0.8 Filter banks and transmultiplexers  (Page 5/23)

Notice that the PR property of a filter bank or transmultiplexer is unchanged if all the analysis and synthesis filters are shifted by the same amount in opposite directions.Also any one analysis/synthesis filter pair can be shifted by multiples of $M$ in opposite directions without affecting the PR property. Using these two properties (and assuming all thefilters are FIR), we can assume without loss of generality that the analysis filters are supported in $[0,N-1]$ (for some integer $N$ ). This also implies that $H_p\left(z\right)$ is a polynomial in $z^{-1}$ , a fact that we will use in the parameterization of an important class of filter banks— unitary filter banks .

All through the discussion of the PR property of filter banks, we have deliberately not said anything about the length of the filters. The bilinear PR constraintsare completely independent of filter lengths and hold for arbitrary sequences. However, if the sequences are infinite then one requires that the infinitesummations over $n$ in [link] and [link] converge. Clearly, assuming that these filter sequences are in ${\ell }^2\left(\mathbf{Z}\right)$ is sufficient to ensure this since inner products are then well-defined.

Unitary filter banks

From [link] it follows that a filter bank can be ensured to be PR if the analysis filters are chosen such that $\mathbf{H}$ is left-unitary, i.e., ${\mathbf{H}}^T\mathbf{H}=\mathbf{I}$ . In this case, the synthesis matrix $\mathbf{G}=\mathbf{H}$ (from [link] ) and therefore ${\mathbf{G}}_i={\mathbf{H}}_i$ for all $i$ . Recall that the rows of ${\mathbf{G}}_i$ contain $g_i$ in natural order while the rows of ${\mathbf{H}}_i$ contains $h_i$ in index-reversed order. Therefore, for such a filter bank, since ${\mathbf{G}}_i={\mathbf{H}}_i$ , the synthesis filters are reflections of the analysis filters about the origin; i.e., $g_i\left(n\right)=h_i(-n)$ . Filter banks where the analysis and synthesis filters satisfy this reflection relationship arecalled unitary (or orthogonal) filter banks for the simple reason that $\mathbf{H}$ is left-unitary. In a similar fashion, it is easy to see that if $\mathbf{H}$ is right-unitary (i.e., $\mathbf{H}{\mathbf{H}}^{\mathbf{T}}=\mathbf{I}$ ), then the transmultiplexer associated with this set of analysis filters is PR with $g_i\left(n\right)=h_i(-n)$ . This defines unitary transmultiplexers.

We now examine how the three ways of viewing PR filter banks and transmultiplexers simplify when we focus on unitary ones. Since $g_i\left(n\right)=h_i(-n)$ , the direct characterization becomes the following:

Theorem 41 A filter bank is unitary iff

A transmultiplexer is unitary iff

If the number of channels is equal to the downsampling factor, then a filter bank is unitary iff the corresponding transmultiplexer is unitary.

The matrix characterization of unitary filter banks/transmultiplexers should be clear from the above discussion:

Theorem 42 A filter bank is unitary iff ${\mathbf{H}}^T\mathbf{H}=\mathbf{I}$ , and a transmultiplexer is unitary iff $\mathbf{H}{\mathbf{H}}^T=\mathbf{I}$ .

The z-transform domain characterization of unitary filter banks and transmultiplexers is given below:

Theorem 43 A filter bank is unitary iff $H_p^T\left(z^{-1}\right)H_p\left(z\right)=I$ , and a transmultiplexer is unitary iff $H_p\left(z\right)H_p^T\left(z^{-1}\right)=I$ .

In this book (as in most of the work in the literature) one primarily considers the situation where the number of channels equals the downsampling factor.For such a unitary filter bank (transmultiplexer), [link] and [link] become: