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Differential Equations Solutions 33

# Differential Equations Solutions 33 - κ between 2 and 6...

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43 0 100 200 300 400 500 600 700 0 5 10 15 20 25 30 Results using exponential windowing Figure 8.2. Results of Problem 5: the true DOA (blue) and the DOA estimated by exponential windowing (red) as a function of time. CHALLENGE 8.6. (a) The sum of the squares of the entries of X is the square of the Frobenius norm of X , and this norm is invariant under multiplication by an orthogonal matrix. Therefore, X 2 F = Σ 2 F = σ 2 1 + . . . + σ 2 m . (b) The expected value of the square of each entry of X is ψ 2 , so the sum of these mn values has expected value ψ 2 mn . (c) The expected value is now m k =1 n j =1 f 2 j E ( x 2 kj ) = m n j =1 f 2 j ψ 2 mf 2 ψ 2 1 f 2 for large n , where E denotes expected value. CHALLENGE 8.7. The software on the website varies
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Unformatted text preview: κ between 2 and 6. For rectangular windowing, a window size of 4 produced fewer d-failures than window sizes of 6 or 8 at a price of increasing the average error to 0 . 75 degrees. As κ increased, the number of d-failures also increased, but the average error when d was correct decreased. For exponential windowing, the fewest d-failures (8) occurred for f = 0 . 7 and κ = 2, but the average error in this case was 1 . 02. As κ increased, the number of d-failures increased but again the average error when d was correct decreased....
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