0.8 Filter banks and transmultiplexers  (Page 17/23)

Theorem 51 $H_p\left(z\right)$ forms an order $K$ , unitary filter bank with PCS symmetry iff

where $\left[\begin{array}{cc}{A}_i& B_i\\ -B_i& A_i\end{array}\right]$ are constant orthogonal matrices. $H_p\left(z\right)$ is characterized by $2K\left(,\begin{array}{c}M/2\\ 2\end{array},\right)$ parameters.

Characterization of unitary $H_p\left(z\right)$ — linear-phase symmetry

For the linear-phase case,

Therefore, we have the following Theorem:

Theorem 52 $H_p\left(z\right)$ of order $K$ , forms a unitary filter bank with linear-phase filters iff

where $\left[\begin{array}{cc}{A}_i& B_i\\ B_i& A_i\end{array}\right]$ are constant orthogonal matrices. $H_p\left(z\right)$ is characterized by $2K\left(,\begin{array}{c}M/2\\ 2\end{array},\right)$ parameters.

Characterization of unitary $H_p\left(z\right)$ — linear phase and pcs symmetry

In this case, $H_p\left(z\right)$ is given by

Therefore we have proved the following Theorem:

Theorem 53 $H_p\left(z\right)$ of order $K$ forms a unitary filter bank with linear-phase and PCS filters iff there exists a unitary, order $K$ , $M/2\times M/2$ matrix $W_0^{\text{'}}\left(z\right)$ such that

In this case $H_p\left(z\right)$ is determined by precisely $(M/2-1)L+\left(,\begin{array}{c}M/2\\ 2\end{array},\right)$ parameters where $L\ge K$ is the McMillan degree of $W_0^{\text{'}}\left(z\right)$ .

Characterization of unitary $H_p\left(z\right)$ — linear phase and ps symmetry

From the previous result we have the following result:

Theorem 54 $H_p\left(z\right)$ of order $K$ forms a unitary filter bank with linear-phase and PS filters iff there exists a unitary, order $K$ , $M/2\times M/2$ matrix $W_0^{\text{'}}\left(z\right)$ such that

$H_p$ is determined by precisely $(M/2-1)L+\left(,\begin{array}{c}M/2\\ 2\end{array},\right)$ parameters where $L\ge K$ is the McMillan degree of $W_0^{\text{'}}\left(z\right)$ .

Notice that Theorems  "Characterization of Unitary H p (z) — PS Symmetry" through Theorem  "Characterization of Unitary H p (z) — Linear Phase and PS Symmetry" give a completeness characterization for unitary filter banks with thesymmetries in question (and the appropriate length restrictions on the filters). However, ifone requires only the matrices $W_0^{\text{'}}\left(z\right)$ and $W_1^{\text{'}}\left(z\right)$ in the above theorems to be invertible on the unit circle (and notunitary), then the above results gives a method to generate nonunitary PR filter banks with the symmetries considered. Notice however, thatin the nonunitary case this is not a complete parameterization of all such filter banks.

Linear-phase wavelet tight frames

A necessary and sufficient condition for a unitary (FIR) filter bank to give rise to a compactly supported wavelet tight frame (WTF) isthat the lowpass filter $h_0$ in the filter bank satisfies the linear constraint [link]

We now examine and characterize how $H_p\left(z\right)$ for unitary filter banks with symmetries can be constrained to give rise to wavelet tight frames (WTFs).First consider the case of PS symmetry in which case $H_p\left(z\right)$ is parameterized in [link] . We have a WTF iff

In [link] , since $P$ permutes the columns, the first row is unaffected. Hence [link] is equivalent to the first rows of both $W_0^{\text{'}}\left(z\right)$ and $W_1^{\text{'}}\left(z\right)$ when $z=1$ is given by

This is precisely the condition to be satisfied by a WTF of multiplicity $M/2$ . Therefore both $W_0^{\text{'}}\left(z\right)$ and $W_1^{\text{'}}\left(z\right)$ give rise to multiplicity $M/2$ compactly supported WTFs. If the McMillan degree of $W_0^{\text{'}}\left(z\right)$ and $W_1^{\text{'}}\left(z\right)$ are $L_0$ and $L_1$ respectively, then they are parameterized respectively by $\left(,\begin{array}{c}M/2-1\\ 2\end{array},\right)+(M/2-1)L_0$ and $\left(,\begin{array}{c}M/2-1\\ 2\end{array},\right)+(M/2-1)L_1$ parameters. In summary, a WTF with PS symmetry can be explicitly parameterized by $2\left(,\begin{array}{c}M/2-1\\ 2\end{array},\right)+(M/2-1)(L_0+L_1)$ parameters. Both $L_0$ and $L_1$ are greater than or equal to $K$ .