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# dddtree

Dual-tree and double-density 1-D wavelet transform

## Syntax

``wt = dddtree(typetree,x,level,fdf,df)``
``wt = dddtree(typetree,x,level,fname)``
``wt = dddtree(typetree,x,level,fname1,fname2)``

## Description

example

````wt = dddtree(typetree,x,level,fdf,df)` returns the `typetree` discrete wavelet transform (DWT) of the 1-D input signal, `x`, down to level, `level`. The wavelet transform uses the decomposition (analysis) filters, `fdf`, for the first level and the analysis filters, `df`, for subsequent levels. Supported wavelet transforms are the critically sampled DWT, double-density, dual-tree complex, and dual-tree double-density complex wavelet transform. The critically sampled DWT is a filter bank decomposition in an orthogonal or biorthogonal basis (nonredundant). The other wavelet transforms are oversampled filter banks.```

example

````wt = dddtree(typetree,x,level,fname)` uses the filters specified by `fname` to obtain the wavelet transform. Valid filter specifications depend on the type of wavelet transform. See `dtfilters` for details.```

example

````wt = dddtree(typetree,x,level,fname1,fname2)` uses the filters specified in `fname1` for the first stage of the dual-tree wavelet transform and the filters specified in `fname2` for subsequent stages of the dual-tree wavelet transform. Specifying different filters for stage 1 is valid and necessary only when `typetree` is `'cplxdt'` or `'cplxdddt'`.```

## Examples

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Obtain the complex dual-tree wavelet transform of the noisy Doppler signal. The FIR filters in the first and subsequent stages result in an approximately analytic wavelet as required.

Create the first-stage analysis filters for the two trees.

``` Faf{1} = [0 0 -0.0884 -0.0112 0.0884 0.0112 0.6959 0.0884 0.6959 0.0884 0.0884 -0.6959 -0.0884 0.6959 0.0112 -0.0884 0.0112 -0.0884 0 0]; Faf{2} = [ 0.0112 0 0.0112 0 -0.0884 -0.0884 0.0884 -0.0884 0.6959 0.6959 0.6959 -0.6959 0.0884 0.0884 -0.0884 0.0884 0 0.0112 0 -0.0112];```

Create the analysis filters for subsequent stages of the multiresolution analysis.

```af{1} = [ 0.0352 0 0 0 -0.0883 -0.1143 0.2339 0 0.7603 0.5875 0.5875 -0.7603 0 0.2339 -0.1143 0.0883 0 0 0 -0.0352]; af{2} = [0 -0.0352 0 0 -0.1143 0.0883 0 0.2339 0.5875 -0.7603 0.7603 0.5875 0.2339 0 -0.0883 -0.1143 0 0 0.0352 0];```

Load the noisy Doppler signal and obtain the complex dual-tree wavelet transform down to level 4.

```load noisdopp; wt = dddtree('cplxdt',noisdopp,4,Faf,af);```

Plot an approximation based on the level-four approximation coefficients.

```xapp = dddtreecfs('r',wt,'scale',{5}); plot(noisdopp); hold on; plot(cell2mat(xapp),'r','linewidth',3); axis tight;```

Obtain the double-density wavelet transform of a signal with two discontinuities. Use the level-one detail coefficients to localize the discontinuities.

Create a signal consisting of a 2-Hz sine wave with a duration of 1 second. The sine wave has discontinuities at 0.3 and 0.72 seconds.

```N = 1024; t = linspace(0,1,1024); x = 4*sin(4*pi*t); x = x - sign(t - .3) - sign(.72 - t); plot(t,x); xlabel('t'); ylabel('x'); title('Original Signal');```

Obtain the double-density wavelet transform of the signal, reconstruct an approximation based on the level-one detail coefficients, and plot the result.

```wt = dddtree('ddt',x,1,'filters1'); wt.cfs{2} = zeros(1,512); xrec = idddtree(wt); plot(t,xrec,'linewidth',2) set(gca,'xtick',[0 0.3 0.72 1]); set(gca,'xgrid','on');```

Obtain the complex dual-tree wavelet transform of a signal with two discontinuities. Use the first-level detail coefficients to localize the discontinuities.

Create a signal consisting of a 2-Hz sine wave with a duration of 1 second. The sine wave has discontinuities at 0.3 and 0.72 seconds.

```N = 1024; t = linspace(0,1,1024); x = 4*sin(4*pi*t); x = x - sign(t - .3) - sign(.72 - t); plot(t,x); xlabel('t'); ylabel('x'); title('Original Signal');```

Obtain the dual-tree wavelet transform of the signal, reconstruct an approximation based on the level-one detail coefficients, and plot the result.

```wt = dddtree('cplxdt',x,1,'FSfarras','qshift06'); wt.cfs{2} = zeros(1,512,2); xrec = idddtree(wt); plot(t,xrec,'linewidth',2) set(gca,'xtick',[0 0.3 0.72 1]); set(gca,'xgrid','on');```

## Input Arguments

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

Input signal, specified as an even-length row or column vector. If L is the value of the `level` of the wavelet decomposition, 2L must divide the length of `x`. Additionally, the length of the signal must be greater than or equal to the product of the maximum length of the decomposition (analysis) filters and 2(L-1).

Data Types: `double`

Level of the wavelet decomposition, specified as an integer. If L is the value of `level`, 2L must divide the length of `x` . Additionally, the length of the signal must be greater than or equal to the product of the maximum length of the decomposition (analysis) filters and 2(L-1).

Data Types: `double`

The level-one analysis filters, specified as a matrix or cell array of matrices. Specify `fdf` as a matrix when `typetree` is `'dwt'` or `'ddt'`. The size and structure of the matrix depend on the `typetree` input as follows:

• `'dwt'` — This is the critically sampled discrete wavelet transform. In this case, `fdf` is a two-column matrix with the lowpass (scaling) filter in the first column and the highpass (wavelet) filter in the second column.

• `'ddt'` — This is the double-density wavelet transform. The double-density DWT is a three-channel perfect reconstruction filter bank. `fdf` is a three-column matrix with the lowpass (scaling) filter in the first column and the two highpass (wavelet) filters in the second and third columns. In the double-density wavelet transform, the single lowpass and two highpass filters constitute a three-channel perfect reconstruction filter bank. This is equivalent to the three filters forming a tight frame. You cannot arbitrarily choose the two wavelet filters in the double-density DWT. The three filters together must form a tight frame.

Specify `fdf` as a 1-by-2 cell array of matrices when `typetree` is a dual-tree transform, `'cplxdt'` or `'cplxdddt'`. The size and structure of the matrix elements depend on the `typetree` input as follows:

• For the dual-tree complex wavelet transform, `'cplxdt'`, `fdf{1}` is a two-column matrix containing the lowpass (scaling) filter and highpass (wavelet) filters for the first tree. The scaling filter is the first column and the wavelet filter is the second column. `fdf{2}` is a two-column matrix containing the lowpass (scaling) and highpass (wavelet) filters for the second tree. The scaling filter is the first column and the wavelet filter is the second column.

• For the double-density dual-tree complex wavelet transform, `'cplxdddt'`, `fdf{1}` is a three-column matrix containing the lowpass (scaling) and two highpass (wavelet) filters for the first tree and `fdf{2}` is a three-column matrix containing the lowpass (scaling) and two highpass (wavelet) filters for the second tree.

Data Types: `double`

Analysis filters for levels > 1, specified as a matrix or cell array of matrices. Specify `df` as a matrix when `typetree` is `'dwt'` or `'ddt'`. The size and structure of the matrix depend on the `typetree` input as follows:

• `'dwt'` — This is the critically sampled discrete wavelet transform. In this case, `df` is a two-column matrix with the lowpass (scaling) filter in the first column and the highpass (wavelet) filter in the second column. For the critically sampled orthogonal or biorthogonal DWT, the filters in `df` and `fdf` must be identical.

• `'ddt'` — This is the double-density wavelet transform. The double-density DWT is a three-channel perfect reconstruction filter bank. `df` is a three-column matrix with the lowpass (scaling) filter in the first column and the two highpass (wavelet) filters in the second and third columns. In the double-density wavelet transform, the single lowpass and two highpass filters must constitute a three-channel perfect reconstruction filter bank. This is equivalent to the three filters forming a tight frame. For the double-density DWT, the filters in `df` and `fdf` must be identical.

Specify `df` as a 1-by-2 cell array of matrices when `typetree` is a dual-tree transform, `'cplxdt'` or `'cplxdddt'`. For dual-tree transforms, the filters in `fdf` and `df` must be different. The size and structure of the matrix elements in the cell array depend on the `typetree` input as follows:

• For the dual-tree complex wavelet transform, `'cplxdt'`, `df{1}` is a two-column matrix containing the lowpass (scaling) and highpass (wavelet) filters for the first tree. The scaling filter is the first column and the wavelet filter is the second column. `df{2}` is a two-column matrix containing the lowpass (scaling) and highpass (wavelet) filters for the second tree. The scaling filter is the first column and the wavelet filter is the second column.

• For the double-density dual-tree complex wavelet transform, `'cplxdddt'`, `df{1}` is a three-column matrix containing the lowpass (scaling) and two highpass (wavelet) filters for the first tree and `df{2}` is a three-column matrix containing the lowpass (scaling) and two highpass (wavelet) filters for the second tree.

Data Types: `double`

Filter name, specified as a character vector. For the critically sampled DWT, specify any valid orthogonal or biorthogonal wavelet filter. See `wfilters` for details. For the double-density wavelet transform, `'ddt'`, valid choices are `'filters1'` and `'filters2'`. For the complex dual-tree wavelet transform, valid choices are `'dtfP'` with P = 1, 2, 3, 4. For the double-density dual-tree wavelet transform, the only valid choice is `'dddtf1'`. See `dtfilters` for more details on valid filter names for the oversampled wavelet filter banks.

Data Types: `char`

First-stage filter name, specified as a character vector. Specifying a different filter for the first stage is valid and necessary only in the dual-tree transforms, `'cplxdt'` and `'cplxddt'`. In the complex dual-tree wavelet transform, you can use any valid wavelet filter for the first stage. In the double-density dual-tree wavelet transform, the first-stage filters must form a three-channel perfect reconstruction filter bank.

Data Types: `char`

Filter name for stages > 1, specified as a character vector. You must specify a first-level filter that is different from the wavelet and scaling filters in subsequent levels when using the dual-tree wavelet transforms, `'cplxdt'` or `'cplxdddt'`. See `dtfilters` for valid choices.

Data Types: `char`

## Output Arguments

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Wavelet transform, returned as a structure with these fields:

Type of wavelet decomposition (filter bank) used in the analysis, returned as one of `'dwt'`, `'ddt'`, `'cplxdt'`, or `'cplxdddt'`. The type, `'dwt'`, gives a critically sampled discrete wavelet transform. The other types correspond to 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, returned as a positive integer.

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

First-stage analysis filters, returned 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, returned 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, returned 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, returned 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, returned 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, `typetree`, 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.