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# 12PSD - MAE 591 RANDOM DATA Spectral Density Functions i...

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MAE 591 RANDOM DATA Spectral Density Functions i) Power in electric circuits: Case of D.C. : P Ri = 2 Case of A.C. : () P T Ri dt T RI f t d t R I av T a T == = ∫∫ 11 2 2 0 2 0 2 sin π I I f T a 2 1 , Arbitrary current : P T Ri dt RI av T T →∞ lim 1 2 0 2 I T id t T T 22 0 1 = lim ii) Power for any time function x t () : P T xtd t av T T = lim ( ) 1 2 0 iii) Wiener-Khinchin Relationship : the ensemble spectral density for stationary process is the Fourier transform of the ensemble correlation. Sf R e d xy xy jf = −∞ ττ πτ 2 RS f e xy xy ( ) τ = 2 d f In particular, when x t y t = , τ τ = τ π d e R f S f j xx xx 2 ) ( ) ( f e xx xx τ = 2 d f 2 + Property: 1) RE x t xx x x x [ () ] 0 2 = ψσµ =≥ d f xx 0 2) d xx xx = 2 τ τ π τ τ τ π τ = d f R j d f R xx xx 2 sin ) ( 2 cos ) ( τ τ π τ = τ τ π τ = d f R d f R xx xx 0 2 cos ) ( 2 2 cos ) ( df f f S f R xx xx τ π = 0 2 cos ) ( 2 ) ( iv) Two-sided power(auto) and cross spectral density for random processes x t ( ) and y t : XfT TT EX fT xx k () l im (,) lim [ ( , ) ] =< > = 2 2 1 Y fT EX fTY fT xy kk (,)(,) lim [ ( , ) ( , )] * * > = 1 1

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MAE 591 RANDOM DATA Proof : Since, for an ergodic process { } xt () , Rx t x t T xtxt d t xx T T T () ( ) l im ) ττ =< + >= +
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12PSD - MAE 591 RANDOM DATA Spectral Density Functions i...

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