# 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-αt0t 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 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.