10.1 Wavelet analysis

Source: Ideas from Computational Physics (Landau, Páez, Bordeianu), §10.1. Rewritten in original wording. Continues 9.8. Next: 10.1b.

Part I — when Fourier is not enough

Chapter 9 told you which frequencies sit in a record. Many real signals also change when those frequencies appear: brain rhythms, markets, gravitational-wave chirps, image textures. Part I of this chapter builds wavelets — localized oscillating packets. Part II (later) covers principal components for huge, correlated space–time data.

The pedagogical signal

$$y(t)= \begin{cases} \sin 2\pi t, & 0\le t\le 2,\\ 5\sin 2\pi t+10\sin 4\pi t, & 2\le t\le 8,\\ 2.5\sin 2\pi t+6\sin 4\pi t+10\sin 6\pi t, & 8\le t\le 12. \end{cases}$$

A global DFT mixes all three regimes into one spectrum: every harmonic looks “always on.” You want a compact picture of which band is active at which time.

Fourier bases are infinite-support sinusoids; their overlaps store correlated, redundant information — awkward for compression. Wavelets are wave packets: finite support (or rapid decay) in time, hence a finite band in frequency. Expand $y$ in a complete family of such packets, each centered at a different time.

Mother → daughters

A mother wavelet $\Psi(t)$ is one localized shape. Daughters come from scale $a$ and translate $\tau$. Pick the daughter formula (and why a plain DFT fails the staged multi-tone) among lookalikes.

Examples of mothers:

• Morlet: $\Psi(t)=e^{2\pi i t}e^{-t^2/(2\sigma^2)}$ (oscillating Gaussian).
• Mexican hat: $\Psi(t)=(1-t^2/\sigma^2)e^{-t^2/(2\sigma^2)}$ (second derivative of a Gaussian).
• Haar: a simple $+1/-1$ step.
• Daubechies: compact, fractal-looking orthogonal filters (details later).

All are large only for a while and carry a limited frequency range — hence joint time–frequency resolution (subject to an uncertainty bound, §10.2).

Quiz

Wavelet motivation

1 / 4

A plain DFT of (10.1) is weak because…

Pick an answer to see if you are right.

Next: 10.1b Wavelet intro labs.