wpfun

Wavelet packet functions

Syntax

[wpws,x] = wpfun(wname,num)

[wpws,x] = wpfun(wname,num,prec)

Description

wpfun is a wavelet packet analysis function.

[wpws,x] = wpfun(wname,num) returns the num+1 wavelet packets associated with the wavelet wname on dyadic intervals of length 2^–7. The wavelet packets are evaluated on the grid x.

[wpws,x] = wpfun(wname,num,prec) uses dyadic intervals of length 2^–prec.

Examples

Visualize Wavelet Packets

Specify the db2 wavelet. Obtain the wavelet packets $$W_n$$, for $$n=0,\dots ,7$$, associated with the wavelet.

wv = "db2";
n = 7;
[wpws,x] = wpfun(wv,7);

Plot the wavelet packets

tiledlayout(2,4)
for k=0:n
    nexttile
    plot(x,wpws(k+1,:))
    title("n = "+num2str(k))
end

Input Arguments

wname — Wavelet

character vector | string scalar

Wavelet, specified as a character vector or string scalar. For more information, see wfilters.

num — Index

nonnegative integer

Index, specified as a nonnegative integer. The wpfun function returns num+1 wavelet packets.

Data Types: double

prec — Length of the dyadic interval

7 (default) | positive integer

Length of the dyadic interval, specified as a positive integer. The length of the interval is 2^–prec.

Data Types: double

Output Arguments

wpws — Wavelet packets

matrix

Wavelet packets, returned as a matrix. wpws has num+1 rows. The wavelet packets are stored row-wise as [W₀; W₁; … ; Wnum].

x — Grid points

vector

Grid points where the wavelet packets are evaluated, returned as a real-valued vector.

More About

Generating Wavelet Packets

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

Next, by induction, define the following sequence of functions (W_n(x) , n = 0,1,2,…) by

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

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

For example, for the Haar wavelet you 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

$$W_{2n+1}(x)=W_n(2x)-W_n(2x-1)$$

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

Then you can obtain W_{2 n} by adding two 1/2-scaled versions of W_n with distinct supports [0,1/2] and [1/2,1], and obtain W_2n+1 by subtracting the same versions of W_n.

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

References

[1] Coifman, R.R., and M.V. Wickerhauser. “Entropy-Based Algorithms for Best Basis Selection.” IEEE Transactions on Information Theory 38, no. 2 (March 1992): 713–18. https://doi.org/10.1109/18.119732.

[2] Jaffard, Stéphane, Yves Meyer, and Robert D. Ryan. Wavelets: Tools for Science & Technology. Philadelphia: Society for Industrial and Applied Mathematics, 2001.

[3] Wickerhauser, M.V. "INRIA lectures on wavelet packet algorithms." Proceedings ondelettes et paquets d'ondes, 17–21 June 1991, Rocquencourt, France, pp. 31–99.

[4] Wickerhauser, Mladen Victor. Adapted Wavelet Analysis from Theory to Software. Wellesley, MA: A.K. Peters, 1994.

Version History

Introduced before R2006a