0.8 Filter banks and transmultiplexers  (Page 15/23)

Therefore

and for linear-phase $H\left(z\right)$ , since $H^R\left(z\right)=\pm H\left(z\right)$ ,

Lemma 2 For even $M$ , $H_p\left(z\right)$ is of the form

• PS Symmetry
  $$\left[\begin{array}{cc}{W}_0\left(z\right)& W_1\left(z\right)\\ J{W}_0\left(z\right)V& {(-1)}^{M/2}J{W}_1\left(z\right)V\end{array}\right]=\left[\begin{array}{cc}I& 0\\ 0& J\end{array}\right]\left[\begin{array}{cc}{W}_0\left(z\right)& W_1\left(z\right)\\ W_0\left(z\right)V& {(-1)}^{M/2}{W}_1\left(z\right)V\end{array}\right]$$
• PCS Symmetry
  $$\left[\begin{array}{cc}{W}_0\left(z\right)& W_1\left(z\right)J\\ J{W}_1^R\left(z\right)V& {(-1)}^{M/2}J{W}_0^R\left(z\right)JV\end{array}\right]=\left[\begin{array}{cc}I& 0\\ 0& J\end{array}\right]\left[\begin{array}{cc}{W}_0\left(z\right)& W_1\left(z\right)J\\ W_1^R\left(z\right)V& {(-1)}^{M/2}{W}_0^R\left(z\right)JV\end{array}\right]$$
• Linear Phase
  $$\left[\begin{array}{cc}{W}_0\left(z\right)& D_0{W}_0^R\left(z\right)J\\ W_1\left(z\right)& D_1{W}_1^R\left(z\right)J\end{array}\right]=\left[\begin{array}{cc}{W}_0\left(z\right)& D_0{W}_0^R\left(z\right)\\ W_1\left(z\right)& D_1{W}_1^R\left(z\right)\end{array}\right]\left[\begin{array}{cc}I& 0\\ 0& J\end{array}\right]$$
  or
  $$Q\left[\begin{array}{cc}{W}_0\left(z\right)& W_0^R\left(z\right)J\\ W_1\left(z\right)& -W_1^R\left(z\right)J\end{array}\right]=Q\left[\begin{array}{cc}{W}_0\left(z\right)& W_0^R\left(z\right)\\ W_1\left(z\right)& -W_1^R\left(z\right)\end{array}\right]\left[\begin{array}{cc}I& 0\\ 0& J\end{array}\right]$$
• Linear Phase and PCS
  $$\left[\begin{array}{cc}{W}_0\left(z\right)& D{W}_0^R\left(z\right)J\\ JD{W}_0\left(z\right)V& {(-1)}^{M/2}J{W}_0^R\left(z\right)JV\end{array}\right]=\left[\begin{array}{cc}I& 0\\ 0& J\end{array}\right]\left[\begin{array}{cc}{W}_0\left(z\right)& D{W}_0^R\left(z\right)J\\ D{W}_0\left(z\right)V& {(-1)}^{M/2}{W}_0^R\left(z\right)JV\end{array}\right]$$
• Linear Phase and PS
  $$\left[\begin{array}{cc}{W}_0\left(z\right)& D{W}_0^R\left(z\right)J\\ J{W}_0\left(z\right)V& {(-1)}^{M/2}JD{W}_0^R\left(z\right)JV\end{array}\right]=\left[\begin{array}{cc}I& 0\\ 0& JD\end{array}\right]\left[\begin{array}{cc}{W}_0\left(z\right)& D{W}_0^R\left(z\right)J\\ D{W}_0\left(z\right)V& {(-1)}^{M/2}{W}_0^R\left(z\right)JV\end{array}\right]$$

Thus in order to generate $H_p\left(z\right)$ for all symmetries other than $PS$ , we need a mechanism that generates a pair of matrices and their reflection (i.e., $W_0\left(z\right)$ , $W_1\left(z\right)$ $W_0^R\left(z\right)$ and $W_1^R\left(z\right)$ ). In the scalar case, there are two well-known lattice structures,that generate such pairs. The first case is the orthogonal lattice [link] , while the second is the linear-prediction lattice [link] . A $K^{th}$ order orthogonal lattice is generated by the product

This lattice is always invertible (unless $a_i$ and $b_i$ are both zero!), and the inverse is anticausal since

As we have seen, this lattice plays a fundamental role in the theory of two-channel FIR unitary modulated filter banks.The hyperbolic lattice of order $K$ generates the product

where $Y_0\left(z\right)$ and $Y_1\left(z\right)$ are of order $K$ . This lattice is invertible only when $a_i^2\ne b_i^2$ (or equivalently $(a_i+b_i)/2$ and $(a_i-b_i)/2$ are nonzero) in which case the inverse is noncausal since

Since the matrix $\left[\begin{array}{cc}{a}_i& b_i\\ b_i& a_i\end{array}\right]$ can be orthogonal iff $\left\{a_i,,,b_i\right\}=\left\{\pm ,1,,,0\right\}$ , or $\left\{a_i,,,b_i\right\}=\left\{0,,,\pm ,1\right\}$ , the $(2\times 2)$ matrix generated by the lattice can never be unitary.

Formally, it is clear that if we replace the scalars $a_i$ and $b_i$ with square matrices of size $M/2\times M/2$ then we would be able to generate matrix versions of these two lattices which can then be used togenerate filter banks with the symmetries we have considered. We will shortly see that both the lattices can generate unitary matrices, and this willlead to a parameterization of FIR unitary $H_p\left(z\right)$ for PCS, linear-phase, and PCS plus linear-phase symmetries. We prefer to call the generalization ofthe orthogonal lattice, the antisymmetric lattice and to call the generalization of the hyperbolic lattice, the symmetric lattice, which should be obvious from the form of the product. The reason for this is that the antisymmetric lattice may not generate a unitary matrix transfer function (in the scalar case, the $2\times 2$ transfer function generated is always unitary). The antisymmetric lattice is defined by the product

where $A_i$ and $B_i$ are constant square matrices of size $M/2\times M/2$ . It is readily verified that $X\left(z\right)$ is of the form

Given $X\left(z\right)$ , its invertibility is equivalent to the invertibility of the constant matrices,