EE 350 Problem Set 9 Cover Sheet Spring 2023 Last Name Print First Name Print ID number Last 4 digits Section Submission deadlines Please submit your solutions as a PDF le to the EE 350 CANVAS page by 11 59 pm on Friday April 21 Problem Weight Score 43 44 45 46 47 Total 20 20 20 20 20 100 The solution submitted for grading represents my own analysis of the problem and not that of another student Signature Neatly print the name s of the students you collaborated with on this assignment Reading assignment Lathi Chapter 6 Sections 6 1 through 6 5 and B 5 pp 24 33 Problem 43 20 points This problem shows how to determine the Fourier transform of the unit step function An attempt to determine F by direct integration leads to an indeterminate result F Z f t e tdt Z 0 e tdt 1 0 e t cid 12 cid 12 because f t u t lim t e t yields an indeterminate answer An alternate method is to start with the Fourier transform pair where a is a positive real valued constant Because it follows that g t e atu t G 1 a f t lim a 0 g t F lim a 0 G where F is the Fourier transform of the unit step function 1 4 points Express G in the form G R X where R and X represent the real and imaginary parts of G respectively 2 2 points Show that 3 8 points Show that lim a 0 X 1 a 2 points R R is an even function of b 4 points The area under R is for any value of a 0 c 2 points In one or two sentences use the results from parts a and b to conclude that 4 2 points Use the results from parts 1 to 3 to obtain an expression for F the Fourier transform of the 5 4 points Use the result from part 4 along with an appropriate Fourier transform property to determine unit step function the Fourier transform of lim a 0 R h t cos ot u t 1 15 points By direct integration nd the Laplace transform F s and the region of convergence of F s for the following signals where a and b are positive real numbers Problem 44 20 points a 2 points t b 2 points u t c 3 points e atu t d 4 points cos bt u t e 4 points sin bt u t Explain while Problem 45 20 points 2 5 points Compare the Fourier and Laplace transforms of the signals f t t g t u t F F s s G 6 G s s 1 Let F s L f t denote the unilateral Laplace transform of f t Prove the following properties of the Laplace transform where to 0 is a real constant and so is a complex constant a 2 points Right shift in time b 3 points Multiplication by t c 3 points Frequency shift L f t to u t to F s e sto to 0 L tf t F s d ds Lnesotf t o F s so 2 Using the elementary transform pairs derived in Problem 44 and the properties derived in part 1 nd the Laplace transform of the following signals where to a and b are positive real parameters a 2 points u t to b 2 points tu t c 2 points te atu t d 3 points e at cos bt u t e 3 points e at sin bt u t Note that this approach particular in the case of the signals considered in parts c and d is much easier than nding the Laplace transform by direct integration Problem 46 20 points The inverse Laplace transform of F s can be calculated by expanding F s into partial fractions whose inverse transform is known Carefully read sections 6 1 3 and B 5 in the text Using the basic transform pairs in Table 6 1 on page 372 of the text and the techniques of partial fraction expansion nd the inverse Laplace transforms of 1 5 points F s s 1 2 s2 s 6 2 5 points F s 2s3 2s2 17s 15 s s 3 s 5 3 5 points F s 2s3 15s2 40s 24 s s 3 s2 4s 8 and 4 5 points F s 4s2 7s 1 s s 1 2 1 6 points If the unilateral Laplace transform of f t is F s show that Problem 47 20 points and L cid 26 df dt cid 27 sF s f 0 L cid 26 d2f dt2 cid 27 s2F s sf 0 f 0 d2y dt2 12 dy dt 20y t 40f t Y s Yzs s Yzi s 2 6 points Consider a LTI system with input f t and output y t that has the ODE representation Using the properties in part 1 determine Y s and express your answer in the form where the zero state response Yzs s depends on f t and not the initial conditions y 0 and y 0 while the zero input response Yzi s depends on the initial conditions y 0 and y 0 but not the input f t A subtle but yet important advantage of the Laplace transform method is that the Laplace transform of the response depends on the initial conditions at time 0 and not 0 Recall that it was rst necessary to determine the initial conditions at time 0 using the initial conditions at time 0 before solving an ODE using the classical method homogeneous plus particular solution 3 8 points In part 2 suppose that f t cid 0 1 2e 2t cid 1 u t y 0 2 and y 0 0 Determine the zero state response yzs t the zero input response yzi t and the total response y t by computing the inverse Laplace transform of the result in part 2 using the method of partial fractions Compare your result against that obtained in Problem Set 3 Problem 12
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