ELEC 533 Professor Behnaam Aazhang Homework 8 Problem 3 continued Consider the

# Elec 533 professor behnaam aazhang homework 8 problem

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ELEC 533 (Professor Behnaam Aazhang): Homework 8 Problem 3 (continued) Consider the following filter : H ( ω ) = 1 ω [ ω 1 , ω 2 ] 0 ω 6∈ [ ω 1 , ω 2 ] Pass X t and Y t through this filter to obtain b X t and b Y t , respectively. Then: S b X ( ω ) = | H ( ω ) | 2 S X ( ω ) and S b Y ( ω ) = | H ( ω ) | 2 S Y ( ω ). Furthermore, S b X b Y ( ω ) = | H ( ω ) | 2 S XY ( ω ) Now we can rewrite the autocorrelations as: R b X b Y (0) = 1 2 π ω 2 R ω 1 S XY ( ω ) , R b X (0) = 1 2 π ω 2 R ω 1 S X ( ω ) and respectively R b Y (0) = 1 2 π ω 2 R ω 1 S Y ( ω ) . Having all of the above we can rewrite the inequality we had for τ = 0: 4 π 2 | R b X b Y (0) | 2 = ω 2 Z ω 1 S XY ( ω ) 2 ω 2 Z ω 1 S X ( ω ) ω 2 Z ω 1 S Y ( ω ) = 4 π 2 R b X (0) R b Y (0) Problem 4 h(t) g(t) X t Y t V t a) E[ Y t V s ] = E[ Z -∞ h ( t - α ) X α Z -∞ g ( s - β ) X β ] = Z -∞ Z -∞ h ( t - α ) g ( s - β ) R X ( α - β ) dαdβ = N 0 2 Z -∞ Z -∞ h ( t - α ) g ( s - β ) δ ( α - β ) dαdβ = N 0 2 Z -∞ h ( γ ) g ( γ - ( t - s )) R Y V ( τ ) = N 0 2 Z -∞ h ( γ ) g ( γ - ( t - s )) = N 0 2 ( h * b g )( τ ) where b g is obtained from g by time reversal.

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• Spring '14
• Aazhang,Behnaam
• Derivative, Trigraph, Professor Behnaam Aazhang, T T0

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