# wavedec

Multilevel 1-D discrete wavelet transform

## Syntax

``[c,l] = wavedec(x,n,wname)``
``[c,l] = wavedec(x,n,LoD,HiD)``
``[c,l] = wavedec(___,Mode=extmode)``

## Description

example

````[c,l] = wavedec(x,n,wname)` returns the wavelet decomposition of the 1-D signal `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 is used to parse `c`.```
````[c,l] = wavedec(x,n,LoD,HiD)` returns the wavelet decomposition using the specified lowpass and highpass wavelet decomposition filters `LoD` and `HiD`, respectively.```

example

````[c,l] = wavedec(___,Mode=extmode)` uses the specified discrete wavelet transform (DWT) extension mode `extmode`. This syntax can be used with either of the previous syntaxes.```

## Examples

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Load and plot a one-dimensional signal.

```load sumsin plot(sumsin) title("Signal")``` Perform a 3-level wavelet decomposition of the signal using the order 2 Daubechies wavelet. Extract the coarse scale approximation coefficients and the detail coefficients from the decomposition.

```[c,l] = wavedec(sumsin,3,"db2"); approx = appcoef(c,l,"db2"); [cd1,cd2,cd3] = detcoef(c,l,[1 2 3]);```

Plot the coefficients.

```tiledlayout(4,1) nexttile plot(approx) title("Approximation Coefficients") nexttile plot(cd3) title("Level 3 Detail Coefficients") nexttile plot(cd2) title("Level 2 Detail Coefficients") nexttile plot(cd1) title("Level 1 Detail Coefficients")``` Refer to GPU Computing Requirements (Parallel Computing Toolbox) to see what GPUs are supported.

Load the noisy Doppler signal. Put the signal on the GPU using `gpuArray`. Save the current global DWT extension mode.

```load noisdopp noisdoppg = gpuArray(noisdopp); origMode = dwtmode("status","nodisp");```

Obtain the three-level DWT of the signal on the GPU using the `db4` wavelet. Even though the `zpd` extension mode is not supported for `gpuArray` input, specify it. The `wavedec` function instead uses the `sym` extension mode.

`[czpd,lzpd] = wavedec(noisdoppg,3,"db4",Mode="zpd");`

Confirm that the `wavedec` function did use the `sym` extension mode. Set the global extension mode to `sym` and obtain the three-level DWT of `noisdoppg` without specifying an extension mode. Confirm this result equals the previous result.

```dwtmode("sym","nodisp") [csym,lsym] = wavedec(noisdoppg,3,"db4"); [max(abs(czpd-csym)) max(abs(lzpd-lsym))]```
```ans = 0 0 ```

Obtain the three-level DWT of `noisdoppg` using the `per` extension mode, which is supported for `gpuArray` input. Confirm this result is different from the `sym` result.

```[cper,lper] = wavedec(noisdoppg,3,"db4",Mode="per"); [length(csym) ; length(cper)]```
```ans = 2×1 1044 1024 ```
`[lsym ; lper]`
```ans = 134 134 261 515 1024 128 128 256 512 1024 ```

Restore the global extension mode to the original setting.

`dwtmode(origMode,"nodisp")`

## Input Arguments

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Input signal, specified as a vector.

Data Types: `single` | `double`
Complex Number Support: Yes

Level of decomposition, specified as a positive integer. `wavedec` does not enforce a maximum level restriction. Use `wmaxlev` to ensure that the wavelet coefficients are free from boundary effects. If boundary effects are not a concern in your application, a good rule is to set `n` less than or equal to `fix(log2(length(x)))`.

Data Types: `single` | `double`

Analyzing wavelet, specified as a character vector or string scalar.

Note

`wavedec` supports only Type 1 (orthogonal) or Type 2 (biorthogonal) wavelets. See `wfilters` for a list of orthogonal and biorthogonal wavelets.

Wavelet decomposition filters, specified as a pair of even-length real-valued vectors. `LoD` is the lowpass decomposition filter, and `HiD` is the highpass decomposition filter. The lengths of `LoD` and `HiD` must be equal. For more information, see `wfilters`.

Data Types: `single` | `double`

Since R2023b

Extension mode to use for the DWT, specified as:

`mode`

DWT Extension Mode

`"zpd"`

Zero extension

`"sp0"`

Smooth extension of order 0

`"spd"` (or``` "sp1"```)

Smooth extension of order 1

`"sym"` or `"symh"`

Symmetric extension (half point): boundary value symmetric replication

`"symw"`

Symmetric extension (whole point): boundary value symmetric replication

`"asym"` or `"asymh"`

Antisymmetric extension (half point): boundary value antisymmetric replication

`"asymw"`

Antisymmetric extension (whole point): boundary value antisymmetric replication

`"ppd"`, `"per"`

Periodized extension

If the signal length is odd and `mode` is `"per"`, an extra sample equal to the last value is added to the right and the extension is performed in `"ppd"` mode. If the signal length is even, `"per"` is equivalent to `"ppd"`.

The global variable managed by `dwtmode` specifies the default extension mode.

## Output Arguments

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Wavelet decomposition vector, returned as a vector.

Data Types: `single` | `double`

Bookkeeping vector, returned as a vector of positive integers. The vector contains the number of coefficients by level and the length of the original signal.

The bookkeeping vector is used to parse the coefficients in the wavelet decomposition vector `c` by level. The decomposition vector and bookkeeping vector are organized as in this level-3 decomposition diagram. Data Types: `single` | `double`

## Algorithms

Given a signal s of length N, the DWT consists of at most log2 N steps. Starting from s, the first step produces two sets of coefficients: approximation coefficients cA1 and detail coefficients cD1. Convolving s with the lowpass filter `LoD` and the highpass filter `HiD`, followed by dyadic decimation (downsampling by 2), results in the approximation and detail coefficients respectively. where

• — Convolve with filter X

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

The length of each filter is equal to 2n. If N = length(s), the signals F and G are of length N + 2n −1 and the coefficients cA1 and cD1 are of length

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

The next step splits the approximation coefficients cA1 in two parts using the same scheme, replacing s by cA1, and producing cA2 and cD2, and so on. The wavelet decomposition of the signal s analyzed at level j has the following structure: [cAj, cDj, ..., cD1].

This structure contains, for j = 3, the terminal nodes of the following tree: Daubechies, I. Ten Lectures on Wavelets, CBMS-NSF Regional Conference Series in Applied Mathematics. Philadelphia, PA: SIAM Ed, 1992.

 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.

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