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ECE594I notes, M. Rodwell, copyrighted References and Citations: Devices State - Solid in Noise : Ziel der Van Physics Thermal : Kroemer and Kittel : Citations / Sources g. ngineeri ns ommunicat f rinciple cobs ozencraft ory) (introduct s Principle Signal Random Variables, Random ty, Probabili : Peebles Z. Peyton ive) comprehens (hard, Variables Random and ity Probabil : Papoulis 982 rca tanford ellman artin otes cture robabilit 1982 circa Stanford, Cover, Thomas : notes lecture theory n Informatio Design c Electroni Noise Low : er Motchenbak ng. Engineeri ions Communicat of s Principle : Jacobs & Wozencraft d f t d circuits. in Noise : Notes ns Applicatio tor Linear Semiconduc National 1982 circa Stanford, Hellman, Martin : notes lecture y Probabilit design) receiver (optical Personik & Smith noise), (device by Fukui Papers Kroemer and Kittel Peebles, Jacobs, & Wozencraft Ziel, der Van study. for references Suggested Theory n Informatio of Elements : Williams and Cover ) (! Notes App. Semi. National
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ECE594I notes, M. Rodwell, copyrighted ndom ariables random variables nd and Expectations
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ECE594I notes, M. Rodwell, copyrighted Recall: Distribution Function of Random Variable is and between lies y that probabilit The . value particular a on takes variable random a , experiment an During 2 2 1 x x x x X x function. on distributi y probabilit the is ) ( ) ( } { 1 2 1 x f dx x f x x x P x X = < < X f X (x) x x 1 x 2
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ECE594I notes, M. Rodwell, copyrighted Mean values and expectations + x X variable random the of g(X) function a of n Expectatio [ ] = dx x f x g x g E X ) ( ) ( ) ( [] + = = = dx x xf X E X X X ) ( X of Value Mean X of value Expected 2 + = = dx x f x X E X X ) ( 2 2 2
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ECE594I notes, M. Rodwell, copyrighted Variance alue erage s om deviation square - mean - root its is X of variance The 2 x σ () [] ) ( value average its from 2 2 2 2 X X dx x f x x x X E x X + = = = variance the of root square the simply is X of deviation standard The x
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ECE594I notes, M. Rodwell, copyrighted Returning to the Gaussian Distribution : on distributi Gaussian the describing notation The ( ) 2 exp 2 1 ) ( 2 2 2 π = x x X x x x f σ clear. be now should f X (x) ~2 σ x x x
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ECE594I notes, M. Rodwell, copyrighted Variance vs Expectation of the Square () ( ) ( ) 2 2 x X x X x X X = = σ ( ) ( ) 2 2 2 2 2 2 x X x X x x X X + = + = 2 2 2 x x x X + = ( ) square the of n expectatio the is variance The 2 2 2 x X X = n. expectatio the of square the minus
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ECE594I notes, M. Rodwell, copyrighted Example of Expectation: Mean Kinetic Energy n istributi elocity thermal ith article ur / where exp 1 1 ) ( : on distributi velocity a thermal with particle Our 2 m kT v v f v x x V = = σ 2 2 2 v v v x = + π [ ] 0 ) ( 2 2 v dv v f v v E x x V x x x + [ ] proof) skip - - - Gaussian a of variance the computes (this ) ( dv v f v v E x x V x x x = ergy inetic o / 2 2 v m kT v = = σ [ ] [ ] [] 2 / 2 / 2 / energy
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This note was uploaded on 12/29/2011 for the course ECE 594C taught by Professor Madhow during the Fall '10 term at UCSB.

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noise_notes_set4 - ECE594I notes, M. Rodwell, copyrighted...

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