Wavelet Toolbox

Example 8: A Second-Derivative Discontinuity

Analyzing wavelet: db1 and db4

Decomposition levels: 2

This figure shows that the regularity can be an important criterion in selecting a wavelet. The basic function is composed of two exponentials that are connected at 0, and the analyzed signal is the sampling of the continuous function with increments of 10^-3. The sampled signal is analyzed using two different wavelets: db1, which is insufficiently regular (shown on the left side of the figure); and db4, which is sufficiently regular (shown on the right side of the figure).

Looking at the figure on the left we notice that the singularity has not been detected in the extent that the details are equal to 0 at 0. The black areas correspond to very rapid oscillations of the details. These values are equal to the difference between the function and an approximation using a constant function. Close to 0, the slow decrease of the details absolute values followed by a slow increase is due to the fact that the function derivative is zero and continuous at 0. The value of the details is very small (close to 10^-3 for db1 and 10^-4 for db4) since the signal is very smooth and does not contain any high frequency. This value is even smaller for db4, since the wavelet is more regular than db1.

However, with db4 (right side of the figure), the discontinuity is well detected: the details are high only close to 0, and are 0 everywhere else. This is the only element that can be derived from the analysis. In this case, as a conclusion, we notice that the selected wavelet must be sufficiently regular, which implies a longer filter impulse response to detect the singularity.

Example 8: A Second-Derivative Discontinuity
Addressed topics	Detecting breakdown points Suppressing signals Identifying a difficult discontinuity Carefully selecting a wavelet to reveal an effect
Further exploration	Calculate the detail values for the Haar wavelet. Be aware of parasitic effects: rapid detail fluctuations may be artifacts. Add noise to the signal until the rupture is no longer visible.

Example 7: Two Proximal Discontinuities

Example 9: A Ramp + White Noise

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