## Wavelet transient reconstruction (WTT)

Purpose

Description

Tips

Macro Synopsis

Modules

Related Functions

References

### Purpose

Wavelet Packet Time-Selective Transient Reconstruction (WTT)

### Description

"Wavelet transient reconstruction (WTT)" can recover a transient from a signal given in wavelet
packet (WPD) representation. The transient is regarded as unknown additive
short-time component within an otherwise stationary background signal.
The basic idea behind "Wavelet transient reconstruction (WTT)" is to analyse whether the energy distribution
in the coefficients of the Wavelet packet decomposition, boxes is uniform or locally concentrated.
In the latter case, it is assumed that these coefficients belong to the
transient.

The algorithm is performed by these steps:

- Divide every box b(i,j) of the WPD input signal into m sub-boxes,
- Choose octave
`i=1` and mark the boxes in this octave,
- Test for all marked boxes of octave
`i` whether the entropy of
the box is below a user-given control threshold `f`.
In this case, leave b(i,j) marked and discard all its child boxes.
If not, set all coefficients of b(i,j) to zero, delete the mark, and
mark the child boxes b(i+1,2j+1) and b(i+1,2j) instead.
- If the last octave has not been reached yet, increase i by 1 and continue with the last step.

The reconstruction is applied to the marked boxes according to this recursion:
- Begin with the unmarked box b(0,1),
- If for an unmarked box b(i,j) the box b(i+1,2i-1) is marked,
set a(i+1) = b(i+1,2i-1),
otherwise calculate a(i+1) recursively from the child boxes of b(i+1,2i-1).
If the box b(i+1,2i) is marked, set d(i+1) = b(i+1,2i), otherwise calculate
d(i+1) recursively from the child boxes of b(i+1,2i).
The box b(i,j) is yielded from the reconstruction formula

where a(i+1) and d(i+1) denote the approximation and the detail signal, respectively.

The parameters are

- number
`m` of subboxes,
- factor
`f` for box selection (control threshold),
- maximum number
`maxoct` of octaves used for reconstruction,
- the
`drop`, determining whether subboxes with low energy are to
be dropped or not. If this option is checked only the two sub-boxes with
the highest fluctuations are taken into account for recontruction.

### Tips

In several cases you will yield better reconstruction results if you
ommit the boxes of the last octave of the WPD. Due to their low time-resolution, they are not well suited for transient reconstruction.

### Macro Synopsis

`y = WTT(x,m,f,maxoct,drop);`

signal x,y;

int m;

float f;

int maxoct;

bool drop;

Note that x must be a wavelet packet decomposition (WPD) signal.

There is also a Dataplore ® macro `TransReconstruct.dpm` that performs transient reconstruction on a standard single-channel signal. It is included as Transient Reconstruction as a standard menu function.

### Modules

Wavelet

### Related Functions

Define wavelet, Load wavelet,
Save wavelet, Decompose,
Reconstruct, Transient Reconstruction, Wavelet packet decomposition

### References

Vandenhouten [21], Desai/Shazeer [61]