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Filterbanks Interpretation of the Discrete Wavelet Transform

Module by: Phil Schniter. E-mail the author

Summary: Introduction to the filterbanks interpretation of the DWT.

Assume that we start with a signal xt 2 x t 2 . Denote the best approximation at the 0 th 0 th level of coarseness by x0 t x0 t . (Recall that x0 t x0 t is the orthogonal projection of xt x t onto V 0 V 0 .) Our goal, for the moment, is to decompose x0 t x0 t into scaling coefficients and wavelet coefficients at higher levels. Since x0 t V 0 x0 t V 0 and V0 = V1 W1 V0 V1 W1 , there exist coefficients c 0 n c 0 n , c 1 n c 1 n , and d 1 n d 1 n such that

x 0 t=nn c 0 n φ 0 , n t=nn c 1 n φ 1 , n t+nn d 1 n ψ 1 , n t 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
(1)
Using the fact that φ 1 , n t nZ φ 1 , n t n is an orthonormal basis for V 1 V 1 , in conjunction with the scaling equation,
c 1 n= x 0 t, φ 1 , n t=mm c 0 m φ 0 , m t, φ 1 , n t=mm c 0 m( φ 0 , m t, φ 1 , n t)=mm c 0 m(φtm,hφt2n)=mm c 0 m h(φtm,φt2n)= m m c 0 mhm2n 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 h φ t 2 n m m c 0 m h φ t m φ t 2 n m m c 0 m h m 2 n
(2)
where δt2n=φtm,φt2n δ t 2 n φ t m φ t 2 n . The previous expression (Equation 2) indicates that c 1 n c 1 n results from convolving c 0 m c 0 m with a time-reversed version of hm h m then downsampling by factor two (Figure 1).

Figure 1
Figure 1 (downsampling1.png)

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

d 1 n= x0 t, ψ 1 , n t=mm 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(φtm, gφt2n)= m m c 0 m g(φtm,φt2n)= m m c 0 mgm2n 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 g φ t 2 n m m c 0 m g φ t m φ t 2 n m m c 0 m g m 2 n
(3)
where δt2n=φtm,φt2n δ t 2 n φ t m φ t 2 n .

The previous expression (Equation 3) indicates that d 1 n d 1 n results from convolving c 0 m c 0 m with a time-reversed version of gm g m then downsampling by factor two (Figure 2).

Figure 2
Figure 2 (downsampling3.png)

Putting these two operations together, we arrive at what looks like the analysis portion of an FIR filterbank (Figure 3):

Figure 3
Figure 3 (downsampling2.png)

We can repeat this process at the next higher level. Since V 1 = W 2 V 2 V 1 W 2 V 2 , there exist coefficients c 2 n c 2 n and d 2 n d 2 n such that

x 1 t=nn c 1 n φ 1 , n t=nn d 2 n ψ 2 , n t+nn c 2 n φ 2 , n t 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
(4)
Using the same steps as before we find that
c 2 n=mm c 1 mhm2n c 2 n m m c 1 m h m 2 n
(5)
d 2 n=mm c 1 mgm2n d 2 n m m c 1 m g m 2 n
(6)
which gives a cascaded analysis filterbank (Figure 4):

Figure 4
Figure 4 (cascade_filterbank.png)

If we use V 0 = W 1 W 2 W 3 W k V k V 0 W 1 W 2 W 3 W k V k to repeat this process up to the k th k th level, we get the iterated analysis filterbank (Figure 5).

Figure 5
Figure 5 (iterated_a_filterbank.png)

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=nn c 1 n φ 1 , n t+nn d 1 n ψ 1 , n t, φ 0 , m t=nn c 1 n( φ 1 , n t, φ 0 , m t)+nn d 1 n( ψ 1 , n t, φ 0 , m t)=nn c 1 nhm2n+nn d 1 ngm2n 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
(7)
where    hm2n= φ 1 , n t, φ 0 , m t where    h m 2 n φ 1 , n t φ 0 , m t and    gm2n= ψ 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 Figure 6.

Figure 6
Figure 6 (filterbanks_block1.png)

The same procedure can be used to derive

c 1 m=nn c 2 nhm2n+nn d 2 ngm2n c 1 m n n c 2 n h m 2 n n n d 2 n g m 2 n
(8)
from which we get the diagram in Figure 7.

Figure 7
Figure 7 (filterbanks_block2.png)

To reconstruct from the k th k th level, we can use the iterated synthesis filterbank (Figure 8).

Figure 8
Figure 8 (iterated_syn_filterbank.png)

The table makes a direct comparison between wavelets and the two-channel orthogonal PR-FIR filterbanks.

Table 1
Discrete Wavelet Transform 2-Channel Orthogonal PR-FIR Filterbank
Analysis-LPF Hz-1 H z -1 H 0 z H 0 z
Power Symmetry HzHz-1+HzHz-1=2 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 H 0 z H 0 z -1 H 0 z H 0 z -1 1
Analysis HPF Gz-1 G z -1 H 1 z H 1 z
Spectral Reverse P,P is odd:Gz=±zPHz-1 P P is odd G z ± z P H z -1 N,N is even: H 1 z=±z(N1) H 0 z-1 N N is even H 1 z ± z N 1 H 0 z -1
Synthesis LPF Hz H z G 0 z=2z(N1) H 0 z-1 G 0 z 2 z N 1 H 0 z -1
Synthesis HPF Gz G z G 1 z=2z(N1) H 1 z-1 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.

  1. Orthogonal PR-FIR filterbanks employ synthesis filters with twice the gain of the analysis filters, whereas in the DWT the gains are equal.
  2. Orthogonal PR-FIR filterbanks employ causal filters of length NN, whereas the DWT filters are not constrained to be causal.
For convenience, however, the wavelet filters Hz H z and Gz G z are usually chosen to be causal. For both to have even impulse response length NN, we require that P=N1 P N 1 .

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