0.8 Filter banks and transmultiplexers  (Page 18/23)

PS symmetry does not reflect itself as any simple property of the scaling function ${\psi }_0\left(t\right)$ and wavelets ${\psi }_i\left(t\right)$ , $i\in \left\{1,,,...,,,M,-,1\right\}$ of the WTF. However, from design and implementation points of view,PS symmetry is useful (because of the reduction in the number of parameters).

Next consider PCS symmetry. From [link] one sees that [link] is equivalent to the first rows of the matrices $A$ and $B$ defined by

are of the form $\left[\begin{array}{ccc}1/\sqrt{M}& ...& 1/\sqrt{M}\end{array}\right].$ Here we only have an implicit parameterization of WTFs, unlike the case of PS symmetry. As in the case of PS symmetry, there is no simple symmetryrelationships between the wavelets.

Now consider the case of linear phase. In this case, it can be seen [link] that the wavelets are also linear phase. If we define

then it can be verified that one of the rows of the matrix $A+B$ has to be of the form $\left[\begin{array}{ccc}\sqrt{2/M}& ...& \sqrt{2/M}\end{array}\right]$ . This is an implicit parameterization of the WTF.

Finally consider the case of linear phase with PCS symmetry. In this case, also the wavelets are linear-phase.From [link] it can be verified that we have a WTF iff the first row of $W_0^{\text{'}}\left(z\right)$ for $z=1$ , evaluates to the vector $\left[\begin{array}{ccc}\sqrt{2/M}& ...& \sqrt{2/M}\end{array}\right].$ Equivalently, $W_0^{\text{'}}\left(z\right)$ gives rise to a multiplicity $M/2$ WTF. In this case, the WTF is parameterized by precisely $\left(,\begin{array}{c}M/2-1\\ 2\end{array},\right)+(M/2-1)L$ parameters where $L\ge K$ is the McMillan degree of $W_0^{\text{'}}\left(z\right)$ .

Linear-phase modulated filter banks

The modulated filter banks we described

1. have filters with nonoverlapping ideal frequency responses as shown in [link] .
2. are associated with DCT III/IV (or equivalently DST III/IV) in their implementation
3. and do not allow for linear phase filters (even though the prototypes could be linear phase).

In trying to overcome 3, Lin and Vaidyanathan introduced a new class of linear-phase modulated filter banks by giving up 1 and 2 [link] . We now introduce a generalization of their results from a viewpointthat unifies the theory of modulated filter banks as seen earlier with the new class of modulated filter banks we introduce here.For a more detailed exposition of this viewpoint see [link] .

The new class of modulated filter banks have $2M$ analysis filters, but $M$ bands—each band being shared by two overlapping filters. The $M$ bands are the $M$ -point Discrete Fourier Transform bands as shown in [link] .

Two broad classes of MFBs (that together are associated with all four DCT/DSTs [link] ) can be defined.

Dct/dst i/ii based $2M$ Channel filter bank

The sets $S_1$ and $S_2$ are defined depending on the parity of $\alpha$ as shown in [link] . When $\alpha$ is even (i.e., Type 1 with odd $M$ or Type 2 with even $M$ ), the MFB is associated with DCT I and DST I. When $\alpha$ is odd (i.e., Type 1 with even $M$ or Type 2 with odd $M$ ), the MFB is associated with DCT II and DST II.The linear-phase MFBs introduced in [link] correspond to the special case where $h\left(n\right)=g\left(n\right)$ and $\alpha$ is even. The other cases above and their corresponding PR results are new.