Problem Set
Problem
1:
Complex-valued Signals
Complex numbers and phasors play a very important role
in electrical engineering. Solving systems for complex
exponentials is much easier than for sinusoids, and
linear systems analysis is particularly easy.
-
Find the phasor representation for each, and
re-express each as the real and imaginary parts of a
complex exponential. What is the frequency (in Hz)
of each? In general, are your answers unique? If
so, prove it; if not, find an alternative answer for
the complex exponential representation.
-
3sin24t
3
24
t
-
2cos2π60t+π4
2
2
60
t
4
-
2cost+π6+4sint-π3
2
t
6
4
t
3
-
Show that for linear systems having real-valued
outputs for real inputs, that when the input is the
real part of a complex exponential, the output is
the real part of the system's output to the complex
exponential (see Figure 1).
SℜAⅇⅈ2πft=ℜSAⅇⅈ2πft
S
A
2
f
t
S
A
2
f
t
Correct!
Incorrect.
Problem
2:
For each of the indicated voltages, write it as the real
part of a complex exponential
(
vt=ℜVⅇst
v
t
V
s
t
).
Explicitly indicate the value of the complex amplitude
VV
and the complex frequency
ss.
Represent each complex amplitude as a vector in the
VV-plane, and indicate the
location of the frequencies in the complex
ss-plane.
-
vt=cos5t
v
t
5
t
-
vt=sin8t+π4
v
t
8
t
4
-
vt=ⅇ-t
v
t
t
-
vt=ⅇ-3tsin4t+3π4
v
t
3t
4
t
3
4
-
vt=5ⅇ2tsin8t+2π
v
t
5
2
t
8
t
2
-
vt=-2
v
t
-2
-
vt=4sin2t+3cos2t
v
t
4
2
t
3
2
t
-
vt=2cos100πt+π6-3sin100πt+π2
v
t
2
100
t
6
3
100
t
2
Correct!
Incorrect.
Problem
3:
Express each of the
following signals
as a linear
combination of delayed and weighted step functions and
ramps (the integral of a step).
Correct!
Incorrect.
Problem
4:
Linear, Time-Invariant Systems
When the input to a linear, time-invariant system is the
signal
xt
xt
,
the output is the signal
yt
y
t
(
Figure 3).
-
Find and sketch this system's output when the input
is the
depicted signal.
-
Find and sketch this system's output when the input
is a unit step.
Correct!
Incorrect.
Problem
5:
Linear Systems
The
depicted input
xt
x
t
to a linear, time-invariant system yields the output
yt
y
t
.
-
What is the system's output to a unit step input
ut
u
t
?
-
What will the output be when the input is the
depicted square wave?
Correct!
Incorrect.
Problem
6:
Communication Channel
A particularly interesting communication channel can be
modeled as a linear, time-invariant system. When the
transmitted signal
xt
xt
is a pulse, the received signal
rt
rt
is
as shown.
-
What will be the received signal when the transmitter
sends the pulse sequence
x
1
t
x
1
t
?
-
What will be the received signal when the transmitter
sends the pulse signal
x
2
t
x
2
t
that has half the duration as the original?
Correct!
Incorrect.
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