Wavelet Toolbox

Example 5: Polynomial + White Noise

Analyzing wavelets: db2 and db3

Decomposition levels: 4

The purpose of this analysis is to illustrate the property that causes the decomposition by dbN of a p-degree polynomial to produce null details as long as N > p. In this case, p=2 and we examine the first four levels of details for two values of N: one is too small, N=2 on the left, and the other is sufficient, N=3 on the right. The approximations are left out since they differ very little from the signal itself.

For db2 (on the left), we obtain the decomposition of t² + b₁(t), since the -t + 1 part of the signal is suppressed by the wavelet. In fact, with the exception of level 1, where noise-generated irregularities can be seen, the details for levels 2 to 4 show a periodic form that is very regular, and which increases with the level. This is because the detail for level j takes into account that the fluctuations of the function around its mean value on dyadic intervals are long. The fluctuations are periodic and very large in relation to the details of the noise decomposition.

On the other hand, for db3 (on the right) we again 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.

Example 5: Polynomial + White Noise
Addressed topics	Suppressing signals Compare the results of the processing for the following wavelets: the short db2 and the longer db3. Explain the regularity that is visible in D3 and D4 in the analysis by db2.
Further exploration	Increase noise intensity and repeat the analysis.

Example 4: Colored AR(3) Noise

Example 6: A Step Signal

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