Figure 1: Signal s with added noise n [8].
This processing with the Haar wavelets can be done using an appropriate An transform matrix to obtain the wavelet coefficients, in this particular case A10 is used. Figure 2 shows a plot of the tenth level of the Haar transform of the signal s.
Figure 2: A plot of the tenth level of the Haar transform of the signal s [8].
The horizontal lines shown in the graph are the hard thresholding levels of 0.25. The effect of this thresholding will set all the values in the filtered signal that have an absolute value less than 0.25 to zero. Since the original signal appears to be a piecewise, constant function, we should hope that most of the detail coefficients would be zero. We can clearly see from the figure that there is some fuzziness around the horizontal axis as well as some outliers even further from zero. These outliers are most likely due to the detail coefficients which heavily contribute to the original signal. Applying the given threshold level yields the new graph of the Haar transform shown in Figure 3.
Figure 3: A plot of the Haar transform after hard thresholding [8].
Notice that the random noise along the horizontal axis is completely gone. After the thresholding is accomplished, reducing a significant amount of noise, a signal must be recomposed. This process is accomplished by computing the inverse transform, the final result is shown in Figure 4 [8].
Figure 4: New signal s' with reduced noise [8].