Find the total transfer function of the block diagram -

Question

Question 1. Convolve the following two functions. Show the resulted function graphically.

Question 2. Evaluate the following. Write the answer in the simplest form.

a) (e^-5t - e^-2t)δ(t-1)

b) The convolution of f(t) = e^-tu(t-1) and g(t) = rect(t).

Question 3. Use phasor to solve the following for y(t):

d³y(t)/dt + 3d²y(t)dt(t) + 4dy(t)/dt + 10y(t) = 3sin(2t -60°)

Question 4. Write a mathematical description of f(t):

Question 5. Find the Laplace transforms of the following functions.

a) f₁(t) = ₀∫¹ e^-2(t-r) sin(3r)dr

b). f₂(t) =  t²u(t-2) x e^-5tu(t)

Question 6. Find the inverse Laplace transforms of the following functions.

a) F₁(s) = s + 2/(s²(s+1)

b). F₂(s) = s + 1.5/(s² + 3s + 4)

Question 7. The transfer function of a causal system is: H(s) = (s³ + 3s + 1)/s²(s² + 6s + 2s) draw the pole-zero plot and decide the BIRO stability of the system.

Question 8. Solve the following equation for y(t).

d²y(t)/dt² + 4dy(t)/dt² + 3y(t) = u(t)

y(0-) =1, y'(0^-)= -4

Question 9. Find the total transfer function of the following block diagram.