This homework deals with applications of concepts related to the Bilateral Laplace Transform, or otherwise known as the BLT 1. For the circuit shown, the voltage source x(t) is the input and the current y(t) y(t) is the output. Write the differential equation of the cireuit relating the input L and the output without assigning x(t) numerical values to the eireuit elements 2. Use the differential equation developed in Prob. 1 to obtain the transfer function H(s) of the system (circuit) 3. Assume now for simplicity that the circuit elements have the unrealistic values of 10 1H, and 1F. Draw the pole-zero plot of the transfer function of the system using these component values. 4. Use graphical methods to evaluate and plot the magnitude of the transfer function H(j@) as a function of o. Show five separate pole-zero plots and the associated vectors. One plot will be done with o=0 rps, the next one with o=0.5 rps , the next one with o-1rps, the next one with -5 rps , and the last one with o->rps. Determine the length of the veetors for each frequency value and use those to estimate the value of | H(jø)| at each of the frequencies. Use those values to plot the magnitude of the transfer function and determine what kind of filter the system implements. This problem must be done by hand. You can use MATLAB to confirm the correctness of your results but MATLAB work for this problem will not be accepted and if turned in will receive no credit 5. Use graphical methods to evaluate and plot the phase of the transfer function Hjø) as a function of o. Show five separate pole-zero plots and the associated vectors. One plot will be done with o 0 rps, the next one with o= 0.5 rps , the next one with o -1rps, the next one with -5rps, and the last one with o-» rps. Determine the angles of the vectors for each frequency value and use those to estimate the value of ZH(jø) at each of the frequencies. Plot the points in order to get a phase plot of the transfer function. This problem must be done bv hand. You can use MATLAB to confirm the correctness of vour results but MATLAB work for this problem will not be acceepted and if turned in will receive no credit 6. Assume a system having input x(t) and output y(t) is described by the transfer function Y(s) H(s) with ROC a-a. Write the differential equation of this system X(s) a relating the input and the output. I think you will agree that this looks like a standard first order differential equation you have seen many times. However, given the ROC and the fact that the transfer function and impulse response are transform pairs, the impulse response of this system must be (look at the tables) ht) = -e""u(-t). In other words a unit impulse applied at time t 0 produces a backward (in time) response. Verify that the h(t) is indeed the solution of the differential equation when a unit impulse is applied at r =0. 7. Suppose that the input to the system of Prob. 6 is a regular unit step u(t). Attempt to obtain the output of the system by convolution for the case where a 0. You are likely to run into trouble. Why? Now attempt to use Laplace Transforms to obtain the output. Does this overcome the problem encountered in using convolution? Look at the ROc's of the Laplace Transforms of the two time functions that are being convolved. Does this explain the difficulties encountered in obtaining the output via convolution? Now repeat this process for a < 0. Obtaining the output via convolution should work. What is the output? Looking at the ROCS of the signals being convolved, does this explain why convolution works now while it did not before? 8. Use the BLT to obtain the output of the system of Prob. 6 when the input is xt)eu(-t Consider both cases, a =b and a #b. Do not assign numerical values to these two constants 9. Use MATLAB to plot the result of Prob. 8, in one case for a =b=-3 and in the other case for a -1 and b1 These problems must be worked on independently and tuned in at the beginning of elass on Tuesday Nov. 26, 2019. You are required to place a cover sheet in front of the papers showing your work. The cover sheet to be used is shown on the next page which can be copied directly or can be your own creation using the exact same format shown. Save the cover page for modification and future use as all homework turned in will have to have the cover page with appropriate dates, etc.