ReviewProblems2_sol

# ReviewProblems2_sol - ENEE630 ADSP 1 REVIEW PROBLEMS II w...

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ENEE630 ADSP REVIEW PROBLEMS II w/ solution December 9, 2009 1. Assume that v ( n ) is a real-valued zero-mean white Gaussian noise with σ 2 v = 1, x ( n ) and y ( n ) are generated by the equations x ( n ) = 0 . 5 x ( n - 1) + v ( n ) , y ( n ) = x ( n - 1) + x ( n ) . (a) Find the power spectrum of sequence x ( n ), and its power. (b) Find the power spectrum of sequence y ( n ), and its power. (c) Calculate r y ( k ) for k = 0 , 1 , 2 , 3. Assume now we don’t know the real model of the signal, and we want to estimate its power spectrum from r y ( k ) obtained in part (c). Estimate power spectrum using the following methods: (d) ARMA(1,1) spectral estimation. (e) AR(2) spectral estimation. (f) Maximum entropy spectral estimation with order 2. (g) Minimum variance spectral estimation with order 1. Solution: (a) x ( n ) can be modeled as AR(1). Hence, P x ( ω ) = σ 2 v (1 - 0 . 5 e - )(1 - 0 . 5 e ) = 1 1 . 25 - cos ω . For AR(1) process, we know that r x ( k ) = (0 . 5) | k | r x (0) and r x (0) = 4 / 3. (b) y ( n ) can be modeled as ARMA(1,1). P x ( ω ) = (1 + e - )(1 + e ) (1 - 0 . 5 e - )(1 - 0 . 5 e ) = 2 + 2cos ω 1 . 25 - cos ω . r y (0) = 2 r x (0) + 2 r x (1) = 4. (c) r y (0) = 4. r y (1) = 2 r x (1) + r x (0) + r x (2) = 3. r y (2) = 3 / 2. r y (3) = 3 / 4. (d) Since the assumed model perfectly matches the real model, the estimated spectrum is exactly the true spectrum P ARMA ( ω ) = (1 + e - )(1 + e ) (1 - 0 . 5 e - )(1 - 0 . 5 e ) = 2 + 2cos ω 1 . 25 - cos ω . (e) Use the Yule-Walker equation, r (0) r (1) r (1) r (0) ‚• a 1 a 2 = - r (1) r (2) . a

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ReviewProblems2_sol - ENEE630 ADSP 1 REVIEW PROBLEMS II w...

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