Wavelet Toolbox |

Multilevel 1-D wavelet decomposition

**Syntax **

**Description **

`wavedec`

performs a multilevel one-dimensional wavelet analysis using either a specific wavelet ('

') or a specific wavelet decomposition filters (*wname*`Lo_D`

and `Hi_D`

, see `wfilters`

).

`[C,L] = wavedec(X,N,`

'

'*wname*`)`

returns the wavelet decomposition of the signal `X`

at level `N`

, using '

'. *wname*`N`

must be a strictly positive integer (see `wmaxlev`

for more information). The output decomposition structure contains the wavelet decomposition vector `C`

and the bookkeeping vector `L`

. The structure is organized as in this level-3 decomposition example:

`[C,L] = wavedec(X,N,Lo_D,Hi_D)`

returns the decomposition structure as above, given the low- and high-pass decomposition filters you specify.

**Examples**

% The current extension mode is zero-padding (see

`dwtmode`

). % Load original one-dimensional signal. load sumsin; s = sumsin; % Perform decomposition at level 3 of s using db1. [c,l] = wavedec(s,3,'db1'); % Using some plotting commands, % the following figure is generated.

**Algorithm **

Given a signal *s* of length *N*, the DWT consists of log_{2} *N* stages at most. The first step produces, starting from *s*, two sets of coefficients: approximation coefficients *CA*_{1}, and detail coefficients *CD*_{1}. These vectors are obtained by convolving *s* with the low-pass filter `Lo_D`

for approximation, and with the high-pass filter `Hi_D`

for detail, followed by dyadic decimation (downsampling).

More precisely, the first step is

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 *cD*_{1} are of length

The next step splits the approximation coefficients *cA*_{1} in two parts using the same scheme, replacing *s* by *cA*_{1}, and producing *cA*_{2} and *cD*_{2}, and so on

The wavelet decomposition of the signal *s* analyzed at level *j* has the following structure: [*cA*_{j}, *cD*_{j}, ..., *cD*_{1}].

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

**See Also**

`dwt`

, `waveinfo`

, `waverec`

, `wfilters`

, `wmaxlev`

**References**

Daubechies, I. (1992), *Ten lectures on wavelets*, CBMS-NSF conference series in applied mathematics. SIAM Ed.

Mallat, S. (1989), "A theory for multiresolution signal decomposition: the wavelet representation," *IEEE Pattern Anal. and Machine Intell*., vol. 11, no. 7, pp 674-693.

Meyer, Y. (1990), *Ondelettes et opérateurs*, Tome 1, Hermann Ed. (English translation: *Wavelets and operators*, Cambridge Univ. Press. 1993.)

upwlev2 | wavedec2 |