Harmonic analysis on Euclidean spaces Definition

Wavelet packet decomposition

Builds on Fast wavelet transform Discrete wavelet transform

Definition

Let $(h_n)$ and $(g_n)$ be the real low-pass and high-pass filters of an orthonormal wavelet, with $g_n = (-1)^n h_{1-n}$ , as in the fast wavelet transform. For a sequence $c = (c_m)_{m \in \Z}$ define the sequences $Hc$ and $Gc$ by $(Hc)_k = \sum_m h_{m-2k} \, c_m, \qquad (Gc)_k = \sum_m g_{m-2k} \, c_m.$ Let $x$ be a sequence and let $J \ge 1$ be an integer. The wavelet packet decomposition of $x$ to depth $J$ is the family of sequences $w_{j,p}$ , for $0 \le j \le J$ and $0 \le p < 2^j$ , defined by

$w_{0,0} = x$ ;
$w_{j+1,2p} = H w_{j,p}$ and $w_{j+1,2p+1} = G w_{j,p}$ for $0 \le j < J$ .

The sequences $w_{j,p}$ are the nodes of a full binary tree, in which $w_{j,p}$ has children $w_{j+1,2p}$ and $w_{j+1,2p+1}$ .

Remarks

The discrete wavelet transform keeps only the nodes $w_{j,0}$ and $w_{j,1}$ , that is, it splits only the low-pass branch again.

Each step $c \mapsto (Hc, Gc)$ is an orthogonal transform of $\ell^2(\Z)$ , by the orthonormality of the filters. So for every admissible subtree, in which each node is either a leaf or has both children, the leaves together determine $x$ and preserve its energy. For a finite signal of length $N = 2^J$ with periodic extension, every level contains $N$ coefficients in total.

Example: with the Haar filters and $x = (1, 1, 1, 1)$ , level $1$ gives $w_{1,0} = (\sqrt{2}, \sqrt{2})$ and $w_{1,1} = (0, 0)$ . Level $2$ gives $w_{2,0} = (2)$ , $w_{2,1} = (0)$ , $w_{2,2} = (0)$ and $w_{2,3} = (0)$ . The energy $4$ of $x$ is kept in each level.

Non-example: if only $H$ is applied at a node and $G$ is dropped, the leaves no longer determine the signal. With the Haar filters, $(1, -1)$ and $(0, 0)$ both have $H$ -output $(0)$ .

Reading path

To understand this page from the ground up, read these 3 pages in order. The foundations come first.

Show the path as a graph

Each page uses the pages above it. The colour of the bar on a box shows how strong the evidence is, and the box also gives it as text. The list has the same pages.

no evidence level

Tag	`2HNN80TMGH`
URI	`m2://2HNN80TMGH`
Kind	definition
Writing	W0 stub
MSC	42C40, 65T60, 94A12
Defines	Wavelet packet decomposition
Uses	Fast wavelet transform, Discrete wavelet transform

Wavelet packet decomposition

Definition

Remarks

Reading path

Further reading

What this page needs for W1

What links here