Mathematics of Signal Representations
Math 4355  Spring 2013  Homework
Assignment 7, due Thursday, March 28
 Review Sections 2.3 and 2.4.
 Do Exercise 5 on p. 128, and compute the FT of
φ * φ in this problem with the help of a suitable
theorem.

Do Exercise 13 on p. 130.
 (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^{ 1}sin(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.