131B_1_final

# 131B_1_final - EE 131B Spring 09 Final June 12th Open book...

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EE 131B Spring 09 Final, June 12th Open book Your Name: Your ID Number: 1

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1. Let φ ( ω ) be uniform [0 , 2 π ]. Show that sin( φ ( ω )) and cos( φ ( ω )) are UNcorrelated. Show that they are NOT independent. 2
2. Show that the random process: x ( t,ω ) = sin( φ ( ω )) cos400 t + cos( φ ( ω )) sin400 t, -∞ < t < is second order stationary by caculating the mean and covariance function. φ ( ω ) is deﬁned in problem 1. What is the spectral density? 3

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3. Let W ( t,ω ) ,t 0 be a Wiener Process with E ( W ( t,ω ) 2 ) = t Is this process stationary? Let y ( t,ω ) = sin( φ ( ω )) sin W ( t,ω ) + cos( φ ( ω )) cos W ( t,ω ) where φ ( ω ) is deﬁned in problem 1 and is independent of the Wiener process. Find the steady state mean, covariance functions and the spectral density. 4
4. Let N ( t,ω ) denote white Gaussian noise with unit spectral density. Calculate the steady state spectral density of the process : x ( t,ω ) = Z t 0 e - 3 s N ( t - s,ω ) ds,t > 0 Calculate steady state output power.

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131B_1_final - EE 131B Spring 09 Final June 12th Open book...

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