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which is zero for even n, and 20/np for odd values of n. The Fourier series can then be written
3
y t
ð Þ ¼
20
p
sint þ
1
3
sin3t þ
1
5
sin5t þ Á Á Á
Figure 2.15 shows the first four partial sums of this function, as they compare with the function
they represent. Note that at the points of discontinuity of the function, the Fourier series takes on the
arithmetic mean of the two function values.
COMMENT As the number of terms in the partial sum increases, the Fourier series approximation of the square wave function becomes even better. For each additional term in the series, the
number of ‘‘humps’’ in the approximate function in each half-cycle corresponds to the number of
terms included in the partial sum.
The Matlab
14 program file FourCoef with the companion software illustrates the behavior of
the partial sums for several waveforms. The LabView
1 program Waveform-Generation.vi creates
signals from trigonometric series.
Example 2.5
As an example of interpreting the frequency content of a given signal, consider the output voltage
from a rectifier. A rectifier functions to ‘‘flip’’ the negative half of an alternating current (AC) into
the positive half plane, resulting in a signal that appears as shown in Figure 2.16. For the AC signal
the voltage is given by
E t
ð Þ ¼ 120 sin 120pt
The period of the signal is 1/60 s, and the frequency is 60 Hz.
KNOWN The rectified signal can be expressed as
E t
ð Þ ¼ 120 sin 120ptj
j
FIND The frequency content of this signal as determined from a Fourier series analysis.
SOLUTION The frequency content of this signal can be determined by expanding the function
in a Fourier series. The coefficients may be determined using the Euler formulas, keeping in mind
3 If we assume that the sum of this series most accurately represents the function y at t ¼ p=2, then
y
p
2
¼ 5 ¼
20
p
1 À
1
3
þ
1
5
À þ Á Á Á
or
p
4
¼ 1 À
1
3
þ
1
5
À
1
7
þ À Á Á Á
This series approximation of p was first obtained by Gottfried Wilhelm Leibniz (1646–1716) in 1673 from geometrical
reasoning.
4 Matlab is a registered trademark of Mathworks, Inc.
2.4 Signal Amplitude And Frequency 61
13:35:19 Page 61
which is zero for even n, and 20/np for odd values of n. The Fourier series can then be written
3
y t
ð Þ ¼
20
p
sint þ
1
3
sin3t þ
1
5
sin5t þ Á Á Á
Figure 2.15 shows the first four partial sums of this function, as they compare with the function
they represent. Note that at the points of discontinuity of the function, the Fourier series takes on the
arithmetic mean of the two function values.
COMMENT As the number of terms in the partial sum increases, the Fourier series approximation of the square wave function becomes even better. For each additional term in the series, the
number of ‘‘humps’’ in the approximate function in each half-cycle corresponds to the number of
terms included in the partial sum.
The Matlab
14 program file FourCoef with the companion software illustrates the behavior of
the partial sums for several waveforms. The LabView
1 program Waveform-Generation.vi creates
signals from trigonometric series.
Example 2.5
As an example of interpreting the frequency content of a given signal, consider the output voltage
from a rectifier. A rectifier functions to ‘‘flip’’ the negative half of an alternating current (AC) into
the positive half plane, resulting in a signal that appears as shown in Figure 2.16. For the AC signal
the voltage is given by
E t
ð Þ ¼ 120 sin 120pt
The period of the signal is 1/60 s, and the frequency is 60 Hz.
KNOWN The rectified signal can be expressed as
E t
ð Þ ¼ 120 sin 120ptj
j
FIND The frequency content of this signal as determined from a Fourier series analysis.
SOLUTION The frequency content of this signal can be determined by expanding the function
in a Fourier series. The coefficients may be determined using the Euler formulas, keeping in mind
3 If we assume that the sum of this series most accurately represents the function y at t ¼ p=2, then
y
p
2
¼ 5 ¼
20
p
1 À
1
3
þ
1
5
À þ Á Á Á
or
p
4
¼ 1 À
1
3
þ
1
5
À
1
7
þ À Á Á Á
This series approximation of p was first obtained by Gottfried Wilhelm Leibniz (1646–1716) in 1673 from geometrical
reasoning.
4 Matlab is a registered trademark of Mathworks, Inc.
2.4 Signal Amplitude And Frequency 61