Wavelet Toolbox

Example 9: A Ramp + White Noise

Analyzing wavelet: db3

Decomposition levels: 6

The signal is built from a trend plus noise. The trend is a slow linear rise from 0 to 3, up to t=500, and becoming constant afterwards. The noise is a uniform zero-mean white noise, varying between -0.5 and 0.5 (see the analyzed signal b₁).

Looking at the figure, in the chart on the right, we again find the decomposition of noise in the details. In the charts on the left, the approximations form increasingly precise estimates of the ramp with less and less noise. These approximations are quite acceptable from level 3, and the ramp is well reconstructed at level 6.

We can, therefore, separate the ramp from the noise. Although the noise affects all scales, its effect decreases sufficiently quickly for the low-resolution approximations to restore the ramp. It should also be noted that the breakdown point of the ramp is shown with good precision. This is due to the fact that the ramp is recovered at too low a resolution.

The uniform noise indicates that the ramp might be best estimated using half sums for the higher and lower portions of the signal. This approach is not applicable for other noises.

Example 9: A Ramp + White Noise
Addressed topics	Detecting breakdown points Processing noise Detecting long-term evolution Splitting signal components Identifying noises and approximations
Further exploration	Compare with the white noise b1(t) shown in Example 3: Uniform White Noise. Identify the number of levels needed to suppress the noise almost entirely. Change the noise.

Example 8: A Second-Derivative Discontinuity

Example 10: A Ramp + Colored Noise

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