    ## Decompose

Purpose
Description
Macro Synopsis
Modules
Related Functions
References

### Purpose

Wavelet decomposition (Discrete wavelet transform)

### Description

"Decompose" computes the discrete wavelet transform of a signal. Its only parameter is
• Octaves: Number p of octaves over which the discrete wavelet transform will be computed.
If the input signal x is then its discrete wavelet transform consists of the signals and .

The signals are computed by the iteration where G is a lowpass and H a highpass filter and denotes a convolution operation. We say that a signal with index i belongs to octave i. Because G is a lowpass filter the a-signals are called approximations. The d-signals are called detail signals because H is a highpass filter. In this terminology the approximation and details at octave i+1 are obtained by applying a filter to the approximation at octave i and subsampling the results with a factor of 2.
If is the mother wavelet and the scaling function of the corresponding multiresolution analysis, we can interpret the discrete wavelet transform in the following manner:
Let f be the square integrable function defined by Then we have For a more detailed discussion of the a- and d-signal in the context of the wavelet transform consult the chapter on multiresolution analysis in the book of Daubechies .

Since the input signal x has finite length, each signal at octave i+1 has half the length of the approximation at octave i. If we do not restrict the number of octaves it may happen that the filters G and H become longer than the signals to which we want to apply G and H. Therefore the maximum number of octaves is such that the last approximation at octave p will have at least the same length as G and H.
The result of "Decompose" is a signal of the DWT type.

At the top of the display window the detail at octave 1 is displayed and at the bottom you find the last approximation. Between these channels the detail signals are in the order of increasing octaves.

Warning: Note that the x-axis offset (shift) of the input signal is not taken into account in the resulting DWT signal.

### Macro Synopsis

y = WaveletDecomp(x,n);
signal x,y;
int n;

Wavelet

### Related Functions

Define wavelet, Load wavelet, Save wavelet, Reconstruct, Continuous wavelet transform, Wavelet packet decomposition

### References

Daubechies , Rioul/Vetterli , Vandenhouten