## 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 [6].

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;

### Modules

Wavelet

### Related Functions

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

### References

Daubechies [6], Rioul/Vetterli [7],
Vandenhouten [21]