Wavelet Toolbox

Suppressing Signals

The purpose of this example is to illustrate the property that causes the decomposition of a polynomial to produce null details, provided the number of vanishing moments of the wavelet (N for a Daubechies wavelet dbN) exceeds the degree of the polynomial. The signal here is a second-degree polynomial combined with a small amount of white noise.

Note that only the noise comes through in the details. The peak-to-peak magnitude of the details is about 2, while the amplitude of the polynomial signal is on the order of 10⁵.

The db3 wavelet, which has three vanishing moments, was used for this analysis. Note that a wavelet of the Daubechies family with fewer vanishing moments would fail to suppress the polynomial signal. For more information, see the section Daubechies Wavelets: dbN.

Here is what the first three details look like when we perform the same analysis with db2.

The peak-to-peak magnitudes of the details D1, D2, and D3 are 2, 10, and 40, respectively. These are much higher detail magnitudes than those obtained using db3.

Discussion

For the db2 analysis, the details for levels 2 to 4 show a periodic form that is very regular, and that increases with the level. This is explained by the fact that the detail for level j takes into account primarily the fluctuations of the polynomial function around its mean value on dyadic intervals that are 2^j long. The fluctuations are periodic and very large in relation to the details of the noise decomposition.

On the other hand, for the db3 analysis, we find the presence of white noise thus indicating that the polynomial does not come into play in any of the details. The wavelet suppresses the polynomial part and analyzes the noise.

Suppressing part of a signal allows us to highlight the remainder.

Vanishing Moments

The ability of a wavelet to suppress a polynomial depends on a crucial mathematical characteristic of the wavelet called its number of vanishing moments. A technical discussion of vanishing moments appears in the sections Frequently Asked Questions and Wavelet Families: Additional Discussion. For the present discussion, it suffices to think of "moment" as an extension of "average." Since a wavelet's average value is zero, it has (at least) one vanishing moment.

More precisely, if the average value of is zero (whereis the wavelet function), for then the wavelet has vanishing moments and polynomials of degree n are suppressed by this wavelet.

Identifying Pure Frequencies

De-Noising Signals

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