hw03su09_soln - GEORGIA INSTITUTE OF TECHNOLOGY ECE 2025...

Info iconThis preview shows pages 1–3. Sign up to view the full content.

View Full Document Right Arrow Icon

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

View Full DocumentRight Arrow Icon
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: GEORGIA INSTITUTE OF TECHNOLOGY ECE 2025 SOLUTIONS Problem Set # 3 Summer 2009 Problem 3.1 (a) Since f i ( t ) = 1 2 d( t ) d t , we find in [0 , 1]: 1 ( t ) = (0) + Z t 2 2 d , All solutions are of the form x ( t ) = cos (0) + 2 t 3 3 , | t | < with x ( t ) = 0 outside the interval. Since, at t = 1, we have x = cos ( (0) 2 3 ) , we may find a continuous solution if cos ( (0) 2 3 ) = 0. This is not possible as we would need simultaneously (0) = 2 3 . Hence all solutions lead to discontinuous signals. (b) If f i ( t ) = 3cos(2 t + / 2), then ( t ) = (0) + 2 Z t 3cos(2 + / 2)d == 3sin(2 t + / 2) + (0) . Hence x ( t ) = cos(3sin(2 t + / 2) + (0)) . Problem 3.2 (a) Periodic. Fundamental frequency is GCD(16 , 20) = 4 rad/sec. Besides a DC compo- nent only the 4-th and 5-th harmonics are present. x =- 3 x 4 = 12 / (2 j )e j/ 6 = 6e j ( / 6- / 2) = 6e- j/ 3 1 When denoting integrals as the one shown, it is important to write the infinitesimal d . 1 x- 4 =- 12 / (2 j )e- j/ 6 = 6e j (- / 6+ / 2) = 6e j/ 3 x 5 =- 6 / (2)e j 5 / 8 = 3e j ( 5 / 8- ) = 3e- j 3 / 8 x- 5 =- 6 / (2)e- j 5 / 8 = 3e j (- 5 / 8+ ) = 3e j 3 / 8 (b) Not periodic. The ratios of the frequencies of the two sinusoidal components is e, which is not a rational number. (c) Periodic. 8cos(- 200 t + / 3)cos(10 t ) = 4cos(- 200 t + / 3 + 10 t ) + 4cos(- 200 t + / 3- 10 t ) = 4cos(- 190 t + / 3) + 4cos(- 210 t + / 3) = 4cos(190 t- / 3) + 4cos(210 t- / 3) Its fundamental frequency is GCD(190 , 210) = 10 rad/sec. Only the 19-th and 21-th harmonics are present....
View Full Document

This note was uploaded on 01/24/2010 for the course ECE 2025 taught by Professor Juang during the Spring '08 term at Georgia Institute of Technology.

Page1 / 7

hw03su09_soln - GEORGIA INSTITUTE OF TECHNOLOGY ECE 2025...

This preview shows document pages 1 - 3. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online