1-D wavelet decomposition

`[`

returns the wavelet decomposition of the 1-D signal `c`

,`l`

] = wavedec(`x`

,`n`

,`wname`

)`x`

at level
`n`

using the wavelet `wname`

. The output
decomposition structure consists of the wavelet decomposition vector
`c`

and the bookkeeping vector `l`

, which
contains the number of coefficients by level. The structure is organized as in this
level-3 decomposition diagram.

Given a signal *s* of length *N*, the DWT consists
of at most log_{2}
*N* steps. Starting from *s*, the first step produces
two sets of coefficients: approximation coefficients
*cA _{1}* and detail coefficients

`LoD`

and the highpass filter
`HiD`

, followed by dyadic decimation (downsampling), results in the
approximation and detail coefficients respectively. where

— Convolve with filter

*X*$$\begin{array}{||}\hline \downarrow 2\\ \hline\end{array}$$ — Downsample (keep the even-indexed elements)

The length of each filter is equal to 2*n*. If *N* =
length(*s*), the signals *F* and
*G* are of length *N* + 2*n* −1
and the coefficients *cA _{1}* and

floor$$\left(\frac{N-1}{2}\right)+n$$.

The next step splits the approximation coefficients
*cA _{1}* in two parts using the same
scheme, replacing

The wavelet decomposition of the signal *s* analyzed at level
*j* has the following structure:
[*cA _{j}*,

This structure contains, for *j* = 3, the terminal nodes of the
following tree:

[1] Daubechies, I. *Ten Lectures on Wavelets*, CBMS-NSF Regional
Conference Series in Applied Mathematics. Philadelphia, PA: SIAM Ed, 1992.

[2] Mallat, S. G. “A Theory for Multiresolution Signal Decomposition: The
Wavelet Representation,” *IEEE Transactions on Pattern Analysis and
Machine Intelligence*. Vol. 11, Issue 7, July 1989, pp.
674–693.

[3] Meyer, Y. *Wavelets and Operators*. Translated by D. H.
Salinger. Cambridge, UK: Cambridge University Press, 1995.