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Unformatted text preview: 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 . 45cos(7 . 1 t ) (b) s 2 ( t ) = 3cos(4 . 3 t ) + 2sin(12 . 9 t ) (c) s 3 ( t ) = 3 . 0cos(2 π 30 t ) 1 . 7cos(2 π 100 t ) (d) s 4 ( t ) = 3 . 0cos(2 π 30 t ) 1 . 7sin(2 π 30 t ) (e) s 5 ( t ) = 20cos(2 π 30 t ) 17cos(2 π 100 t 5 π/ 6) + 8cos(2 π 200 t + π/ 4) Solution (a) s 1 ( t ) = . 45cos(7 . 1 t ) = . 45cos • ( 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 , 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 , 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. ii15105 5 10 15 0.05 0.1 0.15 0.2 0.25 frequency (Hz) S(n) Problem 9.11a: The magnitude spectrum15105 5 10 1510.5 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 ) = 3cos(4 . 3 t ) + 2sin(12 . 9 t ) = 3cos( 2 π 2 π/ 4 . 3 t ) + 2sin( 2 π 2 π/ (3 · 4 . 3) t ) = 3cos( 2 π · 1 2 π/ 4 . 3 t ) + 2sin( 2 π · 3 2 π/ 4 . 3 t ) T = 2 π 4 . 3 = 1 . 4612 sec a k = ‰ 3 ,k = 1 , o.w. b k = ‰ 2 ,k = 3 , 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 , 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.8642 2 4 6 8 0.5 1 1.5 frequency (Hz) S(n) Problem 9.11b: The magnitude spectrum8642 2 4 6 821 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 . 0cos(2 π 30 t ) 1 . 7cos(2 π 100 t ) = 3 . 0cos( 2 π · 3 1 / 10 t ) 1 . 7cos( 2 π · 10 1 / 10 t ) T = 1 10 sec a k = 3 . ,k = 3 1 . 7 ,k = 10 , o.w....
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This note was uploaded on 03/30/2009 for the course ECE 220 taught by Professor Nilson during the Spring '08 term at N.C. State.
 Spring '08
 NILSON
 Frequency

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