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# appcoef

1-D approximation coefficients

## Syntax

``A = appcoef(C,L,wname)``
``A = appcoef(C,L,LoR,HiR)``
``A = appcoef(___,N)``

## Description

````A = appcoef(C,L,wname)` returns the approximation coefficients at the coarsest scale using the wavelet decomposition structure [`C`,`L`] of a 1-D signal and the wavelet specified by `wname`. (See `wavedec` for more information.)```
````A = appcoef(C,L,LoR,HiR)` uses the lowpass reconstruction filter `LoR` and highpass reconstruction filter `HiR`. (See `wfilters` for more information.)```

example

````A = appcoef(___,N)` returns the approximation coefficients at level `N`. If [`C`,`L`] is the `M`-level wavelet decomposition structure of a 1-D signal, then `0 ≤ N ≤ M`.```

## Examples

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This example shows how to extract the level 3 approximation coefficients.

Load the signal consisting of electricity usage data.

```load leleccum; sig = leleccum(1:3920);```

Obtain the DWT down to level 5 with the `'sym4'` wavelet.

`[C,L] = wavedec(sig,5,'sym4');`

Extract the level-3 approximation coefficients. Plot the original signal and the approximation coefficients.

```Lev = 3; a3 = appcoef(C,L,'sym4',Lev); subplot(2,1,1) plot(sig); title('Original Signal'); subplot(2,1,2) plot(a3); title('Level-3 Approximation Coefficients');``` You can substitute any value from 1 to 5 for `Lev` to obtain the approximation coefficients for the corresponding level.

## Input Arguments

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Wavelet decomposition vector of a 1-D signal, specified as a real-valued vector. `C` is the output of `wavedec`. The bookkeeping vector `L` is used to parse the coefficients in the wavelet decomposition vector by level.

Example: `[C,L] = wavedec(randn(1,256),4,'coif1')` returns the 4-level wavelet decomposition of a vector.

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

Bookkeeping vector of the wavelet decomposition of a 1-D signal, specified as a vector of positive integers. The bookkeeping vector is used to parse the coefficients in the wavelet decomposition vector `C` by level.

Example: `[C,L] = wavedec(randn(1,256),4,'coif1')` returns the 4-level wavelet decomposition of a vector.

Data Types: `single` | `double`

Wavelet used to generate the wavelet decomposition of a 1-D signal, specified as a character vector or string scalar. The wavelet is from one of the following wavelet families: Daubechies, Coiflets, Symlets, Fejér-Korovkin, Discrete Meyer, Biorthogonal, and Reverse Biorthogonal. See `wavemngr` for the wavelets available in each family.

Example: `'db4'`

Wavelet lowpass reconstruction filter, specified as an even-length real-valued vector. `LoR` must be the same length as `HiR`. `LoR` must be the lowpass reconstruction filter associated with the wavelet used to create the wavelet decomposition structure [`C`,`L`]. (See `wfilters` for more information.)

Wavelet highpass reconstruction filter, specified as an even-length real-valued vector. `HiR` must be the same length as `LoR`. `HiR` must be the highpass reconstruction filter associated with the wavelet used to create the wavelet decomposition structure [`C`,`L`]. (See `wfilters` for more information.)

Approximation coefficients level, specified as a positive integer. If [`C`,`L`] is the `M`-level wavelet decomposition structure of a 1-D signal, then `0 ≤ N ≤ M`.

## Output Arguments

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Approximation coefficients at level `N`, returned as a real-valued vector.

## Algorithms

The input vectors `C` and `L` contain all the information about the signal decomposition.

Let `NMAX = length(L)-2`; then ```C = [A(NMAX) D(NMAX) ... D(1)]``` where `A` and the `D` are vectors. If `N = NMAX`, then a simple extraction is done; otherwise, `appcoef` computes iteratively the approximation coefficients using the inverse wavelet transform.

## See Also

Introduced before R2006a

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