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Maximal overlap discrete wavelet transform

computes
the MODWT using reflection boundary handling. Other inputs can be
any of the arguments from previous syntaxes. Before computing the
wavelet transform, `w`

= modwt(___,'reflection')`modwt`

extends the signal symmetrically
at the right boundary to twice the signal length, `[x flip(x)]`

.
The number of wavelet and scaling coefficients that `modwt`

returns
is equal to twice the length of the input signal. By default, the
signal is extended periodically.

The standard algorithm for the MODWT implements the circular
convolution directly in the time domain. This implementation of the
MODWT performs the circular convolution in the Fourier domain. The
wavelet and scaling filter coefficients at level j are computed by
taking the inverse discrete Fourier transform (DFT) of a product of
DFTs. The DFTs in the product are the signal’s DFT and the
DFT of the j^{th} level wavelet or scaling
filter.

Let *H _{k}* and

The j^{th} level wavelet filter is defined
by

$$\frac{1}{N}{\displaystyle \sum _{k=0}^{N-1}{H}_{j,k}}{e}^{i2\pi nk/N}$$

where

$${H}_{j,k}={H}_{{2}^{j-1}k\text{mod}N}{\displaystyle \prod _{m=0}^{j-2}{G}_{{2}^{m}k\text{mod}N}}$$

The j^{th} level scaling filter is

$$\frac{1}{N}{\displaystyle \sum _{k=0}^{N-1}{G}_{j,k}}{e}^{i2\pi nk/N}$$

where

$${G}_{j,k}={\displaystyle \prod _{m=0}^{j-1}{G}_{{2}^{m}k\text{mod}N}}$$

[1] Percival, Donald B., and Andrew T. Walden. *Wavelet Methods for Time Series
Analysis*. Cambridge Series in Statistical and Probabilistic Mathematics.
Cambridge ; New York: Cambridge University Press, 2000.

[2] Percival, Donald B., and Harold O. Mofjeld. “Analysis of Subtidal Coastal Sea Level
Fluctuations Using Wavelets.” *Journal of the American
Statistical Association* 92, no. 439 (September 1997): 868–80.
https://doi.org/10.1080/01621459.1997.10474042.