Wavelet Toolbox

Detecting Discontinuities and Breakdown Points I

The purpose of this example is to show how analysis by wavelets can detect the exact instant when a signal changes. The discontinuous signal consists of a slow sine wave abruptly followed by a medium sine wave.

The first- and second-level details (D₁ and D₂) show the discontinuity most clearly, because the rupture contains the high-frequency part. Note that if we were only interested in identifying the discontinuity, db1 would be a more useful wavelet to use for the analysis than db5.

The discontinuity is localized very precisely: only a small domain around time = 500 contains any large first- or second-level details.

Here is a noteworthy example of an important advantage of wavelet analysis over Fourier. If the same signal had been analyzed by the Fourier transform, we would not have been able to detect the instant when the signal's frequency changed, whereas it is clearly observable here.

Details D₃ and D₄ contain the medium sine wave. The slow sine is clearly isolated in approximation A₅, from which the higher-frequency information has been filtered.

Discussion

The deterministic part of the signal may undergo abrupt changes such as a jump, or a sharp change in the first or second derivative. In image processing, one of the major problems is edge detection, which also involves detecting abrupt changes. Also in this category, we find signals with very rapid evolutions such as transient signals in dynamic systems.

The main characteristic of these phenomena is that the change is localized in time or in space.

The purpose of the analysis is to determine

• The site of the change (e.g., time or position)
• The type of change (a rupture of the signal, or an abrupt change in its first or second derivative)
• The amplitude of the change

The local aspects of wavelet analysis are well adapted for processing this type of event, as the processing scales are linked to the speed of the change.

Guidelines for Detecting Discontinuities

Short wavelets are often more effective than long ones in detecting a signal rupture. In the initial analysis scales, the support is small enough to allow fine analysis. The shapes of discontinuities that can be identified by the smallest wavelets are simpler than those that can be identified by the longest wavelets. Therefore, to identify

• A signal discontinuity, use the haar wavelet
• A rupture in the j-th derivative, select a sufficiently regular wavelet with at least j vanishing moments. (See Detecting Discontinuities and Breakdown Points II.)

The presence of noise, which is after all a fairly common situation in signal processing, makes identification of discontinuities more complicated. If the first levels of the decomposition can be used to eliminate a large part of the noise, the rupture is sometimes visible at deeper levels in the decomposition.

Check, for example, the sample analysis FileExample AnalysisBasic Signalsramp + white noise (MAT-file wnoislop). The rupture is visible in the level-six approximation (A₆) of this signal.

Introduction to Wavelet Analysis

Detecting Discontinuities and Breakdown Points II

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