Continuous Wavelet Transform (CWT)

The CWT of $f$ with mother wavelet $\psi$ at scale $a$ and position $b$:

\[W_f(a,b) = \frac{1}{\sqrt{|a|}}\int_{-\infty}^\infty f(t)\,\overline{\psi\!\left(\frac{t-b}{a}\right)}\,dt\]

Scale $a$ controls frequency (small $a$ = high frequency), $b$ controls position. Unlike the STFT, the time-frequency resolution adapts with frequency.

Discrete Wavelet Transform (DWT)

Samples the CWT at dyadic points $(a,b) = (2^j, k2^j)$. Implemented via filter banks:

• Low-pass filter (scaling function $\phi$): captures coarse structure
• High-pass filter (wavelet $\psi$): captures detail at each scale

The Mallat algorithm computes the DWT in $O(N)$ — faster than the FFT for many applications.

Wavelet bases

Wavelet	Properties	Applications
Haar	Discontinuous, simplest	Change-point detection
Daubechies $D_N$	$N$ vanishing moments	General-purpose compression
Morlet	Gaussian × sinusoid	Time-frequency analysis
Mexican hat	Second derivative of Gaussian	Edge detection