Assume that we start with a signal xt∈ ℒ 2 x t ℒ 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 Using the fact that { φ 1 , n t|n∈ℤ} φ 1 , n t n is an orthonormal basis for V 1 V 1 , in conjunction with the scaling equation, where δt-ℓ-2n=<φt-m,φt-ℓ-2n> δ t ℓ 2 n φ t m φ t ℓ 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).

Using the fact that { ψ 1 , n t|n∈ℤ} ψ 1 , n t n is an orthonormal basis for W 1 W 1 , in conjunction with the wavelet scaling equation, where δt-ℓ-2n=<φt-m,φt-ℓ-2n> δ t ℓ 2 n φ t m φ t ℓ 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).

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

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 Using the same steps as before we find that which gives a cascaded analysis filterbank (Figure 4):

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).

As we might expect, signal reconstruction can be accomplished using cascaded two-channel synthesis filterbanks. Using the same assumptions as before, we have: where    hm-2n=< φ 1 , n t, φ 0 , m t> where    h m 2 n φ 1 , n t φ 0 , m t and    gm-2n=< 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.

The same procedure can be used to derive from which we get the diagram in Figure 7.

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

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

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=N-1 P N 1 .