How to visualize the Fourier series

 

visualize the Fourier series Results 

In this blog, of Physics blog we are going to learn how to visualize the Fourier Series. it is a step-by-step guide for how you can visualize your result on a computer. let's start 

To visualize the Fourier series approximation alongside the original function, you can output the results to a file and use a plotting tool like GNUplot, Matplotlib (Python), or even online graphing tools. Here's a step-by-step guide using Matplotlib:

---

Step 1: Modify the C++ Program to Save Data

Update your program to write the original function and Fourier approximation values to a file.

Add this to your main function:

cpp
#include <fstream> // Include for file handling

int main() {
    int N = 10;         // Number of Fourier terms
    int numSteps = 1000; // Steps for numerical integration

    double a0;
    std::vector<double> an(N), bn(N);

    // Compute coefficients
    computeCoefficients(a0, an, bn, N, numSteps);

    // Open a file to save data
    std::ofstream file("fourier_data.csv");
    file << "x,original,approximation\n";

    // Test the Fourier series and write data
    for (double x = -M_PI; x <= M_PI; x += 0.1) {
        double approx = fourierSeries(x, a0, an, bn, N);
        file << x << "," << f(x) << "," << approx << "\n";
    }

    file.close();
    std::cout << "Data saved to fourier_data.csv" << std::endl;

    return 0;
}

---

Step 2: Compile and Run the Program

Compile and run the updated program:

bash
g++ fourier_series.cpp -o fourier_series -lm
./fourier_series

This creates a file named fourier_data.csv with three columns:

• $x$ values
• Original function values
• Fourier approximation values

---

Step 3: Visualize Using Python and Matplotlib

Install Matplotlib if you don’t already have it:

bash
pip install matplotlib

Now, create a Python script (e.g., plot_fourier.py) to visualize the data:

python
import matplotlib.pyplot as plt
import numpy as np

# Load the data
data = np.loadtxt("fourier_data.csv", delimiter=",", skiprows=1)
x = data[:, 0]
original = data[:, 1]
approximation = data[:, 2]

# Plot the original function and the Fourier approximation
plt.figure(figsize=(10, 6))
plt.plot(x, original, label="Original Function", color="blue", linewidth=2)
plt.plot(x, approximation, label="Fourier Approximation", color="red", linestyle="--", linewidth=2)

# Add labels, legend, and title
plt.title("Fourier Series Approximation", fontsize=16)
plt.xlabel("x", fontsize=14)
plt.ylabel("f(x)", fontsize=14)
plt.legend(fontsize=12)
plt.grid(True)

# Show the plot
plt.show()

---

Step 4: Run the Visualization

Run the Python script:

bash
python plot_fourier.py

This will generate a graph showing the original function and the Fourier series approximation.

---

Step 5: Optional Enhancements

• Fine-tune the graph: Adjust line styles, colors, or fonts.
• Compare different $N$: Save Fourier approximations for multiple values of $N$ and plot them together.
• Export the graph: Save it as an image using plt.savefig("fourier_plot.png") before plt.show().

By visualizing, you can see how the Fourier series converges to the original function as $N$ increases