Computer Implementation

Although the computer implementation of the above algorithms seems straightforward enough, I will mention some specific details here that, simple as they may be, are important when dealing with these algorithms

1.

When using non-causal wavelets it is convenient to apply a time shift to the whole matrix so that the complete wavelet can be recovered. This can be easily done at the time of the computation of the filters in the frequency domain. This time shift needs to be compensated for after the filtering, of course.

2.

In the time domain implementation only a few samples of the impulse responses are required to get a satisfactory result. This allows us to speed up the computation enormously because we need to multiply by a matrix that is non zero only near the diagonal as opposed to a dense matrix. We don't even need to store the complete impulse responses.

3.

In the frequency domain similar, even more pronounced savings in computation, can be achieved by realizing that the frequency connection matrix is very nearly diagonal except for wildly varying filters. I found that good results could be obtained with only a few ``traces'' (perhaps 7 or 9) in the frequency domain.

4.

After taking the horizontal Fourier transform in the frequency domain algorithm, it may be necessary to unscramble the traces to get both the positive and negative frequencies, otherwise only the amplitudes above or below the diagonal in Figure 2 will be present and the results will not be satisfactory.