0.8 Filter banks and transmultiplexers  (Page 16/23)

which, in turn is related to the invertibility of the complex matrices $C_i=(A_i+ıB_i)$ and $D_i=(A_i-ıB_i)$ , since,

Moreover, the orthogonality of the matrix is equivalent to the unitariness of the complex matrix $C_i$ (since $D_i$ is just its Hermitian conjugate). Since an arbitrary complex matrix of size $M/2\times M/2$ is determined by precisely $2\left(,\begin{array}{c}M/2\\ 2\end{array},\right)$ parameters, each of the matrices $\left[\begin{array}{cc}{A}_i& B_i\\ -B_i& A_i\end{array}\right]$ has that many degrees of freedom. Clearly when these matrices are orthogonal $X\left(z\right)$ is unitary (on the unit circle) and $X^T\left(z^{-1}\right)X\left(z\right)=I$ . For unitary $X\left(z\right)$ the converse is also true as will be shortly proved.

The symmetric lattice is defined by the product

Once again $A_i$ and $B_i$ are constant square matrices, and it is readily verified that $X\left(z\right)$ written as a product above is of the form

The invertibility of $X\left(z\right)$ is equivalent to the invertibility of

which in turn is equivalent to the invertibility of $C_i=(A_i+B_i)$ and $D_i=(A_i-B_i)$ since

The orthogonality of the constant matrix is equivalent to the orthogonality of the real matrices $C_i$ and $D_i$ , and since each real orthogonal matrix of size $M/2\times M/2$ is determined by $\left(,\begin{array}{c}M/2\\ 2\end{array},\right)$ parameters, the constant orthogonal matrices have $2\left(,\begin{array}{c}M/2\\ 2\end{array},\right)$ degrees of freedom. Clearly when the matrices are orthogonal $X^T\left(z^{-1}\right)X\left(z\right)=I$ . For the hyperbolic lattice too, the converse is true.

We now give a theorem that leads to a parameterization of unitary filter banks with the symmetries we have considered (for a proof, see [link] ).

Theorem 49 Let $X\left(z\right)$ be a unitary $M\times M$ polynomial matrix of degree $K$ . Depending on whether $X\left(z\right)$ is of the form in [link] , or [link] , it is generated by an order $K$ antisymmetric or symmetric lattice.

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

The form of $H_p\left(z\right)$ for PS symmetry in [link] can be simplified by a permutation. Let $P$ be the permutation matrix that exchanges the first column with the last column, the third column with the last but third, etc. That is,

Then the matrix $\left[\begin{array}{cc}{W}_0\left(z\right)& W_1\left(z\right)\\ W_0\left(z\right)V& {(-1)}^{M/2}{W}_1\left(z\right)V\end{array}\right]$ in [link] can be rewritten as $\frac{1}{\sqrt{2}}\left[\begin{array}{cc}{W}_0^{\text{'}}\left(z\right)& W_1^{\text{'}}\left(z\right)\\ -W_0^{\text{'}}\left(z\right)& W_1^{\text{'}}\left(z\right)\end{array}\right]P$ , and therefore

For PS symmetry, one has the following parameterization of unitary filter banks.

Theorem 50 (Unitary PS Symmetry) $H_p\left(z\right)$ of order $K$ forms a unitary PR filter bank with PS symmetry iff there exist unitary, order $K$ , $M/2\times M/2$ matrices $W_0^{\text{'}}\left(z\right)$ and $W_1^{\text{'}}\left(z\right)$ , such that

A unitary $H_p$ , with PS symmetry is determined by precisely $2(M/2-1)(L_0+L_1)+2\left(,\begin{array}{c}M/2\\ 2\end{array},\right)$ parameters where $L_0\ge K$ and $L_1\ge K$ are the McMillan degrees of $W_0^{\text{'}}\left(z\right)$ and $W_1^{\text{'}}\left(z\right)$ respectively.

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

In this case

Hence from Lemma  "Linear Phase Filter Banks" $H_p\left(z\right)$ of unitary filter banks with PCS symmetry can be parameterized as follows: