# Documentation

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# wpfun

Wavelet packet functions

## Syntax

`[WPWS,X] = wpfun('wname',NUM,PREC)[WPWS,X] = wpfun('wname',NUM)[WPWS,X] = wpfun('wname',NUM,7)`

## Description

`wpfun` is a wavelet packet analysis function.

`[WPWS,X] = wpfun('wname',NUM,PREC)` computes the wavelet packets for a wavelet `'wname'` (see `wfilters` for more information), on dyadic intervals of length 2-PREC.

`PREC` must be a positive integer. Output matrix `WPWS` contains the W functions of index from 0 to `NUM`, stored row-wise as `[`W0; W1; ... ; WNUM`]`. Output vector `X` is the corresponding common `X`-grid vector.

`[WPWS,X] = wpfun('wname',NUM)` is equivalent to
`[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 2N, denoted h(n) and g(n), corresponding to the wavelet.

Now by induction let us define the following sequence of functions (Wn(x) , n = 0,1,2,...) by

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

where W0(x) = ϕ (x) is the scaling function and W1(x) = ψ(x) is the wavelet function.

For example for the Haar wavelet we have

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

and

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

The equations become

`${W}_{2n}\left(x\right)={W}_{n}\left(2x\right)+{W}_{n}\left(2x-1\right)$`

and

`$\left({W}_{2n+1}\left(x\right)={W}_{n}\left(2x\right)-{W}_{n}\left(2x-1\right)\right)$`

W0(x) = ϕ(x) is the `haar` scaling function and W1(x) = ψ(x) is the `haar` wavelet, both supported in [0,1].

Then we can obtain W2 n by adding two 1/2-scaled versions of Wn with distinct supports [0,1/2] and [1/2,1], and obtain W2n+1 by subtracting the same versions of Wn.

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, 2N-1].

## Examples

```% 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. ```

## References

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.