ColorGrayscale Jeffrey Schiano 2014. All rights reserved. Rec 7. EE 350Continuous-Time Linear SystemsRecitation 71 Jeffrey Schiano 2014. All rights reserved. Rec 7. Recitation 7 Topics• Solved Problems – Convolution– EE 210 Review: Sinusoidal Steady-State Analysis– Relationship between the zero-state response and the sinusoidal steady-state response• MATLAB Exercises– Approximation of the convolution integral2 Jeffrey Schiano 2014. All rights reserved. Rec 7. Problem 1• A LTI system with impulse response h(t) is driven by the input f(t)1. Determine the zero-state response y(t) using graphical convolution2. Verify that the width property holds for part 13() () ( 2)( ) 2u(t) 2u(t 2)ht ut utft Jeffrey Schiano 2014. All rights reserved. Rec 7. Problem 1 Solution4 Jeffrey Schiano 2014. All rights reserved. Rec 7. Problem 1 Solution5 Jeffrey Schiano 2014. All rights reserved. Rec 7. Problem 1 Solution6 Jeffrey Schiano 2014. All rights reserved. Rec 7. Problem 1 Solution7 Jeffrey Schiano 2014. All rights reserved. Rec 7. Problem 1 Solution8 Jeffrey Schiano 2014. All rights reserved. Rec 7. Numerical Approximation of the Convolution Integral• The zero-state response y(t) of a casual LTI system with impulse response h(t) to the causal input f(t) is• As discussed in lecture, y(t) represents the area under the integrand, and can be approximated by the sum of area under rectangles of width T and height f(kT) h(t-kT) using the Riemann sum90() ( ) ( )tytfht d0(t) (kT) (t )nkyfhkTT Jeffrey Schiano 2014. All rights reserved. Rec 7. MATLAB conv Function• From the Riemann sum, the zero-state response at t = nT is• The built-in MATLAB function conv computes the summation• The syntax of the function conv iswhere s1 is a vector of length N1, s2 is a vector of length N2, and the length of the returned vector is N1+N2-1100(nT) (kT) (nT )nkyTfhkTc = conv(s1, s2) Jeffrey Schiano 2014. All rights reserved. Rec 7. Operation of the conv Function• For example, given the vectorsthe function y = conv(f,h) returns the vectorwhere11[ (0), ( ), (2 )][ (0),h(T),h(2T)]ff fTfThh[ (0), ( ) y(2T), y(3 ), (4 )]yy yT TyT0010(0) ( ) (0 - ) (0) (0),() ( )( - ) (0)() ()(0),kkyfkThkTfhyT f kThT kT f hT f Th Jeffrey Schiano 2014. All rights reserved. Rec 7. Problem 2• As in Problem 1, consider a LTI system with impulse response h(t) is driven by the input f(t)• Write an m-file that1. Uses the conv function to approximate y(t) from t = 0 to t = 4 using T = 0.001 2. Plots the approximate zero-state response y(t)• Does the numerical result approximate the result obtained in Problem 1?12() () ( 2)() 2u(t) 2u(t 2)ht ut utft Jeffrey Schiano 2014. All rights reserved. Rec 7. Problem 2 m-file13 Jeffrey Schiano 2014. All rights reserved. Rec 7. Problem 2 Numerical Results140 0.5 1 1.5 2 2.5 3 3.5 4012345EE 350 Recitation 7 Problem 2y(t)t Jeffrey Schiano 2014. All rights reserved. Rec 7. Problem 3• A LTI system with impulse response h(t) is driven by the input f(t)• Determine the zero-state response y(t) using graphical convolution15|| 2()h( ) u(t 2) u(t)tft et Jeffrey Schiano 2014. All rights reserved. Rec 7. Problem 3 Solution16 Jeffrey Schiano 2014. All rights reserved. Rec 7. Problem 3 Solution17 Jeffrey Schiano 2014. All rights reserved. Rec 7. Problem 3 Solution18 Jeffrey Schiano 2014. All rights reserved. Rec 7. Problem 3 Solution19 Jeffrey Schiano 2014. All rights reserved. Rec 7. • Consider a LTI system with impulse response• Determine if the system is BIBO stable205() 2 2 ()tht e utProblem 4 Jeffrey Schiano 2014. All rights reserved. Rec 7. Problem 4 Solution21 Jeffrey Schiano 2014. All rights reserved. Rec 7. Problem 5• It is known that a certain physical system with input f(t) and output y(t) is represented by an ODE of the form• In order to determine the unknown parameters α and β, the an engineer applies a unit-step input and observes the zero-state response. Using this response, the engineer determines that the impulse response of the system is• Determine the value of the parameters α and β22() ()yyt ft4() 3 ()tht e ut Jeffrey Schiano 2014. All rights reserved. Rec 7. Problem 5 Solution23 Jeffrey Schiano 2014. All rights reserved. Rec 7. Problem 5 Solution24 Jeffrey Schiano 2014. All rights reserved. Rec 7. Problem 5 Solution25 Jeffrey Schiano 2014. All rights reserved. Rec 7. Problem 6• The sinusoidal signalmay be represented by the phasor• The phasor is a complex-valued constant • Using Euler’s identity, show that26() cosft A tjFAe() Rejtft Fe Jeffrey Schiano 2014. All rights reserved. Rec 7. Problem 6 Solution27 Jeffrey Schiano 2014. All rights reserved. Rec 7. Problem 7• Determine the phasor representation of the following signals28(a) ( ) 3sin 10 30(b) ( ) 2cos 90ft tft t Jeffrey Schiano 2014. All rights reserved. Rec 7. Problem 7 Solution29 Jeffrey Schiano 2014. All rights reserved. Rec 7. Problem 8• Consider the RC network with input f(t) and output y(t)1. Determine the zero-state response for the input2. Determine the zero-state response for times much larger than the network RC time constant3. Derive the frequency response function, and use it to determine the sinusoidal steady-state response4. Compare the results from parts 2 and 330f(t)Ry(t)C() cosoft A t Jeffrey Schiano 2014. All rights reserved. Rec 7. Problem 8 Solution31 Jeffrey Schiano 2014. All rights reserved. Rec 7. Problem 8 Solution32 Jeffrey Schiano 2014. All rights reserved. Rec 7. Problem 8 Solution33 Jeffrey Schiano 2014. All rights reserved. Rec 7. Problem 8 Solution34 Jeffrey Schiano 2014. All rights reserved. Rec 7. Problem 8
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