Wavelets & Multiresolution Analysis
Localized, multi-scale alternatives to the Fourier basis.
Fourier analysis uses globally supported sinusoids; wavelets use localized "wavelets" that allow time-frequency decomposition. The continuous wavelet transform:
$$W f(a, b) = \frac{1}{\sqrt{|a|}} \int f(t) \overline{\psi\!\left(\frac{t-b}{a}\right)} dt,$$analyzes the signal at scale $a$ and position $b$ using a "mother wavelet" $\psi$.
Multiresolution analysis (Mallat–Meyer): pick a nested chain of subspaces $\cdots \subset V_{-1} \subset V_0 \subset V_1 \subset \cdots$ of $L^2(\mathbb R)$ whose union is dense, intersection is $\{0\}$, related by dilation, and admitting a scaling function $\varphi$ whose translates form an orthonormal basis of $V_0$. Detail spaces $W_j = V_{j+1} \ominus V_j$ are spanned by wavelets $\psi_{j,k}(t) = 2^{j/2}\psi(2^j t - k)$.
The discrete wavelet transform (DWT) gives a fast $O(N)$ basis change via cascaded quadrature mirror filters — faster than the FFT's $O(N \log N)$. Examples:
- Haar: simplest, $\psi$ is a unit-height square pulse. Discontinuous, poor frequency localization but perfectly time-localized.
- Daubechies $\mathrm{db}_N$: compactly supported, smoothness increases with $N$.
- Mexican hat, Morlet: continuous, used in CWT.
Wavelet bases sparsely represent piecewise-smooth signals — the basis of JPEG 2000, MP3 (partially), and modern signal denoising / compression.