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contwt(2) Scilab Function contwt(2)
NAME
contwt - Continuous L2 wavelet transform with mirroring
Author: Paulo Goncalves
Computes a continuous wavelet transform of a mirrored 1-D signal (real or
complex). The scale operator is unitary with respect to the L2 norm. Two
closed form wavelets are available: the Mexican Hat or the Morlet Wavelet
(real or analytic). For arbitrary analyzing wavelets, numerical approxima-
tion is achieved using a Fast Mellin Transform.
Usage
[wt,scale,f,scalo,wavescaled]=contwtmir(x,[fmin,fmax,N,wvlt_length])
Input parameters
o x : Real or complex vector [1,nt] or [nt,1] Time samples of the sig-
nal to be analyzed.
o fmin : real scalar in [0,0.5] Lower frequency bound of the
analysis. When not specified, this parameter forces the program to
interactive mode.
o fmax : real scalar [0,0.5] and fmax > Upper frequency bound of the
analysis. When not specified, this parameter forces the program to
interactive mode.
o N : positive integer. number of analyzing voices. When not speci-
fied, this parameter forces the program to interactive mode.
o wvlt_length : scalar or vector specifies the analyzing wavelet: 0:
Mexican hat wavelet (real) Positive real integer: real Morlet
wavelet of size 2*wvlt_length+1) at finest scale 1 Positive ima-
ginary integer: analytic Morlet wavelet of size 2*wvlt_length+1) at
finest scale 1 Real valued vector: waveform samples of an arbitrary
bandpass function.
Output parameters
o wt : Real or complex matrix [N,nt] coefficient of the wavelet
transform.
o scale : real vector [1,N] analyzed scales
o f : real vector [1,N] analyzed frequencies
o scalo : real positive valued matrix [N,nt] Scalogram coefficients
(squared magnitude of the wavelet coefficients wt )
o wavescaled : Scalar or real valued matrix [length(wavelet at coarser
scale)+1,N]
Dilated versions of the analyzing wavelet
Description
Parameters
o x : signal to be analyzed. Real or complex vector
o fmin : lower frequency bound of the analysis. fmin is real scalar
comprised in [0,0.5]
o fmax : upper frequency bound of the analysis. fmax is a real scalar
comprised in [0,0.5] and fmax > fmin
o N : number of analyzing voices geometrically sampled between minimum
scale fmax/fmax and maximum scale fmax/fmin.
o wvlt_length : specifies the analyzing wavelet: 0: Mexican hat
wavelet (real). The size of the wavelet is automatically fixed by
the analyzing frequency Positive real integer: real Morlet wavelet
of size 2*wvlt_length+1) at finest scale (1) Positive imaginary
integer: analytic Morlet wavelet of size 2*|wvlt_length|+1) at
finest scale 1. The corresponding wavelet transform is then complex.
May be usefull for event detection purposes. Real valued vector:
corresponds to the time samples waveform of any arbitrary bandpass
function viewed as the analyzing wavelet at any given scale. Then,
an approximation of the scaled wavelet versions is achieved using
the Fast Mellin Transform (see dmt and dilate).
o wt : coefficient of the wavelet transform. X-coordinated
corresponds to time (uniformly sampled), Y-coordinates correspond to
frequency (or scale) voices (geometrically sampled between fmax
(resp. 1) and fmin (resp. fmax / fmin ). First row of wt
corresponds to the highest analyzed frequency (finest scale).
o scale : analyzed scales (geometrically sampled between 1 and fmax
/fmin
o f : analyzed frequencies (geometrically sampled between
fmax and fmin . f corresponds to fmax/scale
o scalo : Scalogram coefficients (squared magnitude of the wavelet
coefficients wt )
o wavescaled : If wvlt_length is a real or Imaginary pure scalar,
then wavescaled equal wvlt_length . If wvlt_length is a vector
(containing the waveform samples of an arbitrary analyzing wavelet),
then wavescaled contains columnwise all scaled version of
wvlt_length used for the analysis. In this latter case, first ele-
ment of each column gives the effective time support of the analyz-
ing wavelet at the corresponding scale. wavescaled can be used for
reconstructing the signal (see icontwt)
Algorithm details
The overall details of the algorithm are similar to those of contwt . The
difference stems from the mirror operation applied to the signal before
computing the wavelet transform to minimize border effects. At each scale
j the analyzed signal is mirrored at its both extremities. The number of
added samples at both sides is equal to scale(j)* wvlt_length (the half
length of the analyzing wavelet at this particular scale). After convolu-
tion of the mirrored signal with the analyzing wavelet, the result is trun-
cated to the actual size of the initial signal.
See also:
contwt, icontwt and cwt
Example:
Signal synthesis x = fbmlevinson(1024,0.8) ;
Regular Wavelet transform [wt_nomirror,scale,f] = contwt(x,2(-6),2(-
1),128,8) ; viewmat(abs(wt_nomirror),[1 1 24]) ;
Compared with a mirrored wavelet transform [wt_mirror,scale,f] =
contwtmir(x,2(-6),2(-1),128,0) ; viewmat(abs(wt_mirror),[1 1 24]) ;
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