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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.
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.
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