ColorGrayscale Jeffrey Schiano 2014. All rights reserved. Rec 10. EE 350Continuous-Time Linear SystemsRecitation 101 Jeffrey Schiano 2014. All rights reserved. Rec 10. Recitation 10 Topics2• Solved Problems – Complex Exponential Fourier series– Sinusoidal-steady-state response of LTI systems to periodic inputs• MATLAB Exercises– Symbolic Math Toolbox Jeffrey Schiano 2014. All rights reserved. Rec 10. Problem 1• Consider the periodic signal• Determine the fundamental period and frequency• Determine the complex exponential Fourier series coefficients• Sketch the Fourier series spectra3() 2 cos3 6sin 43ft t t Jeffrey Schiano 2014. All rights reserved. Rec 10. Problem 1 Solution4 Jeffrey Schiano 2014. All rights reserved. Rec 10. Problem 1 Solution5 Jeffrey Schiano 2014. All rights reserved. Rec 10. Problem 1 Solution6 Jeffrey Schiano 2014. All rights reserved. Rec 10. Problem 2• Consider the periodic signal f(t) where A is a real-valued, positive parameter• Determine and sketch the complex exponential Fourier series spectra of f(t)7tf( )tAAoT Jeffrey Schiano 2014. All rights reserved. Rec 10. Problem 2 Solution8 Jeffrey Schiano 2014. All rights reserved. Rec 10. Problem 2 Solution9 Jeffrey Schiano 2014. All rights reserved. Rec 10. Problem 2 Solution10 Jeffrey Schiano 2014. All rights reserved. Rec 10. Problem 3• Consider the periodic signal f(t)• Sketch f(t)• Determine the fundamental period and frequency• Determine the complex exponential Fourier series coefficients• Sketch the Fourier series spectra111()2nft t n Jeffrey Schiano 2014. All rights reserved. Rec 10. Problem 3 Solution12 Jeffrey Schiano 2014. All rights reserved. Rec 10. Problem 3 Solution13 Jeffrey Schiano 2014. All rights reserved. Rec 10. Problem 4• Consider an asymptotically stable LTI system represented by the ODE• Show that the sinusoidal steady-state response of the system to the inputis 14(D) ( ) ( ) ( )QytPDft()jtft e()y() ( )Q( )jt jtPjteHjej Jeffrey Schiano 2014. All rights reserved. Rec 10. Problem 4 Solution15 Jeffrey Schiano 2014. All rights reserved. Rec 10. Problem 4 Solution16 Jeffrey Schiano 2014. All rights reserved. Rec 10. Problem 5• Consider an asymptotically stable LTI system represented by the ODE• Using the result form Problem 4, show that the sinusoidal steady-state response of the system to the periodic input is 17(D) ( ) ( ) ( )QytPDft()ojn tfnnft Dey( )ojn tynnyfnontDeDHjnD Jeffrey Schiano 2014. All rights reserved. Rec 10. Problem 5 Solution18 Jeffrey Schiano 2014. All rights reserved. Rec 10. Problem 5 Solution19 Jeffrey Schiano 2014. All rights reserved. Rec 10. Review: System Representation• Methods for representing an RC network2011ODE: ( ) f( ); 1impulse resposne: h(t) = ( )1frequency response function: H(j ) =1tdyyt t RCdteutj f(t)Ry(t)C Jeffrey Schiano 2014. All rights reserved. Rec 10. Problem 6• For the RC network in Problem 5, determine the Fourier transform of the impulse response functionand show that it matches the frequency response function• The fact that the Fourier Transform of the impulse response representation of a LTI system is always equivalent to the frequency response function of the network will be proved in lecture 21() ()jtht hte dt Jeffrey Schiano 2014. All rights reserved. Rec 10. Problem 6 Solution22 Jeffrey Schiano 2014. All rights reserved. Rec 10. Problem 6 Solution23 Jeffrey Schiano 2014. All rights reserved. Rec 10. MATLAB Symbolic Variables• x = sym(‘x’) creates a symbolic variable x• t = sym(‘t’); x = symfun(sym('x(t)'), [t]) creates a symbolic function f of a symbolic variable t• syms x f(t) short-cut for constructing symbolic objects• simplify(S) simplifies each element of the symbolic matrix S24 Jeffrey Schiano 2014. All rights reserved. Rec 10. MATLAB Symbolic Variables• syms x real creates a real-valued symbolic variable x• syms x positive creates a real-valued positive variable x• subs(S,NEW) replaces the free symbolic variable in S with NEW, where NEW is a symbolic or numeric variable25 Jeffrey Schiano 2014. All rights reserved. Rec 10. Symbolic Functions• The function solve can be used to solve a single equation or a system of equations• diff(f,‘x') differentiates f with respect to x• int(f,’x’) is the indefinite integral of f with respect to x• int(f,’x’,a, b) is the definite integral of f with respect to the symbolic x variable from a to b26 Jeffrey Schiano 2014. All rights reserved. Rec 10. Problem 7• Using the Symbolic toolbox1. Determine the roots of the quadratic equation2. Determine the derivative of 3. Determine the indefinite integral4. Determine the definite integral2720ax bx c() cos(bt)atft eatedt121tdt Jeffrey Schiano 2014. All rights reserved. Rec 10. Problem 7 Solution28 Jeffrey Schiano 2014. All rights reserved. Rec 10. Heaviside Function• How do we create a symbolic unit-step function?29 Jeffrey Schiano 2014. All rights reserved. Rec 10. Problem 8• F = fourier(f) is the Fourier transform of the sym f with default independent variable x, the default return is a function of w• F = fourier(f, t, w) is the Fourier transform of the symf with independent variable t, the return is a function of w• Verify the result in Problem 6 by using MATLAB to determine the Fourier transform 301h(t) = ( ); 0teut Jeffrey Schiano 2014. All rights reserved. Rec 10. Problem Set 8 Solutions31 Jeffrey Schiano 2014. All rights reserved. Rec 10. EE 350Continuous-Time Linear SystemsRecitation 101 Jeffrey Schiano 2014. All rights reserved. Rec 10. Recitation 10 Topics2• Solved Problems – Complex
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