0.8 Filter banks and transmultiplexers  (Page 4/23)

From this development, we have the following result:

Theorem 39 A filter bank is PR iff

A transmultiplexer is PR iff

Moreover, when $L=M$ , both conditions are equivalent.

One can also write the PR conditions for filter banks and transmultiplexers in the following form, which explicitly shows the formal relationship betweenthe direct and matrix characterizations. For a PR filter bank we have

Correspondingly for a PR transmultiplexer we have

Polyphase (transform-domain) characterization of pr

We finally look at the analysis and synthesis filter banks from a polyphase representation viewpoint. Here subsequences of the inputand output signals and the filters are represented in the z-transform domain. Indeed let the z-transforms of the signals and filters be expressedin terms of the z-transforms of their subsequences as follows:

Then, along each branch of the analysis bank we have

Similarly, from the synthesis bank, we have

and therefore (from [link] )

For $i\in \left\{0,,,1,,,...,,,L,-,1\right\}$ and $k\in \left\{0,,,1,,,...,,,M,-,1\right\}$ , define the polyphase component matrices ${\left(H_p\left(z\right)\right)}_{i,k}=H_{i,k}\left(z\right)$ and ${\left(G_p\left(z\right)\right)}_{i,k}=G_{i,k}\left(z\right)$ . Let $X_p\left(z\right)$ and $Y_p\left(z\right)$ denote the z-transforms of the polyphase signals $x_p\left(n\right)$ and $y_p\left(n\right)$ , and let $D_p\left(z\right)$ be the vector whose components are $D_i\left(z\right)$ . Equations  [link] and [link] can be written compactly as

and

Thus, the analysis filter bank is represented by the multi-input (the polyphase components of $X\left(z\right)$ ), multi-output (the signals $D_i\left(z\right)$ ) linear-shift-invariant system $H_p\left(z\right)$ that takes in $X_p\left(z\right)$ and gives out $D_p\left(z\right)$ . Similarly, the synthesis filter bank can be interpreted as a multi-input (the signals $D_i\left(z\right)$ ), multi-output (the polyphase components of $Y\left(z\right)$ ) system $G_p^T\left(z\right)$ , which maps $D_p\left(z\right)$ to $Y_p\left(z\right)$ . Clearly we have PR iff $Y_p\left(z\right)=X_p\left(z\right)$ . This occurs precisely when $G_p^T\left(z\right)H_p\left(z\right)=I$ .

For the transmultiplexer problem, let $Y_p\left(z\right)$ and $X_p\left(z\right)$ be vectorized versions of the input and output signals respectively and let $D_p\left(z\right)$ be the generalized polyphase representation of the signal $D\left(z\right)$ . Now $D_p\left(z\right)=G_p^T\left(z\right)Y_p\left(z\right)$ and $X_p\left(z\right)=H_p\left(z\right)D_p\left(z\right)$ . Hence $X_p\left(z\right)=H_p\left(z\right)G_p^T\left(z\right)Y_p\left(z\right)$ , and for PR $H_p\left(z\right)G_p^T\left(z\right)=I$ .

Theorem 40 A filter bank has the PR property if and only if

A transmultiplexer has the PR property if and only if

where $H_p\left(z\right)$ and $G_p\left(z\right)$ are as defined above.

Remark: If $G_p^T\left(z\right)H_p\left(z\right)=I$ , then $H_p\left(z\right)$ must have at least as many rows as columns (i.e., $L\ge M$ is necessary for a filter bank to be PR).If $H_p\left(z\right)G_p^T\left(z\right)=I$ then $H_p\left(z\right)$ must have at least as many columns as rows (i.e., $M\ge L$ is necessary for a tranmultiplexer to be PR). If $L=M$ , $G_p^T\left(z\right)H_p\left(z\right)=I=H_p^T\left(z\right)G_p\left(z\right)$ and hence a filter bank is PR iff the corresponding transmultiplexer is PR. This equivalence is trivial with the polyphase representation, whileit is not in the direct and matrix representations.