10.5 Discrete wavelet transforms

Source: Ideas from Computational Physics (Landau, Páez, Bordeianu), §10.5. Rewritten in original wording — no textbook dwt.py dump. Continues 10.4b. Next: 10.5b.

Only $N$ independent coeffs

With $N$ samples $y_m$ you can keep only $N$ independent transform numbers and still hit the sample grid. Stay consistent with $\Delta\omega\Delta t\gtrsim 2\pi$ while doing so. Pick the dyadic lattice, the orthonormal reconstruction, and the Daub4 taps among lookalikes.

Then $Y_{j,k}\simeq\sum_m\psi_{j,k}(t_m)\,y_m\,h$. Orthonormality $\int\psi_{j,k}^*\psi_{j',k'}=\delta_{jj'}\delta_{kk'}$ plus time localization ⇒ low correlation ⇒ good compression.

Low $\omega$: few wide tiles (smooth outline). High $\omega$: many narrow tiles (detail). Same area $\sim 2\pi$.

Filters instead of integrals

Industrial DWTs use multiresolution analysis (Mallat): pass the signal through digital lowpass $L$ and highpass $H$ filters (convolutions = discrete wavelet overlaps), then ↓2 decimate. Recurse on the lowpass branch.

Highpass outputs are details $d^{(level)}$ (store). Lowpass outputs are smooth $s^{(level)}$ (feed the next stage). Stop when two smooth and two detail numbers remain.

Synthesis reverses the arrows: upsample, apply transpose filters, merge until length $N$ returns.

Daub4 coefficients

Daubechies filters are defined by tap weights, not closed forms. Stack $L=(c_0,c_1,c_2,c_3)$ and $H=(c_3,-c_2,c_1,-c_0)$ (and shifted copies) into an orthogonal matrix — the $c_i$ you picked above. $L$ smooths (weighted average); $H$ differences (detail). Larger $N$ builds a banded filter matrix with wrap-around at the edges; pyramid reorders smooth/detail after each stage.

Inverse-transform a unit impulse in coefficient space to see the time-domain Daub4 shape at different scales.

Practice checklist

1. Implement Daub4 forward/inverse on length $2^n$ (chirp $\sin(60t^2)$ is the book’s demo).
2. Stop the pyramid at different nend to watch ↓2 stages.
3. Invert (sign flip) and match $y$.
4. Impulse → mother visualizations at growing widths.

# dwt_daub4_sketch.py — original structure (not a textbook dump)
c0, c1, c2, c3 = ...  # Daub4 closed form
# forward: for each pair, s = L·y_local, d = H·y_local; ↓2; recurse on s
# inverse: upsample; apply transpose taps; merge

Quiz

Discrete wavelets

1 / 4

A dyadic DWT takes scales…

Pick an answer to see if you are right.

Next: 10.5b DWT labs.