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Assume that we start with a signal . Denote the best approximation at the level of coarseness by . (Recall that is the orthogonal projection of onto .) Our goal, for the moment, is to decompose into scaling coefficients and wavelet coefficients at higher levels. Since and , there exist coefficients , , and such that
Using the fact that is an orthonormal basis for , in conjunction with the wavelet scaling equation,
The previous expression ( ) indicates that results from convolving with a time-reversed version of then downsampling by factor two ( ).
Putting these two operations together, we arrive at what looks like the analysis portion of an FIR filterbank ( ):
We can repeat this process at the next higher level. Since , there exist coefficients and such that
If we use to repeat this process up to the level, we get the iterated analysis filterbank ( ).
As we might expect, signal reconstruction can be accomplished using cascaded two-channel synthesis filterbanks. Using thesame assumptions as before, we have:
The same procedure can be used to derive
To reconstruct from the level, we can use the iterated synthesis filterbank ( ).
The table makes a direct comparison between wavelets and the two-channelorthogonal PR-FIR filterbanks.
Discrete Wavelet Transform | 2-Channel Orthogonal PR-FIR Filterbank | |
---|---|---|
Analysis-LPF | ||
Power Symmetry | ||
Analysis HPF | ||
Spectral Reverse | ||
Synthesis LPF | ||
Synthesis HPF |
From the table, we see that the discrete wavelet transform that we have been developing is identical to two-channel orthogonalPR-FIR filterbanks in all but a couple details.
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