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.