0.8 Filter banks and transmultiplexers  (Page 6/23)

and

The matrices ${\mathbf{H}}_i$ are pairwise orthogonal and form a resolution of the identity matrix. In other words,for each $i$ , ${\mathbf{H}}_i^T{\mathbf{H}}_i$ is an orthogonal projection matrix and the filter bank gives an orthogonal decomposition of a given signal.Recall that for a matrix $P$ to be an orthogonal projection matrix, $P^2=P$ and $P\ge 0$ ; in our case, indeed, we do have ${\mathbf{H}}_i^T{\mathbf{H}}_i\ge 0$ and ${\mathbf{H}}_i^T{\mathbf{H}}_i{\mathbf{H}}_i^T{\mathbf{H}}_i={\mathbf{H}}_i^T{\mathbf{H}}_i$ .

Unitarity is a very useful constraint since it leads to orthogonal decompositions. Besides, for a unitary filter bank, one doesnot have to design both the analysis and synthesis filters since $h_i\left(n\right)=g_i(-n)$ . But perhaps the most importantproperty of unitary filter banks and transmultiplexers is that they can be parameterized. As we have already seen, filter bank design is a nonlinearoptimization (of some goodness criterion) problem subject to PR constraints. If the PR constraints are unitary, then a parameterization of unitaryfilters leads to an unconstrained optimization problem. Besides, for designing wavelets with high-order vanishing moments, nonlinearequations can be formulated and solved in this parameter space. A similar parameterization of nonunitary PR filter banks and transmultiplexersseems impossible and it is not too difficult to intuitively see why. Consider the following analogy: a PR filter bank is akin to a left-invertible matrix anda PR transmultiplexer to a right-invertible matrix. If $L=M$ , the PR filter bank is akin to an invertible matrix. A unitary filter bank is akin toa left-unitary matrix, a unitary transmultiplexer to a right-unitary matrix, and when $L=M$ , either of them to a unitary matrix. Left-unitary, right-unitary and in particular unitary matrices can be parameterizedusing Givens' rotations or Householder transformations [link] . However, left-invertible, right-invertible and, in particular,invertible matrices have no general parameterization. Also, unitariness allows explicit parameterization of filter banks and transmultiplexerswhich just PR alone precludes. The analogy is even more appropriate: There are two parameterizations of unitary filter banks and transmultiplexers thatcorrespond to Givens' rotation and Householder transformations, respectively. All our discussions on filter banks and transmultiplexers carry over naturallywith very small notational changes to the multi-dimensional case where downsampling is by some integer matrix [link] . However, the parameterization result we now proceed to develop is not known in themulti-dimensional case. In the two-dimensional case, however, an implicit, and perhaps not too practical (from a filter-design point of view), parameterizationof unitary filter banks is described in [link] .

Consider a unitary filter bank with finite-impulse response filters (i.e., for all $i$ , $h_i$ is a finite sequence). Recall that without loss of generality, the filters can be shifted so that $H_p\left(z\right)$ is a polynomial in $z^{-1}$ . In this case $G_p\left(z\right)=H_p\left(z^{-1}\right)$ is a polynomial in $z$ . Let

That is, $H_p\left(z\right)$ is a matrix polynomial in $z^{-1}$ with coefficients $h_p\left(k\right)$ and degree $K-1$ . Since $H_p^T\left(z^{-1}\right)H_p\left(z\right)=I$ , from [link] we must have $h_p^T\left(0\right)h_p(K-1)=0$ as it is the coefficient of $z^{K-1}$ in the product $H_p^T\left(z^{-1}\right)H_p\left(z\right)$ . Therefore $h_p\left(0\right)$ is singular. Let $P_{K-1}$ be the unique projection matrix onto the range of $h_p(K-1)$ (say of dimension ${\delta }_{K-1}$ ). Then $h_p{\left(0\right)}^T{P}_{K-1}=0=P_{K-1}{h}_p\left(0\right)$ . Also $P_{K-1}h(K-1)=h(K-1)$ and hence $(I-P_{K-1})h(K-1)=0$ . Now $\left[I,-,P_{K-1},+,z,P_{K-1}\right]H_p\left(z\right)$ is a matrix polynomial of degree at most $K-2$ . If $h\left(0\right)$ and $h(K-1)$ are nonzero (an assumption one makes without loss of generality), the degree is precisely $K-2$ . Also it is unitary since $I-P_{K-1}+z{P}_{K-1}$ is unitary. Repeated application of this procedure $(K-1)$ times gives a degree zero (constant) unitary matrix $V_0$ . The discussion above shows that an arbitrary unitary polynomial matrix of degree $K-1$ can be expressed algorithmically uniquely as described in the following theorem: