## Full continuous wavelet transform

Purpose

Description

Macro Synopsis

Modules

Tips

Related Functions

References

### Purpose

Continuous wavelet transform over a nearly arbitrary equidistant
grid of scale and time values

### Description

This function computes a continuous wavelet transform of the
input signal for a finely sampled range of scale and time values. Let
be a square integrable function and a
wavelet. The full continuous wavelet transform consists
of the integrals

with

For the computation we use the following algorithm.

- Spline interpolation of the signal for quadrature
- Numerical integration (Trapezoid or Simpson's Rule) of the
defining integral
- output of the two-dimensional transform (2D signal)

The result is a two-dimensional signal: in the window the transform of
the signal is displayed with increasing scale from bottom
to the top. The time increases along the abscissae. The parameters
are

- the wavelet
- quadrature type
- minimal and maximal scale for wavelet convolution and the number
of samples between them
- number of samples on resulting time axis

The advantage of the wavelet transform is its time resolution,
providing the possibility to analyse nonstationary time series.

### Macro Synopsis

`y = FullContWaveletTrafo(x,wvlttype,quadr,amin,amax,nscale,ntime);`

signal x,y;

option wvlttype, quadr;

float amin, amax;

int nscale, ntime;
Note that y is a 2D signal.

### Modules

Wavelet

### Tips

To simplify the evaluation of the transformed signal some so-called
indicator functions were implemented. They can be used
to detect interesting scales and dates (see Wavelet Indicator).
Further, thresholding the absolute values of the transform result marks
(wavelet related) features of the signal.

### Related Functions

Define wavelet, Load wavelet,
Save wavelet, Decompose,
Wavelet Indicator,
Reconstruct, Wavelet packet decomposition.

### References

Ende et al. [32],
Louis/Maass/Rieder [33],
Mallat [34],
Rioul/Duhamel [22].