ECE
hmw8_sol_s08

# hmw8_sol_s08 - i ECE220 Homework Assignment#8 Solutions...

• Notes
• 12

This preview shows pages 1–5. Sign up to view the full content.

i ECE220 Homework Assignment #8 Solutions Problem 9.11 Without using any integrals, determine the Fourier series co- efficients, a k , b k and α k , of the following functions. Note the frequency that corresponds to each index k . The time variable is measured in seconds. You will need to determine the period, T , over which to determine the coefficients. Use the shortest possible period. (a) s 1 ( t ) = 0 . 45 cos(7 . 1 t ) (b) s 2 ( t ) = 3 cos(4 . 3 t ) + 2 sin(12 . 9 t ) (c) s 3 ( t ) = 3 . 0 cos(2 π 30 t ) - 1 . 7 cos(2 π 100 t ) (d) s 4 ( t ) = 3 . 0 cos(2 π 30 t ) - 1 . 7 sin(2 π 30 t ) (e) s 5 ( t ) = 20 cos(2 π 30 t ) - 17 cos(2 π 100 t - 5 π/ 6) + 8 cos(2 π 200 t + π/ 4) Solution (a) s 1 ( t ) = . 45 cos(7 . 1 t ) = . 45 cos ( 2 π 2 π/ 7 . 1 ) t The period is = T = 2 π 7 . 1 = 0 . 8850 seconds. We have a k = . 45 , k = 1 0 , o.w. b k = 0 , k = 1 , 2 , ... From the conversion equations (9.7)-(9.9), page 445, we have α k = . 45 2 , k = ± 1 0 , o.w. A plot of the α k coefficients is given in Figure 0.1. It was produced using the problem9 11a.m file in the class locker.

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

ii -15 -10 -5 0 5 10 15 0 0.05 0.1 0.15 0.2 0.25 frequency (Hz) |S(n)| Problem 9.11a: The magnitude spectrum -15 -10 -5 0 5 10 15 -1 -0.5 0 0.5 1 frequency (Hz) θ (n) Problem 9.11a: The phase spectrum Figure 0.1. The spectrum in Problem 9.11a. (b) s 2 ( t ) = 3 cos(4 . 3 t ) + 2 sin(12 . 9 t ) = 3 cos( 2 π 2 π/ 4 . 3 t ) + 2 sin( 2 π 2 π/ (3 · 4 . 3) t ) = 3 cos( 2 π · 1 2 π/ 4 . 3 t ) + 2 sin( 2 π · 3 2 π/ 4 . 3 t ) T = 2 π 4 . 3 = 1 . 4612 sec a k = 3 , k = 1 0 , o.w. b k = 2 , k = 3 0 , o.w. From the conversion equations (9.7)-(9.9), page 445, we have α k = 3 2 , k = ± 1 - 2 j 2 = - j , k = 3 j , k = - 3 0 , o.w.
iii A plot of the α k coefficients is given in Figure 0.2. It was produced using the problem9 11b.m file in the class locker. -8 -6 -4 -2 0 2 4 6 8 0 0.5 1 1.5 frequency (Hz) |S(n)| Problem 9.11b: The magnitude spectrum -8 -6 -4 -2 0 2 4 6 8 -2 -1 0 1 2 frequency (Hz) θ (n) Problem 9.11b: The phase spectrum Figure 0.2. The spectrum in Problem 9.11b. (c) s 3 ( t ) = 3 . 0 cos(2 π 30 t ) - 1 . 7 cos(2 π 100 t ) = 3 . 0 cos( 2 π · 3 1 / 10 t ) - 1 . 7 cos( 2 π · 10 1 / 10 t ) T = 1 10 sec a k = 3 . 0 , k = 3 - 1 . 7 , k = 10 0 , o.w. b k = 0 , 2200 k = 1 , 2 , ... From the conversion equations (9.7)-(9.9), page 445, we have

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

iv α k = 3 2 , k = ± 3 - 1 . 7 2 , k = ± 10 0 , o.w. A plot of the α k coefficients is given in Figure 0.3. It was produced using the problem9 11c.m file in the class locker.
This is the end of the preview. Sign up to access the rest of the document.

{[ snackBarMessage ]}

### What students are saying

• As a current student on this bumpy collegiate pathway, I stumbled upon Course Hero, where I can find study resources for nearly all my courses, get online help from tutors 24/7, and even share my old projects, papers, and lecture notes with other students.

Kiran Temple University Fox School of Business ‘17, Course Hero Intern

• I cannot even describe how much Course Hero helped me this summer. It’s truly become something I can always rely on and help me. In the end, I was not only able to survive summer classes, but I was able to thrive thanks to Course Hero.

Dana University of Pennsylvania ‘17, Course Hero Intern

• The ability to access any university’s resources through Course Hero proved invaluable in my case. I was behind on Tulane coursework and actually used UCLA’s materials to help me move forward and get everything together on time.

Jill Tulane University ‘16, Course Hero Intern