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Introduction to the filterbanks interpretation of the DWT.

Assume that we start with a signal x t 2 . Denote the best approximation at the 0 th level of coarseness by x 0 t . (Recall that x 0 t is the orthogonal projection of x t onto V 0 .) Our goal, for the moment, is to decompose x 0 t into scaling coefficients and wavelet coefficients at higher levels. Since x 0 t V 0 and V 0 V 1 W 1 , there exist coefficients c 0 n , c 1 n , and d 1 n such that

x 0 t n n c 0 n 0 n t n n c 1 n 1 n t n n d 1 n 1 n t
Using the fact that 1 n t n is an orthonormal basis for V 1 , in conjunction with the scaling equation,
c 1 n x 0 t 1 n t m m c 0 m 0 m t 1 n t m m c 0 m 0 m t 1 n t m m c 0 m t m h t 2 n m m c 0 m h t m t 2 n m m c 0 m h m 2 n
where t 2 n t m t 2 n . The previous expression ( ) indicates that c 1 n results from convolving c 0 m with a time-reversed version of h m then downsampling by factor two ( ).

Using the fact that 1 n t n is an orthonormal basis for W 1 , in conjunction with the wavelet scaling equation,

d 1 n x 0 t 1 n t m m c 0 m 0 m t 1 n t m m c 0 m 0 m t 1 n t m m c 0 m t m g t 2 n m m c 0 m g t m t 2 n m m c 0 m g m 2 n
where t 2 n t m t 2 n .

The previous expression ( ) indicates that d 1 n results from convolving c 0 m with a time-reversed version of g m 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 V 1 W 2 V 2 , there exist coefficients c 2 n and d 2 n such that

x 1 t n n c 1 n 1 n t n n d 2 n 2 n t n n c 2 n 2 n t
Using the same steps as before we find that
c 2 n m m c 1 m h m 2 n
d 2 n m m c 1 m g m 2 n
which gives a cascaded analysis filterbank ( ):

If we use V 0 W 1 W 2 W 3 W k V k to repeat this process up to the k th 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:

c 0 m x 0 t 0 m t n n c 1 n 1 n t n n d 1 n 1 n t 0 m t n n c 1 n 1 n t 0 m t n n d 1 n 1 n t 0 m t n n c 1 n h m 2 n n n d 1 n g m 2 n
where h m 2 n 1 n t 0 m t and g m 2 n 1 n t 0 m t which can be implemented using the block diagram in .

The same procedure can be used to derive

c 1 m n n c 2 n h m 2 n n n d 2 n g m 2 n
from which we get the diagram in .

To reconstruct from the k th 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 H z -1 H 0 z
Power Symmetry H z H z -1 H z H z -1 2 H 0 z H 0 z -1 H 0 z H 0 z -1 1
Analysis HPF G z -1 H 1 z
Spectral Reverse P P is odd G z z P H z -1 N N is even H 1 z z N 1 H 0 z -1
Synthesis LPF H z G 0 z 2 z N 1 H 0 z -1
Synthesis HPF G z G 1 z 2 z N 1 H 1 z -1

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.

  • Orthogonal PR-FIR filterbanks employ synthesis filters with twice the gain of the analysis filters, whereas in the DWTthe gains are equal.
  • Orthogonal PR-FIR filterbanks employ causal filters of length N , whereas the DWT filters are not constrained to be causal.
For convenience, however, the wavelet filters H z and G z are usually chosen to be causal. For both to have even impulse response length N , we require that P N 1 .

Questions & Answers

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Source:  OpenStax, Fundamentals of signal processing. OpenStax CNX. Nov 26, 2012 Download for free at http://cnx.org/content/col10360/1.4
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