DTWFBREAL Dual-Tree Wavelet FilterBank for real-valued signals
Usage: c=dtwfbreal(f,dualwt);
c=dtwfbreal(f,{dualw,J});
[c,info]=dtwfbreal(...);
Input parameters:
f : Input data.
dualwt : Dual-tree Wavelet Filterbank definition.
Output parameters:
c : Coefficients stored in a cell-array.
info : Additional transform parameters struct.
c=dtwfbtreal(f,dualwt) computes dual-tree complex wavelet coefficients
of the real-valued signal f. The representation is approximately
time-invariant and provides analytic behavior. Due to these facts,
the resulting subbands are nearly aliasing free making them suitable
for severe coefficient modifications. The representation is two times
redundant, provided critical subsampling of all involved filterbanks,
but one half of the coefficients is complex conjugate of the other.
The shape of the filterbank tree and filters used is controlled by
dualwt (for possible formats see below). The output c is a
cell-array with each element containing a single subband. The subbands
are ordered with increasing subband center frequency.
In addition, the function returns struct. info containing transform
parameters. It can be conviniently used for the inverse transform
IDTWFBREAL e.g. fhat = iDTWFBREAL(c,info). It is also required by
the PLOTWAVELETS function.
If f is a matrix, the transform is applied to each column.
Two formats of dualwt are accepted:
1) Cell array of parameters. First two elements of the array are
mandatory {dualw,J}.
dualw
Basic dual-tree filters
J*
Number of levels of the filterbank tree
Possible formats of dualw are the same as in FWTINIT except the
wfiltdt_ prefix is used when searching for function specifying
the actual impulse responses. These filters were designed specially
for the dual-tree filterbank to achieve the half-sample shift
ultimatelly resulting in analytic (complex) behavior of the
transform.
The default shape of the filterbank tree is DWT i.e. only low-pass
output is decomposed further (*J times in total).
Different filterbank tree shapes can be obtained by passing
additional flag in the cell array. Supported flags (mutually
exclusive) are:
'dwt'
Plain DWT tree (default). This gives one band per octave freq.
resolution when using 2 channel basic wavelet filterbank.
'full'
Full filterbank tree. Both (all) basic filterbank outputs are
decomposed further up to depth J achieving linear frequency band
division.
'doubleband','quadband','octaband'
The filterbank is designed such that it mimics 4-band, 8-band or
16-band complex wavelet transform provided the basic filterbank
is 2 channel. In this case, J is treated such that it defines
number of levels of 4-band, 8-band or 16-band transform.
The dual-tree wavelet filterbank can use any basic wavelet
filterbank in the first stage of both trees, provided they are
shifted by 1 sample (done internally). A custom first stage
filterbank can be defined by passing the following
key-value pair in the cell array:
'first',w
w defines a regular basic filterbank. Accepted formats are the
same as in FWTINIT assuming the wfilt_ prefix.
Similarly, when working with a filterbank tree containing
decomposition of high-pass outputs, some filters in both trees must
be replaced by a regular basic filterbank in order to achieve the
aproximatelly analytic behavior. A custom filterbank can be
specified by passing another key-value pair in the cell array:
'leaf',w
w defines a regular basic filterbank. Accepted formats are the
same as in FWTINIT assuming the wfilt_ prefix.
2) Another possibility is to pass directly a struct. returned by
DTWFBINIT and possibly modified by WFBTREMOVE.
Optional args.:
---------------
In addition, the following flag groups are supported:
'freq','nat'
Frequency or natural (Paley) ordering of coefficient subbands.
By default, subbands are ordered according to frequency. The natural
ordering is how the subbands are obtained from the filterbank tree
without modifications. The ordering differ only in non-plain DWT
case.
Boundary handling:
------------------
In contrast with FWT, WFBT and WPFBT, this function supports
periodic boundary handling only.
Examples:
---------
A simple example of calling the DTWFBREAL function using the regular
DWT iterated filterbank. The second figure shows a magnitude frequency
response of an identical filterbank.:
[f,fs] = greasy;
J = 6;
[c,info] = dtwfbreal(f,{'qshift3',J});
figure(1);
plotwavelets(c,info,fs,'dynrange',90);
figure(2);
[g,a] = dtwfb2filterbank({'qshift3',J},'real');
filterbankfreqz(g,a,1024,'plot','linabs');
The second example shows a decomposition using a full filterbank tree
of depth J*:
[f,fs] = greasy;
J = 5;
[c,info] = dtwfbreal(f,{'qshift4',J,'full'});
figure(1);
plotwavelets(c,info,fs,'dynrange',90);
figure(2);
[g,a] = dtwfb2filterbank({'qshift4',J,'full'},'real');
filterbankfreqz(g,a,1024,'plot','linabs');
References:
I. Selesnick, R. Baraniuk, and N. Kingsbury. The dual-tree complex
wavelet transform. Signal Processing Magazine, IEEE, 22(6):123 -- 151,
nov. 2005.
N. Kingsbury. Complex wavelets for shift invariant analysis and
filtering of signals. Applied and Computational Harmonic Analysis,
10(3):234 -- 253, 2001.
I. Bayram and I. Selesnick. On the dual-tree complex wavelet packet and
m-band transforms. Signal Processing, IEEE Transactions on,
56(6):2298--2310, June 2008.
Url: http://ltfat.github.io/doc/wavelets/dtwfbreal.html
See also: dtwfb, idtwfbreal, plotwavelets, dtwfb2filterbank.
Package: ltfat