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Discrete stationary wavelet transform 1-D

Given a signal *s* of length *N*, the first step of
the stationary wavelet transform (SWT) produces, starting from *s*, two
sets of coefficients: approximation coefficients
*cA _{1}* and detail coefficients

`LoD`

for
approximation, and with the highpass filter `HiD`

for detail.More precisely, the first step is

where denotes convolution with the filter *X*.

**Note**

*cA _{1}* and

`N`

instead of `N/2`

as in the DWT
case.The next step splits the approximation coefficients
*cA _{1}* in two parts using the same
scheme, but with modified filters obtained by upsampling the filters used for the
previous step and replacing

where

*F*_{0}=*LoD**G*_{0}=*HiD*— Upsample (insert zeros between elements)

[1] Nason, G. P., and B. W.
Silverman. “The Stationary Wavelet Transform and Some Statistical Applications.” In
*Wavelets and Statistics*, edited by Anestis Antoniadis and
Georges Oppenheim, 103:281–99. New York, NY: Springer New York, 1995.
https://doi.org/10.1007/978-1-4612-2544-7_17.

[2] Coifman, R. R., and D. L.
Donoho. “Translation-Invariant De-Noising.” In *Wavelets and
Statistics*, edited by Anestis Antoniadis and Georges Oppenheim,
103:125–50. New York, NY: Springer New York, 1995.
https://doi.org/10.1007/978-1-4612-2544-7_9.

[3] Pesquet, J.-C., H. Krim, and
H. Carfantan. “Time-Invariant Orthonormal Wavelet Representations.” *IEEE
Transactions on Signal Processing* 44, no. 8 (August 1996): 1964–70.
https://doi.org/10.1109/78.533717.