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

returns the `wt`

= dddtree(`typetree`

,`x`

,`level`

,`fdf`

,`df`

)`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.

uses
the filters specified in `wt`

= dddtree(`typetree`

,`x`

,`level`

,`fname1`

,`fname2`

)`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'`

.

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.

Use `dtfilters`

to create the first-stage Farras analysis filters and 6-tap Kingsbury Q-shift analysis filters for the subsequent stages of the multiresolution analysis.

`df = dtfilters('dtf1');`

The Farras and Kingsbury filters are in `df{1}`

and `df{2}`

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

load noisdopp; wt = dddtree('cplxdt',noisdopp,4,df{1},df{2});

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

Using the output of `dtfilters`

, or the filter name itself, in `dddtree`

is preferable to manually entering truncated filter coefficients. To demonstrate the negative impact on signal reconstruction, create truncated versions of the Farras and Kingsbury analysis filters. Display the differences between the truncated and original filters.

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]; 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]; max(max(abs(df{1}{1}-Faf{1})))

ans = 2.6792e-05

max(max(abs(df{1}{2}-Faf{2})))

ans = 2.6792e-05

max(max(abs(df{2}{1}-af{1})))

ans = 3.6160e-05

max(max(abs(df{2}{2}-af{2})))

ans = 3.6160e-05

Obtain the complex dual-tree wavelet transform down to level 4 using the truncated filters. Take the inverse transform and compare the reconstruction with the original signal.

```
wt = dddtree('cplxdt',noisdopp,4,Faf,af);
xrec = idddtree(wt);
max(abs(noisdopp-xrec))
```

ans = 0.0024

Do the same using the filter name. Confirm the difference is smaller.

wt = dddtree('cplxdt',noisdopp,4,'dtf1'); xrec = idddtree(wt); max(abs(noisdopp-xrec))

ans = 2.1893e-07

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('Time (s)') title('Original Signal') grid on

Obtain the double-density wavelet transform of the signal. Reconstruct an approximation based on the level-one detail coefficients by first setting the lowpass (scaling) coefficients equal to 0. Plot the result. Observe features in the reconstruction align with the signal discontinuities.

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('Time (s)') title('Original Signal') grid on

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')

`typetree`

— Type of wavelet decomposition`'dwt'`

| `'ddt'`

| `'cplxdt'`

| `'cplxdddt'`

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.

`x`

— Input signalvector

Input signal, specified as an even-length row or column vector.
If *L* is the value of the `level`

of
the wavelet decomposition, 2^{L} 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`

— Level of wavelet decompositionpositive integer

Level of the wavelet decomposition, specified as an integer.
If *L* is the value of `level`

,
2^{L} 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`

`fdf`

— Level-one analysis filtersmatrix | cell array

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`

`df`

— Analysis filters for levels > 1matrix | cell array

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`

`fname`

— Filter namecharacter vector | string scalar

Filter name, specified as a character vector or string scalar. 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`

`fname1`

— First-stage filter namecharacter vector | string scalar

First-stage filter name, specified as a character vector or string scalar. 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`

`fname2`

— Filter name for stages > 1character vector | string scalar

Filter name for stages > 1, specified as a character vector or string scalar. 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`

`wt`

— Wavelet transformstructure

Wavelet transform, returned as a structure with these fields:

`type`

— Type of wavelet decomposition (filter bank)`'dwt'`

| `'ddt'`

| `'cplxdt'`

| `'cplxdddt'`

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`

— Level of the wavelet decompositionpositive integer

Level of wavelet decomposition, returned as a positive integer.

`filters`

— Decomposition (analysis) and reconstruction (synthesis) filtersstructure

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

`Fdf`

— First-stage analysis filtersmatrix | cell array

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.

`Df`

— Analysis filters for levels > 1matrix | cell array

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.

`Frf`

— First-level reconstruction filtersmatrix | cell array

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.

`Rf`

— Reconstruction filters for levels > 1matrix | cell array

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.

`cfs`

— Wavelet transform coefficientscell array of matrices

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}(:,:,m)`

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}(:,:,m)`

are the lowpass, or scaling, coefficients.

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