ColorGrayscale Jeffrey Schiano 2014. All rights reserved. Rec 11. EE 350Continuous-Time Linear SystemsRecitation 111 Jeffrey Schiano 2014. All rights reserved. Rec 11. Recitation 11 Topics2• Solved Problems – Complex exponential Fourier series– Fourier Transforms and properties Jeffrey Schiano 2014. All rights reserved. Rec 11. Problem 1• The complex exponential Fourier series coefficients of a periodic signal f(t) are1. Determine the average value of f(t).2. Is the signal f(t) real or complex valued?3. As a function of time, is the signal f(t) even, odd, or neither?3220.5010sin 20()njnnDnenjn Jeffrey Schiano 2014. All rights reserved. Rec 11. Problem 1 Solution4 Jeffrey Schiano 2014. All rights reserved. Rec 11. Problem 1 Solution5 Jeffrey Schiano 2014. All rights reserved. Rec 11. Problem 2• Consider the system below with periodic input xa(t)1. Sketch the signals xb(t) and xc(t)2. Determine the complex exponential Fourier series coefficients of the signal xa(t) through xd(t)3. Find an expression for xd(t)6t()axt1150()axt()Hj21()bxt ()cxt ()dxt180 rad/sec1()180 rad/sec0Hj Jeffrey Schiano 2014. All rights reserved. Rec 11. Problem 2 Solution7 Jeffrey Schiano 2014. All rights reserved. Rec 11. Problem 2 Solution8 Jeffrey Schiano 2014. All rights reserved. Rec 11. Problem 2 Solution9 Jeffrey Schiano 2014. All rights reserved. Rec 11. Problem 2 Solution10 Jeffrey Schiano 2014. All rights reserved. Rec 11. Problem 3• Consider the aperiodic signal f(t)1. Is the Fourier transform of f(t) an even or odd function of frequency?2. Is the Fourier transform of f(t) purely real, purely imaginary, or neither?3. Determine the Fourier transform of f(t) by direct integration 11tf( )tAA Jeffrey Schiano 2014. All rights reserved. Rec 11. Problem 3 Solution12 Jeffrey Schiano 2014. All rights reserved. Rec 11. Problem 3 Solution13 Jeffrey Schiano 2014. All rights reserved. Rec 11. Problem 3 Solution14 Jeffrey Schiano 2014. All rights reserved. Rec 11. Problem 4• An aperiodic signal f(t) has the Fourier transform1. Sketch the Fourier transform as a function of frequency2. Without determining f(t), what conclusions can be drawn regarding f(t)?3. Determine f(t) by direct integration to verify the result in part 2 using the result 152()Fj e22cos( ) cos( ) sin( )axaxeebxdx abxbbxab Jeffrey Schiano 2014. All rights reserved. Rec 11. Problem 4 Solution16 Jeffrey Schiano 2014. All rights reserved. Rec 11. Problem 4 Solution17 Jeffrey Schiano 2014. All rights reserved. Rec 11. Problem 4 Solution18 Jeffrey Schiano 2014. All rights reserved. Rec 11. Problem 4 Solution19 Jeffrey Schiano 2014. All rights reserved. Rec 11. Problem 5• Given the Fourier Transform pairdetermine the Fourier transform of20222( ) , 0 ( ) ,ataft e a Fja216g( )4tt Jeffrey Schiano 2014. All rights reserved. Rec 11. Problem 5 Solution21 Jeffrey Schiano 2014. All rights reserved. Rec 11. Problem 5 Solution22 Jeffrey Schiano 2014. All rights reserved. Rec 11. Problem 6• Given an energy signal f(t), derive Parseval’s theorem:23221() ( )2fEftdt Fjd Jeffrey Schiano 2014. All rights reserved. Rec 11. Problem 6 Solution24 Jeffrey Schiano 2014. All rights reserved. Rec 11. Problem 6 Solution25 Jeffrey Schiano 2014. All rights reserved. Rec 11. EE 350Continuous-Time Linear SystemsRecitation 111 Jeffrey Schiano 2014. All rights reserved. Rec 11. Recitation 11 Topics2• Solved Problems – Complex exponential Fourier series– Fourier Transforms and properties Jeffrey Schiano 2014. All rights reserved. Rec 11. Problem 1• The complex exponential Fourier series coefficients of a periodic signal f(t) are1. Determine the average value of f(t).2. Is the signal f(t) real or complex valued?3. As a function of time, is the signal f(t) even, odd, or neither?3 Jeffrey Schiano 2014. All rights reserved. Rec 11. Problem 1 Solution4 Jeffrey Schiano 2014. All rights reserved. Rec 11. Problem 1 Solution5 Jeffrey Schiano 2014. All rights reserved. Rec 11. Problem 2• Consider the system below with periodic input xa(t)1. Sketch the signals xb(t) and xc(t)2. Determine the complex exponential Fourier series coefficients of the signal xa(t) through xd(t)3. Find an expression for xd(t)6 Jeffrey Schiano 2014. All rights reserved. Rec 11. Problem 2 Solution7 Jeffrey Schiano 2014. All rights reserved. Rec 11. Problem 2 Solution8 Jeffrey Schiano 2014. All rights reserved. Rec 11. Problem 2 Solution9 Jeffrey Schiano 2014. All rights reserved. Rec 11. Problem 2 Solution10 Jeffrey Schiano 2014. All rights reserved. Rec 11. Problem 3• Consider the aperiodic signal f(t)1. Is the Fourier transform of f(t) an even or odd function of frequency?2. Is the Fourier transform of f(t) purely real, purely imaginary, or neither?3. Determine the Fourier transform of f(t) by direct integration 11 Jeffrey Schiano 2014. All rights reserved. Rec 11. Problem 3 Solution12 Jeffrey Schiano 2014. All rights reserved. Rec 11. Problem 3 Solution13 Jeffrey Schiano 2014. All rights reserved. Rec 11. Problem 3 Solution14 Jeffrey Schiano 2014. All rights reserved. Rec 11. Problem 4• An aperiodic signal f(t) has the Fourier transform1. Sketch the Fourier transform as a function of frequency2. Without determining f(t), what conclusions can be drawn regarding f(t)?3. Determine f(t) by direct integration to verify the result in part 2 using the result 15 Jeffrey Schiano 2014. All rights reserved. Rec 11. Problem 4 Solution16 Jeffrey Schiano 2014. All rights reserved. Rec 11. Problem 4 Solution17 Jeffrey Schiano 2014. All rights reserved. Rec 11. Problem 4
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