# 4.2 Perfect reconstruction qmf

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## Perfect reconstruction qmf

The system transfer function for a QMF bank is

$T(z)={H}_{0}(z)^{2}-{H}_{1}(z)^{2}=4z^{-1}{P}_{0}(z^{2}){P}_{1}(z^{2})$
For perfect reconstruction, we need $T(z)=z^{-l}$ for some $l\in \mathbb{N}$ , which implies the equivalent conditions $4z^{-1}{P}_{0}(z^{2}){P}_{1}(z^{2})=z^{-l}$ ${P}_{0}(z^{2}){P}_{1}(z^{2})=\frac{1}{4}z^{-(l-1)}$ ${P}_{0}(z){P}_{1}(z)=\frac{1}{4}z^{-\left(\frac{l-1}{2}\right)}$ For FIR polyphase filters, this can only be satisfied by ${P}_{0}(z)={\beta }_{0}z^{-{n}_{0}}$ ${P}_{1}(z)={\beta }_{1}z^{-{n}_{1}}$ where we have ${n}_{0}()+{n}_{1}=\frac{l-1}{2}$ and ${\beta }_{0}{\beta }_{1}=\frac{1}{4}$ .

In other words, the polyphase filters are trivial, so that the prototype filter ${H}_{0}(z)$ has a two-tap response. With only two taps, ${H}_{0}(z)$ cannot be a very good lowpass filter, meaning that the sub-bandsignals will not be spectrally well-separated. From this we conclude that two-channel

It turns out that $M$ -channel perfect reconstruction QMF banks have more useful responsesfor larger values of $M$ .
perfect reconstruction QMF banks exist but are not very useful.

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