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Use a continuous Fourier series to approximate the sawtooth wave in Fig. P19.4. Plot the first three terms along with the summation.
FIGURE P19.4
A sawtooth wave.
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To calculate: The Fourier series expansion to approximate the sawtooth wave, as shown in the following figure,
Plot the first three terms along with the summation.
Answer to Problem 4P
Solution:
The Fourier series of the sawtooth curve is
Explanation of Solution
Given Information: The sawtooth wave shown as,
Formula used:
Consider
And the coefficients are defined by,
Calculation:
Consider the sawtooth wave as shown in the following figure,
Therefore, the sawtooth wave is a periodic function
;
Therefore, the sawtooth wave,
Therefore, the Fourier series expansion of the function
Here, the coefficients are defined by,
Now, find
Consider,
Thus,
Therefore,
Now, find
Consider,
Thus,
Further,
Therefore,
Therefore, the coefficients of the Fourier series expansions are,
Therefore, the Fourier series expansion defines
Hence,
Graph:
To plot the given sawtooth curve and the approximated Fourier series consider the period,
Therefore, the periodic sawtooth curve is expressed as,
And, the corresponding Fourier series is,
Hence,
Use the following MATLAB code to plot the first three terms along with the summation.
Execute the above code to obtain the plot as,
Interpretation: The above plot shows the comparison between the variation in the first three terms of the series along with the variation in summation.
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Chapter 19 Solutions
Numerical Methods for Engineers
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