Concept explainers
Find the Fourier series for the signal shown in Figure 17.58 and verify it using MATLAB and find the value of
Answer to Problem 20P
The Fourier series
Explanation of Solution
Given data:
Refer to Figure 17.58 in the textbook.
Formula used:
Write the expression to calculate the fundamental angular frequency.
Here,
Write the general expression to calculate trigonometric Fourier series of
Here,
Calculation:
The given waveform is drawn as Figure 1.
Refer to Figure 1, the waveform is symmetrical about the vertical axis. The signal
Write the expressions for Fourier coefficients for an even function.
Refer to the Figure 1. The Fourier series even 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
Assume the following to reduce the equation (5).
Substitute the equations (6) in equation (5) to find
Consider the following integration formula.
Compare the equations (6) and (8) to simplify the equation (6).
Using the equation (8), the equation (6) can be reduced as,
Simplify the above equation to find
Substitute
Substitute
Following includes the MATLAB code to obtain the waveform in Figure 1 using the obtained function
MATLAB Code:
t=0:.01:6;
f=5*ones(size(t));
for n=1:1:99
f=f+(60/(n^2*pi^2))*(cos(2*pi*n/3)-cos(pi*n/3))*cos(n*pi*t/3);
end
plot(t,f)
title('Fourier series plot of f(t)');
xlabel('Time t in sec');
ylabel('Function f(t)');
Output:
The output of the MATLAB is the Fourier series plot of the function
Substitute
Simplify the above equation to find
Write the MATLAB code to find
MATLAB code:
t=2;
f=5*ones(size(t));
for n=1:1:5
f=f+(60/(n^2*pi^2))*(cos(2*pi*n/3)-cos(pi*n/3))*cos(n*pi*t/3);
end
disp(f)
The Output displays the value of
9.5122
This output is approximately satisfied with calculated value of
Conclusion:
Thus, the Fourier series
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Chapter 17 Solutions
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