0.8 Filter banks and transmultiplexers  (Page 19/23)

The PR constraints on the prototype filters $h$ and $g$ (for both versions of the filter banks above) are exactly the same as that for the modulated filter bankstudied earlier [link] . When the prototype filters are linear phase, these filter banks are also linear phase.An interesting consequence is that if one designs an $M$ -channel Class B modulated filter bank, the prototype filter can alsobe used for a Class A $2M$ channel filter bank.

Linear phase modulated wavelet tight frames

Under what conditions do linear phase modulated filter banks give rise to wavelet tight frames (WTFs)? To answer this question,it is convenient to use a slightly different lattice parameterization than that used for Class B modulated filter banks. A seemingly surprising resultis that some Class A unitary MFBs cannot be associated with WTFs. More precisely, a Class A MFB isassociated with a WTF only if it is Type 1.

where

With this parameterization we define ${\text{Θ}}_l$ as follows:

where in the FIR case ${\displaystyle {\text{Θ}}_l=\sum _{k=0}^{k_l-1}{\theta }_{l,k}}$ as before. Type 1 Class A MFBs give rise to a WTF iff ${\text{Θ}}_l=\frac{\pi }{4}$ for all $l\in \mathcal{R}\left(J\right)$ .

Theorem 55 (Modulated Wavelet Tight Frames Theorem) A class A MFB of Type 1 gives rise to a WTF iff ${\text{Θ}}_l=\frac{\pi }{4}$ . A class B MFB (Type 1 or Type 2) gives rise to a WTF iff ${\text{Θ}}_l=\frac{\pi }{4}+\frac{\pi }{2M}(\frac{\alpha }{2}-l)$ .

Time-varying filter bank trees

Filter banks can be applied in cascade to give a rich variety of decompositions. By appropriately designing the filters onecan obtain an arbitrary resolution in frequency. This makes them particularly useful in the analysis of stationary signalswith phenomena residing in various frequency bands or scales. However, for the analysis of nonstationary or piecewise stationarysignals such filter bank trees do not suffice. With this in mind we turn to filter banks for finite-length signals.

If we had filter bank theory for finite-length signals, then, piecewise stationary signals canbe handled by considering each homogeneous segment separately. Several approaches to filter banks for finite-length signals exist andwe follow the one in [link] . If we consider the filter bank tree as a machine that takes an input sample(s) and produces an output sample(s) every instant then one canconsider changing machine every instant (i.e., changing the filter coefficients every instant). Alternatively, we could use a fixedmachine for several instants and then switch to another filter bank tree. The former approach is investigatedin [link] . We follow the latter approach, which, besides leveraging upon powerful methods to design theconstituent filter bank trees switched between, also leads to a theory of wavelet bases on an interval [link] , [link] .

Let $H_p\left(z\right)$ , the polyphase component matrix of the analysis filter bank of a unitary filter bank be of the form (see [link] )

It is convenient to introduce the sequence $\breve{\mathbf{x}}=[\cdots ,x\left(0\right),x(-1),\cdots ,x(-M+1),x\left(M\right),x(M-1),\cdots ]$ obtained from $\mathbf{x}$ by a permutation. Then,