Wavelet Transform

In general, given a complete set of function $\psi(x; \tilde x)$, we can decompose a function $F(\tilde x)$

$$ F(\tilde x) = \int f(x) \psi(x;\tilde x) dx. $$

The choice of $\psi(x;\tilde x)$ gives us different properties.

Fourier Transform

Fourier transform is good for stationary analysis since time is not involved in $F(\omega)$.

$$ F(\omega) = \int_{-\infty}^{\infty} f(t) e^{-i \omega t} dt $$

Short-time Fourier Transform

STFT is a Fourier transform with a moving time window $\tau$,

$$ F(\tau,\omega) = \int_{-\infty}^{\infty} f(t) w(t - \tau) e^{-i\omega t} dt. $$

Moving $\tau$ gives us the ability to investigate Fourier components at different time segments (assuming the window function $w(t-\tau)$ is a step function). It is obvious that the STFT can resolve different things with a different window.

Wavelets

There are many different choices of wavelets.

One of the most used one is the Morlet wavelets or Gabor wavelets,

$$ \psi(t;\omega) = e^{i 2\pi ft - \frac{1}{2}(t/\sigma)^2 }, $$

where $$ \sigma = \frac{n}{2\pi f}. $$