Time-Frequency analysis

Section: Time-Frequency

Short Time Fourier Transform (STFT)

Idea: perform a local spectral analysis of the signal thanks to a sliding window

Let $w(t)$ be a smooth window localized around $t=0$. Let the time-frequency atoms $$ \varphi_{\tau,\nu}(t) = w(t-\tau)e^{i2\pi\nu t} $$ The time-frequency transform of a signal $x$ computes the correlation between $x$ and the time-frequency atoms $\varphi_{\tau,\nu}$ for various time index $\tau$ and various frequency index $\nu$: $$ X(\tau,\nu) = \langle x(t), \varphi_{\tau,\nu}(t) \rangle $$

Spectrogramm

The spectrogramm of $x$ is given by the time-frequency image $|X(\tau,\nu)|$

Parameters of the STFT

  • The shape of the window $w$
  • The length of the window $w$
  • The redundancy in time (hope size between two windows)
  • The redundancy in frequency (length of the frequency transform inside one window)