Fourier Analysis
Fourier analysis decomposes functions into sinusoidal components. It underlies signal processing, PDEs, and many machine learning architectures.
Key Points
- Fourier series represent periodic functions as sums of sines and cosines.
- The Fourier transform extends this to non-periodic functions.
- Convolution in time domain corresponds to multiplication in frequency domain.
Formulas
Fourier transform
$$\hat{f}(\xi) = \int_{-\infty}^{\infty} f(x) e^{-2\pi i x \xi} \, dx$$
Inverse Fourier transform
$$f(x) = \int_{-\infty}^{\infty} \hat{f}(\xi) e^{2\pi i x \xi} \, d\xi$$
Convolution theorem
$$\widehat{f * g} = \hat{f} \cdot \hat{g}$$
Code Example
import numpy as np
x = np.linspace(0, 2*np.pi, 256)
y = np.sin(3*x) + 0.5*np.cos(7*x)
Y = np.fft.fft(y)
print(np.argmax(np.abs(Y[:128]))) # dominant frequency