Fourier Step Function Visualizer

This project demonstrates how to approximate a step function using a Fourier series in Python. It is a great educational example for beginners in signal processing, numerical analysis, or engineering mathematics who want to understand the behavior of Fourier series — especially the Gibbs phenomenon near discontinuities.

🔍 What It Does

• Defines a step function on the interval [-π, π]
• Computes its Fourier series using only sine terms (odd function)
• Plots the original function and its Fourier approximation
• Visualizes how increasing the number of terms (N) improves the approximation

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📊 Example Plot

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📚 Math Background

The step function:

$$f(x) = -1, for x < 0 1, for x ≥ 0$$

Its Fourier series (only sine terms because it's an odd function):

$$f(x) ≈ \sum_{n=1,3,5...}^{N} \frac{4}{n\pi} \sin(nx)$$

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🚀 How to Run

1. Clone the repo or copy the Python script:

   git clone https://github.com/aminomrani/fourier-series-step-function-visualizer.git
   cd fourier-series-step-function-visualizer

2. Make sure you have Python installed with required libraries:

   pip install numpy matplotlib

3. Run the script:

   python fourier_step.py

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🧠 Learning Goals

• Understand Fourier series for periodic functions
• See how sine terms build up a discontinuous signal
• Observe Gibbs phenomenon in action
• Learn basic matplotlib and numpy usage

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📦 Dependencies

• Python 3.x
• numpy
• matplotlib

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📜 License

This project is open-source under the MIT License.

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✍️ Author

Amin Omrani MSc Chemical Engineering Student