Paraunitary Filter Banks

Paraunitary filter banks form an interesting subset of perfect reconstruction (PR) filter banks. We saw above that we get a PR filter bank whenever the synthesis polyphase matrix $\bold{R}(z)$ times the analysis polyphase matrix $\bold{E}(z)$ is the identity matrix $\bold{I}$ , i.e., when

$$\bold{P}(z) \isdefs \bold{R}(z)\bold{E}(z) \eqsp \bold{I}.$$ (12.70)

In particular, if $\bold{R}(z)$ is the paraconjugate of $\bold{E}(z)$ , we say the filter bank is paraunitary.

Paraconjugation is the generalization of the complex conjugate transpose operation from the unit circle to the entire $z$ plane. A paraunitary filter bank is therefore a generalization of an orthogonal filter bank. Recall that an orthogonal filter bank is one in which $\bold{E}(\ejo )$ is an orthogonal (or unitary) matrix, to within a constant scale factor, and $\bold{R}(\ejo )$ is its transpose (or Hermitian transpose).