Fourier Transform

$X(f) = \int_{-\infty}^{\infty} x(t)\, e^{-j2\pi f t}\, dt \qquad x(t) = \int_{-\infty}^{\infty} X(f)\, e^{j2\pi f t}\, df$

Useful properties: linearity, time shift (adds phase), time scaling (shorter in time = wider in frequency), and convolution ↔ multiplication.

For sampled data computers use the DFT, computed efficiently by the FFT in $O(N \log N)$. The frequency resolution is $\Delta f = f_s / N$. Apply a window (Hann, Hamming) to reduce spectral leakage.

import numpy as np
X = np.fft.rfft(x * np.hanning(len(x)))
freqs = np.fft.rfftfreq(len(x), d=1/fs)