# 4.14 Filterbanks interpretation of the discrete wavelet transform

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Introduction to the filterbanks interpretation of the DWT.

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

${x}_{0}(t)=\sum {c}_{0}(n){}_{0n}(t)=\sum {c}_{1}(n){}_{1n}(t)+\sum {d}_{1}(n){}_{1n}(t)$
Using the fact that $\{{}_{1n}(t)\colon n\in \mathbb{Z}\}$ is an orthonormal basis for ${V}_{1}$ , in conjunction with the scaling equation,
${c}_{1}(n)={x}_{0}(t)\dot {}_{1n}(t)=\sum {c}_{0}(m){}_{0m}(t)\dot {}_{1n}(t)=\sum {c}_{0}(m)({}_{0m}(t)\dot {}_{1n}(t))=\sum {c}_{0}(m)((t-m)\dot \sum h()(t--2n))=\sum {c}_{0}(m)\sum h()((t-m)\dot (t--2n))=\sum {c}_{0}(m)h(m-2n)$
where $(t--2n)=(t-m)\dot (t--2n)$ . 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 $\{{}_{1n}(t)\colon n\in \mathbb{Z}\}$ is an orthonormal basis for ${W}_{1}$ , in conjunction with the wavelet scaling equation,

${d}_{1}(n)={x}_{0}(t)\dot {}_{1n}(t)=\sum {c}_{0}(m){}_{0m}(t)\dot {}_{1n}(t)=\sum {c}_{0}(m)({}_{0m}(t)\dot {}_{1n}(t))=\sum {c}_{0}(m)((t-m)\dot \sum g()(t--2n))=\sum {c}_{0}(m)\sum g()((t-m)\dot (t--2n))=\sum {c}_{0}(m)g(m-2n)$
where $(t--2n)=(t-m)\dot (t--2n)$ .

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

${x}_{1}(t)=\sum {c}_{1}(n){}_{1n}(t)=\sum {d}_{2}(n){}_{2n}(t)+\sum {c}_{2}(n){}_{2n}(t)$
Using the same steps as before we find that
${c}_{2}(n)=\sum {c}_{1}(m)h(m-2n)$
${d}_{2}(n)=\sum {c}_{1}(m)g(m-2n)$
which gives a cascaded analysis filterbank ( ):

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:

${c}_{0}(m)={x}_{0}(t)\dot {}_{0m}(t)=\sum {c}_{1}(n){}_{1n}(t)+\sum {d}_{1}(n){}_{1n}(t)\dot {}_{0m}(t)=\sum {c}_{1}(n)({}_{1n}(t)\dot {}_{0m}(t))+\sum {d}_{1}(n)({}_{1n}(t)\dot {}_{0m}(t))=\sum {c}_{1}(n)h(m-2n)+\sum {d}_{1}(n)g(m-2n)$
$\text{where}h(m-2n)={}_{1n}(t)\dot {}_{0m}(t)$ $\text{and}g(m-2n)={}_{1n}(t)\dot {}_{0m}(t)$ which can be implemented using the block diagram in .

The same procedure can be used to derive

${c}_{1}(m)=\sum {c}_{2}(n)h(m-2n)+\sum {d}_{2}(n)g(m-2n)$
from which we get the diagram in .

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.

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 $\forall P, \text{P is odd}\colon G(z)=(z^{-P}H(-z^{-1}))$ $\forall N, \text{N is even}\colon {H}_{1}(z)=(z^{-(N-1)}{H}_{0}(-z^{-1}))$
Synthesis LPF $H(z)$ ${G}_{0}(z)=2z^{-(N-1)}{H}_{0}(z^{-1})$
Synthesis HPF $G(z)$ ${G}_{1}(z)=2z^{-(N-1)}{H}_{1}(z^{-1})$

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 $P=N-1$ .

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