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Unformatted text preview: GEORGIA INSTITUTE OF TECHNOLOGY SCHOOL of ELECTRICAL and COMPUTER ENGINEERING ECE 2025 Summer 2008 Problem Set #2 Assigned: Week of 26-May-08 Due Date: 2/3-Jun-07 = ⇒ Please check the “Bulletin Board” often. All official course announcements are posted there. ALL of the STARRED problems will have to be turned in for grading. A solution to the starred problems will be posted to the web. Your homework is due in recitation at the beginning of class. After the beginning of your assigned recitation time, the homework is considered late and will be given a zero. PROBLEM 2.1 *: (10 pts) Each of the following signals may be simplified, and expressed as a single sinusoid of the form: A cos( ωt + φ ). For each signal, draw a vector diagram of the complex amplitudes (phasors), and use vector addition to estimate the amplitude A and phase φ of the sinusoid. Then use your calculator or Matlab and the phasor addition theorem to find the exact values for A and φ . (a) x a ( t ) = 4cos(30 πt + 7 π )- 6cos(30 πt- 2 . 5 π ) + 7cos(30 πt- 3 π/ 4) (b) x b ( t ) = 17cos(900 πt- π/ 6) + 17sin(900 πt + π ) + 17cos(900 πt- 5 π/ 6) (Note: be sure to notice the minus sign in the third term in part (a) and the sin in part (b)). PROBLEM 2.2 : (a) Consider the equation √ 13cos(150 πt + arctan(2 / 3)) = √ 2cos(150 πt + π/ 4) + A cos( ω πt + φ ) Find the values of A , ω , and φ . Express A as the square root of an integer , and express φ as the arctangent of a rational number . (A rational number is the ratio of two integers.) Make a sketch of the vectors involved which shows your reasoning. Of course, you could do this with the aid of your calculator, but if you appreciate the care with which I picked the numbers, you can do without a calculator and feel extra cool. (b) Simplify the following expression: x ( t ) = 15 X k =0 cos 424 πt + k 8 π You should be able to do this without doing any explicit calculations! Sketch the phasors associated with each of the terms, and give an intuitive explanation for your answer. If you find yourself writing a lot of numbers or pumping anything intoyour answer....
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This note was uploaded on 11/26/2009 for the course ECE 2025 taught by Professor Juang during the Spring '08 term at Georgia Tech.
- Spring '08
- Signal Processing