Wavelet Toolbox

Detecting Discontinuities and Breakdown Points II

The purpose of this example is to show how analysis by wavelets can detect a discontinuity in one of a signal's derivatives. The signal, while apparently a single smooth curve, is actually composed of two separate exponentials that are connected at time = 500. The discontinuity occurs only in the second derivative, at time = 500.

We have zoomed in on the middle part of the signal to show more clearly what happens around time = 500. The details are high only in the middle of the signal and are negligible elsewhere. This suggests the presence of high-frequency information -- a sudden change or discontinuity -- around time = 500.

Discussion

Regularity can be an important criterion in selecting a wavelet. We have chosen to use db4, which is sufficiently regular for this analysis. Had we chosen the haar wavelet, the discontinuity would not have been detected. If you try repeating this analysis using haar at level two, you'll notice that the details are equal to zero at time = 500.

Note that to detect a singularity, the selected wavelet must be sufficiently regular, which implies a longer filter impulse response.

See the sections Frequently Asked Questions and Wavelet Families: Additional Discussion for a discussion of the mathematical meaning of regularity and a comparison of the regularity of various wavelets.

Detecting Discontinuities and Breakdown Points I

Detecting Long-Term Evolution

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