ColorGrayscale Jeffrey Schiano 2014. All rights reserved. Rec 13. EE 350Continuous-Time Linear SystemsRecitation 131 Jeffrey Schiano 2014. All rights reserved. Rec 13. Recitation 13 Topics2• Solved Problems – Laplace Transforms and properties• MATLAB Exercises– Laplace Transforms Jeffrey Schiano 2014. All rights reserved. Rec 13. Problem 1• Determine the unilateral Laplace transform of the following signals and specify the region of convergence3( ) () ()(b) ( ) cos(b ) ( )ataft teutft tut Jeffrey Schiano 2014. All rights reserved. Rec 13. Problem 1 Solution4 Jeffrey Schiano 2014. All rights reserved. Rec 13. Problem 1 Solution5 Jeffrey Schiano 2014. All rights reserved. Rec 13. Problem 1 Solution6 Jeffrey Schiano 2014. All rights reserved. Rec 13. Problem 1 Solution7 Jeffrey Schiano 2014. All rights reserved. Rec 13. Problem 2• Derive the frequency shift property8() ()() ( )ostoft Fseft Fss Jeffrey Schiano 2014. All rights reserved. Rec 13. Problem 2 Solution9 Jeffrey Schiano 2014. All rights reserved. Rec 13. Problem 3• Using the transform pair developed in Problem 1(b) and the frequency shift property derived in Problem 2, determine the Laplace transform of the 10() cos(b) ()atft e tut Jeffrey Schiano 2014. All rights reserved. Rec 13. Problem 3 Solution11 Jeffrey Schiano 2014. All rights reserved. Rec 13. Problem 3 Solution12 Jeffrey Schiano 2014. All rights reserved. Rec 13. Problem 4• Derive the time shift property for to> 0 (right shift in time)1300() ()()u(t) ()eostft Fsft t t Fs Jeffrey Schiano 2014. All rights reserved. Rec 13. Problem 4 Solution14 Jeffrey Schiano 2014. All rights reserved. Rec 13. Problem 5• Given that f(t) has the Laplace transform F(s), show that 152220() (0)( ) (0 ) f(0 )()()tdfsF s fdtdfsFs sfdtFsfds Jeffrey Schiano 2014. All rights reserved. Rec 13. Problem 5 Solution16 Jeffrey Schiano 2014. All rights reserved. Rec 13. Problem 5 Solution17 Jeffrey Schiano 2014. All rights reserved. Rec 13. Problem 5 Solution18 Jeffrey Schiano 2014. All rights reserved. Rec 13. Problem 6• Determine the causal function f(t) for each of the following Laplace transforms19221(a) ( )5648(b) ( )(1)(3)sFssssFsss Jeffrey Schiano 2014. All rights reserved. Rec 13. Problem 6 Solution20 Jeffrey Schiano 2014. All rights reserved. Rec 13. Problem 6 Solution21 Jeffrey Schiano 2014. All rights reserved. Rec 13. Problem 6 Solution22 Jeffrey Schiano 2014. All rights reserved. Rec 13. Problem 6 Solution23 Jeffrey Schiano 2014. All rights reserved. Rec 13. MATLAB Laplace Transform Tools• Polynomial Construction– The MATLAB command poly generates polynomial from its roots• Partial Fraction Expansion– The MATLAB command residue determines the partial fraction expansion for any rational function in s• Symbolic Computation– MATLAB determines the unilateral Laplace and inverse Laplace transform of symbolic functions24 Jeffrey Schiano 2014. All rights reserved. Rec 13. MATLAB poly Function25 Jeffrey Schiano 2014. All rights reserved. Rec 13. Problem 7• Use MATLAB to place the polynomialin the formby providing a vector of the coefficients 262(1)(3)ss1110nnnsas asa 1101, , , ,naaa Jeffrey Schiano 2014. All rights reserved. Rec 13. MATLAB residue Function27 Jeffrey Schiano 2014. All rights reserved. Rec 13. Problem 8• Use MATLAB to generate the partial fraction expansion for each of the following Laplace transforms28221(a) ( )5648(b) ( )(1)(3)sFssssFsss Jeffrey Schiano 2014. All rights reserved. Rec 13. MATLAB Symbolic Functions29 Jeffrey Schiano 2014. All rights reserved. Rec 13. Problem 9• Use MATLAB to generate the Laplace transform for each of the following signals• Use the MATLAB function ilaplace to recover f(t) from F(s) for each of the signals above30( ) () ()(b) ( ) cos(b ) ( )atataft teutft e tut Jeffrey Schiano 2014. All rights reserved. Rec 13. EE 350Continuous-Time Linear SystemsRecitation 131 Jeffrey Schiano 2014. All rights reserved. Rec 13. Recitation 13 Topics2• Solved Problems – Laplace Transforms and properties• MATLAB Exercises– Laplace Transforms Jeffrey Schiano 2014. All rights reserved. Rec 13. Problem 1• Determine the unilateral Laplace transform of the following signals and specify the region of convergence3 Jeffrey Schiano 2014. All rights reserved. Rec 13. Problem 1 Solution4 Jeffrey Schiano 2014. All rights reserved. Rec 13. Problem 1 Solution5 Jeffrey Schiano 2014. All rights reserved. Rec 13. Problem 1 Solution6 Jeffrey Schiano 2014. All rights reserved. Rec 13. Problem 1 Solution7 Jeffrey Schiano 2014. All rights reserved. Rec 13. Problem 2• Derive the frequency shift property8 Jeffrey Schiano 2014. All rights reserved. Rec 13. Problem 2 Solution9 Jeffrey Schiano 2014. All rights reserved. Rec 13. Problem 3• Using the transform pair developed in Problem 1(b) and the frequency shift property derived in Problem 2, determine the Laplace transform of the 10 Jeffrey Schiano 2014. All rights reserved. Rec 13. Problem 3 Solution11 Jeffrey Schiano 2014. All rights reserved. Rec 13. Problem 3 Solution12 Jeffrey Schiano 2014. All rights reserved. Rec 13. Problem 4• Derive the time shift property for to> 0 (right shift in time)13 Jeffrey Schiano 2014. All rights reserved. Rec 13. Problem 4 Solution14 Jeffrey Schiano 2014. All rights reserved. Rec 13. Problem 5• Given that f(t) has the Laplace transform F(s), show that 15 Jeffrey Schiano 2014. All rights reserved. Rec 13. Problem 5 Solution16 Jeffrey Schiano 2014. All rights reserved. Rec 13. Problem
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