SIGNAL SYSTEM

PROBLEM 2.4*: Compute the following convolutions.2 Since we have unit step functions available to express functions that are 0 for t < 0, you should be able to easily express your answers for parts (a) and (b) a single line, without a giant “conditional statement” like in Problem 2.1. Part (c) has three regions; you should still be able to express your answer in one line by employing a difference of unit step functions.

(a) exp(−2t)u(t) ∗ exp(−4t)u(t)

(b) exp(−2t)u(t) ∗ exp(−2t)u(t)

(c) [u(t) −u(t− 2π)] ∗ sin(t)u(t) (Hint: Remember that a sinusoid integrated over a full, complete period results in 0. I recommend “flipping and shifting” the [u(t) − u(t − 2π)] function and leaving sin(t)u(t) fixed to help see what is going on.)

PROBLEM 2.5: For each pair of signals on the left, select the signal on the right that is the convolution of the two signals. Not all of the signals on the right will be used, and some may be used more than once.

2In traditional mathematical notation, multiplication takes precedence over convolution, but convolution takes precedence over addition.