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
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
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.