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Unformatted text preview: EE 562a Midterm Solution October, 2009 1 1. Let’s approach this problem algebraically first, and then review it geometrically. There are many references to the vector b in different situations, so let’s generally define b θ ◦ = cos θ ◦ sin θ ◦ . to compress notation, note that we are giving the arguments of the sine and cosine function in degrees rather than the standard radians. (Be careful if you are evaluating these numerically using a computer.) The problem statement tells you that E { b t 30 ◦ x ( u )  2 } = 10 = b t 30 ◦ K x b 30 ◦ and E { b t 75 ◦ x ( u )  2 } = 6 = b t 75 ◦ K x b 75 ◦ Since there are three unknowns in the K x matrix (using symmetry), the above two equations by themselves are not enough to determine K x . You must also use the fact that the maximum meansquared projection is achieved when using b 30 ◦ , and hence b 30 ◦ = e 1 is a normalized eigenvector of K x with eigenvalue λ 1 = 10. Furthermore, the remaining orthonormal eigen vector must be e 2 = b 120 ◦ or its negative. Hence there is really only one unknown in K x , namely the second eigenvalue λ 2 . K x = h b 30 ◦ . . . b 120 ◦ i 10 λ 2 h b 30 ◦ . . . b 120 ◦ i t Now using information about the second measurement gives 6 = b t 75 ◦ h b 30 ◦ . . . b 120 ◦ i 10 λ 2 h b 30 ◦ . . . b 120 ◦ i t b 75 ◦ since all the inner products of b vectors involve 45 ◦ angles, each of the inner products is 1 / √ 2. The above equation reduces to 6 = (10 + λ 2 ) / 2 = ⇒ λ 2 = 2 . So the results are as follows: (a) E { x ( u )  2 } = Tr { K x } = λ 1 + λ 2 = 12 . (b) Summarizing, e 1 = b 30 ◦ = 1 2 √ 3 1 , λ 1 = 10, e 2 = b 120 ◦ = 1 2 1 √ 3 , λ 2 = 2. (c) Because m x = , it follows that the covariance matrix and correlation matrix of x ( u ) are identical. Using the finitedimensional equivalent of Mercer’s expansion gives R x = λ 1 b 30 ◦ b t 30 ◦ + λ 2 b 120 ◦ b t 120 ◦ = 10 · 1 4 3 √ 3 √ 3 1 + 2 · 1 4 1 √ 3 √ 3 3 = 8 2 √ 3 2 √ 3 4 (d) The eigenvector with minimum eigenvalue points the way. The minimizing choices for b are ± b 120 ◦ and the corresponding angles are 120 ◦ and 60 ◦ = 300 ◦ . 2 EE 562a Midterm Solution October 15, 2009 (e) Here are two ways to write the answer. b θ ◦ : E { y ( u )  2 } = b t θ ◦ K x b θ ◦ = 8cos 2 θ + 4sin 2 θ + 4 √ 3cos θ sin θ b θ ◦ : E { y ( u )  2 } = b t θ ◦ K x b θ ◦ = b t θ ◦ h b 30 ◦ . . . b 120 ◦ i 10 0 2 h b 30 ◦ . . . b 120 ◦ i t b θ ◦ = 10cos 2 ( θ ◦ 30 ◦ ) + 2cos 2 ( θ ◦ 120 ◦ ) and you can get other equivalent expressions by applying trig identities. Now let’s take a geometric view. Basic geometry (the pythagorean theorem) tells us that the sums of squares of the projections of a vector x ( u ) onto an orthonormal basis is  x ( u )  2 ....
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 Fall '07
 ToddBrun
 Variance, Probability theory, Eigenvalue, eigenvector and eigenspace, emin

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