0.8 Filter banks and transmultiplexers  (Page 7/23)

Theorem 44 For a polynomial matrix $H_p\left(z\right)$ , unitary on the unit circle (i.e., $H_p^T\left(z^{-1}\right)H_p\left(z\right)=I$ ), and of polynomial degree $K-1$ , there exists a unique set of projection matrices $P_k$ (each of rank some integer ${\delta }_k$ ), such that

Remark: Since the projection $P_k$ is of rank ${\delta }_k$ , it can be written as $v_1{v}_1^T+...+v_{{\delta }_k}{v}_{{\delta }_k}^T$ , for a nonunique set of orthonormal vectors $v_i$ . Using the fact that

defining $\text{Δ}={\sum }_k{\delta }_k$ and collecting all the $v_j$ 's that define the $P_k$ 's into a single pool (and reindexing) we get the following factorization:

If $H_p\left(z\right)$ is the analysis bank of a filter bank, then notice that $\text{Δ}$ (from [link] ) is the number of storage elements required to implement the analysis bank. The minimum number of storage elements to implementany transfer function is called the McMillan degree and in this case $\text{Δ}$ is indeed the McMillan degree [link] . Recall that $P_K$ is chosen to be the projection matrix onto the range of $h_p(K-1)$ . Instead we could have chosen $P_K$ to be the projection onto the nullspace of $h_p\left(0\right)$ (which contains the range of $h_p(K-1)$ ) or any space sandwiched between the two. Each choice leads to a different sequence of factors $P_K$ and corresponding ${\delta }_k$ (except when the range and nullspaces in question coincide at some stage during the order reduction process). However, $\text{Δ}$ , the McMillan degree is constant.

Equation [link] can be used as a starting point for filter bank design. It parameterizes all unitary filter banks with McMillan degree $\text{Δ}$ . If $\text{Δ}=K$ , then all unitary filter banks with filters of length $N\le MK$ are parameterized using a collection of $K-1$ unitary vectors, $v_k$ , and a unitary matrix, $V_0$ . Each unitary vector has $(M-1)$ free parameters, while the unitary matrix has $M(M-1)/2$ free parameters for a total of $(K-1)(M-1)+\left(,\begin{array}{c}M\\ 2\end{array},\right)$ free parameters for $H_p\left(z\right)$ . The filter bank design problem is to choose these free parameters to optimize the “usefulness” criterionof the filter bank.

If $L>M$ , and $H_p\left(z\right)$ is left-unitary, a similar analysis leads to exactly the same factorization as before except that $V_0$ is a left unitary matrix. In this case, the number of free parameters is given by $(K-1)(L-1)+\left(,\begin{array}{c}L\\ 2\end{array},\right)-\left(,\begin{array}{c}M\\ 2\end{array},\right)$ . For a transmultiplexer with $L<M$ , one can use the same factorization above for $H_p^T\left(z\right)$ (which is left unitary). Even for a filter bank or transmultiplexer with $L=M$ , factorizations of left-/right-unitary $H_p\left(z\right)$ is useful for the following reason. Let us assume that a subset of the analysis filters has beenpredesigned (for example in wavelet theory one sometimes independently designs $h_0$ to be a $K$ -regular scaling filter, as in Chapter: Regularity, Moments, and Wavelet System Design ). The submatrix of $H_p\left(z\right)$ corresponding to this subset of filters is right-unitary, hence its transpose can be parameterized asabove with a collection of vectors $v_i$ and a left-unitary $V_0$ . Each choice for the remaining columns of $V_0$ gives a choice for the remaining filters in the filter bank. In fact, all possible completions of the original subset with fixed McMillan degree are given this way.

Orthogonal filter banks are sometimes referred to as lossless filter banks because the collective energy of the subband signals is the same asthat of the original signal. If $U$ is an orthogonal matrix, then the signals $x\left(n\right)$ and $Ux\left(n\right)$ have the same energy. If $P$ is an orthogonal projection matrix, then