# dwpt

Multisignal 1-D wavelet packet transform

## Syntax

``wpt = dwpt(X)``
``wpt = dwpt(X,wname)``
``wpt = dwpt(X,LoD,HiD)``
``[wpt,l] = dwpt(___)``
``[wpt,l,packetlevels] = dwpt(___)``
``[wpt,l,packetlevels,f] = dwpt(___)``
``[wpt,l,packetlevels,f,re] = dwpt(___)``
``[___] = dwpt(___,Name,Value)``

## Description

````wpt = dwpt(X)` returns the terminal (final-level) nodes of the discrete wavelet packet transform (DWPT) of `X`. The input `X` is a real-valued vector, matrix, or timetable. By default, the `fk18` wavelet is used, and the decomposition level is `floor(log2(Ns))`, where Ns is the number of data samples. The wavelet packet transform `wpt` is a 1-by-N cell array, where ```N = 2^floor(log2(Ns))```.```
````wpt = dwpt(X,wname)` uses the wavelet specified by `wname` for the DWPT. `wname` must be recognized by `wavemngr`.```
````wpt = dwpt(X,LoD,HiD)` uses the scaling (lowpass) filter, `LoD`, and wavelet (highpass) filter, `HiD`.```
````[wpt,l] = dwpt(___)` also returns the bookkeeping vector using any of the previous syntaxes. The vector `l` contains the length of the input signal and the number of coefficients by level. The bookkeeping vector is required for perfect reconstruction.```
````[wpt,l,packetlevels] = dwpt(___)` also returns the transform levels of the nodes of `wpt` using any of the previous syntaxes.```
````[wpt,l,packetlevels,f] = dwpt(___)` also returns the center frequencies of the approximate passbands in cycles per sample using any of the previous syntaxes.```
````[wpt,l,packetlevels,f,re] = dwpt(___)` also returns the relative energy for the wavelet packets in `wpt` using any of the previous syntaxes. The relative energy is the proportion of energy contained in each wavelet packet by level.```

example

````[___] = dwpt(___,Name,Value)` specifies options using name-value pair arguments in addition to the input arguments in the previous syntaxes. For example, `'Level',4` specifies the decomposition level.```

## Examples

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Load the 23-channel EEG data `Espiga3` [3]. The data is sampled at 200 Hz.

`load Espiga3`

Compute the 1-D DWPT of the data using the `sym3` wavelet down to level 4. Obtain the terminal wavelet packet nodes, bookkeeping vector, and center frequencies of the approximate passbands.

`[wpt,bk,~,f] = dwpt(Espiga3,'sym3','Level',4);`

The output `wpt` is a 1-by-${2}^{4}$ cell array. Every element of `wpt` is a matrix. Choose any terminal node, and confirm the size of the matrix is 23-by-M, where M is the last element of the bookkeeping vector `bk`.

```nd = 13; size(wpt{nd})```
```ans = 1×2 23 66 ```
`bk(end)`
```ans = 66 ```

Extract the final-level coefficients of the fifth channel.

```p5 = cell2mat(cellfun(@(x) x(5,:).',wpt,'UniformOutput',false)); size(p5)```
```ans = 1×2 66 16 ```

The terminal nodes are sequency-ordered. Plot the center frequencies of the approximate passbands in hertz, and confirm they are in order of increasing frequency.

```plot(200*f,'x') title('Center Frequencies') ylabel('Hz')```

## Input Arguments

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Input data, specified as a real-valued vector, matrix, or timetable. If `X` is a matrix, the transform is applied to each column of `X`. If `X` is a timetable, `X` must either contain a matrix in a single variable or column vectors in separate variables. `X` must be uniformly sampled.

Data Types: `single` | `double`

Wavelet to use in the DWPT, specified as a character vector or string scalar. `wname` must be recognized by `wavemngr`.

You cannot specify both `wname` and a filter pair, `LoD` and `HiD`.

Example: `wpt = dwpt(data,"sym4")` specifies the `sym4` wavelet.

Wavelet analysis (decomposition) filters to use in the DWPT, specified as a pair of real-valued vectors. `LoD` is the scaling (lowpass) analysis filter, and `HiD` is the wavelet (highpass) analysis filter. You cannot specify both `wname` and a filter pair, `LoD` and `HiD`. See `wfilters` for additional information.

Note

`dwpt` does not check that `LoD` and `HiD` satisfy the requirements for a perfect reconstruction wavelet packet filter bank. To confirm your filter pair satisfies the requirements, use `isorthwfb` or `isbiorthwfb`.

### Name-Value Arguments

Specify optional pairs of arguments as `Name1=Value1,...,NameN=ValueN`, where `Name` is the argument name and `Value` is the corresponding value. Name-value arguments must appear after other arguments, but the order of the pairs does not matter.

Before R2021a, use commas to separate each name and value, and enclose `Name` in quotes.

Example: `wpt = dwpt(x,'sym4','Level',4)` specifies a level 4 decomposition using the `sym4` wavelet.

Wavelet decomposition level, specified as a positive integer less than or equal to `floor(log2(Ns))`, where Ns is the number of samples in the data. If unspecified, `Level` defaults to `floor(log2(Ns))`.

Wavelet packet tree handling, specified as a numeric or logical `1` (`true`) or `0` (`false`). When set to `true`, `wpt` contains the full packet tree. When set to `false`, `wpt` contains only the terminal nodes. If unspecified, `FullTree` defaults to `false`.

Wavelet packet transform boundary handling, specified as `'reflection'` or `'periodic'`. Setting to `'reflection'` or `'periodic'`, the wavelet packet coefficients are extended at each level based on the `'sym'` or `'per'` mode in `dwtmode`, respectively. If unspecified, `Boundary` defaults to `'reflection'`.

## Output Arguments

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Wavelet packet transform, returned as a 1-by-M cell array. If taking the DWPT of one signal, each element of `wpt` is a vector. Otherwise, each element is a matrix. The coefficients in the jth row of the matrix correspond to the signal in the jth column of `X`. The packets are sequency-ordered.

If returning the terminal nodes of a level N decomposition, `wpt` is a 1-by-2N cell array. If returning the full wavelet packet tree, `wpt` is a 1-by-(2N+1−2) cell array.

Bookkeeping vector, returned as a vector of positive integers. The vector `l` contains the length of the input signal and the number of coefficients by level, and is required for perfect reconstruction.

Transform levels, returned as a vector of positive integers. The ith element of `packetlevels` corresponds to the ith element of `wpt`. If `wpt` contains only the terminal nodes, `packetlevels` is a vector with each element equal to the terminal level. If `wpt` contains the full wavelet packet tree, then `packetlevels` is a vector with 2j elements for each level j.

Center frequencies of the approximate passbands in cycles per sample, returned as a real-valued vector. The jthe element of `f` corresponds to the jth wavelet packet node of `wpt`. You can multiply the elements in `f` by a sampling frequency to convert to cycles per unit time.

Relative energy for the wavelet packets in `wpt`, returned as a cell array. The relative energy is the proportion of energy contained in each wavelet packet by level. The jth element of `re` corresponds to the jth wavelet packet node of `wpt`.

Each element of `re` is a scalar when taking the DWPT of one signal. Otherwise, when taking the DWPT of M signals, each element of `re` is a M-by-1 vector, where the ith element is the relative energy of the ith signal channel. For each channel, the sum of relative energies in the wavelet packets at a given level is equal to 1.

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### Wavelet Packet Decomposition

The wavelet packet method is a generalization of wavelet decomposition that offers a richer signal analysis. Wavelet packet atoms are waveforms indexed by three naturally interpreted parameters: position and scale as in wavelet decomposition, and frequency.

For a given orthogonal wavelet function, a library of wavelet packets bases is generated. Each of these bases offers a particular way of coding signals, preserving global energy and reconstructing exact features. The wavelet packets can then be used for numerous expansions of a given signal.

Simple and efficient algorithms exist for both wavelet packets decomposition and optimal decomposition selection. Adaptive filtering algorithms with direct applications in optimal signal coding and data compression can then be produced.

In the orthogonal wavelet decomposition procedure, the generic step splits the approximation coefficients into two parts. After splitting we obtain a vector of approximation coefficients and a vector of detail coefficients, both at a coarser scale. The information lost between two successive approximations is captured in the detail coefficients. The next step consists in splitting the new approximation coefficient vector; successive details are never re-analyzed.

In the corresponding wavelet packets situation, each detail coefficient vector is also decomposed into two parts using the same approach as in approximation vector splitting. This offers the richest analysis: the complete binary tree is produced in the one-dimensional case or a quaternary tree in the two-dimensional case.

## Algorithms

The `dwpt` function performs a discrete wavelet packet transform and produces a sequency-ordered wavelet packet tree. Compare the sequency-ordered and normal (Paley)-ordered trees. $\stackrel{˜}{G}\left(f\right)$ is the scaling (lowpass) analysis filter, and $\stackrel{˜}{H}\left(f\right)$ represents the wavelet (highpass) analysis filter. The labels at the bottom show the partition of the frequency axis [0, ½].

## References

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

[2] Percival, D. B., and A. T. Walden. Wavelet Methods for Time Series Analysis. Cambridge, UK: Cambridge University Press, 2000.

[3] Mesa, Hector. “Adapted Wavelets for Pattern Detection.” In Progress in Pattern Recognition, Image Analysis and Applications, edited by Alberto Sanfeliu and Manuel Lazo Cortés, 3773:933–44. Berlin, Heidelberg: Springer Berlin Heidelberg, 2005. https://doi.org/10.1007/11578079_96.

## Version History

Introduced in R2020a