Filterbanks Interpretation of the Discrete Wavelet Transform 2.6 2002/01/28 2005/09/30 16:03:35.219 GMT-5 Phil Schniter schniter@ee.eng.ohio-state.edu Charlet Reedstrom charlet@rice.edu Adan Galvan jago@rice.edu Phil Schniter schniter@ee.eng.ohio-state.edu DWT filterbanks wavelet Introduction to the filterbanks interpretation of the DWT. Assume that we start with a signal x t ℒ 2 . Denote the best approximation at the 0 th level of coarseness by x0 t . (Recall that x0 t is the orthogonal projection of x t onto V 0 .) Our goal, for the moment, is to decompose x0 t into scaling coefficients and wavelet coefficients at higher levels. Since x0 t V 0 and V0 V1 W1 , there exist coefficients c 0 n , c 1 n , and d 1 n such that x 0 t n n c 0 n φ 0 , n t n n c 1 n φ 1 , n t n n d 1 n ψ 1 , n t Using the fact that φ 1 , n t n is an orthonormal basis for V 1 , in conjunction with the scaling equation, c 1 n x 0 t φ 1 , n t m m c 0 m φ 0 , m t φ 1 , n t m m c 0 m φ 0 , m t φ 1 , n t m m c 0 m φ t m ℓ ℓ h ℓ φ t ℓ 2 n m m c 0 m ℓ ℓ h ℓ φ t m φ t ℓ 2 n m m c 0 m h m 2 n where δ t ℓ 2 n φ t m φ t ℓ 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 ψ 1 , n t n is an orthonormal basis for W 1 , in conjunction with the wavelet scaling equation, d 1 n x0 t ψ 1 , n t m m c 0 m φ 0 , m t ψ 1 , n t m m c 0 m φ 0 , m t ψ 1 , n t m m c 0 m φ t m ℓ ℓ g ℓ φ t ℓ 2 n m m c 0 m ℓ ℓ g ℓ φ t m φ t ℓ 2 n m m c 0 m g m 2 n where δ t ℓ 2 n φ t m φ t ℓ 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 φ 1 , n t n n d 2 n ψ 2 , n t n n c 2 n φ 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 the same assumptions as before, we have: c 0 m x 0 t φ 0 , m t n n c 1 n φ 1 , n t n n d 1 n ψ 1 , n t φ 0 , m t n n c 1 n φ 1 , n t φ 0 , m t n n d 1 n ψ 1 , n t φ 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 φ 1 , n t φ 0 , m t and    g m 2 n ψ 1 , n t φ 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-channel orthogonal 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 orthogonal PR-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 DWT the 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 .