# hwB20 - c = c c 1 c 2 c 3 T such that s = Vc S 20.2 P 3.1...

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ENEE 241 02* HOMEWORK ASSIGNMENT 20 Due Wed 03/26 Exam 2 (Monday 03/31) will cover Assignments 10-20. Let V = £ v (0) v (1) v (2) v (3) v (4) v (5) / be the matrix of Fourier sinusoids of length N = 6. (This is the same matrix as in the last example of Reading Assignment 20.) (i) (7 pts.) If x = £ 4 4 1 - 2 - 2 1 / T , use projections to represent x in the form x = Vc . Verify that x is a linear combination of three columns of V (only). (ii) (7 pts.) Repeat for y = £ 4 2 - 2 - 4 10 - 10 / T , expressing it as y = Vd . Verify that y is also a linear combination of three columns of V . (iii) (2 pts.) Verify your results in (i) and (ii) using the FFT command in MATLAB (which will generate the vectors 6 c and 6 d ). (iv) (4 pts.) If s = x + y , use your results from (i) and (ii) to obtain the least squares approxi- mation ˆ s of s in terms of v (1) , v (3) and v (5) . Display the entries of ˆ s . Solved Examples S 20.1 ( P 3.4 in textbook). The columns of the matrix V = 1 1 1 1 1 j - 1 - j 1 - 1 1 - 1 1 - j - 1 j are the complex Fourier sinusoids of length N = 4. Express the vector s = £ 1 4 - 2 5 / T as a linear combination of the above sinusoids. In other words, ﬁnd a vector

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Unformatted text preview: c = [ c c 1 c 2 c 3 ] T such that s = Vc . S 20.2 ( P 3.1 in textbook). Let α = 1 2 and β = √ 3 2 (i) Determine the complex number z such that the vector v = £ 1 α + jβ-α + jβ-1-α-jβ α-jβ / T equals £ 1 z z 2 z 3 z 4 z 5 / T (ii) If s = £ 3 2-1 0-1 2 / T determine the least-squares approximation ˆ s of s in the form of a linear combination of 1 (i.e., the all-ones vector), v and v * . Clearly show the numerical values of the elements of ˆ s . S 20.3 ( P 3.2 in textbook). Let v (0) , v (1) and v (7) denote the complex Fourier sinusoids of length N = 8 at frequencies ω = 0, ω = π/ 4 and ω = 7 π/ 4, respectively. Determine the least-squares approximation ˆ s of s = £ 4 3 2 1 0 1 2 3 / based on v (0) , v (1) and v (7) . Compute the squared approximation error k ˆ s-s k 2 ....
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hwB20 - c = c c 1 c 2 c 3 T such that s = Vc S 20.2 P 3.1...

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