Concept explainers
Given that Fourier coefficients a0, an, and bn of the waveform in Fig. 17.47. Plot the amplitude and phase spectra.
Figure 17.47
Find the Fourier coefficients
Answer to Problem 3P
The Fourier coefficients
Explanation of Solution
Given data:
Refer to Figure 17.47 in the textbook.
Formula used:
Write the expression to calculate the fundamental angular frequency.
Here,
Write the expression to calculate the dc component of the function
Write the expression to calculate Fourier coefficients
Here,
Calculation:
The given waveform is drawn as Figure 1.
Refer to the Figure 1. The Fourier series function of the waveform is defined as,
The time period of the function in Figure 1 is,
Substitute
Substitute
Simplify the above equation to find
Substitute
Simplify the above equation to find
Simplify the above equation to find
From above equation, it is clear that
For odd
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Simplify the above equation to find
Simplify the above equation to find
Substitute
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Write the expression to calculate the amplitude spectra of the signal.
Substitute
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Substitute
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Substitute
Substitute
Substitute
Substitute
Substitute
From above calculations, the amplitude spectrum of the waveform in Figure 1 is drawn as Figure 2.
Write the expression to calculate the phase spectra of the signal.
Substitute
Substitute
Substitute
Substitute
Substitute
Substitute
Substitute
Substitute
Substitute
Substitute
Substitute
Substitute
Substitute
Substitute
Substitute
Substitute
From above calculations, the phase spectrum of the waveform in Figure 1 is drawn as Figure 3.
Conclusion:
Thus, the Fourier coefficients
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Chapter 17 Solutions
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