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13:35:23 Page 78
c. Musical score, as input to a musician.
d. Flashing neon sign.
e. Telephone conversation.
f. Fax transmission.
2.33 Describe the signal defined by the amplitude spectrum and its phase shift of Figure 2.27 in terms of its
Fourier series. What is the frequency resolution of these plots? What was the sample time increment used?
2.34 For the even-functioned triangle wave signal defined by
y t
ð Þ ¼ 4C=T
ð
Þt þ C ÀT=2 t 0
y t
ð Þ ¼ À4C=T
ð
Þ t þ C 0 t T=2
where C is an amplitude and T is the period:
a. Show that this signal can be represented by the Fourier series
y t
ð Þ ¼
X 1
n¼1
4C 1 À cosnp
ð
Þ
pn
ð Þ
2
cos
2pnt
T
b. Expand the first three nonzero terms of the series. Identify the terms containing the fundamental
frequency and its harmonics.
c. Plot each of the first three terms on a separate graph. Then plot the sum of the three. For this, set
C ¼ 1 V and T ¼ 1 s. Note: The program FourCoef or Waveform Generation is useful in this
problem.
d. Sketch the amplitude spectrum for the first three terms of this series, first separately for each term
and then combined.
2.35 Figure 2.15 illustrates how the inclusion of higher frequency terms in a Fourier series refine the
accuracy of the series representation of the original function. However, measured signals often
display an amplitude spectrum for which amplitudes decrease with frequency. Using Figure 2.15 as a
resource, discuss the effect of high-frequency, low-amplitude noise on a measured signal. What
signal characteristics would be unaffected? Consider both periodic and random noise.
2.36 The program Sound.vi samples the ambient room sounds using your laptop computer microphone
and sound board and returns the amplitude spectrum of the sounds. Experiment with different sounds
(e.g., tapping, whistling, talking, humming) and describe your findings in a brief report.
5
4
3
2
1
0
5
4
3
2
1
0
f (Hz)
f (Hz)
0.4
0.3
0.2
0.1
0.0
Phase (rad)
5
4
3
2
1
0
C(f ) [V]
Figure 2.27 Spectrum for Problem 2.33.
78 Chapter 2 Static and Dynamic Characteristics of Signals
13:35:23 Page 78
c. Musical score, as input to a musician.
d. Flashing neon sign.
e. Telephone conversation.
f. Fax transmission.
2.33 Describe the signal defined by the amplitude spectrum and its phase shift of Figure 2.27 in terms of its
Fourier series. What is the frequency resolution of these plots? What was the sample time increment used?
2.34 For the even-functioned triangle wave signal defined by
y t
ð Þ ¼ 4C=T
ð
Þt þ C ÀT=2 t 0
y t
ð Þ ¼ À4C=T
ð
Þ t þ C 0 t T=2
where C is an amplitude and T is the period:
a. Show that this signal can be represented by the Fourier series
y t
ð Þ ¼
X 1
n¼1
4C 1 À cosnp
ð
Þ
pn
ð Þ
2
cos
2pnt
T
b. Expand the first three nonzero terms of the series. Identify the terms containing the fundamental
frequency and its harmonics.
c. Plot each of the first three terms on a separate graph. Then plot the sum of the three. For this, set
C ¼ 1 V and T ¼ 1 s. Note: The program FourCoef or Waveform Generation is useful in this
problem.
d. Sketch the amplitude spectrum for the first three terms of this series, first separately for each term
and then combined.
2.35 Figure 2.15 illustrates how the inclusion of higher frequency terms in a Fourier series refine the
accuracy of the series representation of the original function. However, measured signals often
display an amplitude spectrum for which amplitudes decrease with frequency. Using Figure 2.15 as a
resource, discuss the effect of high-frequency, low-amplitude noise on a measured signal. What
signal characteristics would be unaffected? Consider both periodic and random noise.
2.36 The program Sound.vi samples the ambient room sounds using your laptop computer microphone
and sound board and returns the amplitude spectrum of the sounds. Experiment with different sounds
(e.g., tapping, whistling, talking, humming) and describe your findings in a brief report.
5
4
3
2
1
0
5
4
3
2
1
0
f (Hz)
f (Hz)
0.4
0.3
0.2
0.1
0.0
Phase (rad)
5
4
3
2
1
0
C(f ) [V]
Figure 2.27 Spectrum for Problem 2.33.
78 Chapter 2 Static and Dynamic Characteristics of Signals