# Documentation

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# idddtree

Inverse dual-tree and double-density 1-D wavelet transform

## Syntax

``xrec = idddtree(wt)``

## Description

example

````xrec = idddtree(wt)` returns the inverse wavelet transform of the wavelet decomposition (analysis filter bank), `wt`. `wt` is the output of `dddtree`.```

## Examples

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Demonstrate perfect reconstruction of a signal using a dual-tree double-density wavelet transform.

Load the noisy Doppler signal. Obtain the dual-tree double-density wavelet transform down to level 5. Invert the transform and demonstrate perfect reconstruction.

```load noisdopp; wt = dddtree('cplxdddt',noisdopp,5,'FSdoubledualfilt',... 'doubledualfilt'); xrec = idddtree(wt); max(abs(noisdopp-xrec))```
```ans = 1.9291e-12 ```

## Input Arguments

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Wavelet transform, returned as a structure from `dddtree` with these fields:

Type of wavelet decomposition (filter bank), specified as one of `'dwt'`, `'ddt'`, `'cplxdt'`, or `'cplxdddt'`. The type,`'dwt'`, gives a critically sampled discrete wavelet transform. The other types are oversampled wavelet transforms. `'ddt'` is a double-density wavelet transform, `'cplxdt'` is a dual-tree complex wavelet transform, and `'cplxdddt'` is a double-density dual-tree complex wavelet transform.

Level of wavelet decomposition, specified as a positive integer.

Decomposition (analysis) and reconstruction (synthesis) filters, specified as a structure with these fields:

First-stage analysis filters, specified as an N-by-2 or N-by-3 matrix for single-tree wavelet transforms, or a cell array of two N-by-2 or N-by-3 matrices for dual-tree wavelet transforms. The matrices are N-by-3 for the double-density wavelet transforms. For an N-by-2 matrix, the first column of the matrix is the scaling (lowpass) filter and the second column is the wavelet (highpass) filter. For an N-by-3 matrix, the first column of the matrix is the scaling (lowpass) filter and the second and third columns are the wavelet (highpass) filters. For the dual-tree transforms, each element of the cell array contains the first-stage analysis filters for the corresponding tree.

Analysis filters for levels > 1, specified as an N-by-2 or N-by-3 matrix for single-tree wavelet transforms, or a cell array of two N-by-2 or N-by-3 matrices for dual-tree wavelet transforms. The matrices are N-by-3 for the double-density wavelet transforms. For an N-by-2 matrix, the first column of the matrix is the scaling (lowpass) filter and the second column is the wavelet (highpass) filter. For an N-by-3 matrix, the first column of the matrix is the scaling (lowpass) filter and the second and third columns are the wavelet (highpass) filters. For the dual-tree transforms, each element of the cell array contains the analysis filters for the corresponding tree.

First-level reconstruction filters, specified as an N-by-2 or N-by-3 matrix for single-tree wavelet transforms, or a cell array of two N-by-2 or N-by-3 matrices for dual-tree wavelet transforms. The matrices are N-by-3 for the double-density wavelet transforms. For an N-by-2 matrix, the first column of the matrix is the scaling (lowpass) filter and the second column is the wavelet (highpass) filter. For an N-by-3 matrix, the first column of the matrix is the scaling (lowpass) filter and the second and third columns are the wavelet (highpass) filters. For the dual-tree transforms, each element of the cell array contains the first-stage synthesis filters for the corresponding tree.

Reconstruction filters for levels > 1, specified as an N-by-2 or N-by-3 matrix for single-tree wavelet transforms, or a cell array of two N-by-2 or N-by-3 matrices for dual-tree wavelet transforms. The matrices are N-by-3 for the double-density wavelet transforms. For an N-by-2 matrix, the first column of the matrix is the scaling (lowpass) filter and the second column is the wavelet (highpass) filter. For an N-by-3 matrix, the first column of the matrix is the scaling (lowpass) filter and the second and third columns are the wavelet (highpass) filters. For the dual-tree transforms, each element of the cell array contains the synthesis filters for the corresponding tree.

Wavelet transform coefficients, specified as a 1-by-(`level`+1) cell array of matrices. The size and structure of the matrix elements of the cell array depend on the type of wavelet transform as follows:

• `'dwt'``cfs{j}`

• j = 1,2,... `level` is the level.

• `cfs{level+1}` are the lowpass, or scaling, coefficients.

• `'ddt'``cfs{j}(:,:,k)`

• j = 1,2,... `level` is the level.

• k = 1,2 is the wavelet filter.

• `cfs{level+1}(:,:)` are the lowpass, or scaling, coefficients.

• `'cplxdt'``cfs{j}(:,:,m)`

• j = 1,2,... `level` is the level.

• m = 1,2 are the real and imaginary parts.

• `cfs{level+1}(:,:)` are the lowpass, or scaling, coefficients.

• `'cplxdddt'``cfs{j}(:,:,k,m)`

• j = 1,2 `level` is the level.

• k = 1,2 is the wavelet filter.

• m = 1,2 are the real and imaginary parts.

• `cfs{level+1}(:,:)` are the lowpass, or scaling, coefficients.

## Output Arguments

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Synthesized 1-D signal, returned as a vector. The row or column orientation of `xrec` depends on the row or column orientation of the 1-D signal input to `dddtree`.

Data Types: `double`