10.4 Wavelet transforms

Source: Ideas from Computational Physics (Landau, Páez, Bordeianu), §10.4. Rewritten in original wording — no textbook cwt.py dump. Continues 10.3b. Next: 10.4b.

Definition

Replace the STFT’s eternal complex exponential by a localized wavelet packet. Pick the CWT integral and the daughter formula among lookalikes.

$Y(s,\tau)$ is the overlap of $y$ with a wavelet of scale $s$ centered at time $\tau$. Scale maps to frequency as $\omega=2\pi/s$: small $s$ $\Leftrightarrow$ high $\omega$ $\Leftrightarrow$ fine time detail.

Mothers and daughters

Start from a mother, e.g. $\Psi(t)=\sin(8t)\,e^{-t^{2}/2}$, and build the daughters you picked above. Explicitly,

$$\psi_{s,\tau}(t) =\frac{1}{\sqrt{s}}\sin\!\left[\frac{8(t-\tau)}{s}\right] \exp\!\left[-\frac{(t-\tau)^{2}}{2s^{2}}\right].$$

s<1 squeezes the packet (higher frequency); s>1 stretches it; $\tau$ slides the center. The $1/\sqrt{s}$ factor equalizes power across scales (other normalizations exist). An inverse integrates over $\tau$ and s>0 with a mother-dependent constant $C$ (finite $(s,\tau)$ grids reconstruct up to edge effects).

Mother requirements

1. Real-valued $\Psi$.
2. Zero mean: $\int\Psi=0$ (must oscillate).
3. Localized / square-integrable packet.
4. Vanishing moments $\int t^{k}\Psi=0$ for $k=0,\ldots,p-1$ — kills smooth polynomials so the transform highlights detail.

Self-similarity across scales is why fractal language shows up later (Chapter 14).

Continuous wavelet transform (CWT) practice

Implement mothers (Morlet, Mexican hat, Haar), form daughters, and numerically integrate (10.14) on a $(s,\tau)$ grid for:

• pure / multi-tone sines,
• the staged signal (10.1),
• a half-wave.

Invert approximately and compare to $y$ (normalize). Expect near-overlap except near edges.

For the staged multi-tone, a Morlet CWT scalogram should show one dominant band early, then a second (smaller $s$), then a third — time-resolved where a global DFT is not.

Discrete / dyadic choices of $(s,\tau)$ come in §10.5. Prefer library CWTs for production after one slow reference integrator.

Quiz

Wavelet transforms

1 / 4

In ω = 2π/s, small scale s means…

Pick an answer to see if you are right.

Next: 10.4b Wavelet transform labs.