The filter bank has a wide variety of applications. Such as extraction of the required frequency from a modulated signal, denoising of a signal, removing redundancies present in a signal, etc.
One of the applications we have
implemented is DENOISING.
The method of denoising relies on the fact that noise commonly manifests itself as a fine-grained structure, and the wavelet transform provides a scaled based decomposition. Thus, most of the noise tends to be represented by wavelet coefficients at the finer scales. Discarding these coefficients would result in a natural filtering out of the noise on the basis of scale.
By removing coefficients containing the noise, we are also removing a part of the primary carriers thus degrading the signal. Since these coefficients at such scales also tend to be the primary carriers of edge information, they cannot be removed. So this edge related coefficients are threshold to a certain value. Here a very low amount of noise is left behind and some edge information is lost. To over come this, a method where in the values are zeroed of the wavelet coefficients if they are below certain threshold. These coefficients are mainly corresponding to noise. The edge related coefficients are above the threshold.
We have followed the former method to denoise a signal. We have threshold the detail scale at certain levels to a certain range in order to save the primary carriers but losing some edge information.
Consider the signal shown below.
The above figure shows the original signal, reconstructed signal and a denoised signal, where one particular frequency assumed to be noise has been successfully eliminated by thresholding the wavelet coefficient in that relative region of frequency to a low value. The equivalent power spectral density is shown adjacent to the signal. It can bee seen that the PSD has reduced to a large extent in the denoised signal
In the above figure we find the outputs of three level decomposition. Therefore the filter has 2 band pass, one low pass and one high pass filter.
The figure shows the approximations (output of low pass filter) at different levels of decompositions. One can see that the approximation at level 3 (A3) has been smoothened.
The figure shows the details (Output of high pass filter) at different levels of decompositions.
The figure shows the multiresolution graph of the original input signal.
The figure shows the multiresolution graph of the denoised signal. Here the details of level 1 and 2 decomposition have been thresholded to levels of magnitude 0.01.