Concept explainers
With respect to the periodic waveform sketched in Fig. 17.30, let gn(t) represent the Fourier series representation of f(t) truncated at n. [For example, if n = 1, g1(t) has three terms, defined through a0, a1 and b1.] (a) Sketch g2(t), g3(t), and g5(t), along with f(t). (b) Calculate f (2.5), g2(2.5), g3(2.5), and g5(2.5).
■ FIGURE 17.30
(a)
Sketch
Answer to Problem 8E
The sketch for
Explanation of Solution
Given data:
Refer to Figure 17.30 in the textbook.
Formula used:
Write the general expression for Fourier series expansion.
Write the general expression for Fourier series coefficient
Write the general expression for Fourier series coefficient
Write the general expression for Fourier series coefficient
Write the expression to calculate the fundamental angular frequency.
Here,
Calculation:
In the given Figure 17.29, the time period is
Substitute 6 for T in equation (5) to find
Substitute 6 for T in equation (2) to find the value of coefficient
Simplify the above equation as follows,
Substitute 6 for T in equation (3) to find the value of coefficient
The above equation as follows,
Substitute equation (6) in equation (7) as follows,
Now finding the Fourier coefficient
Substitute 6 for T in equation (4) to find the value of coefficient
The above equation as follows,
Substitute equation (6) in equation (9) as follows,
Substituting the value of
The function
For
Therefore, equation (11) will be as follows,
Simplify the above equation as follows,
Similarly, for
Therefore, equation (11) will be as follows,
From equation (12), the above equation is written as,
Similarly, for
Therefore, equation (11) will be as follows,
From equation (13), the above equation is written as,
MATLAB code to sketch for
t=-5:0.01:5;
g2=1.667-0.275*cos(3.141*t/3)+0.159*sin(3.141*t/3)+0.137*cos(2*3.141*t/3)-0.238*sin(2*3.141*t/3);
g3=1.667-0.275*cos(3.141*t/3)+0.159*sin(3.141*t/3)+0.137*cos(2*3.141*t/3)-0.238*sin(2*3.141*t/3)+0.212*sin(pi*t);
g5=1.667- 0.275*cos(3.141*t/3)+0.159*sin(3.141*t/3)+0.137*cos(2*3.141*t/3)-0.238*sin(2*3.141*t/3)+0.212*sin(pi*t)-0.069*cos(4*pi*t/3)-0.119*sin(4*pi*t/3)+0.055*cos(5*pi*t/3)+0.0318*sin(5*pi*t/3);
plot(t,g2,t,g3,t,g5)
legend({'g2','g3','g5'},'Location','best')
xlabel('Time t in sec')
ylabel('The values g2, g3, and g5')
title('Plots for g2,g3, and g5')
MATLAB output: The MATLAB output shown in Figure 1.
MATLAB code to sketch for
t=linspace(-5,5,1000); % vector for time over 1000 points.
T=6; % Period
w0=2*pi/T; % natural frequency, is w0=2*pi.
f0=1.667; % constant.
N=40;
for i=1:1000;
sum=0;
for n=1:N;
sum=sum+(1/n*pi)*(sin(n*pi) -sin(2*n*pi/3))*cos(n*pi*t(i)/3) + (1/n*pi)*(cos(2*n*pi/3) -cos(n*pi))*sin(n*pi*t(i)/3);
end
f40(i)=f0+sum;
end
plot(t,f40)
xlabel('Time t in seconds')
ylabel('Value of function f(t)')
plot_ttle = ['Fourier Series representation of function f(t) for N = ',num2str(N)];
title(plot_ttle);
MATLAB output: The MATLAB output shown in Figure 2.
MATLAB code to sketch for
t=linspace(-5,5,1000); % vector for time over 1000 points.
T=6; % Period
w0=2*pi/T; % natural frequency, is w0=2*pi.
f0=1.667; % constant.
N=40; % consider N=40 for instant.
for i=1:1000;
g2=1.667-0.275*cos(3.141*t(i)/3)+0.159*sin(3.141*t(i)/3)+0.137*cos(2*3.141*t(i)/3)-0.238*sin(2*3.141*t(i)/3);
g3=1.667-0.275*cos(3.141*t(i)/3)+0.159*sin(3.141*t(i)/3)+0.137*cos(2*3.141*t(i)/3)-0.238*sin(2*3.141*t(i)/3)+0.212*sin(pi*t(i));
g5=1.667-0.275*cos(3.141*t(i)/3)+0.159*sin(3.141*t(i)/3)+0.137*cos(2*3.141*t(i)/3)-0.238*sin(2*3.141*t/3)+0.212*sin(pi*t)-0.069*cos(4*pi*t(i)/3)-0.119*sin(4*pi*t(i)/3)+0.055*cos(5*pi*t(i)/3)+0.0318*sin(5*pi*t(i)/3);
end
for i=1:1000;
sum=0;
for n=1:N;
sum=sum+(1/n*pi)*(sin(n*pi) -sin(2*n*pi/3))*cos(n*pi*t(i)/3) + (1/n*pi)*(cos(2*n*pi/3) -cos(n*pi))*sin(n*pi*t(i)/3);
end
f40(i)=f0+sum;
end
plot(t,g2,t,g3,t,g5,t,f40)
legend({'g2','g3','g5','f40'},'Location','best')
xlabel('Time t in sec')
ylabel('The values g2, g3, g5 and f40')
title('Plots for g2, g3, g5 and f40')
MATLAB output:
Conclusion:
Thus, the sketch for
(b)
Find the function
Answer to Problem 8E
The value of
Explanation of Solution
Given data:
Refer to Figure 17.30 in the textbook.
Calculation:
From Part (a), the function
Finding
From Part (a),
Finding
From Part (a),
Finding
From Part (a),
Finding
Conclusion:
Thus, the value of
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