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.

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.

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'.

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.

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.

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.

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, 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).}

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).}

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.

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.

Filter name, specified as a string. 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 strings for the oversampled wavelet filter banks.

First-stage filter name, specified as a string. 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.

Filter name for stages > 1, specified as a string. 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.

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.

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.