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Maximal overlap discrete wavelet packet transform

`wpt = modwpt(x)`

`wpt = modwpt(x,wname)`

`wpt = modwpt(x,lo,hi)`

`wpt = modwpt(___,lev)`

```
[wpt,packetlevs]
= modwpt(___)
```

```
[wpt,packetlevs,cfreq]
= modwpt(___)
```

```
[wpt,packetlevs,cfreq,energy]
= modwpt(___)
```

```
[wpt,packetlevs,cfreq,energy,relenergy]
= modwpt(___)
```

`[___] = modwpt(___,Name,Value)`

returns
the terminal nodes for the maximal overlap discrete wavelet packet
transform (MODWPT) for the 1-D real-valued signal, `wpt`

= modwpt(`x`

)`x`

.

The output of the MODWPT is time-delayed compared to the input signal. Most filters used to obtain the MODWPT have a nonlinear phase response, which makes compensating for the time delay difficult. This is true for all orthogonal scaling and wavelet filters, except the Haar wavelet. It is possible to time-align the coefficients with the signal features, but the result is an approximation, not an exact alignment with the original signal. The MODWPT partitions the energy among the wavelet packets at each level. The sum of the energy over all the packets equals the total energy of the input signal. The output of MODWPT is useful for applications where you want to analyze the energy levels in different packets.

The MODWPT details (`modwptdetails`

)
are the result of zero-phase filtering of the signal. The features
in the MODWPT details align exactly with features in the input signal.
For a given level, summing the details for each sample returns the
exact original signal. The output of the MODWPT details is useful
for applications that require time-alignment, such as nonparametric
regression analysis.

`[`

returns a vector of transform
levels corresponding to the rows of `wpt`

,`packetlevs`

]
= modwpt(___)`wpt`

.

`[`

returns the center frequencies
of the approximate passbands corresponding to the rows of `wpt`

,`packetlevs`

,`cfreq`

]
= modwpt(___)`wpt`

.

`[`

returns the energy (squared L2
norm) of the wavelet packet coefficients for the nodes in `wpt`

,`packetlevs`

,`cfreq`

,`energy`

]
= modwpt(___)`wpt`

.

`[___] = modwpt(___,`

returns
the MODWPT with additional options specified by one or more `Name,Value`

)`Name,Value`

pair
arguments.

The `modwpt`

performs a discrete wavelet
packet transform and produces a sequency-ordered wavelet packet tree.
Compare the sequency-ordered and normal (Paley)-ordered trees.

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

[2] Walden, A.T., and A. Contreras Cristan. “The phase-corrected
undecimated discrete wavelet packet transform and its application
to interpreting the timing of events.” *Proceedings
of the Royal Society of London A*. Vol. 454, Issue 1976,
1998, pp. 2243-2266.