E1C02 09/14/2010
13:35:18 Page 53
The following definitions relate to Fourier analysis:
1. A function y(t) is a periodic function if there is some positive number T such that
y t þ T
ð
Þ¼y t
ð Þ
The period of y(t) is T. If both y 1 (t) and y 2 (t) have period T, then
ay 1 t
ð Þ þ by 2 t
ð Þ
also has a period of T (a and b are constants).
2. A trigonometric series is given by
A 0 þ A 1 cos t þ B 1 sin t þ A 2 cos 2t þ B 2 sin 2t þ . . .
where A n and B n are the coefficients of the series.
Example 2.2
As a physical (instead of mathematical) example of frequency content of a signal, consider stringed
musical instruments, such as guitars and violins. When a string is caused to vibrate by plucking or
bowing, the sound is a result of the motion of the string and the resonance of the instrument. (The
concept of resonance is explored in Chapter 3.) The musical pitch for such an instrument is the
lowest frequency of the string vibrations. Our ability to recognize differences in musical instruments
is primarily a result of the higher frequencies present in the sound, which are usually integer
multiples of the fundamental frequency. These higher frequencies are called harmonics.
The motion of a vibrating string is best thought of as composed of several basic motions that
together create a musical tone. Figure 2.11 illustrates the vibration modes associated with a string
plucked at its center. The string vibrates with a fundamental frequency and odd-numbered harmonics,
each having a specific phase relationship with the fundamental. The relative strength of each harmonic
is graphically illustrated through its amplitude in Figure 2.11. Figure 2.12 shows the motion, which is
caused by plucking a string one-fifth of the distance from a fixed end. The resulting frequencies are
illustrated in Figure 2.13. Notice that the fifth harmonic is missing from the resulting sound.
Musical sound from a vibrating string is analogous to a measurement system input or output,
which contains many frequency components. Fourier analysis and frequency spectra provide
insightful and practical means of reducing such complex signals into a combination of simple
waveforms. Next, we explore the frequency and amplitude analysis of complex signals.
Available on the companion software site, Program Sound.vi uses your computer’s microphone and
sound board to sample ambient sounds and to decompose them into harmonics (try humming a tune).
Fourier Series and Coefficients
A periodic function y(t) with a period T ¼ 2p is to be represented by a trigonometric series, such that
for any t,
y t
ð Þ ¼ A 0 þ
X 1
n¼1
A n cos nt þ B n sin nt
ð
Þ
ð 2:12Þ
With y(t) known, the coefficients A n and B n are to be determined; this requires a well-established
mathematical procedure, but not one that we need to reinvent. For A 0 to be determined,
2.4 Signal Amplitude And Frequency 53
13:35:18 Page 53
The following definitions relate to Fourier analysis:
1. A function y(t) is a periodic function if there is some positive number T such that
y t þ T
ð
Þ¼y t
ð Þ
The period of y(t) is T. If both y 1 (t) and y 2 (t) have period T, then
ay 1 t
ð Þ þ by 2 t
ð Þ
also has a period of T (a and b are constants).
2. A trigonometric series is given by
A 0 þ A 1 cos t þ B 1 sin t þ A 2 cos 2t þ B 2 sin 2t þ . . .
where A n and B n are the coefficients of the series.
Example 2.2
As a physical (instead of mathematical) example of frequency content of a signal, consider stringed
musical instruments, such as guitars and violins. When a string is caused to vibrate by plucking or
bowing, the sound is a result of the motion of the string and the resonance of the instrument. (The
concept of resonance is explored in Chapter 3.) The musical pitch for such an instrument is the
lowest frequency of the string vibrations. Our ability to recognize differences in musical instruments
is primarily a result of the higher frequencies present in the sound, which are usually integer
multiples of the fundamental frequency. These higher frequencies are called harmonics.
The motion of a vibrating string is best thought of as composed of several basic motions that
together create a musical tone. Figure 2.11 illustrates the vibration modes associated with a string
plucked at its center. The string vibrates with a fundamental frequency and odd-numbered harmonics,
each having a specific phase relationship with the fundamental. The relative strength of each harmonic
is graphically illustrated through its amplitude in Figure 2.11. Figure 2.12 shows the motion, which is
caused by plucking a string one-fifth of the distance from a fixed end. The resulting frequencies are
illustrated in Figure 2.13. Notice that the fifth harmonic is missing from the resulting sound.
Musical sound from a vibrating string is analogous to a measurement system input or output,
which contains many frequency components. Fourier analysis and frequency spectra provide
insightful and practical means of reducing such complex signals into a combination of simple
waveforms. Next, we explore the frequency and amplitude analysis of complex signals.
Available on the companion software site, Program Sound.vi uses your computer’s microphone and
sound board to sample ambient sounds and to decompose them into harmonics (try humming a tune).
Fourier Series and Coefficients
A periodic function y(t) with a period T ¼ 2p is to be represented by a trigonometric series, such that
for any t,
y t
ð Þ ¼ A 0 þ
X 1
n¼1
A n cos nt þ B n sin nt
ð
Þ
ð 2:12Þ
With y(t) known, the coefficients A n and B n are to be determined; this requires a well-established
mathematical procedure, but not one that we need to reinvent. For A 0 to be determined,
2.4 Signal Amplitude And Frequency 53