241 ENEE 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 , 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 that y is also a linear combination of three columns of V.
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, expressing it as y = Vd. Verify
(iii) (2 pts.) Verify your results in (i) and (ii) using the FFT command in MATLAB (which will generate the vectors 6c and 6d). (iv) (4 pts.) If s = x + y, use your results from (i) and (ii) to obtain the least squares approximation of s in terms of v(1) , v(3) and v(5) . Display the entries of . s s Solved Examples S 20.1 (P 3.4 in textbook). The columns of matrix the 1 1 1 1 1 j 1 j V= 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
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as a linear combination of the above sinusoids. In other words, nd a vector c = [c0 c1 c2 c3 ]T such that s = Vc. S 20.2 (P 3.1 in textbook). Let 1 = 2 and 3 = 2
(i) Determine the complex number z such that the vector v= equals 1 z z2 z3 z4 z5 1 + j + j 1 j j
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(ii) If s= 3 2 1 0 1 2
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determine the least-squares approximation of s in the form of a linear combination of 1 (i.e., the s . Clearly show the numerical values of the elements of . all-ones vector), v and v 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 = 0, = /4 and = 7/4, respectively. Determine the least-squares approximation of s s= 43210123
based on v(0) , v(1) and v(7) . Compute the squared approximation error s 2 . s