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+---
+title: "Fourier Analysis: From Math to Code"
+date: "2025-03-10"
+excerpt: "Bridging the gap between Fourier's mathematical theory and practical implementation."
+---
+
+Fourier analysis reveals a profound truth: any periodic function can be decomposed into a sum of simple sine and cosine waves.
+
+## The Math
+
+The Fourier series of a function $f(x)$ with period $2\pi$ is:
+
+$$f(x) = \frac{a_0}{2} + \sum_{n=1}^{\infty} \left( a_n \cos(nx) + b_n \sin(nx) \right)$$
+
+where:
+
+$$a_n = \frac{1}{\pi} \int_{-\pi}^{\pi} f(x)\cos(nx)\,dx$$
+
+$$b_n = \frac{1}{\pi} \int_{-\pi}^{\pi} f(x)\sin(nx)\,dx$$
+
+## Python Implementation
+
+Let's approximate a square wave using the first few terms:
+
+```python
+import numpy as np
+import matplotlib.pyplot as plt
+
+def fourier_square_wave(x, n_terms):
+    """Approximate a square wave using Fourier series."""
+    result = np.zeros_like(x)
+    for n in range(1, n_terms + 1, 2):  # odd harmonics only
+        result += (4 / (n * np.pi)) * np.sin(n * x)
+    return result
+
+x = np.linspace(-np.pi, np.pi, 1000)
+plt.figure(figsize=(10, 6))
+plt.plot(x, np.where(x >= 0, 1, -1), 'k--', label='Square wave')
+
+for terms in [1, 3, 7, 15]:
+    plt.plot(x, fourier_square_wave(x, terms), label=f'{terms} terms')
+
+plt.legend()
+plt.title('Fourier Series Approximation of a Square Wave')
+plt.grid(True, alpha=0.3)
+plt.savefig('fourier.png', dpi=150)
+```
+
+## Understanding Convergence
+
+The Gibbs phenomenon shows that near discontinuities, the approximation overshoots by approximately:
+
+$$9\%$$
+
+This is mathematically expressed as:
+
+$$\lim_{N \to \infty} \text{overshoot} \approx 1.08949 \times \text{jump}$$
+
+## Conclusion
+
+The connection between the elegant mathematics and practical code demonstrates how theory informs implementation. Fourier analysis is the foundation of signal processing, image compression, and countless other applications.