Wavelet packet functions

`[WPWS,X] = wpfun(`

* 'wname'*,NUM,PREC)

[WPWS,X] = wpfun(

`'wname'`

[WPWS,X] = wpfun(

`'wname'`

`wpfun`

is a wavelet packet
analysis function.

`[WPWS,X] = wpfun(`

computes
the wavelet packets for a wavelet * 'wname'*,NUM,PREC)

`'wname'`

`wfilters`

for more information),
on dyadic intervals of length 2`PREC`

must be a positive integer. Output matrix `WPWS`

contains
the *W* functions of index from 0 to `NUM`

,
stored row-wise as `[`

*W _{0}*;

`]`

.
Output vector `X`

is the corresponding common `X`

-grid
vector. `[WPWS,X] = wpfun(`

is
equivalent to * 'wname'*,NUM)

`[WPWS,X] = wpfun(``'wname'`

,NUM,7)

. The computation scheme for wavelet packets generation is easy
when using an orthogonal wavelet. We start with the two filters of
length 2*N*, denoted *h*(*n*)
and *g*(*n*), corresponding to the
wavelet.

Now by induction let us define the following sequence of functions
(*W _{n}*(

$$\begin{array}{l}{W}_{2n}(x)=\sqrt{2}{\displaystyle \sum _{k=0,\dots ,2N-1}h(k){W}_{n}(2x-k)}\\ {W}_{2n+1}(x)=\sqrt{2}{\displaystyle \sum _{k=0,\dots ,2N-1}g(k){W}_{n}(2x-k)}\end{array}$$

where *W _{0}*(

For example for the Haar wavelet we have

$$N=1,h(0)=h(1)=\frac{1}{\sqrt{2}}$$

and

$$g(0)=-g(1)=\frac{1}{\sqrt{2}}$$

The equations become

$${W}_{2n}(x)={W}_{n}(2x)+{W}_{n}(2x-1)$$

and

$$({W}_{2n+1}(x)={W}_{n}(2x)-{W}_{n}(2x-1))$$

*W _{0}*(

`haar`

scaling
function and `haar`

wavelet,
both supported in [0,1].Then we can obtain *W _{2}*

Starting from more regular original wavelets, using a similar
construction, we obtain smoothed versions of this system of *W*-functions,
all with support in the interval [0, 2*N*-1].

% Compute the db2 Wn functions for n = 0 to 7, generating % the db2 wavelet packets. [wp,x] = wpfun('db2',7); % Using some plotting commands, % the following figure is generated.

Coifman, R.R.; M.V. Wickerhauser (1992), “Entropy-based
Algorithms for best basis selection,” *IEEE Trans.
on Inf. Theory*, vol. 38, 2, pp. 713–718.

Meyer, Y. (1993), *Les ondelettes. Algorithmes et
applications*, Colin Ed., Paris, 2nd edition. (English translation: *Wavelets:
Algorithms and applications*, SIAM).

Wickerhauser, M.V. (1991), “INRIA lectures on wavelet
packet algorithms,” *Proceedings ondelettes et paquets
d'ondes*, 17–21 June, Rocquencourt, France, pp. 31–99.

Wickerhauser, M.V. (1994), *Adapted wavelet analysis
from theory to software algorithms*, A.K. Peters.