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# hw8 - EE 562a Homework Set 8 Due Wednesday 25 April 2007...

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EE 562a Homework Set 8 Due Wednesday 25 April 2007 1 (The following well-defined problems come from different sources, and the notation used may vary. Don’t let that bother you!) 1. Consider an LTI system, M , characterized by the following differential equation: ˜ y = M ˜ x ⇐⇒ ¨ y ( t ) + 3 ˙ y ( t ) + 2 y ( t ) = 3 x ( t ) - ˙ x ( t ) . (1) A continuous time, wss random process, x ( u, t ), with correlation function R x ( τ ) = 1 8 e - 4 | τ | is passed through the above LTI system, with the output denoted by y ( u, t ). (a) What is the frequency response of the system - i.e. M ( f )? (b) Determine S y ( f ) and S xy ( f ). (c) Determine the optimal (Wiener) causal filter for estimating x ( u, t ) from y ( u, t ); specify the frequency response of this filter. (d) What is the PSD of the best estimate, ˆ x ( u, t ), in terms of S x ( f )? Hint: Determine the frequency response of the cascade of M ( f ) and the Wiener filter found in the previous part. (e) What is the associated MMSE of this estimator? (f) Explain how your solution would change if the right-hand side of (1) were 3 x ( t ) + ˙ x ( t ). What is the system characteristic which is changed by this sign change. 2. Let w ( u, t ), x ( u, t ), y ( u, t ), and z ( u, t ), be zero-mean wide-sense stationary random processes. (a) Supply the correlation function or power spectral density, which ever is not given. Here α and σ are positive constants. i. R w ( τ ) = σ 2 e - ατ 2 , τ R . ii. R x ( τ ) = σ 2 sin 2 τ τ , τ R . iii. R y ( τ ) = σ 2 1 + ατ 2 exp( i 60 πτ ) , τ R . iv. S z ( f ) = sin 2 ( πfT ) T ( πf ) 2 , f R . (b) Which of these four processes have a power spectral densities that are factorable in the form of a causal system function times its conjugate? (c) Which of these four processes are differentiable in the mean square sense? (d) Which of these four processes are mean-ergodic?

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2 EE 562a Homework Set 8 Due Wednesday 25 April 2007 3. A clock signal detector (Final Exam, Spring 1990) The following circuit is designed to produce a periodic signal with period T from a data signal x ( u, t ) of the form x ( u, t ) = n = -∞ a n ( u ) p ( t - nT ) where { a n ( u ) } is a sequence of independent identically distributed random variables, equally
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hw8 - EE 562a Homework Set 8 Due Wednesday 25 April 2007...

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