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part40 - ECE 863 Analysis of Stochastic Systems Fall 2001...

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1 ECE 863 – Analysis of Stochastic Systems Fall 2001 Solutions for Homework Set #9 Michigan State University Department of Electrical & Computer Engineering
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2 6.78 () [] EXt EA 0 == ()() 2 12 EXt Xt 1   2 T X T 2T 2T 0 u 1 VAR X t 1 C u du 2T 2T 2u 1d u 2T 2T 1  <> =   =− = Since the variance does not converge to zero as T gets large, process is not mean-ergodic 6.79 X RA 1 , 1 τ= −τ τ≤ X R τ t - 1 1 2T X T 2T u 1 VAR X t 1 R u du 2T 2T = 2T X 2T 1 Rud u 2T < 2T 2T 1 A1 u du 2T 1A for T 1 2T 2 => 0 a s T →→ X(t) is mean-ergodic
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3 7.3 () () YX 0 St F TR c o s 2f π τ   () j2 f j2 f 00 X ee FT R 2 πτ − πτ +  j2 f j2 f XX 11 FT R e FT R e 22 + τ X0 Sff St f =− + + w h e r e Sf F 7.8 Since X(t) and Y(t) are independent, then the expected value of the product is the product of the expected values: [] XY EZt EXtYt EXt EYt mm == = Z Rt , t EX tY tX t Y t EXtXt EYtYt RR +τ = =+ τ+ τ =ττ Multiplication in the “time” domain is equivaluent to convolution in the “frequency” domain. Therefore. () () () ZX Y X Y TR R Sf Sf τ =
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4 7.11 () n 0 Xc o s 2 f n + Θ ( ) ( ) n X0 0 nk 00 0 0 R n,n k E X X E cos 2 fn cos 2 f n k 11 E cos2 f k cos 4 fn 2 f k 2 22 1 cos2 f k 2 +  + = = π π +  + π + π + Θ  Therefore, the autocorrelation function R X is a function of the time-index difference k (only). Consequently, we can evaluate the power-spectral- density function by taking the Fourier transform of R X (k).
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part40 - ECE 863 Analysis of Stochastic Systems Fall 2001...

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