# hw11 - c = c c 1 c 2 c 3 T such that s = Vc S 11.2 P 3.1 in...

This preview shows pages 1–2. Sign up to view the full content.

ENEE 241 02* HOMEWORK ASSIGNMENT 11 Due Thu 03/26 Exam 2 (Tue 03/31) will cover Assignments 6–11. Let V = £ v (0) v (1) v (2) v (3) v (4) v (5) / be the matrix of Fourier sinusoids of length N = 6. (i) (7 pts.) If x = £ 6 - 2 6 - 2 6 - 2 / T , use projections to represent x in the form x = Vc . Verify that x is a linear combination of two columns of V (only). (ii) (7 pts.) Repeat for y = £ 0 8 2 0 - 2 - 8 / T , expressing it as y = Vd . Verify that y is a linear combination of four 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 = 2 x - y , use your results from (i) and (ii) to obtain the least squares approximation ˆ s of s in terms of v (0) , v (1) and v (5) . Display the entries of ˆ s . Solved Examples S 11.1 ( 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

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: c = [ c c 1 c 2 c 3 ] T such that s = Vc . S 11.2 ( 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 11.3 ( 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 ....
View Full Document

{[ snackBarMessage ]}

### Page1 / 2

hw11 - c = c c 1 c 2 c 3 T such that s = Vc S 11.2 P 3.1 in...

This preview shows document pages 1 - 2. Sign up to view the full document.

View Full Document
Ask a homework question - tutors are online