Fourier Analysis

Fourier Analysis allows us to decompose complex, periodic signals into a sum of simple sine and cosine waves.

Fourier Series

Any periodic function $f(x)$ with period $2L$ can be represented as: $f(x) = \frac{a_0}{2} + \sum_{n=1}^{\infty} \left( a_n \cos\frac{n\pi x}{L} + b_n \sin\frac{n\pi x}{L} \right)$

import numpy as np

def square_wave_fourier(x, terms):
  res = 0
  for n in range(1, terms + 1, 2):
      # f(x) = (4/pi) * summation sin(n*x)/n
      res += (4 / np.pi) * (np.sin(n * x) / n)
  return res

x_val = np.pi/2
print(f"Square wave approx (5 terms) at pi/2: {square_wave_fourier(x_val, 5)}")
print(f"Square wave approx (100 terms) at pi/2: {square_wave_fourier(x_val, 100)}")

The Fourier Transform

For non-periodic functions, the Fourier Transform $\hat{f}(\xi)$ maps a function from the time (or space) domain to the frequency domain: $\hat{f}(\xi) = \int_{-\infty}^{\infty} f(x) e^{-2\pi i x \xi} dx$

import numpy as np

# Fast Fourier Transform (FFT)
signal = np.array([1, 0, 1, 0, 1, 0, 1, 0])
fft_res = np.fft.fft(signal)

print(f"Signal: {signal}")
print(f"FFT Magnitudes: {np.abs(fft_res).round(2)}")
print("The spike at index 4 represents the fundamental frequency of the oscillation.")

Parseval’s Theorem

Parseval’s Theorem states that the total energy of a signal in the time domain is equal to the total energy in the frequency domain: $\int |f(x)|^2 dx = \int |\hat{f}(\xi)|^2 d\xi$