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Maximal overlap discrete wavelet packet transform details

`w = modwptdetails(x)`

`w = modwptdetails(x,wname)`

`w = modwptdetails(x,lo,hi)`

`w = modwptdetails(___,lev)`

```
[w,packetlevs]
= modwptdetails(___)
```

```
[w,packetlevs,cfreq]
= modwptdetails(___)
```

`[___] = modwptdetails(___,'FullTree',tf)`

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

= modwptdetails(`x`

)`x`

. The MODWPT
details provide zero-phase filtering of the signal. By default, `modwptdetails`

returns
only the terminal nodes, which are at level 4 or at level `floor(log2(numel(x)))`

,
whichever is smaller.

To decide whether to use `modwptdetails`

or `modwpt`

, consider the type of data analysis
you need to perform. For applications that require time alignment,
such as nonparametric regression analysis, use `modwptdetails`

.
For applications where you want to analyze the energy levels in different
packets, use `modwpt`

. For more information, see Algorithms

`[`

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

,`packetlevs`

]
= modwptdetails(___)`w`

.

`[`

returns `w`

,`packetlevs`

,`cfreq`

]
= modwptdetails(___)`cfreq`

, the center frequencies of the approximate
passbands corresponding to the MODWPT details in `w`

.

`[___] = modwptdetails(___,'FullTree',`

,
where `tf`

)`tf`

is `false`

, returns details about
only the terminal (final-level) wavelet packet nodes. If you specify
`true`

, then `modwptdetails`

returns
details about the full wavelet packet tree down to the default or specified
level. The default for `tf`

is
`false`

.

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 (`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. All orthogonal scaling and wavelet filters
have this response, 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.

[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.