S
OLUTIONS
TO
ECE2025 P
ROBLEM
S
ET
#4, J
ANUARY
30, 2009
PROBLEM 4.1*.
From the picture:
(a)
Concepts
— Periodic signals, fundamental frequency, harmonic frequencies, GCD
Approach
— A sum of sinuoids is periodic if the sinusoid frequencies are harmonically
related; i.e., if they can all be expressed as an integer multiple of a common frequency
called the fundamental frequency. When this is true, the fundamental frequency can be
found using the greatest-common-divisor (GCD) as shown below.
Solution
— From the picture we see two nonzero frequencies:
3.2
π
rad
/
s, and
8.4
π
rad
/
s.
The fundamental frequency can be computed as:
ω
0
= 0.1
π
GCD
{
32
,
84
}
=
0.1
π
(4)
=
0.4
π
rad
/
s.
Note how
0.1
π
was factored out from both frequencies to ensure that the arguments to
GCD were integers.
(b)
Concepts
— Periodic signals, fundamental frequency, fundamental period
Approach
— The fundamental period
T
0
is related to
ω
0
by
T
0
=
2
π
/
ω
0
Solution
— Using the answer from part (a) yields:
T
0
=
2
π
/
ω
0
= 2
π
/0.4
π
=
5
seconds.
(c)
Concepts
— DC value, constant value, average value, zero frequency
Approach
— From the formula
A
cos(
ω
t
+
φ
)
of a generic sinusoid, we see that a
“sinusoid” with zero frequency (
ω
= 0
) reduces to a
constant
value that is independent of
time. The DC value is precisely specified by the spectrum at zero frequency.
Solution
— From the picture we see that the spectrum shows an amplitude of
5
at zero
frequency; therefore, the DC value is
5
.
(d)
Concepts
— Periodic signals, Fourier series representation, Spectrum of Fourier Series
Approach
— The Fourier series coefficients of a periodic signal may be determined easily
from the signal’s spectrum; specifically, the
k
-th Fourier series coefficient
a
k
is the
complex amplitude at frequency
k
ω
0
.
Solution
— Translating the spectrum (from left to right) into an equation for the signal
waveform yields:
x
(
t
) = 2
e
j
π
/3
e
–
j
ω
0
t
+ 3
e
j
3
π
/4
e
–
j
ω
0
t
+ 5 + 3
e
–
j
3
π
/4
e
j
ω
0
t
+ 2
e
–
j
π
/3
e
j
ω
0
t
.
This has the desired form of a Fourier series, i.e.
x
(
t
) =
a
k
e
jk
ω
0
t
, where
a
–
= 2
e
j
π
/3
,
a
–
= 3
e
j
3
π
/4
,
a
0
= 5
,
a
8
= 3
e
–
j
3
π
/4
,
a
= 2
e
–
j
π
/3
.