a We wish to form an estimate of X 3 of the form ˆ X 3 aX 2 b where a and b are

A we wish to form an estimate of x 3 of the form ˆ x

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a) We wish to form an estimate of X 3 of the form ˆ X 3 = aX 2 + b where a and b are some constants. Find the value of a and b that minimize the mean squared error (MSE) given X 2 = x 2 . What is the corresponding MSE value? b) Find the minimum mean square estimator of X 3 given X 1 = x 1 and X 2 = x 2 . Find the corresponding value of MSE and compare that with the value in Part (a). Solution2 a) ˆ X 3 = E[ X 3 ] + Cov ( X 3 ,X 2 ) V ar ( X 2 ) ( X 2 -E[ X 2 ]) = ρ ( x 2 - 1) Var( ˆ X 3 ) = Cov ( X 3 , X 2 ) - Cov ( X 3 ,X 2 ) V ar ( X 2 ) = 1 - ρ 2 b) Similarly ˆ X 3 = ( 0 ρ ) 1 ρ ρ 1 - 1 x 1 - 1 x 2 - 1 Var( ˆ X 3 ) = 1 - ( 0 ρ ) 1 ρ ρ 1 - 1 0 ρ = (1 - ρ 2 1 - ρ 2 ) 1 - ρ 2 Page 3 of 5
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ELEC 533 (Behnaam Aazhang ): Midterm Examination 2017 Problem 2 Problem 3 Suppose { X t ; t R } is a Gauss-Markov process, meaning that X t is a zero-mean wide sense stationary (WSS) Gaussian process with autocorrelation R X ( t, s ) = e -| t - s | . 1. Show that if you sample X t at three different points of time t 1 , t 2 , t 3 , then E [ X t 3 | X t 1 , X t 2 ] depends only on X t 2 , and not on X t 1 . 2. Show that X t 1 , X t 2 , and X t 3 form a Markov chain; that is, X t 3 conditioned on X t 2 is independent of X t 1 . Solution3 a) We will assume that: t 1 < t 2 < t 3 R x ( t, s ) = e -| t - s | = E [ X t X s ] because the mean is zero. Therefore we can write the following: R x ( t 1 , t 2 ) = R x ( t 2 , t 1 ) = e t 1 - t 2 and similar we can write: R x ( t 2 , t 3 ) = e t 2 - t 3 , R x ( t 1 , t 3 ) = e t 1 - t 3 . X (1) = X t 1 X t 2 ∼ N 0 0 , Σ 11 Σ 11 = R x ( t 1 , t 1 ) R x ( t 1 , t 2 ) R x ( t 2 , t 1 ) R x ( t 2 , t 2 ) = 1 e t 1 - t 2 e t 1 - t 2 1 det(Σ 11 ) = 1 - e 2( t 1 - t 2 ) Σ - 1 11 = 1 e t 1 - t 2 e t 1 - t 2 1 × 1 1 - e 2( t 1 - t 2 ) X (2) = X t 3 ∼ N (0 , Σ 22 ) with Σ 22 = R x ( t 3 , t 3 ) = 1.
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  • Spring '14
  • Aazhang,Behnaam
  • Probability theory, random process Xt, Behnaam Aazhang, different points of time t1

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