Concept explainers
Construct amplitude and phase line spectra for Prob. 19.4.
To graph: The amplitude and phase line spectra for the sawtooth wave as shown in the following figure,
Explanation of Solution
Given Information: The sawtooth wave given in the following figure,
Formula used:
Consider
then the Fourier series expansion of the function,
And the coefficients are defined by,
Alternatively, the Fourier series can also be written as,
Here, the amplitude
Plot
Graph:
Consider the sawtooth wave given in the following figure,
Therefore, the sawtooth wave is a periodic function
Therefore, the sawtooth wave,
Therefore, the Fourier series expansion of this function is,
In the above expression, the coefficients are defined by,
Now, find
Consider,
Hence,
Further,
Therefore,
Now, find
Consider,
Hence,
Further,
Thus,
Hence, the coefficients of the Fourier series expansions are,
That is,
Consider,
Thus, the amplitude of the
Furthermore, consider,
As
As
Therefore,
Thus, the phases corresponding to
Use the following MATLAB code to construct the amplitude plot.
Execute the above code to obtain the amplitude plot as,
Interpretation: The above plot shows the amplitude plot for the sawtooth wave as shown in the figure provided.
Use the following MATLAB file can be used to construct the phase plot.
Execute the above code to obtain the plot as,
Interpretation: The above plot shows the phase line spectra for the sawtooth wave as shown in the figure provided.
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Chapter 19 Solutions
Numerical Methods For Engineers, 7 Ed
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