Wavelet Toolbox

Example 7: Two Proximal Discontinuities

Analyzing wavelet: db2 and db7

Decomposition levels: 5

The signal is formed of two straight lines with identical slopes, extending across a very short plateau. On the initial signal, the plateau is in fact barely visible to the naked eye. Two analyses are thus carried out: one on a well localized wavelet with the short filter (db2, shown on the left side of the figure); and the other on a wavelet having a longer filter (db7, shown on the right side of the figure).

In both analyses, the plateau is detected clearly. With the exception of a fairly limited domain, D₁ is equal to zero. The regularity of the signal in the plateau, however, is clearly distinguished for db2 (for which plateau beginning and end time are distinguished), whereas for db7 both discontinuities are fused and only the entire plateau can be said to be visible.

This example suggests that the selected wavelets should be associated with short filters to distinguish proximal discontinuities of the first derivative. A look at the other detail levels again shows the lack of precision when detecting at low resolutions. The wavelet filters the straight line and analyzes the discontinuities.

Example 7: Two Proximal Discontinuities
Addressed topics	Detecting breakdown points
Further exploration	Move the discontinuities closer together and further apart. Add noise to the signal until the rupture is no longer visible. Try using other wavelets, haar for instance.

Example 6: A Step Signal

Example 8: A Second-Derivative Discontinuity

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