ECE 303: Signals & Systems
Homework #1
Due: Tuesday, September 7 at 11:55 PM
Submit your handwritten work and plots as a single PDF to the Homework 1 assignment in Isidore. Include
MATLAB code as *.m files in a ZIP file.
1. For the following signals plotted in (a), (b), and (c) below:
x2(t)
x1(t)
B
x3(t)
A
B
…
2
t
(a)
…
2
t
1
4
t
(c)
(b)
(a) Derive an equation that fully describes each signal.
(b) Write a MATLAB script that will plot each signal for t = [−6, 6].
2. Determine the following for the signal x(t) = e−2t · u(t):
(a) The even and odd components of x(t).
(b) Show that the energy of x(t) is equal to the sum of the energies of x(t)’s even and odd components.
(c) Generate a plot similar to that of Figure 1.15 in the notes in MATLAB for this x(t).
3. For the systems below, x(t) is the input while y(t) is the output. Determine whether each system is linear
or nonlinear. Fully justify your answer for each system.
(a)
d
dt y(t)
d
dt y(t)
+ 2 · y(t) = x2 (t)
(b)
+ 3t · y(t) = t2 · x(t)
(c) 3 · y(t) + 2 = x(t)
d
(d) dt
y(t) + y 2 (t) = x(t)
4. For the systems below, x(t) is the input while y(t) is the output. Determine whether each system is
time-invariant or time-varying. Fully justify your answer for each system.
(a)
(b)
(c)
(d)
y(t) = x(t − 2)
y(t) = x(−t)
y(t) = x(at)
y(t) = t · x(t − 2)
5. Suppose a linear, time-invariant system is described by the following equation:
d
1
d2
y(t) + 3 · y(t) + 2 · y(t) = · x(t)
d2 t
dt
3
(a) Find the zero-input response of the system if y(0) = 1 and
(b) Is the system stable? Why?
(c) Plot the zero-input response in MATLAB for t = [−1, 10].
d
dt
· y(0) = 0.