12.9 Filterbanks interpretation of the discrete wavelet transform

Assume that we start with a signal $x(t)\in {}_2$ . Denote the best approximation at the $0^{\mathrm{th}}$ 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 $x_0(t)\in V_0$ and $V_0=V_1\mathop{\mathrm{xor}}{W}_1$ , there exist coefficients $\{c_0(n)\}$ , $\{c_1(n)\}$ , and $\{d_1(n)\}$ such that

Using the fact that $\{{}_{1n}(t)\colon n\in \mathbb{Z}\}$ is an orthonormal basis for $W_1$ , in conjunction with the wavelet scaling equation,

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 $V_1=W_2\mathop{\mathrm{xor}}{V}_2$ , there exist coefficients $\{c_2(n)\}$ and $\{d_2(n)\}$ such that

If we use $V_0=W_1\mathop{\mathrm{xor}}{W}_2\mathop{\mathrm{xor}}{W}_3\mathop{\mathrm{xor}}\mathop{\mathrm{xor}}{W}_k\mathop{\mathrm{xor}}{V}_k$ to repeat this process up to the $k^{\mathrm{th}}$ 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:

The same procedure can be used to derive

To reconstruct from the $k^{\mathrm{th}}$ level, we can use the iterated synthesis filterbank ( ).

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

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.