Mathematics of Signal Representations

Assignment 6, due Wednesday, April 8

• Do Exercise §2.6: 5, and compute the FT of φ * φ in this problem with the help of a suitable theorem. 
  
• (Butterworth filter.) You may use tables of integrals or computer algebra systems to obtain indefinite integrals used in solving the problem below. Otherwise, please do the calculations by hand.
  
  Let α > 0. Set h(t) = α e^{-α t} for t ≥ 0 and set h(t) = 0 for t < 0. We introduce an abbreviation L[f]=h*f and call L[f] the signal f filtered by convolving it with h.
  
  Show that, for any signal f(t) that is 0 when t < 0, one has for t ≥ 0
  L[f](t) = h*f(t) = α e^-αt ∫₀^t e^αu f(u)du
  
  
  Take f(t) = e^-t/2 (sin(t) + 10 ^-1sin(50t)) for t ≥ 0 and f(t) = 0 for t < 0, so the formula above applies. Assuming α ≠ 1/2, find L[f] using the formula.
  
  Examine the analytical expression for L[f]. What is a good choice for α in order that the corrsponding L will filter out the high frequency term e^-t/2 sin(50t)/10 and leave the term e^-t/2 sin(t) roughly unmodified?
  
  Using Matlab, plot f(t) and the analytical expression L[f] for t = 0 to 20 and for α = 1, 4, 8, along with your choice for α from the previous part. Attach the plot and your matlab script to the homework.