mixed_signal_notes_set_6

mixed_signal_notes_set_6 - ECE145C /218C notes, M. Rodwell,...

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ECE145C /218C notes, M. Rodwell, copyrighted 2007 Mixed Signal IC Design Notes set 6: Mathematics of Electrical Noise Mark Rodwell University of California, Santa Barbara [email protected] 805-893-3244, 805-893-3262 fax

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ECE145C /218C notes, M. Rodwell, copyrighted 2007 Strategy ive. comprehens is book s Zeil' der Van . literature the in or web), the (on notes class noise my in found be can detail More rates. error ns, correlatio signal functions, n correlatio densities, spectral SNR, of n calculatio correct for sufficient backround give : Strategy detail. in subject this develop to 145c/218c in not time is There
ECE145C /218C notes, M. Rodwell, copyrighted 2007 Topics Math : distributions, random variables, expectations, pairs of RV, joint distributions, covariance and correlations. Random processes, stationarity, ergodicity, correlation functions, autocorrelation function, power spectral density. Noise models of devices: thermal and shot noise. Models of resistors, diodes, transitors, antennas. Circuit noise analysis: network representation. Solution. Total output noise. Total input noise. 2 generator model. En/In model. Noise figure, noise temperature. Signal / noise ratio.

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random variables
ECE145C /218C notes, M. Rodwell, copyrighted 2007 The first step: random Variables function. on distributi y probabilit the is ) ( ) ( } { is and between lies y that probabilit The . value particular a on takes variable random a , experiment an During 2 1 2 1 2 1 x f dx x f x x x P x x x x X X x x X = < < f X (x) x x 1 x 2

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ECE145C /218C notes, M. Rodwell, copyrighted 2007 Example: The Gaussian Distribution ( ) Gaussian. the of that to close ons distributi y probabilit have effects small many of sum the from arising processes random physical rem*, limit theo central * the of Because ). ( deviation standard the and ) ( mean e shortly th define will We 2 exp 2 1 ) ( : on distributi Gaussian The 2 2 2 2 x x x X x x x x f σ π = f X (x) x x ~2 σ x
ECE145C /218C notes, M. Rodwell, copyrighted 2007 Mean values and expectations [ ] [ ] [ ] + + + = = = = = = dx x f x X E X X dx x xf X E X X dx x f x g x g E X X X ) ( of value Expected ) ( X of Value Mean ) ( ) ( ) ( X variable random the of g(X) function a of n Expectatio 2 2 2 2

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ECE145C /218C notes, M. Rodwell, copyrighted 2007 Variance ( ) ( ) [ ] ( ) variance the of root square the simply is X of deviation standard The ) ( value average its from deviation square - mean - root its is X of variance The 2 2 2 2 2 x X X x dx x f x x x X E x X σ + = = =
ECE145C /218C notes, M. Rodwell, copyrighted 2007 Returning to the Gaussian Distribution f X (x) x x ~2 σ x ( ) clear. be now should 2 exp 2 1 ) ( : on distributi Gaussian the describing notation The 2 2 2 π = x x X x x x f σ

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ECE145C /218C notes, M. Rodwell, copyrighted 2007 Variance vs Expectation of the Square ( ) ( ) ( ) ( ) ( ) ( ) ( ) n. expectatio the of square the minus square the of n expectatio the is variance The 2 2 2 2 2 2 2 2 2 2 2 2 2 2 x X x x x X x X x X x x X X x X x X x X X X = + = + = + = = = σ
ECE145C /218C notes, M. Rodwell, copyrighted 2007 Pairs of Random Variables ∫∫ = < < < < D C B A XY XY dxdy y x f D y C B x A

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This note was uploaded on 12/29/2011 for the course ECE 594A taught by Professor Rodwell during the Fall '09 term at UCSB.

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mixed_signal_notes_set_6 - ECE145C /218C notes, M. Rodwell,...

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