The Fourier series is a series representation of a periodic function in terms of sines and cosines. The Fourier series representation of the function

The Fourier series is a series representation of a periodic function in terms of sines and cosines. The Fourier series representation of the function

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Plot on the same graph the function f (x) and its series representation, using the four terms shown.

Answer 1

[Umar]

Hi @Egypt,

The provided MATLAB code effectively demonstrates the Fourier series representation of a periodic function, specifically a square wave defined by the function ( f(x) ).

% Define the function f(x)
f = @(x) (x > 0 & x < pi) - (x < 0 & x > -pi);

% Define the Fourier series representation
fourier_series = @(x) (4/pi) * (sin(x)/1 + sin(3*x)/3 + sin(5*x)/5 + sin(7*x)/7);

% Define the range for x
x = linspace(-pi, pi, 1000);

% Calculate the function values and Fourier series values
f_values = f(x);
fourier_values = fourier_series(x);

% Plotting the results
figure;
plot(x, f_values, 'r', 'LineWidth', 2); % Original function
hold on;
plot(x, fourier_values, 'b', 'LineWidth', 2); % Fourier series
hold off;

% Adding labels and title
xlabel('x');
ylabel('f(x) and Fourier Series');
title('Fourier Series Representation of f(x)');
legend('f(x)', 'Fourier Series', 'Location', 'Best');
grid on;

Please see attached.

The function is defined as ( f(x) = 1 ) for ( 0 < x < pi ) and ( f(x) = -1 ) for ( -pi < x < 0 ). The Fourier series approximation is constructed using the first four odd harmonics of sine functions.

Function Definition: The function ( f(x) ) is defined using an anonymous function, which allows for concise representation and evaluation over a range of ( x ) values.

Fourier Series Calculation: The Fourier series is computed by summing the first four terms of the sine series, scaled by their respective coefficients. This is encapsulated in another anonymous function.

Plotting: The code generates a range of ( x ) values from (-pi) to (pi) and computes both the original function and its Fourier series values. The results are plotted on the same graph, with distinct colors for clarity.

This code serves as a practical example of how Fourier series can approximate complex periodic functions, which is essential in signal processing, acoustics, and electrical engineering. By visualizing the original function alongside its Fourier series, one can observe how the series converges to the function as more terms are added, illustrating the power of Fourier analysis in practical applications.

Please let me know if you have any further questions.

[John D'Errico]

@Umar

Please. By now, you should know better. Answers is not a service where we do student homework assignments for them, with no effort made.

This does not help the student, as it teaches them nothing more than that there is some sucker out there, willing to do theirwork for them.

This does not help the forum, as it teaches that same student to continue posting assignments with no effort made.

Worse, it actively hurts the forum, as it then teaches other students to do the same.

Please, if you feel you want to help a student who has made no effort at all, consider trying to push them in the proper direction.

[Umar]

Hi @John D'Errico,

Thank you for your feedback regarding our approach to assisting students. I appreciate your insights and understand the concerns you have raised about the importance of encouraging independent learning.

I fully agree that our role should be to guide students towards finding solutions themselves rather than providing direct answers to their assignments. It is crucial for us to foster an environment where students are motivated to engage with the material and develop their skills.

Moving forward, I will make sure to emphasize this philosophy in our interactions. Encouraging students to explore resources and think critically about their work will be a priority, as it aligns with our commitment to promoting genuine learning experiences at Mathworks.

Thank you once again for bringing this matter to my attention. Your input is invaluable in helping us improve our practices.