# wrcoef

Reconstruct single branch from 1-D wavelet coefficients

## Description

reconstructs the coefficients vector of type `x`

= wrcoef(`type`

,`c`

,`l`

,`wname`

)`type`

based on the
wavelet decomposition structure `[c,l]`

of a 1-D signal (see
`wavedec`

for more information)
using the wavelet specified by `wname`

. The coefficients at the
maximum decomposition level are reconstructed. The length of `x`

is equal to the length of the original 1-D signal.

## Examples

### Reconstruct Wavelet Coefficients

Load a 1-D signal.

```
load sumsin
s = sumsin;
```

Perform a level 5 wavelet decomposition of the signal using the `sym4`

wavelet.

`[c,l] = wavedec(s,5,'sym4');`

Reconstruct the approximation coefficients at level 5 from the wavelet decomposition structure `[c,l]`

.

a5 = wrcoef('a',c,l,'sym4');

Reconstruct the detail coefficients at level 2.

d2 = wrcoef('d',c,l,'sym4',2);

Plot the original signal and reconstructed coefficients.

subplot(3,1,1) plot(s) title('Original Signal') subplot(3,1,2) plot(a5) title('Reconstructed Approximation At Level 5') subplot(3,1,3) plot(d2) title('Reconstructed Details At Level 2')

### Multiresolution Analysis Based on Decimated Discrete Wavelet Transform

This example shows how, starting with a multilevel 1-D discrete wavelet decomposition of a signal, you can obtain projections of the signal onto wavelet subspaces at successive scales and a scaling subspace. These projections are at the same time scale as the original signal. In other words, you can obtain a *multiresolution analysis* (MRA) based on the decimated discrete wavelet transform (DWT). You can recover the signal by summing the projections. For more information, see Practical Introduction to Multiresolution Analysis.

Load and plot a signal.

load noissin plot(noissin) title("Original Signal")

Use the `wavedec`

function to obtain the discrete wavelet decomposition of the signal down to level 3 using the `db4`

wavelet.

```
level = 3;
wv = "db4";
[C,L] = wavedec(noissin,level,wv);
```

Preallocate a matrix to save the MRA. The number of rows is one more than the level of decomposition, and the number of columns equals the length of the signal.

mra = zeros(level+1,numel(noissin));

Use the `wrcoef`

function to obtain the projections of the signal onto the three wavelet (detail) subspaces. Then obtain the projection onto the final scaling (coarse or approximation) subspace.

for k=1:level mra(k,:) = wrcoef("d",C,L,wv,k); end mra(end,:) = wrcoef("a",C,L,wv,level);

Confirm the sum along the rows of the MRA equals the original signal.

mraSum = sum(mra,1); max(abs(mraSum-noissin))

ans = 1.6591e-12

Plot the MRA.

tiledlayout(level+1,1) for k=1:level nexttile plot(mra(k,:)) title("Projection Onto Detail Subspace "+num2str(k)) end nexttile plot(mra(end,:)) title("Projection Onto Approximation Subspace")

## Input Arguments

`type`

— Coefficients to reconstruct

`'a'`

| `'d'`

Coefficients to reconstruct, specified as `'a'`

or
`'d'`

, for approximation or detail coefficients,
respectively.

`wname`

— Analyzing wavelet

character vector | string scalar

Analyzing wavelet used to create the wavelet decomposition structure
`[c,l]`

, specified as a character vector or string
scalar. `wrcoef`

supports only orthogonal or biorthogonal
wavelets. See `wfilters`

.

`LoR,HiR`

— Wavelet reconstruction filters

even-length real-valued vectors

Wavelet reconstruction filters, specified as a pair of even-length
real-valued vectors. `LoR`

is the lowpass reconstruction
filter, and `HiR`

is the highpass reconstruction filter.
The lengths of `LoR`

and `HiR`

must be
equal. See `wfilters`

for additional
information.

## Version History

**Introduced before R2006a**

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