Problems: You may use tables of integrals or computer algebra
systems to obtain indefinite integrals used in solving the
problems below. Otherwise, I want you to do the calculations
by hand.
§ 2.6, problem 8.
(Butterworth filter.) Let α > 0. Set h(t)
= α e-α t for t ≥ 0 and set h(t) = 0
for t < 0.
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
∫0t eαu 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 it.
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?
Plot f(t) and L[f] for t = 0 to 20 and for α = 1, 4, 8,
along with your choice for α from the previous part. To do
these, either use Matlab's plot or try
fplot. (See the help entry for
fplot.)