Homework 9 Solution

Homework 9 Solution - MEEN 260 Introduction to Engineering...

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Unformatted text preview: MEEN 260 Introduction to Engineering Experimentation Homework 9: Complex Numbers, Fourier Series, and Fourier Transform Solution Assigned: Thursday, 2 Apr. 2009 Due: Thursday, 9 Apr. 2009, 5:00pm Learning Objectives: After completing this homework assignment, you should be able to: 1) 2) 3) Determine the magnitude and phase angle of a complex number, and manipulate functions of a complex variable Determine the Fourier Series of a periodic signal, in polar, trigonometric, or exponential form Determine the Fourier Transform of a non-periodic signal using the definition, tables, or properties of the Fourier Transform Homework Problems: Problem 1) Complex Numbers Given the imaginary number: a) Determine the magnitude of b) Determine the phase angle of c) Manipulate into the form Aejφ Solution: a) The magnitude is given by the following: || √3 √2 4 2 1.77 0.171 b) The phase angle is given by the following: 4 2 tan tan 3 2 c) Manipulating the values obtained in (a) and (b), we get: 1.77 . Problem 2) Plotting Functions of a Complex Variable Given the function of a complex variable: a) Using Matlab, Excel, or a similar program, plot the magnitude of ? b) Plot the phase of using a log scale for ? Solution a) Using the code: x=0:.001:1000; y=3./sqrt(25+4.*x.^2); z=-atan(2*x/5); semilogx(x,y) using a log scale for we get: b) Using the code: x=0:.001:1000; y=3./sqrt(25+4.*x.^2); z=-atan(2*x/5); semilogx(x,z) we get: Problem 3) Digital Signals and Signal Reconstruction For the following signal: a) Find the Fourier Series in trigonometric form b) Find the Fourier Series in polar form c) Find the Fourier Series in exponential form Solution: The triangle wave may be considered for one period as follows: 4 2, 0 4 6, 2 2 For the sake of simplicity, we begin with the exponential form from part (c) to obtain ak, bk, ck, and φk a) The trigonometric form is given by: cos 2 Where 1 2 2 0 2· 1 1 sin 2 cos 2 sin 2 Plugging in the values for ak and bk, we get the trigonometric form: 2· 1 1 cos sin b) The polar form is given by: cos 2 Where 1 √4 0 | | tan √4 We find ck and φk from ak and bk c) The exponential form is given by: tan cos 2 tan 1 2 1 Where 1 1 0 1 1 2 2 Due to the simple integration of dk, we begin our derivation with it. 4 1 4 2 6 2 Using: 1, 1 We arrive at 41 2 4 From this, we derive all other constants. Problem 4) Fourier Transform Definition 1 2 Find the Fourier Transform for the following signal using the integral definition for the Fourier Transform (you may assume the signal is zero for the time periods not shown in the graph).: Solution: The function x(t) may be decompsed as follows: 0 1 2.5 · 1 3 3 2.5 2.5 · The Fourier Transform is given by: · 5 · 2 Using Maple, we can evaluate the function as follows: Problem 5) Fourier Transform Tables and Properties Using the table, find the Fourier Transform for the following: π a) 4 cos(6t + ) + sin(4t ) 4 2 tj b) 2 + e (Generalized) Fourier Transform Pairs ⇔ F{x(t )} = X (ω ) ⇔ 2πδ (ω ) 1, −∞ < t < ∞ ⇔ 1 πδ (ω ) + u (t ) jω ⇔ δ (t ) 1 − jω c ⇔ δ (t − c ) , c any real number e ⇔ 1 , b>0 e−bt u(t ) jω + b ⇔ 2πδ (ω − ω0 ) , ω0 any real number e jω0t ⇔ cos(ω0t ) π [δ (ω + ω0 ) + δ (ω − ω0 )] x(t ) cos(ω0t + θ ) sin(ω0t ) sin(ω0t + θ ) Solution: a) Using ⇔ ⇔ ⇔ π [e− jθ δ (ω + ω0 ) + e jθ δ (ω − ω0 )] jπ [δ (ω + ω0 ) − δ (ω − ω0 )] jπ [e− jθ δ (ω + ω0 ) − e jθ δ (ω − ω0 )] cos(ω0t + θ ) sin(ω0t ) We get: 4 cos 6 b) Using 4 sin 4 ⇔ ⇔ 4 4 ⇔ ⇔ 2 4 π [e− jθ δ (ω + ω0 ) + e jθ δ (ω − ω0 )] jπ [δ (ω + ω0 ) − δ (ω − ω0 )] 6 4 2πδ (ω ) 6 1, −∞ < t < ∞ e jω0t 2πδ (ω − ω0 ) , ω0 any real number 2 2 We get: ...
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This note was uploaded on 07/28/2010 for the course MEEN 260 taught by Professor Langari during the Fall '08 term at Texas A&M.

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