0.8 Filter banks and transmultiplexers  (Page 20/23)

and $\mathbf{H}$ is unitary iff $\breve{\mathbf{H}}$ is unitary. [link] induces a factorization of $\breve{\mathbf{H}}$ (and hence $\mathbf{H}$ ). If ${\mathbf{V}}_0=diag\left(V_0\right)$ and

The factors ${\mathbf{V}}_i$ , with appropriate modifications, will be used as fundamental building blocks for filter banks forfinite-length signals.

Now consider a finite input signal $\mathbf{x}=\left[x\right(0),x(1),\cdots ,x(L-1\left)\right]$ , where $L$ is a multiple of $M$ and let $\breve{\mathbf{x}}=[x(M-1),\cdots ,x\left(0\right),x\left(M\right),\cdots ,x(L-1),\cdots ,x(L-M)]$ . Then, the finite vector $\mathbf{d}$ (the output signal) is given by

$\breve{\mathbf{H}}$ is an $(L-N+M)\times L$ matrix, where $N=MK$ is the length of the filters. Now since the rows of $\breve{\mathbf{H}}$ are mutually orthonormal (i.e., has rank $L$ ), one has to append $N-M=M(K-1)$ rows from the orthogonal complement of $\breve{\mathbf{H}}$ to make the map from $\mathbf{x}$ to an augmented $\mathbf{d}$ unitary. To get a complete description of these rows, we turn to thefactorization of $H_p\left(z\right)$ . Define the $L\times L$ matrix ${\mathbf{V}}_0=diag\left(V_0\right)$ and for $i\in \left\{1,,,...,,,K,-,1\right\}$ the $(L-Mi)\times (L-Mi+M)$ matrices

then $\breve{\mathbf{H}}$ is readily verified by induction to be ${\displaystyle \prod _{i=K-1}^0{\mathbf{V}}_i}$ . Since each of the factors (except ${\mathbf{V}}_0$ ) has $M$ more columns than rows, they can be made unitary by appending appropriate rows. Indeed, $\left[\begin{array}{c}{B}_i\\ {\mathbf{V}}_i\\ C_i\end{array}\right]$ is unitary where, $B_j=\left[\begin{array}{cccc}{\text{Υ}}_j^T(I-P_j)& 0& ...& 0\end{array}\right]$ , and $C_j=\left[\begin{array}{cccc}0& ...& 0& {\text{Ξ}}_j^T{P}_j.\end{array}\right].$ Here $\text{Ξ}$ is the ${\delta }_i\times M$ left unitary matrix that spans the range of the $P_i$ ; i.e., $P_i=\text{Ξ}{\text{Ξ}}^T$ , and $\text{Υ}$ is the $(M-{\delta }_i)\times M$ left unitary matrix that spans the range of the $I-P_i$ ; i.e., $I-P_i=\text{Υ}{\text{Υ}}^T$ . Clearly $\left[{\text{Υ}}_i,{\text{Ξ}}_i\right]$ is unitary. Moreover, if we define ${\mathbf{T}}_0={\mathbf{V}}_0$ and for $i\in \left\{1,,,...,,,K,-,1\right\}$ ,

then each of the factors ${\mathbf{T}}_i$ is a square unitary matrix of size $L-N+M$ and

is the unitary matrix that acts on the data. The corresponding unitary matrix that acts on $\mathbf{x}$ (rather than $\breve{\mathbf{x}}$ ) is of the form $\left[\begin{array}{c}\mathbf{U}\\ \mathbf{H}\\ \mathbf{W}\end{array}\right]$ , where $\mathbf{U}$ has $MK-M-\text{Δ}$ rows of entry filters in $(K-1)$ sets given by [link] , while $\mathbf{W}$ has $\text{Δ}$ rows of exit filters in $(K-1)$ given by [link] :

where $J$ is the exchange matrix (i.e., permutation matrix of ones along the anti-diagonal) and

The rows of $\mathbf{U}$ and $\mathbf{W}$ form the entry and exit filters respectively. Clearly they are nonunique.The input/output behavior is captured in

For example, in the four-coefficient Daubechies' filters in [link] case, there is one entry filter and exit filter.

If the input signal is right-sided (i.e., supported in $\left\{0,,,1,,,...\right\}$ ), then the corresponding filter bank would only have entry filters. Ifthe filter bank is for left-sided signals one would only have exit filters. Based on the above, we can consider switching betweenfilter banks (that operate on infinite extent input signals). Consider switching from a one-channel to an $M$ channel filter bank. Until instant $n=-1$ , the input is the same as the output. At $n=0$ , one switches into an $M$ -channel filter bank as quickly as possible. The transition is accomplished by the entry filters(hence the name entry) of the $M$ -channel filter bank. The input/output of this time-varying filter bank is