High & Low Pass Wavelet Filters

The first step of denoising is the application of filters.  H denotes the low pass filter, while G denotes the high pass filter.  For a signal si where i = 1,…,2^n, H and G are defined by

and

.

The  hj and gj are called filter coefficients.  For the Haar wavelets, the above equations simplify to the following:

and

since h0, h1, g0, and g1 are the only non-zero filter coefficients.  Note that for the Haar wavelets

and

.

The radicals come from normalizing the father and mother wavelets.  In the case of the Haar wavelets, filtering simply averages and differences.  The low pass filter does the following to a signal s = [a, b, c, d, e, f]

.

The high pass filter applied to s yields

.

Thus, the low pass filter computes averages while the high pass filter accomplishes differencing.  The process of differencing detects the noise in the signal.  If some detail coefficients are small compared to the others, making them zero will not alter the signal too much.  If the noise is located in these areas of the signal, denoising will have a positive effect on the signal.  The art is to choose a threshold level that will eliminate most of the noise, but preserve the other qualities of the signal.

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