Determine the differential equation describing a system that would give this transfer function, i.e. perform the

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Consider the signal h(t) = 58(t – 6) - 2-5(t-6) uſt - 6) where u(t) is the unit step function. Perform the tasks below, always specifying all the signals on - st soo using u(t) as necessary, possibly shifted in time, to cut off the tails of exponential functions, as in Example 3.1.2, p. 13, in the errata list.

1. Starting from the graph of signal 8(t) and u(t), plot h(t) showing graphically the details of arriving at h(t)

2. Analytically derive the Laplace Transform of h(t) without the use of Laplace Transform tables. Show all the details - you will not get points for simply writing the answer directly.

3. Assume now that h(t) is the impulse response of a system. Find the transfer function H(s) of this system and put it in the form a(s)/b(s), i.e. as the ratio of two polynomials, possibly with additional factor accounting for the time delay, by adding separate terms, if necessary.

4. Determine the differential equation describing a system that would give this transfer function, i.e. perform the inverse Laplace Transform of H(s) = Y(s)/X(s) where Y(s) and X(s) are Laplace Transforms of the output and the input of the system, respectively.

5. Find the poles and the zeros of this system and draw them on the s-plane.

6. Derive analytically, using the noncausal convolution integral, the response of this system to a step input 4u(t), where u(t) is the unit step. Sketch the resulting function.

7. Sketch graphically the detailed sequence of convolution calculations of the response of this system to a step input 4u(t), where u(t) is the unit step. Show graphically, using folding and shifting, how the integrand product defines the integration limits. Sketch the resulting function.

8. Use the Laplace Transform multiplication theorem to find the response of this system to a step input 4u(t) in the Laplace domain. Then take the inverse of this response and show that you obtain the same result as in 6. Show all the details - you will not get points for simply writing the answer directly.

9. Analytically derive and sketch the magnitude response of this system both in linear scale or logarithmic scale (Bode) plots.

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