0.8 Filter banks and transmultiplexers  (Page 13/23)

The result says that $P_l,P_{\alpha -l},Q_l$ and $Q_{\alpha -l}$ form analysis and synthesis filters of a two-channel PR filter bank( [link] in Z-transform domain).

Modulated filter bank design involves choosing $h$ and $g$ to optimize some goodness criterion while subject to the constraints in the theorem above.

Unitary modulated filter bank

In a unitary bank, the filters satisfy $g_i\left(n\right)=h_i(-n)$ . From [link] and [link] , it is clear that in a modulated filter bank if $g\left(n\right)=h(-n)$ , then $g_i\left(n\right)=h_i(-n)$ . Imposing this restriction (that the analysis and synthesis prototype filters are reflections of each other)gives PR conditions for unitary modulated filter banks. That $g\left(n\right)=h(-n)$ means that $P_l\left(z\right)=Q_l\left(z^{-1}\right)$ and therefore ${\mathcal{Q}}_l\left(z\right)={\mathcal{P}}_l\left(z^{-1}\right)$ . Indeed, for PR, we require

This condition is equivalent to requiring that $P_l$ and $P_{\alpha -l}$ are analysis filters of a two-channel unitary filter bank. Equivalently, for $l\in \mathcal{R}\left(M\right)$ , $P_{l,0}$ and $P_{l,1}$ are power-complementary.

Corollary 6 (Unitary MFB PR Theorem) A modulated filter bank (Type 1 or Type 2) is unitary iff for $l\in \mathcal{R}\left(J\right)$ , $P_{l,0}\left(z\right)$ and $P_{l,1}\left(z\right)$ are power complementary.

Furthermore, when $\alpha$ is even $P_{\frac{\alpha }{2},0}\left(z\right)P_{\frac{\alpha }{2},0}\left(z^{-1}\right)=\frac{1}{M}$ (i.e., $P_{\frac{\alpha }{2},0}\left(z\right)$ has to be $\frac{1}{\sqrt{M}}{z}^k$ for some integer $k$ ). In the Type 2 case, we further require $P_{M-1}\left(z\right)P_{M-1}\left(z^{-1}\right)=\frac{2}{M}$ (i.e., $P_{M-1}\left(z\right)$ has to be $\frac{2}{\sqrt{M}}{z}^k$ for some integer $k$ ).

Unitary modulated filter bank design entails the choice of $h$ , the analysis prototype filter. There are $J$ associated two-channel unitary filter banks each of which can be parameterized using the lattice parameterization.Besides, depending on whether the filter is Type 2 and/or $alpha$ is even one has to choose the locations of the delays.

For the prototype filter of a unitary MFB to be linear phase, it is necessary that

for some integer $k$ . In this case, the prototype filter (if FIR) is of length $2Mk$ and symmetric about $(Mk-\frac{1}{2})$ in the Type 1 case and of length $2Mk-1$ and symmetric about $(Mk-1)$ (for both Class A and Class B MFBs).In the FIR case, one can obtain linear-phase prototype filters by using the lattice parameterization [link] of two-channel unitary filter banks. Filter banks with FIR linear-phase prototype filterswill be said to be canonical . In this case, $P_l\left(z\right)$ is typically a filter of length $2k$ for all $l$ . For canonical modulated filter banks, one has to check power complementarity only for $l\in \mathcal{R}\left(J\right)$ .

Modulated wavelet tight frames

For all $M$ , there exist $M$ -band modulated WTFs. The simple linear constraint on $h_0$ becomes a set of $J$ linear constraints, one each, on each of the $J$ two-channel unitary lattices associated with the MFB.

Theorem 48 (Modulated Wavelet Tight Frames Theorem) Every compactly supported modulated WTF is associated with an FIR unitary MFB and is parameterized by $J$ unitary lattices such that the sum of the angles in the lattices satisfy (for $l\in \mathcal{R}\left(J\right)$ ) Eqn.  [link] .

If a canonical MFB has $Jk$ parameters, the corresponding WTF has $J(k-1)$ parameters.

Notice that even though the PR conditions for MFBs depended on whether it is Type 1 or Type 2, the MWTF conditions are identical. Now consider aType 1 or Type 2 MFB with one angle parameter per lattice; i.e., $N=2M$ (Type 1) or $N=2M-1$ (Type 2). This angle parameter is specified by the MWTF theorem above if we want associated wavelets. This choiceof angle parameters leads to a particularly simple form for the prototype filter.